From 8dd5186a0d3df3412bbb5ad8893a047b12ed8bcd Mon Sep 17 00:00:00 2001 From: Akos Kiss Date: Thu, 17 Mar 2016 10:42:00 +0100 Subject: [PATCH] Re-style fdlibm to conform to jerry guidelines * First re-style was done automatically by indent to minimize the chance of errors during rewrite. * Manual changes were applied to non-critical places only (comments and spaces): * Replaced all tabs with spaces. * Fixed tab stops in formulae in function comments. (Note: ASCII art for math formulae (especially for super- and subscripts) is a terrible idea.) * Unified the style of function comments. * Moved some in-code comments to their right places, which indent couldn't handle. * Added spaces to formulae of in-code comments to make them more readable. * Added braces mandated by jerry style guidelines. * Added parentheses to multiline #ifdef. JerryScript-DCO-1.0-Signed-off-by: Akos Kiss akiss@inf.u-szeged.hu --- third-party/fdlibm/include/fdlibm-math.h | 30 +- third-party/fdlibm/include/fdlibm.h | 36 +- third-party/fdlibm/s_acos.c | 125 +- third-party/fdlibm/s_asin.c | 156 +-- third-party/fdlibm/s_atan.c | 133 ++- third-party/fdlibm/s_atan2.c | 231 ++-- third-party/fdlibm/s_ceil.c | 140 ++- third-party/fdlibm/s_copysign.c | 15 +- third-party/fdlibm/s_exp.c | 235 ++-- third-party/fdlibm/s_fabs.c | 14 +- third-party/fdlibm/s_finite.c | 17 +- third-party/fdlibm/s_floor.c | 140 ++- third-party/fdlibm/s_fmod.c | 293 +++-- third-party/fdlibm/s_isnan.c | 23 +- third-party/fdlibm/s_log.c | 222 ++-- third-party/fdlibm/s_pow.c | 631 ++++++---- third-party/fdlibm/s_scalbn.c | 81 +- third-party/fdlibm/s_sqrt.c | 708 +++++------ third-party/fdlibm/s_trig.c | 1383 +++++++++++++--------- 19 files changed, 2726 insertions(+), 1887 deletions(-) diff --git a/third-party/fdlibm/include/fdlibm-math.h b/third-party/fdlibm/include/fdlibm-math.h index 0f3db3f7c..cd0fb05ba 100644 --- a/third-party/fdlibm/include/fdlibm-math.h +++ b/third-party/fdlibm/include/fdlibm-math.h @@ -52,29 +52,29 @@ extern "C" #define M_2_SQRTPI 1.1283791670955125738961589031215452 // Trigonometric functions -double cos(double); -double sin(double); -double tan(double); -double acos(double); -double asin(double); -double atan(double); -double atan2(double, double); +double cos (double); +double sin (double); +double tan (double); +double acos (double); +double asin (double); +double atan (double); +double atan2 (double, double); // Exponential and logarithmic functions -double exp(double); -double log(double); +double exp (double); +double log (double); // Power functions -double pow(double, double); -double sqrt(double); +double pow (double, double); +double sqrt (double); // Rounding and remainder functions -double ceil(double); -double floor(double); +double ceil (double); +double floor (double); // Other functions -double fabs(double); -double fmod(double, double); +double fabs (double); +double fmod (double, double); #ifdef __cplusplus } diff --git a/third-party/fdlibm/include/fdlibm.h b/third-party/fdlibm/include/fdlibm.h index 680138bd0..de1d3050f 100644 --- a/third-party/fdlibm/include/fdlibm.h +++ b/third-party/fdlibm/include/fdlibm.h @@ -13,43 +13,43 @@ /* Sometimes it's necessary to define __LITTLE_ENDIAN explicitly but these catch some common cases. */ -#if defined(i386) || defined(__i386) || defined(__i386__) || \ - defined(i486) || defined(__i486) || defined(__i486__) || \ - defined(intel) || defined(x86) || defined(i86pc) || \ - defined(__alpha) || defined(__osf__) || \ - defined(__x86_64__) || defined(__arm__) +#if (defined (i386) || defined (__i386) || defined (__i386__) || \ + defined (i486) || defined (__i486) || defined (__i486__) || \ + defined (intel) || defined (x86) || defined (i86pc) || \ + defined (__alpha) || defined (__osf__) || \ + defined (__x86_64__) || defined (__arm__)) #define __LITTLE_ENDIAN #endif #ifdef __LITTLE_ENDIAN -#define __HI(x) *(1+(int*)&x) -#define __LO(x) *(int*)&x +#define __HI(x) *(1 + (int *) &x) +#define __LO(x) *(int *) &x #else -#define __HI(x) *(int*)&x -#define __LO(x) *(1+(int*)&x) +#define __HI(x) *(int *) &x +#define __LO(x) *(1 + (int *) &x) #endif /* * ANSI/POSIX */ -#define MAXFLOAT ((float)3.40282346638528860e+38) +#define MAXFLOAT ((float) 3.40282346638528860e+38) -#define HUGE MAXFLOAT +#define HUGE MAXFLOAT /* * set X_TLOSS = pi*2**52, which is possibly defined in * (one may replace the following line by "#include ") */ -#define X_TLOSS 1.41484755040568800000e+16 +#define X_TLOSS 1.41484755040568800000e+16 -#define DOMAIN 1 -#define SING 2 -#define OVERFLOW 3 -#define UNDERFLOW 4 -#define TLOSS 5 -#define PLOSS 6 +#define DOMAIN 1 +#define SING 2 +#define OVERFLOW 3 +#define UNDERFLOW 4 +#define TLOSS 5 +#define PLOSS 6 /* * ANSI/POSIX diff --git a/third-party/fdlibm/s_acos.c b/third-party/fdlibm/s_acos.c index bd6ec1259..cf2237f64 100644 --- a/third-party/fdlibm/s_acos.c +++ b/third-party/fdlibm/s_acos.c @@ -6,31 +6,32 @@ * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice + * software is freely granted, provided that this notice * is preserved. * ==================================================== */ /* acos(x) - * Method : - * acos(x) = pi/2 - asin(x) - * acos(-x) = pi/2 + asin(x) + * + * Method: + * acos(x) = pi/2 - asin(x) + * acos(-x) = pi/2 + asin(x) * For |x|<=0.5 - * acos(x) = pi/2 - (x + x*x^2*R(x^2)) (see asin.c) + * acos(x) = pi/2 - (x + x*x^2*R(x^2)) (see asin.c) * For x>0.5 - * acos(x) = pi/2 - (pi/2 - 2asin(sqrt((1-x)/2))) - * = 2asin(sqrt((1-x)/2)) - * = 2s + 2s*z*R(z) ...z=(1-x)/2, s=sqrt(z) - * = 2f + (2c + 2s*z*R(z)) + * acos(x) = pi/2 - (pi/2 - 2asin(sqrt((1-x)/2))) + * = 2asin(sqrt((1-x)/2)) + * = 2s + 2s*z*R(z) ...z=(1-x)/2, s=sqrt(z) + * = 2f + (2c + 2s*z*R(z)) * where f=hi part of s, and c = (z-f*f)/(s+f) is the correction term * for f so that f+c ~ sqrt(z). * For x<-0.5 - * acos(x) = pi - 2asin(sqrt((1-|x|)/2)) - * = pi - 0.5*(s+s*z*R(z)), where z=(1-|x|)/2,s=sqrt(z) + * acos(x) = pi - 2asin(sqrt((1-|x|)/2)) + * = pi - 0.5*(s+s*z*R(z)), where z=(1-|x|)/2,s=sqrt(z) * * Special cases: - * if x is NaN, return x itself; - * if |x|>1, return NaN with invalid signal. + * if x is NaN, return x itself; + * if |x|>1, return NaN with invalid signal. * * Function needed: sqrt */ @@ -52,44 +53,62 @@ #define qS3 -6.88283971605453293030e-01 /* 0xBFE6066C, 0x1B8D0159 */ #define qS4 7.70381505559019352791e-02 /* 0x3FB3B8C5, 0xB12E9282 */ -double acos(double x) +double +acos (double x) { - double z,p,q,r,w,s,c,df; - int hx,ix; - hx = __HI(x); - ix = hx&0x7fffffff; - if(ix>=0x3ff00000) { /* |x| >= 1 */ - if(((ix-0x3ff00000)|__LO(x))==0) { /* |x|==1 */ - if(hx>0) return 0.0; /* acos(1) = 0 */ - else return pi+2.0*pio2_lo; /* acos(-1)= pi */ - } - return (x-x)/(x-x); /* acos(|x|>1) is NaN */ - } - if(ix<0x3fe00000) { /* |x| < 0.5 */ - if(ix<=0x3c600000) return pio2_hi+pio2_lo;/*if|x|<2**-57*/ - z = x*x; - p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5))))); - q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4))); - r = p/q; - return pio2_hi - (x - (pio2_lo-x*r)); - } else if (hx<0) { /* x < -0.5 */ - z = (one+x)*0.5; - p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5))))); - q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4))); - s = sqrt(z); - r = p/q; - w = r*s-pio2_lo; - return pi - 2.0*(s+w); - } else { /* x > 0.5 */ - z = (one-x)*0.5; - s = sqrt(z); - df = s; - __LO(df) = 0; - c = (z-df*df)/(s+df); - p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5))))); - q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4))); - r = p/q; - w = r*s+c; - return 2.0*(df+w); - } -} + double z, p, q, r, w, s, c, df; + int hx, ix; + + hx = __HI (x); + ix = hx & 0x7fffffff; + if (ix >= 0x3ff00000) /* |x| >= 1 */ + { + if (((ix - 0x3ff00000) | __LO (x)) == 0) /* |x| == 1 */ + { + if (hx > 0) /* acos(1) = 0 */ + { + return 0.0; + } + else /* acos(-1) = pi */ + { + return pi + 2.0 * pio2_lo; + } + } + return (x - x) / (x - x); /* acos(|x|>1) is NaN */ + } + if (ix < 0x3fe00000) /* |x| < 0.5 */ + { + if (ix <= 0x3c600000) /* if |x| < 2**-57 */ + { + return pio2_hi + pio2_lo; + } + z = x * x; + p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5))))); + q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4))); + r = p / q; + return pio2_hi - (x - (pio2_lo - x * r)); + } + else if (hx < 0) /* x < -0.5 */ + { + z = (one + x) * 0.5; + p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5))))); + q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4))); + s = sqrt (z); + r = p / q; + w = r * s - pio2_lo; + return pi - 2.0 * (s + w); + } + else /* x > 0.5 */ + { + z = (one - x) * 0.5; + s = sqrt (z); + df = s; + __LO (df) = 0; + c = (z - df * df) / (s + df); + p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5))))); + q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4))); + r = p / q; + w = r * s + c; + return 2.0 * (df + w); + } +} /* acos */ diff --git a/third-party/fdlibm/s_asin.c b/third-party/fdlibm/s_asin.c index 8f8a83b47..1802e410a 100644 --- a/third-party/fdlibm/s_asin.c +++ b/third-party/fdlibm/s_asin.c @@ -6,42 +6,41 @@ * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice + * software is freely granted, provided that this notice * is preserved. * ==================================================== */ /* asin(x) - * Method : - * Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ... - * we approximate asin(x) on [0,0.5] by - * asin(x) = x + x*x^2*R(x^2) - * where - * R(x^2) is a rational approximation of (asin(x)-x)/x^3 - * and its remez error is bounded by - * |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75) * - * For x in [0.5,1] - * asin(x) = pi/2-2*asin(sqrt((1-x)/2)) - * Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2; - * then for x>0.98 - * asin(x) = pi/2 - 2*(s+s*z*R(z)) - * = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo) - * For x<=0.98, let pio4_hi = pio2_hi/2, then - * f = hi part of s; - * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z) - * and - * asin(x) = pi/2 - 2*(s+s*z*R(z)) - * = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo) - * = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c)) + * Method: + * Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ... + * we approximate asin(x) on [0,0.5] by + * asin(x) = x + x*x^2*R(x^2) + * where + * R(x^2) is a rational approximation of (asin(x)-x)/x^3 + * and its remez error is bounded by + * |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75) + * + * For x in [0.5,1] + * asin(x) = pi/2-2*asin(sqrt((1-x)/2)) + * Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2; + * then for x>0.98 + * asin(x) = pi/2 - 2*(s+s*z*R(z)) + * = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo) + * For x<=0.98, let pio4_hi = pio2_hi/2, then + * f = hi part of s; + * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z) + * and + * asin(x) = pi/2 - 2*(s+s*z*R(z)) + * = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo) + * = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c)) * * Special cases: - * if x is NaN, return x itself; - * if |x|>1, return NaN with invalid signal. - * + * if x is NaN, return x itself; + * if |x|>1, return NaN with invalid signal. */ - #include "fdlibm.h" #define one 1.00000000000000000000e+00 /* 0x3FF00000, 0x00000000 */ @@ -49,7 +48,7 @@ #define pio2_hi 1.57079632679489655800e+00 /* 0x3FF921FB, 0x54442D18 */ #define pio2_lo 6.12323399573676603587e-17 /* 0x3C91A626, 0x33145C07 */ #define pio4_hi 7.85398163397448278999e-01 /* 0x3FE921FB, 0x54442D18 */ - /* coefficient for R(x^2) */ +/* coefficient for R(x^2) */ #define pS0 1.66666666666666657415e-01 /* 0x3FC55555, 0x55555555 */ #define pS1 -3.25565818622400915405e-01 /* 0xBFD4D612, 0x03EB6F7D */ #define pS2 2.01212532134862925881e-01 /* 0x3FC9C155, 0x0E884455 */ @@ -61,44 +60,67 @@ #define qS3 -6.88283971605453293030e-01 /* 0xBFE6066C, 0x1B8D0159 */ #define qS4 7.70381505559019352791e-02 /* 0x3FB3B8C5, 0xB12E9282 */ -double asin(double x) +double +asin (double x) { - double t = 0,w,p,q,c,r,s; - int hx,ix; - hx = __HI(x); - ix = hx&0x7fffffff; - if(ix>= 0x3ff00000) { /* |x|>= 1 */ - if(((ix-0x3ff00000)|__LO(x))==0) - /* asin(1)=+-pi/2 with inexact */ - return x*pio2_hi+x*pio2_lo; - return (x-x)/(x-x); /* asin(|x|>1) is NaN */ - } else if (ix<0x3fe00000) { /* |x|<0.5 */ - if(ix<0x3e400000) { /* if |x| < 2**-27 */ - if(huge+x>one) return x;/* return x with inexact if x!=0*/ - } else - t = x*x; - p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5))))); - q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4))); - w = p/q; - return x+x*w; - } - /* 1> |x|>= 0.5 */ - w = one-fabs(x); - t = w*0.5; - p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5))))); - q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4))); - s = sqrt(t); - if(ix>=0x3FEF3333) { /* if |x| > 0.975 */ - w = p/q; - t = pio2_hi-(2.0*(s+s*w)-pio2_lo); - } else { - w = s; - __LO(w) = 0; - c = (t-w*w)/(s+w); - r = p/q; - p = 2.0*s*r-(pio2_lo-2.0*c); - q = pio4_hi-2.0*w; - t = pio4_hi-(p-q); - } - if(hx>0) return t; else return -t; -} + double t = 0, w, p, q, c, r, s; + int hx, ix; + + hx = __HI (x); + ix = hx & 0x7fffffff; + if (ix >= 0x3ff00000) /* |x| >= 1 */ + { + if (((ix - 0x3ff00000) | __LO (x)) == 0) /* asin(1) = +-pi/2 with inexact */ + { + return x * pio2_hi + x * pio2_lo; + } + return (x - x) / (x - x); /* asin(|x|>1) is NaN */ + } + else if (ix < 0x3fe00000) /* |x| < 0.5 */ + { + if (ix < 0x3e400000) /* if |x| < 2**-27 */ + { + if (huge + x > one) /* return x with inexact if x != 0 */ + { + return x; + } + } + else + { + t = x * x; + } + p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5))))); + q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4))); + w = p / q; + return x + x * w; + } + /* 1 > |x| >= 0.5 */ + w = one - fabs (x); + t = w * 0.5; + p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5))))); + q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4))); + s = sqrt (t); + if (ix >= 0x3FEF3333) /* if |x| > 0.975 */ + { + w = p / q; + t = pio2_hi - (2.0 * (s + s * w) - pio2_lo); + } + else + { + w = s; + __LO (w) = 0; + c = (t - w * w) / (s + w); + r = p / q; + p = 2.0 * s * r - (pio2_lo - 2.0 * c); + q = pio4_hi - 2.0 * w; + t = pio4_hi - (p - q); + } + if (hx > 0) + { + return t; + } + else + { + return -t; + } +} /* asin */ diff --git a/third-party/fdlibm/s_atan.c b/third-party/fdlibm/s_atan.c index 1af2ab835..691303cf1 100644 --- a/third-party/fdlibm/s_atan.c +++ b/third-party/fdlibm/s_atan.c @@ -6,14 +6,14 @@ * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice + * software is freely granted, provided that this notice * is preserved. * ==================================================== - * */ /* atan(x) - * Method + * + * Method: * 1. Reduce x to positive by atan(x) = -atan(-x). * 2. According to the integer k=4t+0.25 chopped, t=x, the argument * is further reduced to one of the following intervals and the @@ -34,14 +34,16 @@ #include "fdlibm.h" -static const double atanhi[] = { +static const double atanhi[] = +{ 4.63647609000806093515e-01, /* atan(0.5)hi 0x3FDDAC67, 0x0561BB4F */ 7.85398163397448278999e-01, /* atan(1.0)hi 0x3FE921FB, 0x54442D18 */ 9.82793723247329054082e-01, /* atan(1.5)hi 0x3FEF730B, 0xD281F69B */ 1.57079632679489655800e+00, /* atan(inf)hi 0x3FF921FB, 0x54442D18 */ }; -static const double atanlo[] = { +static const double atanlo[] = +{ 2.26987774529616870924e-17, /* atan(0.5)lo 0x3C7A2B7F, 0x222F65E2 */ 3.06161699786838301793e-17, /* atan(1.0)lo 0x3C81A626, 0x33145C07 */ 1.39033110312309984516e-17, /* atan(1.5)lo 0x3C700788, 0x7AF0CBBD */ @@ -63,48 +65,83 @@ static const double atanlo[] = { #define one 1.0 #define huge 1.0e300 -double atan(double x) +double +atan (double x) { - double w,s1,s2,z; - int ix,hx,id; + double w, s1, s2, z; + int ix, hx, id; - hx = __HI(x); - ix = hx&0x7fffffff; - if(ix>=0x44100000) { /* if |x| >= 2^66 */ - if(ix>0x7ff00000|| - (ix==0x7ff00000&&(__LO(x)!=0))) - return x+x; /* NaN */ - if(hx>0) return atanhi[3]+atanlo[3]; - else return -atanhi[3]-atanlo[3]; - } if (ix < 0x3fdc0000) { /* |x| < 0.4375 */ - if (ix < 0x3e200000) { /* |x| < 2^-29 */ - if(huge+x>one) return x; /* raise inexact */ - } - id = -1; - } else { - x = fabs(x); - if (ix < 0x3ff30000) { /* |x| < 1.1875 */ - if (ix < 0x3fe60000) { /* 7/16 <=|x|<11/16 */ - id = 0; x = (2.0*x-one)/(2.0+x); - } else { /* 11/16<=|x|< 19/16 */ - id = 1; x = (x-one)/(x+one); - } - } else { - if (ix < 0x40038000) { /* |x| < 2.4375 */ - id = 2; x = (x-1.5)/(one+1.5*x); - } else { /* 2.4375 <= |x| < 2^66 */ - id = 3; x = -1.0/x; - } - }} - /* end of argument reduction */ - z = x*x; - w = z*z; - /* break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly */ - s1 = z*(aT0+w*(aT2+w*(aT4+w*(aT6+w*(aT8+w*aT10))))); - s2 = w*(aT1+w*(aT3+w*(aT5+w*(aT7+w*aT9)))); - if (id<0) return x - x*(s1+s2); - else { - z = atanhi[id] - ((x*(s1+s2) - atanlo[id]) - x); - return (hx<0)? -z:z; - } -} + hx = __HI (x); + ix = hx & 0x7fffffff; + if (ix >= 0x44100000) /* if |x| >= 2^66 */ + { + if (ix > 0x7ff00000 || (ix == 0x7ff00000 && (__LO (x) != 0))) + { + return x + x; /* NaN */ + } + if (hx > 0) + { + return atanhi[3] + atanlo[3]; + } + else + { + return -atanhi[3] - atanlo[3]; + } + } + if (ix < 0x3fdc0000) /* |x| < 0.4375 */ + { + if (ix < 0x3e200000) /* |x| < 2^-29 */ + { + if (huge + x > one) /* raise inexact */ + { + return x; + } + } + id = -1; + } + else + { + x = fabs (x); + if (ix < 0x3ff30000) /* |x| < 1.1875 */ + { + if (ix < 0x3fe60000) /* 7/16 <= |x| < 11/16 */ + { + id = 0; + x = (2.0 * x - one) / (2.0 + x); + } + else /* 11/16 <= |x| < 19/16 */ + { + id = 1; + x = (x - one) / (x + one); + } + } + else + { + if (ix < 0x40038000) /* |x| < 2.4375 */ + { + id = 2; + x = (x - 1.5) / (one + 1.5 * x); + } + else /* 2.4375 <= |x| < 2^66 */ + { + id = 3; + x = -1.0 / x; + } + } + } + /* end of argument reduction */ + z = x * x; + w = z * z; + /* break sum from i=0 to 10 aT[i] z**(i+1) into odd and even poly */ + s1 = z * (aT0 + w * (aT2 + w * (aT4 + w * (aT6 + w * (aT8 + w * aT10))))); + s2 = w * (aT1 + w * (aT3 + w * (aT5 + w * (aT7 + w * aT9)))); + if (id < 0) + { + return x - x * (s1 + s2); + } + else + { + z = atanhi[id] - ((x * (s1 + s2) - atanlo[id]) - x); + return (hx < 0) ? -z : z; + } +} /* atan */ diff --git a/third-party/fdlibm/s_atan2.c b/third-party/fdlibm/s_atan2.c index 1fd92e8b1..209b932e7 100644 --- a/third-party/fdlibm/s_atan2.c +++ b/third-party/fdlibm/s_atan2.c @@ -6,31 +6,30 @@ * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice + * software is freely granted, provided that this notice * is preserved. * ==================================================== - * */ /* atan2(y,x) - * Method : - * 1. Reduce y to positive by atan2(y,x)=-atan2(-y,x). - * 2. Reduce x to positive by (if x and y are unexceptional): - * ARG (x+iy) = arctan(y/x) ... if x > 0, - * ARG (x+iy) = pi - arctan[y/(-x)] ... if x < 0, + * + * Method: + * 1. Reduce y to positive by atan2(y,x)=-atan2(-y,x). + * 2. Reduce x to positive by (if x and y are unexceptional): + * ARG (x+iy) = arctan(y/x) ... if x > 0, + * ARG (x+iy) = pi - arctan[y/(-x)] ... if x < 0, * * Special cases: - * - * ATAN2((anything), NaN ) is NaN; - * ATAN2(NAN , (anything) ) is NaN; - * ATAN2(+-0, +(anything but NaN)) is +-0 ; - * ATAN2(+-0, -(anything but NaN)) is +-pi ; - * ATAN2(+-(anything but 0 and NaN), 0) is +-pi/2; - * ATAN2(+-(anything but INF and NaN), +INF) is +-0 ; - * ATAN2(+-(anything but INF and NaN), -INF) is +-pi; - * ATAN2(+-INF,+INF ) is +-pi/4 ; - * ATAN2(+-INF,-INF ) is +-3pi/4; - * ATAN2(+-INF, (anything but,0,NaN, and INF)) is +-pi/2; + * ATAN2((anything), NaN ) is NaN; + * ATAN2(NAN , (anything) ) is NaN; + * ATAN2(+-0, +(anything but NaN)) is +-0 ; + * ATAN2(+-0, -(anything but NaN)) is +-pi ; + * ATAN2(+-(anything but 0 and NaN), 0) is +-pi/2; + * ATAN2(+-(anything but INF and NaN), +INF) is +-0 ; + * ATAN2(+-(anything but INF and NaN), -INF) is +-pi; + * ATAN2(+-INF,+INF ) is +-pi/4 ; + * ATAN2(+-INF,-INF ) is +-3pi/4; + * ATAN2(+-INF, (anything but,0,NaN, and INF)) is +-pi/2; * * Constants: * The hexadecimal values are the intended ones for the following @@ -48,66 +47,142 @@ #define pi 3.1415926535897931160E+00 /* 0x400921FB, 0x54442D18 */ #define pi_lo 1.2246467991473531772E-16 /* 0x3CA1A626, 0x33145C07 */ -double atan2(double y, double x) -{ - double z; - int k,m,hx,hy,ix,iy; - unsigned lx,ly; +double +atan2 (double y, double x) +{ + double z; + int k, m, hx, hy, ix, iy; + unsigned lx, ly; - hx = __HI(x); ix = hx&0x7fffffff; - lx = __LO(x); - hy = __HI(y); iy = hy&0x7fffffff; - ly = __LO(y); - if(((ix|((lx|-lx)>>31))>0x7ff00000)|| - ((iy|((ly|-ly)>>31))>0x7ff00000)) /* x or y is NaN */ - return x+y; - if((hx-0x3ff00000|lx)==0) return atan(y); /* x=1.0 */ - m = ((hy>>31)&1)|((hx>>30)&2); /* 2*sign(x)+sign(y) */ + hx = __HI (x); + ix = hx & 0x7fffffff; + lx = __LO (x); + hy = __HI (y); + iy = hy & 0x7fffffff; + ly = __LO (y); + if (((ix | ((lx | -lx) >> 31)) > 0x7ff00000) || ((iy | ((ly | -ly) >> 31)) > 0x7ff00000)) /* x or y is NaN */ + { + return x + y; + } + if ((hx - 0x3ff00000 | lx) == 0) /* x = 1.0 */ + { + return atan (y); + } + m = ((hy >> 31) & 1) | ((hx >> 30) & 2); /* 2 * sign(x) + sign(y) */ - /* when y = 0 */ - if((iy|ly)==0) { - switch(m) { - case 0: - case 1: return y; /* atan(+-0,+anything)=+-0 */ - case 2: return pi+tiny;/* atan(+0,-anything) = pi */ - case 3: return -pi-tiny;/* atan(-0,-anything) =-pi */ - } - } - /* when x = 0 */ - if((ix|lx)==0) return (hy<0)? -pi_o_2-tiny: pi_o_2+tiny; - - /* when x is INF */ - if(ix==0x7ff00000) { - if(iy==0x7ff00000) { - switch(m) { - case 0: return pi_o_4+tiny;/* atan(+INF,+INF) */ - case 1: return -pi_o_4-tiny;/* atan(-INF,+INF) */ - case 2: return 3.0*pi_o_4+tiny;/*atan(+INF,-INF)*/ - case 3: return -3.0*pi_o_4-tiny;/*atan(-INF,-INF)*/ - } - } else { - switch(m) { - case 0: return zero ; /* atan(+...,+INF) */ - case 1: return -zero ; /* atan(-...,+INF) */ - case 2: return pi+tiny ; /* atan(+...,-INF) */ - case 3: return -pi-tiny ; /* atan(-...,-INF) */ - } - } - } - /* when y is INF */ - if(iy==0x7ff00000) return (hy<0)? -pi_o_2-tiny: pi_o_2+tiny; + /* when y = 0 */ + if ((iy | ly) == 0) + { + switch (m) + { + case 0: + case 1: + { + return y; /* atan(+-0,+anything) = +-0 */ + } + case 2: + { + return pi + tiny; /* atan(+0,-anything) = pi */ + } + case 3: + { + return -pi - tiny; /* atan(-0,-anything) = -pi */ + } + } + } + /* when x = 0 */ + if ((ix | lx) == 0) + { + return (hy < 0) ? -pi_o_2 - tiny : pi_o_2 + tiny; + } - /* compute y/x */ - k = (iy-ix)>>20; - if(k > 60) z=pi_o_2+0.5*pi_lo; /* |y/x| > 2**60 */ - else if(hx<0&&k<-60) z=0.0; /* |y|/x < -2**60 */ - else z=atan(fabs(y/x)); /* safe to do y/x */ - switch (m) { - case 0: return z ; /* atan(+,+) */ - case 1: __HI(z) ^= 0x80000000; - return z ; /* atan(-,+) */ - case 2: return pi-(z-pi_lo);/* atan(+,-) */ - default: /* case 3 */ - return (z-pi_lo)-pi;/* atan(-,-) */ - } -} + /* when x is INF */ + if (ix == 0x7ff00000) + { + if (iy == 0x7ff00000) + { + switch (m) + { + case 0: /* atan(+INF,+INF) */ + { + return pi_o_4 + tiny; + } + case 1: /* atan(-INF,+INF) */ + { + return -pi_o_4 - tiny; + } + case 2: /* atan(+INF,-INF) */ + { + return 3.0 * pi_o_4 + tiny; + } + case 3: /* atan(-INF,-INF) */ + { + return -3.0 * pi_o_4 - tiny; + } + } + } + else + { + switch (m) + { + case 0: /* atan(+...,+INF) */ + { + return zero; + } + case 1: /* atan(-...,+INF) */ + { + return -zero; + } + case 2: /* atan(+...,-INF) */ + { + return pi + tiny; + } + case 3: /* atan(-...,-INF) */ + { + return -pi - tiny; + } + } + } + } + /* when y is INF */ + if (iy == 0x7ff00000) + { + return (hy < 0) ? -pi_o_2 - tiny : pi_o_2 + tiny; + } + + /* compute y / x */ + k = (iy - ix) >> 20; + if (k > 60) /* |y / x| > 2**60 */ + { + z = pi_o_2 + 0.5 * pi_lo; + } + else if (hx < 0 && k < -60) /* |y| / x < -2**60 */ + { + z = 0.0; + } + else /* safe to do y / x */ + { + z = atan (fabs (y / x)); + } + switch (m) + { + case 0: /* atan(+,+) */ + { + return z; + } + case 1: /* atan(-,+) */ + { + __HI (z) ^= 0x80000000; + return z; + } + case 2: /* atan(+,-) */ + { + return pi - (z - pi_lo); + } + /* case 3: */ + default: /* atan(-,-) */ + { + return (z - pi_lo) - pi; + } + } +} /* atan2 */ diff --git a/third-party/fdlibm/s_ceil.c b/third-party/fdlibm/s_ceil.c index 93d0fde87..e5b5db08e 100644 --- a/third-party/fdlibm/s_ceil.c +++ b/third-party/fdlibm/s_ceil.c @@ -6,64 +6,110 @@ * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice + * software is freely granted, provided that this notice * is preserved. * ==================================================== */ -/* - * ceil(x) +/* ceil(x) * Return x rounded toward -inf to integral value + * * Method: - * Bit twiddling. + * Bit twiddling. + * * Exception: - * Inexact flag raised if x not equal to ceil(x). + * Inexact flag raised if x not equal to ceil(x). */ #include "fdlibm.h" #define huge 1.0e300 -double ceil(double x) +double +ceil (double x) { - int i0,i1,j0; - unsigned i,j; - i0 = __HI(x); - i1 = __LO(x); - j0 = ((i0>>20)&0x7ff)-0x3ff; - if(j0<20) { - if(j0<0) { /* raise inexact if x != 0 */ - if(huge+x>0.0) {/* return 0*sign(x) if |x|<1 */ - if(i0<0) {i0=0x80000000;i1=0;} - else if((i0|i1)!=0) { i0=0x3ff00000;i1=0;} - } - } else { - i = (0x000fffff)>>j0; - if(((i0&i)|i1)==0) return x; /* x is integral */ - if(huge+x>0.0) { /* raise inexact flag */ - if(i0>0) i0 += (0x00100000)>>j0; - i0 &= (~i); i1=0; - } - } - } else if (j0>51) { - if(j0==0x400) return x+x; /* inf or NaN */ - else return x; /* x is integral */ - } else { - i = ((unsigned)(0xffffffff))>>(j0-20); - if((i1&i)==0) return x; /* x is integral */ - if(huge+x>0.0) { /* raise inexact flag */ - if(i0>0) { - if(j0==20) i0+=1; - else { - j = i1 + (1<<(52-j0)); - if(j> 20) & 0x7ff) - 0x3ff; + if (j0 < 20) + { + if (j0 < 0) /* raise inexact if x != 0 */ + { + if (huge + x > 0.0) /* return 0 * sign(x) if |x| < 1 */ + { + if (i0 < 0) + { + i0 = 0x80000000; + i1 = 0; + } + else if ((i0 | i1) != 0) + { + i0 = 0x3ff00000; + i1 = 0; + } + } + } + else + { + i = (0x000fffff) >> j0; + if (((i0 & i) | i1) == 0) /* x is integral */ + { + return x; + } + if (huge + x > 0.0) /* raise inexact flag */ + { + if (i0 > 0) + { + i0 += (0x00100000) >> j0; + } + i0 &= (~i); + i1 = 0; + } + } + } + else if (j0 > 51) + { + if (j0 == 0x400) /* inf or NaN */ + { + return x + x; + } + else /* x is integral */ + { + return x; + } + } + else + { + i = ((unsigned) (0xffffffff)) >> (j0 - 20); + if ((i1 & i) == 0) /* x is integral */ + { + return x; + } + if (huge + x > 0.0) /* raise inexact flag */ + { + if (i0 > 0) + { + if (j0 == 20) + { + i0 += 1; + } + else + { + j = i1 + (1 << (52 - j0)); + if (j < i1) /* got a carry */ + { + i0 += 1; + } + i1 = j; + } + } + i1 &= (~i); + } + } + __HI (x) = i0; + __LO (x) = i1; + return x; +} /* ceil */ diff --git a/third-party/fdlibm/s_copysign.c b/third-party/fdlibm/s_copysign.c index ba7ea2522..5dd6c0fc1 100644 --- a/third-party/fdlibm/s_copysign.c +++ b/third-party/fdlibm/s_copysign.c @@ -6,21 +6,20 @@ * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice + * software is freely granted, provided that this notice * is preserved. * ==================================================== */ -/* - * copysign(double x, double y) - * copysign(x,y) returns a value with the magnitude of x and +/* copysign(x,y) returns a value with the magnitude of x and * with the sign bit of y. */ #include "fdlibm.h" -double copysign(double x, double y) +double +copysign (double x, double y) { - __HI(x) = (__HI(x)&0x7fffffff)|(__HI(y)&0x80000000); - return x; -} + __HI (x) = (__HI (x) & 0x7fffffff) | (__HI (y) & 0x80000000); + return x; +} /* copysign */ diff --git a/third-party/fdlibm/s_exp.c b/third-party/fdlibm/s_exp.c index b98baab21..b2fe87fa2 100644 --- a/third-party/fdlibm/s_exp.c +++ b/third-party/fdlibm/s_exp.c @@ -5,7 +5,7 @@ * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved. * * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice + * software is freely granted, provided that this notice * is preserved. * ==================================================== */ @@ -13,58 +13,58 @@ /* exp(x) * Returns the exponential of x. * - * Method + * Method: * 1. Argument reduction: * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658. - * Given x, find r and integer k such that + * Given x, find r and integer k such that * * x = k*ln2 + r, |r| <= 0.5*ln2. * * Here r will be represented as r = hi-lo for better - * accuracy. + * accuracy. * * 2. Approximation of exp(r) by a special rational function on - * the interval [0,0.34658]: - * Write - * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ... + * the interval [0,0.34658]: + * Write + * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ... * We use a special Remes algorithm on [0,0.34658] to generate - * a polynomial of degree 5 to approximate R. The maximum error - * of this polynomial approximation is bounded by 2**-59. In - * other words, - * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5 - * (where z=r*r, and the values of P1 to P5 are listed below) - * and - * | 5 | -59 - * | 2.0+P1*z+...+P5*z - R(z) | <= 2 - * | | - * The computation of exp(r) thus becomes + * a polynomial of degree 5 to approximate R. The maximum error + * of this polynomial approximation is bounded by 2**-59. In + * other words, + * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5 + * (where z=r*r, and the values of P1 to P5 are listed below) + * and + * | 5 | -59 + * | 2.0+P1*z+...+P5*z - R(z) | <= 2 + * | | + * The computation of exp(r) thus becomes * 2*r - * exp(r) = 1 + ------- - * R - r + * exp(r) = 1 + ------- + * R - r * r*R1(r) - * = 1 + r + ----------- (for better accuracy) - * 2 - R1(r) - * where - * 2 4 10 - * R1(r) = r - (P1*r + P2*r + ... + P5*r ). + * = 1 + r + ----------- (for better accuracy) + * 2 - R1(r) + * where + * 2 4 10 + * R1(r) = r - (P1*r + P2*r + ... + P5*r ). * * 3. Scale back to obtain exp(x): - * From step 1, we have - * exp(x) = 2^k * exp(r) + * From step 1, we have + * exp(x) = 2^k * exp(r) * * Special cases: - * exp(INF) is INF, exp(NaN) is NaN; - * exp(-INF) is 0, and - * for finite argument, only exp(0)=1 is exact. + * exp(INF) is INF, exp(NaN) is NaN; + * exp(-INF) is 0, and + * for finite argument, only exp(0)=1 is exact. * * Accuracy: - * according to an error analysis, the error is always less than - * 1 ulp (unit in the last place). + * according to an error analysis, the error is always less than + * 1 ulp (unit in the last place). * - * Misc. info. - * For IEEE double - * if x > 7.09782712893383973096e+02 then exp(x) overflow - * if x < -7.45133219101941108420e+02 then exp(x) underflow + * Misc. info: + * For IEEE double + * if x > 7.09782712893383973096e+02 then exp(x) overflow + * if x < -7.45133219101941108420e+02 then exp(x) underflow * * Constants: * The hexadecimal values are the intended ones for the following @@ -75,73 +75,118 @@ #include "fdlibm.h" -static const double -halF[2] = {0.5,-0.5,}, -ln2HI[2] = { 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */ - -6.93147180369123816490e-01,}, /* 0xbfe62e42, 0xfee00000 */ -ln2LO[2] = { 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */ - -1.90821492927058770002e-10,}; /* 0xbdea39ef, 0x35793c76 */ +static const double halF[2] = +{ + 0.5, + -0.5, +}; +static const double ln2HI[2] = +{ + 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */ + -6.93147180369123816490e-01, /* 0xbfe62e42, 0xfee00000 */ +}; +static const double ln2LO[2] = +{ + 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */ + -1.90821492927058770002e-10, /* 0xbdea39ef, 0x35793c76 */ +}; #define one 1.0 #define huge 1.0e+300 #define twom1000 9.33263618503218878990e-302 /* 2**-1000=0x01700000,0 */ -#define o_threshold 7.09782712893383973096e+02 /* 0x40862E42, 0xFEFA39EF */ -#define u_threshold -7.45133219101941108420e+02 /* 0xc0874910, 0xD52D3051 */ -#define invln2 1.44269504088896338700e+00 /* 0x3ff71547, 0x652b82fe */ -#define P1 1.66666666666666019037e-01 /* 0x3FC55555, 0x5555553E */ -#define P2 -2.77777777770155933842e-03 /* 0xBF66C16C, 0x16BEBD93 */ -#define P3 6.61375632143793436117e-05 /* 0x3F11566A, 0xAF25DE2C */ -#define P4 -1.65339022054652515390e-06 /* 0xBEBBBD41, 0xC5D26BF1 */ -#define P5 4.13813679705723846039e-08 /* 0x3E663769, 0x72BEA4D0 */ +#define o_threshold 7.09782712893383973096e+02 /* 0x40862E42, 0xFEFA39EF */ +#define u_threshold -7.45133219101941108420e+02 /* 0xc0874910, 0xD52D3051 */ +#define invln2 1.44269504088896338700e+00 /* 0x3ff71547, 0x652b82fe */ +#define P1 1.66666666666666019037e-01 /* 0x3FC55555, 0x5555553E */ +#define P2 -2.77777777770155933842e-03 /* 0xBF66C16C, 0x16BEBD93 */ +#define P3 6.61375632143793436117e-05 /* 0x3F11566A, 0xAF25DE2C */ +#define P4 -1.65339022054652515390e-06 /* 0xBEBBBD41, 0xC5D26BF1 */ +#define P5 4.13813679705723846039e-08 /* 0x3E663769, 0x72BEA4D0 */ -double exp(double x) /* default IEEE double exp */ +double +exp (double x) /* default IEEE double exp */ { - double y,hi,lo,c,t; - int k = 0,xsb; - unsigned hx; + double y, hi, lo, c, t; + int k = 0, xsb; + unsigned hx; - hx = __HI(x); /* high word of x */ - xsb = (hx>>31)&1; /* sign bit of x */ - hx &= 0x7fffffff; /* high word of |x| */ + hx = __HI (x); /* high word of x */ + xsb = (hx >> 31) & 1; /* sign bit of x */ + hx &= 0x7fffffff; /* high word of |x| */ - /* filter out non-finite argument */ - if(hx >= 0x40862E42) { /* if |x|>=709.78... */ - if(hx>=0x7ff00000) { - if(((hx&0xfffff)|__LO(x))!=0) - return x+x; /* NaN */ - else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */ - } - if(x > o_threshold) return huge*huge; /* overflow */ - if(x < u_threshold) return twom1000*twom1000; /* underflow */ - } + /* filter out non-finite argument */ + if (hx >= 0x40862E42) /* if |x| >= 709.78... */ + { + if (hx >= 0x7ff00000) + { + if (((hx & 0xfffff) | __LO (x)) != 0) /* NaN */ + { + return x + x; + } + else /* exp(+-inf) = {inf,0} */ + { + return (xsb == 0) ? x : 0.0; + } + } + if (x > o_threshold) /* overflow */ + { + return huge * huge; + } + if (x < u_threshold) /* underflow */ + { + return twom1000 * twom1000; + } + } - /* argument reduction */ - if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ - if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ - hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb; - } else { - k = (int)(invln2*x+halF[xsb]); - t = k; - hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */ - lo = t*ln2LO[0]; - } - x = hi - lo; - } - else if(hx < 0x3e300000) { /* when |x|<2**-28 */ - if(huge+x>one) return one+x;/* trigger inexact */ - } - else k = 0; + /* argument reduction */ + if (hx > 0x3fd62e42) /* if |x| > 0.5 ln2 */ + { + if (hx < 0x3FF0A2B2) /* and |x| < 1.5 ln2 */ + { + hi = x - ln2HI[xsb]; + lo = ln2LO[xsb]; + k = 1 - xsb - xsb; + } + else + { + k = (int) (invln2 * x + halF[xsb]); + t = k; + hi = x - t * ln2HI[0]; /* t * ln2HI is exact here */ + lo = t * ln2LO[0]; + } + x = hi - lo; + } + else if (hx < 0x3e300000) /* when |x| < 2**-28 */ + { + if (huge + x > one) /* trigger inexact */ + { + return one + x; + } + } + else + { + k = 0; + } - /* x is now in primary range */ - t = x*x; - c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); - if(k==0) return one-((x*c)/(c-2.0)-x); - else y = one-((lo-(x*c)/(2.0-c))-hi); - if(k >= -1021) { - __HI(y) += (k<<20); /* add k to y's exponent */ - return y; - } else { - __HI(y) += ((k+1000)<<20);/* add k to y's exponent */ - return y*twom1000; - } -} + /* x is now in primary range */ + t = x * x; + c = x - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5)))); + if (k == 0) + { + return one - ((x * c) / (c - 2.0) - x); + } + else + { + y = one - ((lo - (x * c) / (2.0 - c)) - hi); + } + if (k >= -1021) + { + __HI (y) += (k << 20); /* add k to y's exponent */ + return y; + } + else + { + __HI (y) += ((k + 1000) << 20); /* add k to y's exponent */ + return y * twom1000; + } +} /* exp */ diff --git a/third-party/fdlibm/s_fabs.c b/third-party/fdlibm/s_fabs.c index e7377d821..c3864ae92 100644 --- a/third-party/fdlibm/s_fabs.c +++ b/third-party/fdlibm/s_fabs.c @@ -6,19 +6,19 @@ * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice + * software is freely granted, provided that this notice * is preserved. * ==================================================== */ -/* - * fabs(x) returns the absolute value of x. +/* fabs(x) returns the absolute value of x. */ #include "fdlibm.h" -double fabs(double x) +double +fabs (double x) { - __HI(x) &= 0x7fffffff; - return x; -} + __HI (x) &= 0x7fffffff; + return x; +} /* fabs */ diff --git a/third-party/fdlibm/s_finite.c b/third-party/fdlibm/s_finite.c index 9f8613707..7a6ed175c 100644 --- a/third-party/fdlibm/s_finite.c +++ b/third-party/fdlibm/s_finite.c @@ -6,21 +6,22 @@ * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice + * software is freely granted, provided that this notice * is preserved. * ==================================================== */ -/* - * finite(x) returns 1 is x is finite, else 0; +/* finite(x) returns 1 is x is finite, else 0; * no branching! */ #include "fdlibm.h" -int finite(double x) +int +finite (double x) { - int hx; - hx = __HI(x); - return (unsigned)((hx&0x7fffffff)-0x7ff00000)>>31; -} + int hx; + + hx = __HI (x); + return (unsigned) ((hx & 0x7fffffff) - 0x7ff00000) >> 31; +} /* finite */ diff --git a/third-party/fdlibm/s_floor.c b/third-party/fdlibm/s_floor.c index 911dbfb64..e4c3db916 100644 --- a/third-party/fdlibm/s_floor.c +++ b/third-party/fdlibm/s_floor.c @@ -6,65 +6,109 @@ * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice + * software is freely granted, provided that this notice * is preserved. * ==================================================== */ -/* - * floor(x) +/* floor(x) * Return x rounded toward -inf to integral value + * * Method: - * Bit twiddling. + * Bit twiddling. + * * Exception: - * Inexact flag raised if x not equal to floor(x). + * Inexact flag raised if x not equal to floor(x). */ #include "fdlibm.h" #define huge 1.0e300 -double floor(double x) +double +floor (double x) { - int i0,i1,j0; - unsigned i,j; - i0 = __HI(x); - i1 = __LO(x); - j0 = ((i0>>20)&0x7ff)-0x3ff; - if(j0<20) { - if(j0<0) { /* raise inexact if x != 0 */ - if(huge+x>0.0) {/* return 0*sign(x) if |x|<1 */ - if(i0>=0) {i0=i1=0;} - else if(((i0&0x7fffffff)|i1)!=0) - { i0=0xbff00000;i1=0;} - } - } else { - i = (0x000fffff)>>j0; - if(((i0&i)|i1)==0) return x; /* x is integral */ - if(huge+x>0.0) { /* raise inexact flag */ - if(i0<0) i0 += (0x00100000)>>j0; - i0 &= (~i); i1=0; - } - } - } else if (j0>51) { - if(j0==0x400) return x+x; /* inf or NaN */ - else return x; /* x is integral */ - } else { - i = ((unsigned)(0xffffffff))>>(j0-20); - if((i1&i)==0) return x; /* x is integral */ - if(huge+x>0.0) { /* raise inexact flag */ - if(i0<0) { - if(j0==20) i0+=1; - else { - j = i1+(1<<(52-j0)); - if(j> 20) & 0x7ff) - 0x3ff; + if (j0 < 20) + { + if (j0 < 0) /* raise inexact if x != 0 */ + { + if (huge + x > 0.0) /* return 0 * sign(x) if |x| < 1 */ + { + if (i0 >= 0) + { + i0 = i1 = 0; + } + else if (((i0 & 0x7fffffff) | i1) != 0) + { + i0 = 0xbff00000; + i1 = 0; + } + } + } + else + { + i = (0x000fffff) >> j0; + if (((i0 & i) | i1) == 0) /* x is integral */ + { + return x; + } + if (huge + x > 0.0) /* raise inexact flag */ + { + if (i0 < 0) + { + i0 += (0x00100000) >> j0; + } + i0 &= (~i); + i1 = 0; + } + } + } + else if (j0 > 51) + { + if (j0 == 0x400) /* inf or NaN */ + { + return x + x; + } + else /* x is integral */ + { + return x; + } + } + else + { + i = ((unsigned) (0xffffffff)) >> (j0 - 20); + if ((i1 & i) == 0) /* x is integral */ + { + return x; + } + if (huge + x > 0.0) /* raise inexact flag */ + { + if (i0 < 0) + { + if (j0 == 20) + { + i0 += 1; + } + else + { + j = i1 + (1 << (52 - j0)); + if (j < i1) /* got a carry */ + { + i0 += 1; + } + i1 = j; + } + } + i1 &= (~i); + } + } + __HI (x) = i0; + __LO (x) = i1; + return x; +} /* floor */ diff --git a/third-party/fdlibm/s_fmod.c b/third-party/fdlibm/s_fmod.c index 2d6cbc437..7e488c713 100644 --- a/third-party/fdlibm/s_fmod.c +++ b/third-party/fdlibm/s_fmod.c @@ -11,124 +11,209 @@ * ==================================================== */ -/* - * fmod(x,y) +/* fmod(x,y) * Return x mod y in exact arithmetic + * * Method: shift and subtract */ #include "fdlibm.h" -static const double -Zero[] = {0.0, -0.0,}; +static const double Zero[] = { 0.0, -0.0, }; #define one 1.0 -double fmod(double x, double y) +double +fmod (double x, double y) { - int n,hx,hy,hz,ix,iy,sx,i; - unsigned lx,ly,lz; + int n, hx, hy, hz, ix, iy, sx, i; + unsigned lx, ly, lz; - hx = __HI(x); /* high word of x */ - lx = __LO(x); /* low word of x */ - hy = __HI(y); /* high word of y */ - ly = __LO(y); /* low word of y */ - sx = hx&0x80000000; /* sign of x */ - hx ^=sx; /* |x| */ - hy &= 0x7fffffff; /* |y| */ + hx = __HI (x); /* high word of x */ + lx = __LO (x); /* low word of x */ + hy = __HI (y); /* high word of y */ + ly = __LO (y); /* low word of y */ + sx = hx & 0x80000000; /* sign of x */ + hx ^= sx; /* |x| */ + hy &= 0x7fffffff; /* |y| */ - /* purge off exception values */ - if((hy|ly)==0||(hx>=0x7ff00000)|| /* y=0,or x not finite */ - ((hy|((ly|-ly)>>31))>0x7ff00000)) /* or y is NaN */ - return (x*y)/(x*y); - if(hx<=hy) { - if((hx>31]; /* |x|=|y| return x*0*/ - } + /* purge off exception values */ + if ((hy | ly) == 0 || (hx >= 0x7ff00000) || /* y = 0, or x not finite */ + ((hy | ((ly | -ly) >> 31)) > 0x7ff00000)) /* or y is NaN */ + { + return (x * y) / (x * y); + } + if (hx <= hy) + { + if ((hx < hy) || (lx < ly)) /* |x| < |y| return x */ + { + return x; + } + if (lx == ly) /* |x| = |y| return x * 0 */ + { + return Zero[(unsigned) sx >> 31]; + } + } - /* determine ix = ilogb(x) */ - if(hx<0x00100000) { /* subnormal x */ - if(hx==0) { - for (ix = -1043, i=lx; i>0; i<<=1) ix -=1; - } else { - for (ix = -1022,i=(hx<<11); i>0; i<<=1) ix -=1; - } - } else ix = (hx>>20)-1023; + /* determine ix = ilogb(x) */ + if (hx < 0x00100000) /* subnormal x */ + { + if (hx == 0) + { + for (ix = -1043, i = lx; i > 0; i <<= 1) + { + ix -= 1; + } + } + else + { + for (ix = -1022, i = (hx << 11); i > 0; i <<= 1) + { + ix -= 1; + } + } + } + else + { + ix = (hx >> 20) - 1023; + } - /* determine iy = ilogb(y) */ - if(hy<0x00100000) { /* subnormal y */ - if(hy==0) { - for (iy = -1043, i=ly; i>0; i<<=1) iy -=1; - } else { - for (iy = -1022,i=(hy<<11); i>0; i<<=1) iy -=1; - } - } else iy = (hy>>20)-1023; + /* determine iy = ilogb(y) */ + if (hy < 0x00100000) /* subnormal y */ + { + if (hy == 0) + { + for (iy = -1043, i = ly; i > 0; i <<= 1) + { + iy -= 1; + } + } + else + { + for (iy = -1022, i = (hy << 11); i > 0; i <<= 1) + { + iy -= 1; + } + } + } + else + { + iy = (hy >> 20) - 1023; + } - /* set up {hx,lx}, {hy,ly} and align y to x */ - if(ix >= -1022) - hx = 0x00100000|(0x000fffff&hx); - else { /* subnormal x, shift x to normal */ - n = -1022-ix; - if(n<=31) { - hx = (hx<>(32-n)); - lx <<= n; - } else { - hx = lx<<(n-32); - lx = 0; - } - } - if(iy >= -1022) - hy = 0x00100000|(0x000fffff&hy); - else { /* subnormal y, shift y to normal */ - n = -1022-iy; - if(n<=31) { - hy = (hy<>(32-n)); - ly <<= n; - } else { - hy = ly<<(n-32); - ly = 0; - } - } + /* set up {hx,lx}, {hy,ly} and align y to x */ + if (ix >= -1022) + { + hx = 0x00100000 | (0x000fffff & hx); + } + else /* subnormal x, shift x to normal */ + { + n = -1022 - ix; + if (n <= 31) + { + hx = (hx << n) | (lx >> (32 - n)); + lx <<= n; + } + else + { + hx = lx << (n - 32); + lx = 0; + } + } + if (iy >= -1022) + { + hy = 0x00100000 | (0x000fffff & hy); + } + else /* subnormal y, shift y to normal */ + { + n = -1022 - iy; + if (n <= 31) + { + hy = (hy << n) | (ly >> (32 - n)); + ly <<= n; + } + else + { + hy = ly << (n - 32); + ly = 0; + } + } - /* fix point fmod */ - n = ix - iy; - while(n--) { - hz=hx-hy;lz=lx-ly; if(lx>31); lx = lx+lx;} - else { - if((hz|lz)==0) /* return sign(x)*0 */ - return Zero[(unsigned)sx>>31]; - hx = hz+hz+(lz>>31); lx = lz+lz; - } - } - hz=hx-hy;lz=lx-ly; if(lx=0) {hx=hz;lx=lz;} + /* fix point fmod */ + n = ix - iy; + while (n--) + { + hz = hx - hy; + lz = lx - ly; + if (lx < ly) + { + hz -= 1; + } + if (hz < 0) + { + hx = hx + hx + (lx >> 31); + lx = lx + lx; + } + else + { + if ((hz | lz) == 0) /* return sign(x) * 0 */ + { + return Zero[(unsigned) sx >> 31]; + } + hx = hz + hz + (lz >> 31); + lx = lz + lz; + } + } + hz = hx - hy; + lz = lx - ly; + if (lx < ly) + { + hz -= 1; + } + if (hz >= 0) + { + hx = hz; + lx = lz; + } - /* convert back to floating value and restore the sign */ - if((hx|lx)==0) /* return sign(x)*0 */ - return Zero[(unsigned)sx>>31]; - while(hx<0x00100000) { /* normalize x */ - hx = hx+hx+(lx>>31); lx = lx+lx; - iy -= 1; - } - if(iy>= -1022) { /* normalize output */ - hx = ((hx-0x00100000)|((iy+1023)<<20)); - __HI(x) = hx|sx; - __LO(x) = lx; - } else { /* subnormal output */ - n = -1022 - iy; - if(n<=20) { - lx = (lx>>n)|((unsigned)hx<<(32-n)); - hx >>= n; - } else if (n<=31) { - lx = (hx<<(32-n))|(lx>>n); hx = sx; - } else { - lx = hx>>(n-32); hx = sx; - } - __HI(x) = hx|sx; - __LO(x) = lx; - x *= one; /* create necessary signal */ - } - return x; /* exact output */ -} + /* convert back to floating value and restore the sign */ + if ((hx | lx) == 0) /* return sign(x) * 0 */ + { + return Zero[(unsigned) sx >> 31]; + } + while (hx < 0x00100000) /* normalize x */ + { + hx = hx + hx + (lx >> 31); + lx = lx + lx; + iy -= 1; + } + if (iy >= -1022) /* normalize output */ + { + hx = ((hx - 0x00100000) | ((iy + 1023) << 20)); + __HI (x) = hx | sx; + __LO (x) = lx; + } + else /* subnormal output */ + { + n = -1022 - iy; + if (n <= 20) + { + lx = (lx >> n) | ((unsigned) hx << (32 - n)); + hx >>= n; + } + else if (n <= 31) + { + lx = (hx << (32 - n)) | (lx >> n); + hx = sx; + } + else + { + lx = hx >> (n - 32); + hx = sx; + } + __HI (x) = hx | sx; + __LO (x) = lx; + x *= one; /* create necessary signal */ + } + return x; /* exact output */ +} /* fmod */ diff --git a/third-party/fdlibm/s_isnan.c b/third-party/fdlibm/s_isnan.c index 2b9c65976..30724a315 100644 --- a/third-party/fdlibm/s_isnan.c +++ b/third-party/fdlibm/s_isnan.c @@ -6,24 +6,25 @@ * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice + * software is freely granted, provided that this notice * is preserved. * ==================================================== */ -/* - * isnan(x) returns 1 is x is nan, else 0; +/* isnan(x) returns 1 is x is nan, else 0; * no branching! */ #include "fdlibm.h" -int isnan(double x) +int +isnan (double x) { - int hx,lx; - hx = (__HI(x)&0x7fffffff); - lx = __LO(x); - hx |= (unsigned)(lx|(-lx))>>31; - hx = 0x7ff00000 - hx; - return ((unsigned)(hx))>>31; -} + int hx, lx; + + hx = (__HI (x) & 0x7fffffff); + lx = __LO (x); + hx |= (unsigned) (lx | (-lx)) >> 31; + hx = 0x7ff00000 - hx; + return ((unsigned) (hx)) >> 31; +} /* isnan */ diff --git a/third-party/fdlibm/s_log.c b/third-party/fdlibm/s_log.c index 6cf97ca39..1e6d11f25 100644 --- a/third-party/fdlibm/s_log.c +++ b/third-party/fdlibm/s_log.c @@ -6,7 +6,7 @@ * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice + * software is freely granted, provided that this notice * is preserved. * ==================================================== */ @@ -16,44 +16,44 @@ * * Method : * 1. Argument Reduction: find k and f such that - * x = 2^k * (1+f), - * where sqrt(2)/2 < 1+f < sqrt(2) . + * x = 2^k * (1+f), + * where sqrt(2)/2 < 1+f < sqrt(2) . * * 2. Approximation of log(1+f). - * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) - * = 2s + 2/3 s**3 + 2/5 s**5 + ....., - * = 2s + s*R + * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) + * = 2s + 2/3 s**3 + 2/5 s**5 + ....., + * = 2s + s*R * We use a special Reme algorithm on [0,0.1716] to generate - * a polynomial of degree 14 to approximate R The maximum error - * of this polynomial approximation is bounded by 2**-58.45. In - * other words, - * 2 4 6 8 10 12 14 - * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s - * (the values of Lg1 to Lg7 are listed in the program) - * and - * | 2 14 | -58.45 - * | Lg1*s +...+Lg7*s - R(z) | <= 2 - * | | - * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. - * In order to guarantee error in log below 1ulp, we compute log - * by - * log(1+f) = f - s*(f - R) (if f is not too large) - * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy) - * - * 3. Finally, log(x) = k*ln2 + log(1+f). - * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) - * Here ln2 is split into two floating point number: - * ln2_hi + ln2_lo, - * where n*ln2_hi is always exact for |n| < 2000. + * a polynomial of degree 14 to approximate R The maximum error + * of this polynomial approximation is bounded by 2**-58.45. In + * other words, + * 2 4 6 8 10 12 14 + * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s + * (the values of Lg1 to Lg7 are listed in the program) + * and + * | 2 14 | -58.45 + * | Lg1*s +...+Lg7*s - R(z) | <= 2 + * | | + * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. + * In order to guarantee error in log below 1ulp, we compute log + * by + * log(1+f) = f - s*(f - R) (if f is not too large) + * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy) + * + * 3. Finally, log(x) = k*ln2 + log(1+f). + * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) + * Here ln2 is split into two floating point number: + * ln2_hi + ln2_lo, + * where n*ln2_hi is always exact for |n| < 2000. * * Special cases: - * log(x) is NaN with signal if x < 0 (including -INF) ; - * log(+INF) is +INF; log(0) is -INF with signal; - * log(NaN) is that NaN with no signal. + * log(x) is NaN with signal if x < 0 (including -INF) ; + * log(+INF) is +INF; log(0) is -INF with signal; + * log(NaN) is that NaN with no signal. * * Accuracy: - * according to an error analysis, the error is always less than - * 1 ulp (unit in the last place). + * according to an error analysis, the error is always less than + * 1 ulp (unit in the last place). * * Constants: * The hexadecimal values are the intended ones for the following @@ -65,64 +65,108 @@ #include "fdlibm.h" #define zero 0.0 -#define ln2_hi 6.93147180369123816490e-01 /* 3fe62e42 fee00000 */ -#define ln2_lo 1.90821492927058770002e-10 /* 3dea39ef 35793c76 */ -#define two54 1.80143985094819840000e+16 /* 43500000 00000000 */ -#define Lg1 6.666666666666735130e-01 /* 3FE55555 55555593 */ -#define Lg2 3.999999999940941908e-01 /* 3FD99999 9997FA04 */ -#define Lg3 2.857142874366239149e-01 /* 3FD24924 94229359 */ -#define Lg4 2.222219843214978396e-01 /* 3FCC71C5 1D8E78AF */ -#define Lg5 1.818357216161805012e-01 /* 3FC74664 96CB03DE */ -#define Lg6 1.531383769920937332e-01 /* 3FC39A09 D078C69F */ -#define Lg7 1.479819860511658591e-01 /* 3FC2F112 DF3E5244 */ +#define ln2_hi 6.93147180369123816490e-01 /* 3fe62e42 fee00000 */ +#define ln2_lo 1.90821492927058770002e-10 /* 3dea39ef 35793c76 */ +#define two54 1.80143985094819840000e+16 /* 43500000 00000000 */ +#define Lg1 6.666666666666735130e-01 /* 3FE55555 55555593 */ +#define Lg2 3.999999999940941908e-01 /* 3FD99999 9997FA04 */ +#define Lg3 2.857142874366239149e-01 /* 3FD24924 94229359 */ +#define Lg4 2.222219843214978396e-01 /* 3FCC71C5 1D8E78AF */ +#define Lg5 1.818357216161805012e-01 /* 3FC74664 96CB03DE */ +#define Lg6 1.531383769920937332e-01 /* 3FC39A09 D078C69F */ +#define Lg7 1.479819860511658591e-01 /* 3FC2F112 DF3E5244 */ -double log(double x) +double +log (double x) { - double hfsq,f,s,z,R,w,t1,t2,dk; - int k,hx,i,j; - unsigned lx; + double hfsq, f, s, z, R, w, t1, t2, dk; + int k, hx, i, j; + unsigned lx; - hx = __HI(x); /* high word of x */ - lx = __LO(x); /* low word of x */ + hx = __HI (x); /* high word of x */ + lx = __LO (x); /* low word of x */ - k=0; - if (hx < 0x00100000) { /* x < 2**-1022 */ - if (((hx&0x7fffffff)|lx)==0) - return -two54/zero; /* log(+-0)=-inf */ - if (hx<0) return (x-x)/zero; /* log(-#) = NaN */ - k -= 54; x *= two54; /* subnormal number, scale up x */ - hx = __HI(x); /* high word of x */ - } - if (hx >= 0x7ff00000) return x+x; - k += (hx>>20)-1023; - hx &= 0x000fffff; - i = (hx+0x95f64)&0x100000; - __HI(x) = hx|(i^0x3ff00000); /* normalize x or x/2 */ - k += (i>>20); - f = x-1.0; - if((0x000fffff&(2+hx))<3) { /* |f| < 2**-20 */ - if(f==zero) if(k==0) return zero; else {dk=(double)k; - return dk*ln2_hi+dk*ln2_lo;} - R = f*f*(0.5-0.33333333333333333*f); - if(k==0) return f-R; else {dk=(double)k; - return dk*ln2_hi-((R-dk*ln2_lo)-f);} - } - s = f/(2.0+f); - dk = (double)k; - z = s*s; - i = hx-0x6147a; - w = z*z; - j = 0x6b851-hx; - t1= w*(Lg2+w*(Lg4+w*Lg6)); - t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7))); - i |= j; - R = t2+t1; - if(i>0) { - hfsq=0.5*f*f; - if(k==0) return f-(hfsq-s*(hfsq+R)); else - return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f); - } else { - if(k==0) return f-s*(f-R); else - return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f); - } -} + k = 0; + if (hx < 0x00100000) /* x < 2**-1022 */ + { + if (((hx & 0x7fffffff) | lx) == 0) /* log(+-0) = -inf */ + { + return -two54 / zero; + } + if (hx < 0) /* log(-#) = NaN */ + { + return (x - x) / zero; + } + k -= 54; + x *= two54; /* subnormal number, scale up x */ + hx = __HI (x); /* high word of x */ + } + if (hx >= 0x7ff00000) + { + return x + x; + } + k += (hx >> 20) - 1023; + hx &= 0x000fffff; + i = (hx + 0x95f64) & 0x100000; + __HI (x) = hx | (i ^ 0x3ff00000); /* normalize x or x / 2 */ + k += (i >> 20); + f = x - 1.0; + if ((0x000fffff & (2 + hx)) < 3) /* |f| < 2**-20 */ + { + if (f == zero) + { + if (k == 0) + { + return zero; + } + else + { + dk = (double) k; + return dk * ln2_hi + dk * ln2_lo; + } + } + R = f * f * (0.5 - 0.33333333333333333 * f); + if (k == 0) + { + return f - R; + } + else + { + dk = (double) k; + return dk * ln2_hi - ((R - dk * ln2_lo) - f); + } + } + s = f / (2.0 + f); + dk = (double) k; + z = s * s; + i = hx - 0x6147a; + w = z * z; + j = 0x6b851 - hx; + t1 = w * (Lg2 + w * (Lg4 + w * Lg6)); + t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7))); + i |= j; + R = t2 + t1; + if (i > 0) + { + hfsq = 0.5 * f * f; + if (k == 0) + { + return f - (hfsq - s * (hfsq + R)); + } + else + { + return dk * ln2_hi - ((hfsq - (s * (hfsq + R) + dk * ln2_lo)) - f); + } + } + else + { + if (k == 0) + { + return f - s * (f - R); + } + else + { + return dk * ln2_hi - ((s * (f - R) - dk * ln2_lo) - f); + } + } +} /* log */ diff --git a/third-party/fdlibm/s_pow.c b/third-party/fdlibm/s_pow.c index 7066eefd3..10fa7eb2d 100644 --- a/third-party/fdlibm/s_pow.c +++ b/third-party/fdlibm/s_pow.c @@ -5,50 +5,50 @@ * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved. * * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice + * software is freely granted, provided that this notice * is preserved. * ==================================================== */ /* pow(x,y) return x**y * - * n + * n * Method: Let x = 2 * (1+f) - * 1. Compute and return log2(x) in two pieces: - * log2(x) = w1 + w2, - * where w1 has 53-24 = 29 bit trailing zeros. - * 2. Perform y*log2(x) = n+y' by simulating muti-precision - * arithmetic, where |y'|<=0.5. - * 3. Return x**y = 2**n*exp(y'*log2) + * 1. Compute and return log2(x) in two pieces: + * log2(x) = w1 + w2, + * where w1 has 53-24 = 29 bit trailing zeros. + * 2. Perform y*log2(x) = n+y' by simulating muti-precision + * arithmetic, where |y'|<=0.5. + * 3. Return x**y = 2**n*exp(y'*log2) * * Special cases: - * 1. (anything) ** 0 is 1 - * 2. (anything) ** 1 is itself - * 3. (anything) ** NAN is NAN - * 4. NAN ** (anything except 0) is NAN - * 5. +-(|x| > 1) ** +INF is +INF - * 6. +-(|x| > 1) ** -INF is +0 - * 7. +-(|x| < 1) ** +INF is +0 - * 8. +-(|x| < 1) ** -INF is +INF - * 9. +-1 ** +-INF is NAN - * 10. +0 ** (+anything except 0, NAN) is +0 - * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 - * 12. +0 ** (-anything except 0, NAN) is +INF - * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF - * 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) - * 15. +INF ** (+anything except 0,NAN) is +INF - * 16. +INF ** (-anything except 0,NAN) is +0 - * 17. -INF ** (anything) = -0 ** (-anything) - * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) - * 19. (-anything except 0 and inf) ** (non-integer) is NAN + * 1. (anything) ** 0 is 1 + * 2. (anything) ** 1 is itself + * 3. (anything) ** NAN is NAN + * 4. NAN ** (anything except 0) is NAN + * 5. +-(|x| > 1) ** +INF is +INF + * 6. +-(|x| > 1) ** -INF is +0 + * 7. +-(|x| < 1) ** +INF is +0 + * 8. +-(|x| < 1) ** -INF is +INF + * 9. +-1 ** +-INF is NAN + * 10. +0 ** (+anything except 0, NAN) is +0 + * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 + * 12. +0 ** (-anything except 0, NAN) is +INF + * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF + * 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) + * 15. +INF ** (+anything except 0,NAN) is +INF + * 16. +INF ** (-anything except 0,NAN) is +0 + * 17. -INF ** (anything) = -0 ** (-anything) + * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) + * 19. (-anything except 0 and inf) ** (non-integer) is NAN * * Accuracy: - * pow(x,y) returns x**y nearly rounded. In particular - * pow(integer,integer) - * always returns the correct integer provided it is - * representable. + * pow(x,y) returns x**y nearly rounded. In particular + * pow(integer,integer) + * always returns the correct integer provided it is + * representable. * - * Constants : + * Constants: * The hexadecimal values are the intended ones for the following * constants. The decimal values may be used, provided that the * compiler will convert from decimal to binary accurately enough @@ -57,242 +57,371 @@ #include "fdlibm.h" -static const double -one = 1.0, -bp[] = {1.0, 1.5,}, -dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */ -dp_l[] = { 0.0, 1.35003920212974897128e-08,}; /* 0x3E4CFDEB, 0x43CFD006 */ +static const double one = 1.0; +static const double bp[] = +{ + 1.0, + 1.5, +}; +static const double dp_h[] = +{ + 0.0, + 5.84962487220764160156e-01, /* 0x3FE2B803, 0x40000000 */ +}; +static const double dp_l[] = +{ + 0.0, + 1.35003920212974897128e-08, /* 0x3E4CFDEB, 0x43CFD006 */ +}; #define zero 0.0 #define two 2.0 -#define two53 9007199254740992.0 /* 0x43400000, 0x00000000 */ +#define two53 9007199254740992.0 /* 0x43400000, 0x00000000 */ #define huge 1.0e300 #define tiny 1.0e-300 - /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ -#define L1 5.99999999999994648725e-01 /* 0x3FE33333, 0x33333303 */ -#define L2 4.28571428578550184252e-01 /* 0x3FDB6DB6, 0xDB6FABFF */ -#define L3 3.33333329818377432918e-01 /* 0x3FD55555, 0x518F264D */ -#define L4 2.72728123808534006489e-01 /* 0x3FD17460, 0xA91D4101 */ -#define L5 2.30660745775561754067e-01 /* 0x3FCD864A, 0x93C9DB65 */ -#define L6 2.06975017800338417784e-01 /* 0x3FCA7E28, 0x4A454EEF */ -#define P1 1.66666666666666019037e-01 /* 0x3FC55555, 0x5555553E */ -#define P2 -2.77777777770155933842e-03 /* 0xBF66C16C, 0x16BEBD93 */ -#define P3 6.61375632143793436117e-05 /* 0x3F11566A, 0xAF25DE2C */ -#define P4 -1.65339022054652515390e-06 /* 0xBEBBBD41, 0xC5D26BF1 */ -#define P5 4.13813679705723846039e-08 /* 0x3E663769, 0x72BEA4D0 */ -#define lg2 6.93147180559945286227e-01 /* 0x3FE62E42, 0xFEFA39EF */ -#define lg2_h 6.93147182464599609375e-01 /* 0x3FE62E43, 0x00000000 */ -#define lg2_l -1.90465429995776804525e-09 /* 0xBE205C61, 0x0CA86C39 */ +/* poly coefs for (3/2) * (log(x) - 2s - 2/3 * s**3 */ +#define L1 5.99999999999994648725e-01 /* 0x3FE33333, 0x33333303 */ +#define L2 4.28571428578550184252e-01 /* 0x3FDB6DB6, 0xDB6FABFF */ +#define L3 3.33333329818377432918e-01 /* 0x3FD55555, 0x518F264D */ +#define L4 2.72728123808534006489e-01 /* 0x3FD17460, 0xA91D4101 */ +#define L5 2.30660745775561754067e-01 /* 0x3FCD864A, 0x93C9DB65 */ +#define L6 2.06975017800338417784e-01 /* 0x3FCA7E28, 0x4A454EEF */ +#define P1 1.66666666666666019037e-01 /* 0x3FC55555, 0x5555553E */ +#define P2 -2.77777777770155933842e-03 /* 0xBF66C16C, 0x16BEBD93 */ +#define P3 6.61375632143793436117e-05 /* 0x3F11566A, 0xAF25DE2C */ +#define P4 -1.65339022054652515390e-06 /* 0xBEBBBD41, 0xC5D26BF1 */ +#define P5 4.13813679705723846039e-08 /* 0x3E663769, 0x72BEA4D0 */ +#define lg2 6.93147180559945286227e-01 /* 0x3FE62E42, 0xFEFA39EF */ +#define lg2_h 6.93147182464599609375e-01 /* 0x3FE62E43, 0x00000000 */ +#define lg2_l -1.90465429995776804525e-09 /* 0xBE205C61, 0x0CA86C39 */ #define ovt 8.0085662595372944372e-0017 /* -(1024-log2(ovfl+.5ulp)) */ -#define cp 9.61796693925975554329e-01 /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */ -#define cp_h 9.61796700954437255859e-01 /* 0x3FEEC709, 0xE0000000 =(float)cp */ -#define cp_l -7.02846165095275826516e-09 /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/ -#define ivln2 1.44269504088896338700e+00 /* 0x3FF71547, 0x652B82FE =1/ln2 */ -#define ivln2_h 1.44269502162933349609e+00 /* 0x3FF71547, 0x60000000 =24b 1/ln2*/ -#define ivln2_l 1.92596299112661746887e-08 /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/ +#define cp 9.61796693925975554329e-01 /* 0x3FEEC709, 0xDC3A03FD = 2 / (3 ln2) */ +#define cp_h 9.61796700954437255859e-01 /* 0x3FEEC709, 0xE0000000 = (float) cp */ +#define cp_l -7.02846165095275826516e-09 /* 0xBE3E2FE0, 0x145B01F5 = tail of cp_h */ +#define ivln2 1.44269504088896338700e+00 /* 0x3FF71547, 0x652B82FE = 1 / ln2 */ +#define ivln2_h 1.44269502162933349609e+00 /* 0x3FF71547, 0x60000000 = 24b 1 / ln2 */ +#define ivln2_l 1.92596299112661746887e-08 /* 0x3E54AE0B, 0xF85DDF44 = 1 / ln2 tail */ -double pow(double x, double y) +double +pow (double x, double y) { - double z,ax,z_h,z_l,p_h,p_l; - double y1,t1,t2,r,s,t,u,v,w; - int i0,i1,i,j,k,yisint,n; - int hx,hy,ix,iy; - unsigned lx,ly; + double z, ax, z_h, z_l, p_h, p_l; + double y1, t1, t2, r, s, t, u, v, w; + int i0, i1, i, j, k, yisint, n; + int hx, hy, ix, iy; + unsigned lx, ly; - i0 = ((*(int*)&one)>>29)^1; i1=1-i0; - hx = __HI(x); lx = __LO(x); - hy = __HI(y); ly = __LO(y); - ix = hx&0x7fffffff; iy = hy&0x7fffffff; + i0 = ((*(int *) &one) >> 29) ^ 1; + i1 = 1 - i0; + hx = __HI (x); + lx = __LO (x); + hy = __HI (y); + ly = __LO (y); + ix = hx & 0x7fffffff; + iy = hy & 0x7fffffff; - /* y==zero: x**0 = 1 */ - if((iy|ly)==0) return one; + /* y == zero: x**0 = 1 */ + if ((iy | ly) == 0) + { + return one; + } - /* +-NaN return x+y */ - if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) || - iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0))) - return x+y; + /* +-NaN return x + y */ + if (ix > 0x7ff00000 || ((ix == 0x7ff00000) && (lx != 0)) || iy > 0x7ff00000 || ((iy == 0x7ff00000) && (ly != 0))) + { + return x + y; + } - /* determine if y is an odd int when x < 0 - * yisint = 0 ... y is not an integer - * yisint = 1 ... y is an odd int - * yisint = 2 ... y is an even int - */ - yisint = 0; - if(hx<0) { - if(iy>=0x43400000) yisint = 2; /* even integer y */ - else if(iy>=0x3ff00000) { - k = (iy>>20)-0x3ff; /* exponent */ - if(k>20) { - j = ly>>(52-k); - if((j<<(52-k))==ly) yisint = 2-(j&1); - } else if(ly==0) { - j = iy>>(20-k); - if((j<<(20-k))==iy) yisint = 2-(j&1); - } - } - } + /* determine if y is an odd int when x < 0 + * yisint = 0 ... y is not an integer + * yisint = 1 ... y is an odd int + * yisint = 2 ... y is an even int + */ + yisint = 0; + if (hx < 0) + { + if (iy >= 0x43400000) /* even integer y */ + { + yisint = 2; + } + else if (iy >= 0x3ff00000) + { + k = (iy >> 20) - 0x3ff; /* exponent */ + if (k > 20) + { + j = ly >> (52 - k); + if ((j << (52 - k)) == ly) + { + yisint = 2 - (j & 1); + } + } + else if (ly == 0) + { + j = iy >> (20 - k); + if ((j << (20 - k)) == iy) + { + yisint = 2 - (j & 1); + } + } + } + } - /* special value of y */ - if(ly==0) { - if (iy==0x7ff00000) { /* y is +-inf */ - if(((ix-0x3ff00000)|lx)==0) - return y - y; /* inf**+-1 is NaN */ - else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */ - return (hy>=0)? y: zero; - else /* (|x|<1)**-,+inf = inf,0 */ - return (hy<0)?-y: zero; - } - if(iy==0x3ff00000) { /* y is +-1 */ - if(hy<0) return one/x; else return x; - } - if(hy==0x40000000) return x*x; /* y is 2 */ - if(hy==0x3fe00000) { /* y is 0.5 */ - if(hx>=0) /* x >= +0 */ - return sqrt(x); - } - } + /* special value of y */ + if (ly == 0) + { + if (iy == 0x7ff00000) /* y is +-inf */ + { + if (((ix - 0x3ff00000) | lx) == 0) /* inf**+-1 is NaN */ + { + return y - y; + } + else if (ix >= 0x3ff00000) /* (|x|>1)**+-inf = inf,0 */ + { + return (hy >= 0) ? y : zero; + } + else /* (|x|<1)**-,+inf = inf,0 */ + { + return (hy < 0) ? -y : zero; + } + } + if (iy == 0x3ff00000) /* y is +-1 */ + { + if (hy < 0) + { + return one / x; + } + else + { + return x; + } + } + if (hy == 0x40000000) /* y is 2 */ + { + return x * x; + } + if (hy == 0x3fe00000) /* y is 0.5 */ + { + if (hx >= 0) /* x >= +0 */ + { + return sqrt (x); + } + } + } - ax = fabs(x); - /* special value of x */ - if(lx==0) { - if(ix==0x7ff00000||ix==0||ix==0x3ff00000){ - z = ax; /*x is +-0,+-inf,+-1*/ - if(hy<0) z = one/z; /* z = (1/|x|) */ - if(hx<0) { - if(((ix-0x3ff00000)|yisint)==0) { - z = (z-z)/(z-z); /* (-1)**non-int is NaN */ - } else if(yisint==1) - z = -z; /* (x<0)**odd = -(|x|**odd) */ - } - return z; - } - } + ax = fabs (x); + /* special value of x */ + if (lx == 0) + { + if (ix == 0x7ff00000 || ix == 0 || ix == 0x3ff00000) + { + z = ax; /* x is +-0,+-inf,+-1 */ + if (hy < 0) + { + z = one / z; /* z = (1 / |x|) */ + } + if (hx < 0) + { + if (((ix - 0x3ff00000) | yisint) == 0) + { + z = (z - z) / (z - z); /* (-1)**non-int is NaN */ + } + else if (yisint == 1) + { + z = -z; /* (x<0)**odd = -(|x|**odd) */ + } + } + return z; + } + } - n = (hx>>31)+1; + n = (hx >> 31) + 1; - /* (x<0)**(non-int) is NaN */ - if((n|yisint)==0) return (x-x)/(x-x); + /* (x<0)**(non-int) is NaN */ + if ((n | yisint) == 0) + { + return (x - x) / (x - x); + } - s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ - if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */ + s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ + if ((n | (yisint - 1)) == 0) + { + s = -one; /* (-ve)**(odd int) */ + } - /* |y| is huge */ - if(iy>0x41e00000) { /* if |y| > 2**31 */ - if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */ - if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny; - if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny; - } - /* over/underflow if x is not close to one */ - if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny; - if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny; - /* now |1-x| is tiny <= 2**-20, suffice to compute - log(x) by x-x^2/2+x^3/3-x^4/4 */ - t = ax-one; /* t has 20 trailing zeros */ - w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25)); - u = ivln2_h*t; /* ivln2_h has 21 sig. bits */ - v = t*ivln2_l-w*ivln2; - t1 = u+v; - __LO(t1) = 0; - t2 = v-(t1-u); - } else { - double ss,s2,s_h,s_l,t_h,t_l; - n = 0; - /* take care subnormal number */ - if(ix<0x00100000) - {ax *= two53; n -= 53; ix = __HI(ax); } - n += ((ix)>>20)-0x3ff; - j = ix&0x000fffff; - /* determine interval */ - ix = j|0x3ff00000; /* normalize ix */ - if(j<=0x3988E) k=0; /* |x| 0x41e00000) /* if |y| > 2**31 */ + { + if (iy > 0x43f00000) /* if |y| > 2**64, must o/uflow */ + { + if (ix <= 0x3fefffff) + { + return (hy < 0) ? huge * huge : tiny * tiny; + } + if (ix >= 0x3ff00000) + { + return (hy > 0) ? huge * huge : tiny * tiny; + } + } + /* over/underflow if x is not close to one */ + if (ix < 0x3fefffff) + { + return (hy < 0) ? s * huge * huge : s * tiny * tiny; + } + if (ix > 0x3ff00000) + { + return (hy > 0) ? s * huge * huge : s * tiny * tiny; + } + /* now |1 - x| is tiny <= 2**-20, suffice to compute + log(x) by x - x^2 / 2 + x^3 / 3 - x^4 / 4 */ + t = ax - one; /* t has 20 trailing zeros */ + w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25)); + u = ivln2_h * t; /* ivln2_h has 21 sig. bits */ + v = t * ivln2_l - w * ivln2; + t1 = u + v; + __LO (t1) = 0; + t2 = v - (t1 - u); + } + else + { + double ss, s2, s_h, s_l, t_h, t_l; - /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ - u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ - v = one/(ax+bp[k]); - ss = u*v; - s_h = ss; - __LO(s_h) = 0; - /* t_h=ax+bp[k] High */ - t_h = zero; - __HI(t_h)=((ix>>1)|0x20000000)+0x00080000+(k<<18); - t_l = ax - (t_h-bp[k]); - s_l = v*((u-s_h*t_h)-s_h*t_l); - /* compute log(ax) */ - s2 = ss*ss; - r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); - r += s_l*(s_h+ss); - s2 = s_h*s_h; - t_h = 3.0+s2+r; - __LO(t_h) = 0; - t_l = r-((t_h-3.0)-s2); - /* u+v = ss*(1+...) */ - u = s_h*t_h; - v = s_l*t_h+t_l*ss; - /* 2/(3log2)*(ss+...) */ - p_h = u+v; - __LO(p_h) = 0; - p_l = v-(p_h-u); - z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */ - z_l = cp_l*p_h+p_l*cp+dp_l[k]; - /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */ - t = (double)n; - t1 = (((z_h+z_l)+dp_h[k])+t); - __LO(t1) = 0; - t2 = z_l-(((t1-t)-dp_h[k])-z_h); - } + n = 0; + /* take care subnormal number */ + if (ix < 0x00100000) + { + ax *= two53; + n -= 53; + ix = __HI (ax); + } + n += ((ix) >> 20) - 0x3ff; + j = ix & 0x000fffff; + /* determine interval */ + ix = j | 0x3ff00000; /* normalize ix */ + if (j <= 0x3988E) /* |x| < sqrt(3/2) */ + { + k = 0; + } + else if (j < 0xBB67A) /* |x| < sqrt(3) */ + { + k = 1; + } + else + { + k = 0; + n += 1; + ix -= 0x00100000; + } + __HI (ax) = ix; - /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ - y1 = y; - __LO(y1) = 0; - p_l = (y-y1)*t1+y*t2; - p_h = y1*t1; - z = p_l+p_h; - j = __HI(z); - i = __LO(z); - if (j>=0x40900000) { /* z >= 1024 */ - if(((j-0x40900000)|i)!=0) /* if z > 1024 */ - return s*huge*huge; /* overflow */ - else { - if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */ - } - } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */ - if(((j-0xc090cc00)|i)!=0) /* z < -1075 */ - return s*tiny*tiny; /* underflow */ - else { - if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */ - } - } - /* - * compute 2**(p_h+p_l) - */ - i = j&0x7fffffff; - k = (i>>20)-0x3ff; - n = 0; - if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */ - n = j+(0x00100000>>(k+1)); - k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */ - t = zero; - __HI(t) = (n&~(0x000fffff>>k)); - n = ((n&0x000fffff)|0x00100000)>>(20-k); - if(j<0) n = -n; - p_h -= t; - } - t = p_l+p_h; - __LO(t) = 0; - u = t*lg2_h; - v = (p_l-(t-p_h))*lg2+t*lg2_l; - z = u+v; - w = v-(z-u); - t = z*z; - t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); - r = (z*t1)/(t1-two)-(w+z*w); - z = one-(r-z); - j = __HI(z); - j += (n<<20); - if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */ - else __HI(z) += (n<<20); - return s*z; -} + /* compute ss = s_h + s_l = (x - 1) / (x + 1) or (x - 1.5) / (x + 1.5) */ + u = ax - bp[k]; /* bp[0] = 1.0, bp[1] = 1.5 */ + v = one / (ax + bp[k]); + ss = u * v; + s_h = ss; + __LO (s_h) = 0; + /* t_h = ax + bp[k] High */ + t_h = zero; + __HI (t_h) = ((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18); + t_l = ax - (t_h - bp[k]); + s_l = v * ((u - s_h * t_h) - s_h * t_l); + /* compute log(ax) */ + s2 = ss * ss; + r = s2 * s2 * (L1 + s2 * (L2 + s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6))))); + r += s_l * (s_h + ss); + s2 = s_h * s_h; + t_h = 3.0 + s2 + r; + __LO (t_h) = 0; + t_l = r - ((t_h - 3.0) - s2); + /* u + v = ss * (1 + ...) */ + u = s_h * t_h; + v = s_l * t_h + t_l * ss; + /* 2 / (3 * log2) * (ss + ...) */ + p_h = u + v; + __LO (p_h) = 0; + p_l = v - (p_h - u); + z_h = cp_h * p_h; /* cp_h + cp_l = 2 / (3 * log2) */ + z_l = cp_l * p_h + p_l * cp + dp_l[k]; + /* log2(ax) = (ss + ...) * 2 / (3 * log2) = n + dp_h + z_h + z_l */ + t = (double) n; + t1 = (((z_h + z_l) + dp_h[k]) + t); + __LO (t1) = 0; + t2 = z_l - (((t1 - t) - dp_h[k]) - z_h); + } + + /* split up y into y1 + y2 and compute (y1 + y2) * (t1 + t2) */ + y1 = y; + __LO (y1) = 0; + p_l = (y - y1) * t1 + y * t2; + p_h = y1 * t1; + z = p_l + p_h; + j = __HI (z); + i = __LO (z); + if (j >= 0x40900000) /* z >= 1024 */ + { + if (((j - 0x40900000) | i) != 0) /* if z > 1024 */ + { + return s * huge * huge; /* overflow */ + } + else + { + if (p_l + ovt > z - p_h) + { + return s * huge * huge; /* overflow */ + } + } + } + else if ((j & 0x7fffffff) >= 0x4090cc00) /* z <= -1075 */ + { + if (((j - 0xc090cc00) | i) != 0) /* z < -1075 */ + { + return s * tiny * tiny; /* underflow */ + } + else + { + if (p_l <= z - p_h) + { + return s * tiny * tiny; /* underflow */ + } + } + } + /* + * compute 2**(p_h + p_l) + */ + i = j & 0x7fffffff; + k = (i >> 20) - 0x3ff; + n = 0; + if (i > 0x3fe00000) /* if |z| > 0.5, set n = [z + 0.5] */ + { + n = j + (0x00100000 >> (k + 1)); + k = ((n & 0x7fffffff) >> 20) - 0x3ff; /* new k for n */ + t = zero; + __HI (t) = (n & ~(0x000fffff >> k)); + n = ((n & 0x000fffff) | 0x00100000) >> (20 - k); + if (j < 0) + { + n = -n; + } + p_h -= t; + } + t = p_l + p_h; + __LO (t) = 0; + u = t * lg2_h; + v = (p_l - (t - p_h)) * lg2 + t * lg2_l; + z = u + v; + w = v - (z - u); + t = z * z; + t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5)))); + r = (z * t1) / (t1 - two) - (w + z * w); + z = one - (r - z); + j = __HI (z); + j += (n << 20); + if ((j >> 20) <= 0) /* subnormal output */ + { + z = scalbn (z, n); + } + else + { + __HI (z) += (n << 20); + } + return s * z; +} /* pow */ diff --git a/third-party/fdlibm/s_scalbn.c b/third-party/fdlibm/s_scalbn.c index 1f5dff570..d7b437404 100644 --- a/third-party/fdlibm/s_scalbn.c +++ b/third-party/fdlibm/s_scalbn.c @@ -6,14 +6,12 @@ * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice + * software is freely granted, provided that this notice * is preserved. * ==================================================== */ -/* - * scalbn (double x, int n) - * scalbn(x,n) returns x* 2**n computed by exponent +/* scalbn(x,n) returns x* 2**n computed by exponent * manipulation rather than by actually performing an * exponentiation or a multiplication. */ @@ -25,29 +23,54 @@ #define huge 1.0e+300 #define tiny 1.0e-300 -double scalbn (double x, int n) +double +scalbn (double x, int n) { - int k,hx,lx; - hx = __HI(x); - lx = __LO(x); - k = (hx&0x7ff00000)>>20; /* extract exponent */ - if (k==0) { /* 0 or subnormal x */ - if ((lx|(hx&0x7fffffff))==0) return x; /* +-0 */ - x *= two54; - hx = __HI(x); - k = ((hx&0x7ff00000)>>20) - 54; - if (n< -50000) return tiny*x; /*underflow*/ - } - if (k==0x7ff) return x+x; /* NaN or Inf */ - k = k+n; - if (k > 0x7fe) return huge*copysign(huge,x); /* overflow */ - if (k > 0) /* normal result */ - {__HI(x) = (hx&0x800fffff)|(k<<20); return x;} - if (k <= -54) - if (n > 50000) /* in case integer overflow in n+k */ - return huge*copysign(huge,x); /*overflow*/ - else return tiny*copysign(tiny,x); /*underflow*/ - k += 54; /* subnormal result */ - __HI(x) = (hx&0x800fffff)|(k<<20); - return x*twom54; -} + int k, hx, lx; + + hx = __HI (x); + lx = __LO (x); + k = (hx & 0x7ff00000) >> 20; /* extract exponent */ + if (k == 0) /* 0 or subnormal x */ + { + if ((lx | (hx & 0x7fffffff)) == 0) /* +-0 */ + { + return x; + } + x *= two54; + hx = __HI (x); + k = ((hx & 0x7ff00000) >> 20) - 54; + if (n < -50000) /*underflow */ + { + return tiny * x; + } + } + if (k == 0x7ff) /* NaN or Inf */ + { + return x + x; + } + k = k + n; + if (k > 0x7fe) /* overflow */ + { + return huge * copysign (huge, x); + } + if (k > 0) /* normal result */ + { + __HI (x) = (hx & 0x800fffff) | (k << 20); + return x; + } + if (k <= -54) + { + if (n > 50000) /* in case integer overflow in n + k */ + { + return huge * copysign (huge, x); /*overflow */ + } + else + { + return tiny * copysign (tiny, x); /*underflow */ + } + } + k += 54; /* subnormal result */ + __HI (x) = (hx & 0x800fffff) | (k << 20); + return x * twom54; +} /* scalbn */ diff --git a/third-party/fdlibm/s_sqrt.c b/third-party/fdlibm/s_sqrt.c index 2baf9bc58..82afe2138 100644 --- a/third-party/fdlibm/s_sqrt.c +++ b/third-party/fdlibm/s_sqrt.c @@ -6,79 +6,80 @@ * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice + * software is freely granted, provided that this notice * is preserved. * ==================================================== */ /* sqrt(x) * Return correctly rounded sqrt. + * * ------------------------------------------ - * | Use the hardware sqrt if you have one | + * | Use the hardware sqrt if you have one | * ------------------------------------------ + * * Method: * Bit by bit method using integer arithmetic. (Slow, but portable) * 1. Normalization - * Scale x to y in [1,4) with even powers of 2: - * find an integer k such that 1 <= (y=x*2^(2k)) < 4, then - * sqrt(x) = 2^k * sqrt(y) + * Scale x to y in [1,4) with even powers of 2: + * find an integer k such that 1 <= (y=x*2^(2k)) < 4, then + * sqrt(x) = 2^k * sqrt(y) * 2. Bit by bit computation - * Let q = sqrt(y) truncated to i bit after binary point (q = 1), - * i 0 + * Let q = sqrt(y) truncated to i bit after binary point (q = 1), + * i 0 * i+1 2 - * s = 2*q , and y = 2 * ( y - q ). (1) - * i i i i + * s = 2*q , and y = 2 * ( y - q ). (1) + * i i i i * - * To compute q from q , one checks whether - * i+1 i + * To compute q from q , one checks whether + * i+1 i * - * -(i+1) 2 - * (q + 2 ) <= y. (2) - * i - * -(i+1) - * If (2) is false, then q = q ; otherwise q = q + 2 . - * i+1 i i+1 i + * -(i+1) 2 + * (q + 2 ) <= y. (2) + * i + * -(i+1) + * If (2) is false, then q = q ; otherwise q = q + 2 . + * i+1 i i+1 i * - * With some algebric manipulation, it is not difficult to see - * that (2) is equivalent to + * With some algebric manipulation, it is not difficult to see + * that (2) is equivalent to * -(i+1) - * s + 2 <= y (3) - * i i + * s + 2 <= y (3) + * i i * - * The advantage of (3) is that s and y can be computed by - * i i - * the following recurrence formula: - * if (3) is false + * The advantage of (3) is that s and y can be computed by + * i i + * the following recurrence formula: + * if (3) is false * - * s = s , y = y ; (4) - * i+1 i i+1 i + * s = s , y = y ; (4) + * i+1 i i+1 i * - * otherwise, + * otherwise, * -i -(i+1) - * s = s + 2 , y = y - s - 2 (5) + * s = s + 2 , y = y - s - 2 (5) * i+1 i i+1 i i * - * One may easily use induction to prove (4) and (5). - * Note. Since the left hand side of (3) contain only i+2 bits, - * it does not necessary to do a full (53-bit) comparison - * in (3). + * One may easily use induction to prove (4) and (5). + * Note. Since the left hand side of (3) contain only i+2 bits, + * it does not necessary to do a full (53-bit) comparison + * in (3). * 3. Final rounding - * After generating the 53 bits result, we compute one more bit. - * Together with the remainder, we can decide whether the - * result is exact, bigger than 1/2ulp, or less than 1/2ulp - * (it will never equal to 1/2ulp). - * The rounding mode can be detected by checking whether - * huge + tiny is equal to huge, and whether huge - tiny is - * equal to huge for some floating point number "huge" and "tiny". + * After generating the 53 bits result, we compute one more bit. + * Together with the remainder, we can decide whether the + * result is exact, bigger than 1/2ulp, or less than 1/2ulp + * (it will never equal to 1/2ulp). + * The rounding mode can be detected by checking whether + * huge + tiny is equal to huge, and whether huge - tiny is + * equal to huge for some floating point number "huge" and "tiny". * * Special cases: - * sqrt(+-0) = +-0 ... exact - * sqrt(inf) = inf - * sqrt(-ve) = NaN ... with invalid signal - * sqrt(NaN) = NaN ... with invalid signal for signaling NaN + * sqrt(+-0) = +-0 ... exact + * sqrt(inf) = inf + * sqrt(-ve) = NaN ... with invalid signal + * sqrt(NaN) = NaN ... with invalid signal for signaling NaN * - * Other methods : see the appended file at the end of the program below. - *--------------- + * Other methods: see the appended file at the end of the program below. */ #include "fdlibm.h" @@ -86,103 +87,143 @@ #define one 1.0 #define tiny 1.0e-300 -double sqrt(double x) +double +sqrt (double x) { - double z; - int sign = (int)0x80000000; - unsigned r,t1,s1,ix1,q1; - int ix0,s0,q,m,t,i; + double z; + int sign = (int) 0x80000000; + unsigned r, t1, s1, ix1, q1; + int ix0, s0, q, m, t, i; - ix0 = __HI(x); /* high word of x */ - ix1 = __LO(x); /* low word of x */ + ix0 = __HI (x); /* high word of x */ + ix1 = __LO (x); /* low word of x */ - /* take care of Inf and NaN */ - if((ix0&0x7ff00000)==0x7ff00000) { - return x*x+x; /* sqrt(NaN)=NaN, sqrt(+inf)=+inf - sqrt(-inf)=sNaN */ - } - /* take care of zero */ - if(ix0<=0) { - if(((ix0&(~sign))|ix1)==0) return x;/* sqrt(+-0) = +-0 */ - else if(ix0<0) - return (x-x)/(x-x); /* sqrt(-ve) = sNaN */ - } - /* normalize x */ - m = (ix0>>20); - if(m==0) { /* subnormal x */ - while(ix0==0) { - m -= 21; - ix0 |= (ix1>>11); ix1 <<= 21; - } - for(i=0;(ix0&0x00100000)==0;i++) ix0<<=1; - m -= i-1; - ix0 |= (ix1>>(32-i)); - ix1 <<= i; - } - m -= 1023; /* unbias exponent */ - ix0 = (ix0&0x000fffff)|0x00100000; - if(m&1){ /* odd m, double x to make it even */ - ix0 += ix0 + ((ix1&sign)>>31); - ix1 += ix1; - } - m >>= 1; /* m = [m/2] */ + /* take care of Inf and NaN */ + if ((ix0 & 0x7ff00000) == 0x7ff00000) + { + return x * x + x; /* sqrt(NaN) = NaN, sqrt(+inf) = +inf, sqrt(-inf) = sNaN */ + } + /* take care of zero */ + if (ix0 <= 0) + { + if (((ix0 & (~sign)) | ix1) == 0) /* sqrt(+-0) = +-0 */ + { + return x; + } + else if (ix0 < 0) /* sqrt(-ve) = sNaN */ + { + return (x - x) / (x - x); + } + } + /* normalize x */ + m = (ix0 >> 20); + if (m == 0) /* subnormal x */ + { + while (ix0 == 0) + { + m -= 21; + ix0 |= (ix1 >> 11); + ix1 <<= 21; + } + for (i = 0; (ix0 & 0x00100000) == 0; i++) + { + ix0 <<= 1; + } + m -= i - 1; + ix0 |= (ix1 >> (32 - i)); + ix1 <<= i; + } + m -= 1023; /* unbias exponent */ + ix0 = (ix0 & 0x000fffff) | 0x00100000; + if (m & 1) /* odd m, double x to make it even */ + { + ix0 += ix0 + ((ix1 & sign) >> 31); + ix1 += ix1; + } + m >>= 1; /* m = [m / 2] */ - /* generate sqrt(x) bit by bit */ - ix0 += ix0 + ((ix1&sign)>>31); - ix1 += ix1; - q = q1 = s0 = s1 = 0; /* [q,q1] = sqrt(x) */ - r = 0x00200000; /* r = moving bit from right to left */ + /* generate sqrt(x) bit by bit */ + ix0 += ix0 + ((ix1 & sign) >> 31); + ix1 += ix1; + q = q1 = s0 = s1 = 0; /* [q,q1] = sqrt(x) */ + r = 0x00200000; /* r = moving bit from right to left */ - while(r!=0) { - t = s0+r; - if(t<=ix0) { - s0 = t+r; - ix0 -= t; - q += r; - } - ix0 += ix0 + ((ix1&sign)>>31); - ix1 += ix1; - r>>=1; - } + while (r != 0) + { + t = s0 + r; + if (t <= ix0) + { + s0 = t + r; + ix0 -= t; + q += r; + } + ix0 += ix0 + ((ix1 & sign) >> 31); + ix1 += ix1; + r >>= 1; + } - r = sign; - while(r!=0) { - t1 = s1+r; - t = s0; - if((t>31); - ix1 += ix1; - r>>=1; - } + r = sign; + while (r != 0) + { + t1 = s1 + r; + t = s0; + if ((t < ix0) || ((t == ix0) && (t1 <= ix1))) + { + s1 = t1 + r; + if (((t1 & sign) == sign) && (s1 & sign) == 0) + { + s0 += 1; + } + ix0 -= t; + if (ix1 < t1) + { + ix0 -= 1; + } + ix1 -= t1; + q1 += r; + } + ix0 += ix0 + ((ix1 & sign) >> 31); + ix1 += ix1; + r >>= 1; + } - /* use floating add to find out rounding direction */ - if((ix0|ix1)!=0) { - z = one-tiny; /* trigger inexact flag */ - if (z>=one) { - z = one+tiny; - if (q1==(unsigned)0xffffffff) { q1=0; q += 1;} - else if (z>one) { - if (q1==(unsigned)0xfffffffe) q+=1; - q1+=2; - } else - q1 += (q1&1); - } - } - ix0 = (q>>1)+0x3fe00000; - ix1 = q1>>1; - if ((q&1)==1) ix1 |= sign; - ix0 += (m <<20); - __HI(z) = ix0; - __LO(z) = ix1; - return z; -} + /* use floating add to find out rounding direction */ + if ((ix0 | ix1) != 0) + { + z = one - tiny; /* trigger inexact flag */ + if (z >= one) + { + z = one + tiny; + if (q1 == (unsigned) 0xffffffff) + { + q1 = 0; + q += 1; + } + else if (z > one) + { + if (q1 == (unsigned) 0xfffffffe) + { + q += 1; + } + q1 += 2; + } + else + { + q1 += (q1 & 1); + } + } + } + ix0 = (q >> 1) + 0x3fe00000; + ix1 = q1 >> 1; + if ((q & 1) == 1) + { + ix1 |= sign; + } + ix0 += (m << 20); + __HI (z) = ix0; + __LO (z) = ix1; + return z; +} /* sqrt */ /* Other methods (use floating-point arithmetic) @@ -190,253 +231,252 @@ Other methods (use floating-point arithmetic) (This is a copy of a drafted paper by Prof W. Kahan and K.C. Ng, written in May, 1986) - Two algorithms are given here to implement sqrt(x) - (IEEE double precision arithmetic) in software. - Both supply sqrt(x) correctly rounded. The first algorithm (in - Section A) uses newton iterations and involves four divisions. - The second one uses reciproot iterations to avoid division, but - requires more multiplications. Both algorithms need the ability - to chop results of arithmetic operations instead of round them, - and the INEXACT flag to indicate when an arithmetic operation - is executed exactly with no roundoff error, all part of the - standard (IEEE 754-1985). The ability to perform shift, add, - subtract and logical AND operations upon 32-bit words is needed - too, though not part of the standard. + Two algorithms are given here to implement sqrt(x) + (IEEE double precision arithmetic) in software. + Both supply sqrt(x) correctly rounded. The first algorithm (in + Section A) uses newton iterations and involves four divisions. + The second one uses reciproot iterations to avoid division, but + requires more multiplications. Both algorithms need the ability + to chop results of arithmetic operations instead of round them, + and the INEXACT flag to indicate when an arithmetic operation + is executed exactly with no roundoff error, all part of the + standard (IEEE 754-1985). The ability to perform shift, add, + subtract and logical AND operations upon 32-bit words is needed + too, though not part of the standard. A. sqrt(x) by Newton Iteration - (1) Initial approximation + (1) Initial approximation - Let x0 and x1 be the leading and the trailing 32-bit words of - a floating point number x (in IEEE double format) respectively + Let x0 and x1 be the leading and the trailing 32-bit words of + a floating point number x (in IEEE double format) respectively - 1 11 52 ...widths - ------------------------------------------------------ - x: |s| e | f | - ------------------------------------------------------ - msb lsb msb lsb ...order + 1 11 52 ...widths + ------------------------------------------------------ + x: |s| e | f | + ------------------------------------------------------ + msb lsb msb lsb ...order - ------------------------ ------------------------ - x0: |s| e | f1 | x1: | f2 | - ------------------------ ------------------------ + ------------------------ ------------------------ + x0: |s| e | f1 | x1: | f2 | + ------------------------ ------------------------ - By performing shifts and subtracts on x0 and x1 (both regarded - as integers), we obtain an 8-bit approximation of sqrt(x) as - follows. + By performing shifts and subtracts on x0 and x1 (both regarded + as integers), we obtain an 8-bit approximation of sqrt(x) as + follows. - k := (x0>>1) + 0x1ff80000; - y0 := k - T1[31&(k>>15)]. ... y ~ sqrt(x) to 8 bits - Here k is a 32-bit integer and T1[] is an integer array containing - correction terms. Now magically the floating value of y (y's - leading 32-bit word is y0, the value of its trailing word is 0) - approximates sqrt(x) to almost 8-bit. + k := (x0>>1) + 0x1ff80000; + y0 := k - T1[31&(k>>15)]. ... y ~ sqrt(x) to 8 bits + Here k is a 32-bit integer and T1[] is an integer array containing + correction terms. Now magically the floating value of y (y's + leading 32-bit word is y0, the value of its trailing word is 0) + approximates sqrt(x) to almost 8-bit. - Value of T1: - static int T1[32]= { - 0, 1024, 3062, 5746, 9193, 13348, 18162, 23592, - 29598, 36145, 43202, 50740, 58733, 67158, 75992, 85215, - 83599, 71378, 60428, 50647, 41945, 34246, 27478, 21581, - 16499, 12183, 8588, 5674, 3403, 1742, 661, 130,}; + Value of T1: + static int T1[32]= { + 0, 1024, 3062, 5746, 9193, 13348, 18162, 23592, + 29598, 36145, 43202, 50740, 58733, 67158, 75992, 85215, + 83599, 71378, 60428, 50647, 41945, 34246, 27478, 21581, + 16499, 12183, 8588, 5674, 3403, 1742, 661, 130,}; - (2) Iterative refinement + (2) Iterative refinement - Apply Heron's rule three times to y, we have y approximates - sqrt(x) to within 1 ulp (Unit in the Last Place): + Apply Heron's rule three times to y, we have y approximates + sqrt(x) to within 1 ulp (Unit in the Last Place): - y := (y+x/y)/2 ... almost 17 sig. bits - y := (y+x/y)/2 ... almost 35 sig. bits - y := y-(y-x/y)/2 ... within 1 ulp + y := (y+x/y)/2 ... almost 17 sig. bits + y := (y+x/y)/2 ... almost 35 sig. bits + y := y-(y-x/y)/2 ... within 1 ulp - Remark 1. - Another way to improve y to within 1 ulp is: + Remark 1. + Another way to improve y to within 1 ulp is: - y := (y+x/y) ... almost 17 sig. bits to 2*sqrt(x) - y := y - 0x00100006 ... almost 18 sig. bits to sqrt(x) + y := (y+x/y) ... almost 17 sig. bits to 2*sqrt(x) + y := y - 0x00100006 ... almost 18 sig. bits to sqrt(x) - 2 - (x-y )*y - y := y + 2* ---------- ...within 1 ulp - 2 - 3y + x + 2 + (x-y )*y + y := y + 2* ---------- ...within 1 ulp + 2 + 3y + x - This formula has one division fewer than the one above; however, - it requires more multiplications and additions. Also x must be - scaled in advance to avoid spurious overflow in evaluating the - expression 3y*y+x. Hence it is not recommended uless division - is slow. If division is very slow, then one should use the - reciproot algorithm given in section B. + This formula has one division fewer than the one above; however, + it requires more multiplications and additions. Also x must be + scaled in advance to avoid spurious overflow in evaluating the + expression 3y*y+x. Hence it is not recommended uless division + is slow. If division is very slow, then one should use the + reciproot algorithm given in section B. (3) Final adjustment - By twiddling y's last bit it is possible to force y to be - correctly rounded according to the prevailing rounding mode - as follows. Let r and i be copies of the rounding mode and - inexact flag before entering the square root program. Also we - use the expression y+-ulp for the next representable floating - numbers (up and down) of y. Note that y+-ulp = either fixed - point y+-1, or multiply y by nextafter(1,+-inf) in chopped - mode. + By twiddling y's last bit it is possible to force y to be + correctly rounded according to the prevailing rounding mode + as follows. Let r and i be copies of the rounding mode and + inexact flag before entering the square root program. Also we + use the expression y+-ulp for the next representable floating + numbers (up and down) of y. Note that y+-ulp = either fixed + point y+-1, or multiply y by nextafter(1,+-inf) in chopped + mode. - I := FALSE; ... reset INEXACT flag I - R := RZ; ... set rounding mode to round-toward-zero - z := x/y; ... chopped quotient, possibly inexact - If(not I) then { ... if the quotient is exact - if(z=y) { - I := i; ... restore inexact flag - R := r; ... restore rounded mode - return sqrt(x):=y. - } else { - z := z - ulp; ... special rounding - } - } - i := TRUE; ... sqrt(x) is inexact - If (r=RN) then z=z+ulp ... rounded-to-nearest - If (r=RP) then { ... round-toward-+inf - y = y+ulp; z=z+ulp; - } - y := y+z; ... chopped sum - y0:=y0-0x00100000; ... y := y/2 is correctly rounded. - I := i; ... restore inexact flag - R := r; ... restore rounded mode - return sqrt(x):=y. + I := FALSE; ... reset INEXACT flag I + R := RZ; ... set rounding mode to round-toward-zero + z := x/y; ... chopped quotient, possibly inexact + If(not I) then { ... if the quotient is exact + if(z=y) { + I := i; ... restore inexact flag + R := r; ... restore rounded mode + return sqrt(x):=y. + } else { + z := z - ulp; ... special rounding + } + } + i := TRUE; ... sqrt(x) is inexact + If (r=RN) then z=z+ulp ... rounded-to-nearest + If (r=RP) then { ... round-toward-+inf + y = y+ulp; z=z+ulp; + } + y := y+z; ... chopped sum + y0:=y0-0x00100000; ... y := y/2 is correctly rounded. + I := i; ... restore inexact flag + R := r; ... restore rounded mode + return sqrt(x):=y. - (4) Special cases + (4) Special cases - Square root of +inf, +-0, or NaN is itself; - Square root of a negative number is NaN with invalid signal. + Square root of +inf, +-0, or NaN is itself; + Square root of a negative number is NaN with invalid signal. B. sqrt(x) by Reciproot Iteration - (1) Initial approximation + (1) Initial approximation - Let x0 and x1 be the leading and the trailing 32-bit words of - a floating point number x (in IEEE double format) respectively - (see section A). By performing shifs and subtracts on x0 and y0, - we obtain a 7.8-bit approximation of 1/sqrt(x) as follows. + Let x0 and x1 be the leading and the trailing 32-bit words of + a floating point number x (in IEEE double format) respectively + (see section A). By performing shifs and subtracts on x0 and y0, + we obtain a 7.8-bit approximation of 1/sqrt(x) as follows. - k := 0x5fe80000 - (x0>>1); - y0:= k - T2[63&(k>>14)]. ... y ~ 1/sqrt(x) to 7.8 bits + k := 0x5fe80000 - (x0>>1); + y0:= k - T2[63&(k>>14)]. ... y ~ 1/sqrt(x) to 7.8 bits - Here k is a 32-bit integer and T2[] is an integer array - containing correction terms. Now magically the floating - value of y (y's leading 32-bit word is y0, the value of - its trailing word y1 is set to zero) approximates 1/sqrt(x) - to almost 7.8-bit. + Here k is a 32-bit integer and T2[] is an integer array + containing correction terms. Now magically the floating + value of y (y's leading 32-bit word is y0, the value of + its trailing word y1 is set to zero) approximates 1/sqrt(x) + to almost 7.8-bit. - Value of T2: - static int T2[64]= { - 0x1500, 0x2ef8, 0x4d67, 0x6b02, 0x87be, 0xa395, 0xbe7a, 0xd866, - 0xf14a, 0x1091b,0x11fcd,0x13552,0x14999,0x15c98,0x16e34,0x17e5f, - 0x18d03,0x19a01,0x1a545,0x1ae8a,0x1b5c4,0x1bb01,0x1bfde,0x1c28d, - 0x1c2de,0x1c0db,0x1ba73,0x1b11c,0x1a4b5,0x1953d,0x18266,0x16be0, - 0x1683e,0x179d8,0x18a4d,0x19992,0x1a789,0x1b445,0x1bf61,0x1c989, - 0x1d16d,0x1d77b,0x1dddf,0x1e2ad,0x1e5bf,0x1e6e8,0x1e654,0x1e3cd, - 0x1df2a,0x1d635,0x1cb16,0x1be2c,0x1ae4e,0x19bde,0x1868e,0x16e2e, - 0x1527f,0x1334a,0x11051,0xe951, 0xbe01, 0x8e0d, 0x5924, 0x1edd,}; + Value of T2: + static int T2[64]= { + 0x1500, 0x2ef8, 0x4d67, 0x6b02, 0x87be, 0xa395, 0xbe7a, 0xd866, + 0xf14a, 0x1091b,0x11fcd,0x13552,0x14999,0x15c98,0x16e34,0x17e5f, + 0x18d03,0x19a01,0x1a545,0x1ae8a,0x1b5c4,0x1bb01,0x1bfde,0x1c28d, + 0x1c2de,0x1c0db,0x1ba73,0x1b11c,0x1a4b5,0x1953d,0x18266,0x16be0, + 0x1683e,0x179d8,0x18a4d,0x19992,0x1a789,0x1b445,0x1bf61,0x1c989, + 0x1d16d,0x1d77b,0x1dddf,0x1e2ad,0x1e5bf,0x1e6e8,0x1e654,0x1e3cd, + 0x1df2a,0x1d635,0x1cb16,0x1be2c,0x1ae4e,0x19bde,0x1868e,0x16e2e, + 0x1527f,0x1334a,0x11051,0xe951, 0xbe01, 0x8e0d, 0x5924, 0x1edd,}; - (2) Iterative refinement + (2) Iterative refinement - Apply Reciproot iteration three times to y and multiply the - result by x to get an approximation z that matches sqrt(x) - to about 1 ulp. To be exact, we will have - -1ulp < sqrt(x)-z<1.0625ulp. + Apply Reciproot iteration three times to y and multiply the + result by x to get an approximation z that matches sqrt(x) + to about 1 ulp. To be exact, we will have + -1ulp < sqrt(x)-z<1.0625ulp. - ... set rounding mode to Round-to-nearest - y := y*(1.5-0.5*x*y*y) ... almost 15 sig. bits to 1/sqrt(x) - y := y*((1.5-2^-30)+0.5*x*y*y)... about 29 sig. bits to 1/sqrt(x) - ... special arrangement for better accuracy - z := x*y ... 29 bits to sqrt(x), with z*y<1 - z := z + 0.5*z*(1-z*y) ... about 1 ulp to sqrt(x) + ... set rounding mode to Round-to-nearest + y := y*(1.5-0.5*x*y*y) ... almost 15 sig. bits to 1/sqrt(x) + y := y*((1.5-2^-30)+0.5*x*y*y)... about 29 sig. bits to 1/sqrt(x) + ... special arrangement for better accuracy + z := x*y ... 29 bits to sqrt(x), with z*y<1 + z := z + 0.5*z*(1-z*y) ... about 1 ulp to sqrt(x) - Remark 2. The constant 1.5-2^-30 is chosen to bias the error so that - (a) the term z*y in the final iteration is always less than 1; - (b) the error in the final result is biased upward so that - -1 ulp < sqrt(x) - z < 1.0625 ulp - instead of |sqrt(x)-z|<1.03125ulp. + Remark 2. The constant 1.5-2^-30 is chosen to bias the error so that + (a) the term z*y in the final iteration is always less than 1; + (b) the error in the final result is biased upward so that + -1 ulp < sqrt(x) - z < 1.0625 ulp + instead of |sqrt(x)-z|<1.03125ulp. - (3) Final adjustment + (3) Final adjustment - By twiddling y's last bit it is possible to force y to be - correctly rounded according to the prevailing rounding mode - as follows. Let r and i be copies of the rounding mode and - inexact flag before entering the square root program. Also we - use the expression y+-ulp for the next representable floating - numbers (up and down) of y. Note that y+-ulp = either fixed - point y+-1, or multiply y by nextafter(1,+-inf) in chopped - mode. + By twiddling y's last bit it is possible to force y to be + correctly rounded according to the prevailing rounding mode + as follows. Let r and i be copies of the rounding mode and + inexact flag before entering the square root program. Also we + use the expression y+-ulp for the next representable floating + numbers (up and down) of y. Note that y+-ulp = either fixed + point y+-1, or multiply y by nextafter(1,+-inf) in chopped + mode. - R := RZ; ... set rounding mode to round-toward-zero - switch(r) { - case RN: ... round-to-nearest - if(x<= z*(z-ulp)...chopped) z = z - ulp; else - if(x<= z*(z+ulp)...chopped) z = z; else z = z+ulp; - break; - case RZ:case RM: ... round-to-zero or round-to--inf - R:=RP; ... reset rounding mod to round-to-+inf - if(x=(z+ulp)*(z+ulp) ...rounded up) z = z+ulp; - break; - case RP: ... round-to-+inf - if(x>(z+ulp)*(z+ulp)...chopped) z = z+2*ulp; else - if(x>z*z ...chopped) z = z+ulp; - break; - } + R := RZ; ... set rounding mode to round-toward-zero + switch(r) { + case RN: ... round-to-nearest + if(x<= z*(z-ulp)...chopped) z = z - ulp; else + if(x<= z*(z+ulp)...chopped) z = z; else z = z+ulp; + break; + case RZ:case RM: ... round-to-zero or round-to--inf + R:=RP; ... reset rounding mod to round-to-+inf + if(x=(z+ulp)*(z+ulp) ...rounded up) z = z+ulp; + break; + case RP: ... round-to-+inf + if(x>(z+ulp)*(z+ulp)...chopped) z = z+2*ulp; else + if(x>z*z ...chopped) z = z+ulp; + break; + } - Remark 3. The above comparisons can be done in fixed point. For - example, to compare x and w=z*z chopped, it suffices to compare - x1 and w1 (the trailing parts of x and w), regarding them as - two's complement integers. + Remark 3. The above comparisons can be done in fixed point. For + example, to compare x and w=z*z chopped, it suffices to compare + x1 and w1 (the trailing parts of x and w), regarding them as + two's complement integers. - ...Is z an exact square root? - To determine whether z is an exact square root of x, let z1 be the - trailing part of z, and also let x0 and x1 be the leading and - trailing parts of x. + ...Is z an exact square root? + To determine whether z is an exact square root of x, let z1 be the + trailing part of z, and also let x0 and x1 be the leading and + trailing parts of x. - If ((z1&0x03ffffff)!=0) ... not exact if trailing 26 bits of z!=0 - I := 1; ... Raise Inexact flag: z is not exact - else { - j := 1 - [(x0>>20)&1] ... j = logb(x) mod 2 - k := z1 >> 26; ... get z's 25-th and 26-th - fraction bits - I := i or (k&j) or ((k&(j+j+1))!=(x1&3)); - } - R:= r ... restore rounded mode - return sqrt(x):=z. + If ((z1&0x03ffffff)!=0) ... not exact if trailing 26 bits of z!=0 + I := 1; ... Raise Inexact flag: z is not exact + else { + j := 1 - [(x0>>20)&1] ... j = logb(x) mod 2 + k := z1 >> 26; ... get z's 25-th and 26-th + fraction bits + I := i or (k&j) or ((k&(j+j+1))!=(x1&3)); + } + R:= r ... restore rounded mode + return sqrt(x):=z. - If multiplication is cheaper then the foregoing red tape, the - Inexact flag can be evaluated by + If multiplication is cheaper then the foregoing red tape, the + Inexact flag can be evaluated by - I := i; - I := (z*z!=x) or I. + I := i; + I := (z*z!=x) or I. - Note that z*z can overwrite I; this value must be sensed if it is - True. + Note that z*z can overwrite I; this value must be sensed if it is + True. - Remark 4. If z*z = x exactly, then bit 25 to bit 0 of z1 must be - zero. + Remark 4. If z*z = x exactly, then bit 25 to bit 0 of z1 must be + zero. - -------------------- - z1: | f2 | - -------------------- - bit 31 bit 0 + -------------------- + z1: | f2 | + -------------------- + bit 31 bit 0 - Further more, bit 27 and 26 of z1, bit 0 and 1 of x1, and the odd - or even of logb(x) have the following relations: + Further more, bit 27 and 26 of z1, bit 0 and 1 of x1, and the odd + or even of logb(x) have the following relations: - ------------------------------------------------- - bit 27,26 of z1 bit 1,0 of x1 logb(x) - ------------------------------------------------- - 00 00 odd and even - 01 01 even - 10 10 odd - 10 00 even - 11 01 even - ------------------------------------------------- - - (4) Special cases (see (4) of Section A). + ------------------------------------------------- + bit 27,26 of z1 bit 1,0 of x1 logb(x) + ------------------------------------------------- + 00 00 odd and even + 01 01 even + 10 10 odd + 10 00 even + 11 01 even + ------------------------------------------------- + (4) Special cases (see (4) of Section A). */ diff --git a/third-party/fdlibm/s_trig.c b/third-party/fdlibm/s_trig.c index 9041dc547..06bfef4f6 100644 --- a/third-party/fdlibm/s_trig.c +++ b/third-party/fdlibm/s_trig.c @@ -13,7 +13,7 @@ * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice + * software is freely granted, provided that this notice * is preserved. * ==================================================== */ @@ -26,12 +26,11 @@ #define two24 1.67772160000000000000e+07 /* 0x41700000, 0x00000000 */ #define twon24 5.96046447753906250000e-08 /* 0x3E700000, 0x00000000 */ -/* - * __kernel_rem_pio2(x,y,e0,nx,prec) +/* __kernel_rem_pio2(x,y,e0,nx,prec) * double x[],y[]; int e0,nx,prec; * * __kernel_rem_pio2 return the last three digits of N with - * y = x - N*pi/2 + * y = x - N*pi/2 * so that |y| < pi/2. * * The method is to compute the integer (mod 8) and fraction parts of @@ -43,97 +42,92 @@ * (2/pi) is represented by an array of 24-bit integers in ipio2[]. * * Input parameters: - * x[] The input value (must be positive) is broken into nx - * pieces of 24-bit integers in double precision format. - * x[i] will be the i-th 24 bit of x. The scaled exponent - * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0 - * match x's up to 24 bits. + * x[] The input value (must be positive) is broken into nx + * pieces of 24-bit integers in double precision format. + * x[i] will be the i-th 24 bit of x. The scaled exponent + * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0 + * match x's up to 24 bits. * - * Example of breaking a double positive z into x[0]+x[1]+x[2]: - * e0 = ilogb(z)-23 - * z = scalbn(z,-e0) - * for i = 0,1,2 - * x[i] = floor(z) - * z = (z-x[i])*2**24 + * Example of breaking a double positive z into x[0]+x[1]+x[2]: + * e0 = ilogb(z)-23 + * z = scalbn(z,-e0) + * for i = 0,1,2 + * x[i] = floor(z) + * z = (z-x[i])*2**24 * + * y[] ouput result in an array of double precision numbers. + * The dimension of y[] is: + * 24-bit precision 1 + * 53-bit precision 2 + * 64-bit precision 2 + * 113-bit precision 3 + * The actual value is the sum of them. Thus for 113-bit + * precison, one may have to do something like: * - * y[] ouput result in an array of double precision numbers. - * The dimension of y[] is: - * 24-bit precision 1 - * 53-bit precision 2 - * 64-bit precision 2 - * 113-bit precision 3 - * The actual value is the sum of them. Thus for 113-bit - * precison, one may have to do something like: + * long double t,w,r_head, r_tail; + * t = (long double)y[2] + (long double)y[1]; + * w = (long double)y[0]; + * r_head = t+w; + * r_tail = w - (r_head - t); * - * long double t,w,r_head, r_tail; - * t = (long double)y[2] + (long double)y[1]; - * w = (long double)y[0]; - * r_head = t+w; - * r_tail = w - (r_head - t); + * e0 The exponent of x[0] * - * e0 The exponent of x[0] + * nx dimension of x[] * - * nx dimension of x[] - * - * prec an integer indicating the precision: - * 0 24 bits (single) - * 1 53 bits (double) - * 2 64 bits (extended) - * 3 113 bits (quad) + * prec an integer indicating the precision: + * 0 24 bits (single) + * 1 53 bits (double) + * 2 64 bits (extended) + * 3 113 bits (quad) * * External function: - * double scalbn(), floor(); - * + * double scalbn(), floor(); * * Here is the description of some local variables: * - * ipio2[] - * integer array, contains the (24*i)-th to (24*i+23)-th - * bit of 2/pi after binary point. The corresponding - * floating value is + * ipio2[] integer array, contains the (24*i)-th to (24*i+23)-th + * bit of 2/pi after binary point. The corresponding + * floating value is * - * ipio2[i] * 2^(-24(i+1)). + * ipio2[i] * 2^(-24(i+1)). * - * jk jk+1 is the initial number of terms of ipio2[] needed - * in the computation. The recommended value is 2,3,4, - * 6 for single, double, extended,and quad. + * jk jk+1 is the initial number of terms of ipio2[] needed + * in the computation. The recommended value is 2,3,4, + * 6 for single, double, extended,and quad. * - * jz local integer variable indicating the number of - * terms of ipio2[] used. + * jz local integer variable indicating the number of + * terms of ipio2[] used. * - * jx nx - 1 + * jx nx - 1 * - * jv index for pointing to the suitable ipio2[] for the - * computation. In general, we want - * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8 - * is an integer. Thus - * e0-3-24*jv >= 0 or (e0-3)/24 >= jv - * Hence jv = max(0,(e0-3)/24). + * jv index for pointing to the suitable ipio2[] for the + * computation. In general, we want + * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8 + * is an integer. Thus + * e0-3-24*jv >= 0 or (e0-3)/24 >= jv + * Hence jv = max(0,(e0-3)/24). * - * jp jp+1 is the number of terms in PIo2[] needed, jp = jk. + * jp jp+1 is the number of terms in PIo2[] needed, jp = jk. * - * q[] double array with integral value, representing the - * 24-bits chunk of the product of x and 2/pi. + * q[] double array with integral value, representing the + * 24-bits chunk of the product of x and 2/pi. * - * q0 the corresponding exponent of q[0]. Note that the - * exponent for q[i] would be q0-24*i. + * q0 the corresponding exponent of q[0]. Note that the + * exponent for q[i] would be q0-24*i. * - * PIo2[] double precision array, obtained by cutting pi/2 - * into 24 bits chunks. + * PIo2[] double precision array, obtained by cutting pi/2 + * into 24 bits chunks. * - * f[] ipio2[] in floating point + * f[] ipio2[] in floating point * - * iq[] integer array by breaking up q[] in 24-bits chunk. + * iq[] integer array by breaking up q[] in 24-bits chunk. * - * fq[] final product of x*(2/pi) in fq[0],..,fq[jk] - * - * ih integer. If >0 it indicates q[] is >= 0.5, hence - * it also indicates the *sign* of the result. + * fq[] final product of x*(2/pi) in fq[0],..,fq[jk] * + * ih integer. If >0 it indicates q[] is >= 0.5, hence + * it also indicates the *sign* of the result. */ - /* * Constants: * The hexadecimal values are the intended ones for the following @@ -142,9 +136,14 @@ * to produce the hexadecimal values shown. */ -static const int init_jk[] = {2,3,4,6}; /* initial value for jk */ +/* initial value for jk */ +static const int init_jk[] = +{ + 2, 3, 4, 6 +}; -static const double PIo2[] = { +static const double PIo2[] = +{ 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */ 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */ 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */ @@ -158,183 +157,288 @@ static const double PIo2[] = { /* * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi */ -static const int ipio2[] = { -0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62, -0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A, -0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129, -0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41, -0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8, -0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF, -0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5, -0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08, -0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3, -0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880, -0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B, +static const int ipio2[] = +{ + 0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62, + 0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A, + 0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129, + 0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41, + 0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8, + 0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF, + 0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5, + 0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08, + 0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3, + 0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880, + 0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B, }; -static int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec) +static int +__kernel_rem_pio2 (double *x, double *y, int e0, int nx, int prec) { - int jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih; - double z,fw,f[20],fq[20],q[20]; + int jz, jx, jv, jp, jk, carry, n, iq[20], i, j, k, m, q0, ih; + double z, fw, f[20], fq[20], q[20]; - /* initialize jk*/ - jk = init_jk[prec]; - jp = jk; + /* initialize jk */ + jk = init_jk[prec]; + jp = jk; - /* determine jx,jv,q0, note that 3>q0 */ - jx = nx-1; - jv = (e0-3)/24; if(jv<0) jv=0; - q0 = e0-24*(jv+1); + /* determine jx, jv, q0, note that 3 > q0 */ + jx = nx - 1; + jv = (e0 - 3) / 24; + if (jv < 0) + { + jv = 0; + } + q0 = e0 - 24 * (jv + 1); - /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */ - j = jv-jx; m = jx+jk; - for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j]; + /* set up f[0] to f[jx + jk] where f[jx + jk] = ipio2[jv + jk] */ + j = jv - jx; + m = jx + jk; + for (i = 0; i <= m; i++, j++) + { + f[i] = (j < 0) ? zero : (double) ipio2[j]; + } - /* compute q[0],q[1],...q[jk] */ - for (i=0;i<=jk;i++) { - for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw; - } + /* compute q[0], q[1], ... q[jk] */ + for (i = 0; i <= jk; i++) + { + for (j = 0, fw = 0.0; j <= jx; j++) + { + fw += x[j] * f[jx + i - j]; + } + q[i] = fw; + } - jz = jk; + jz = jk; recompute: - /* distill q[] into iq[] reversingly */ - for(i=0,j=jz,z=q[jz];j>0;i++,j--) { - fw = (double)((int)(twon24* z)); - iq[i] = (int)(z-two24*fw); - z = q[j-1]+fw; - } + /* distill q[] into iq[] reversingly */ + for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--) + { + fw = (double) ((int) (twon24 * z)); + iq[i] = (int) (z - two24 * fw); + z = q[j - 1] + fw; + } - /* compute n */ - z = scalbn(z,q0); /* actual value of z */ - z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */ - n = (int) z; - z -= (double)n; - ih = 0; - if(q0>0) { /* need iq[jz-1] to determine n */ - i = (iq[jz-1]>>(24-q0)); n += i; - iq[jz-1] -= i<<(24-q0); - ih = iq[jz-1]>>(23-q0); - } - else if(q0==0) ih = iq[jz-1]>>23; - else if(z>=0.5) ih=2; + /* compute n */ + z = scalbn (z, q0); /* actual value of z */ + z -= 8.0 * floor (z * 0.125); /* trim off integer >= 8 */ + n = (int) z; + z -= (double) n; + ih = 0; + if (q0 > 0) /* need iq[jz - 1] to determine n */ + { + i = (iq[jz - 1] >> (24 - q0)); + n += i; + iq[jz - 1] -= i << (24 - q0); + ih = iq[jz - 1] >> (23 - q0); + } + else if (q0 == 0) + { + ih = iq[jz - 1] >> 23; + } + else if (z >= 0.5) + { + ih = 2; + } - if(ih>0) { /* q > 0.5 */ - n += 1; carry = 0; - for(i=0;i0) { /* rare case: chance is 1 in 12 */ - switch(q0) { - case 1: - iq[jz-1] &= 0x7fffff; break; - case 2: - iq[jz-1] &= 0x3fffff; break; - } - } - if(ih==2) { - z = one - z; - if(carry!=0) z -= scalbn(one,q0); - } - } + if (ih > 0) /* q > 0.5 */ + { + n += 1; + carry = 0; + for (i = 0; i < jz; i++) /* compute 1 - q */ + { + j = iq[i]; + if (carry == 0) + { + if (j != 0) + { + carry = 1; + iq[i] = 0x1000000 - j; + } + } + else + { + iq[i] = 0xffffff - j; + } + } + if (q0 > 0) /* rare case: chance is 1 in 12 */ + { + switch (q0) + { + case 1: + { + iq[jz - 1] &= 0x7fffff; + break; + } + case 2: + { + iq[jz - 1] &= 0x3fffff; + break; + } + } + } + if (ih == 2) + { + z = one - z; + if (carry != 0) + { + z -= scalbn (one, q0); + } + } + } - /* check if recomputation is needed */ - if(z==zero) { - j = 0; - for (i=jz-1;i>=jk;i--) j |= iq[i]; - if(j==0) { /* need recomputation */ - for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */ + /* check if recomputation is needed */ + if (z == zero) + { + j = 0; + for (i = jz - 1; i >= jk; i--) + { + j |= iq[i]; + } + if (j == 0) /* need recomputation */ + { + for (k = 1; iq[jk - k] == 0; k++) /* k = no. of terms needed */ + { + } - for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */ - f[jx+i] = (double) ipio2[jv+i]; - for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; - q[i] = fw; - } - jz += k; - goto recompute; - } - } + for (i = jz + 1; i <= jz + k; i++) /* add q[jz + 1] to q[jz + k] */ + { + f[jx + i] = (double) ipio2[jv + i]; + for (j = 0, fw = 0.0; j <= jx; j++) + { + fw += x[j] * f[jx + i - j]; + } + q[i] = fw; + } + jz += k; + goto recompute; + } + } - /* chop off zero terms */ - if(z==0.0) { - jz -= 1; q0 -= 24; - while(iq[jz]==0) { jz--; q0-=24;} - } else { /* break z into 24-bit if necessary */ - z = scalbn(z,-q0); - if(z>=two24) { - fw = (double)((int)(twon24*z)); - iq[jz] = (int)(z-two24*fw); - jz += 1; q0 += 24; - iq[jz] = (int) fw; - } else iq[jz] = (int) z ; - } + /* chop off zero terms */ + if (z == 0.0) + { + jz -= 1; + q0 -= 24; + while (iq[jz] == 0) + { + jz--; + q0 -= 24; + } + } + else + { /* break z into 24-bit if necessary */ + z = scalbn (z, -q0); + if (z >= two24) + { + fw = (double) ((int) (twon24 * z)); + iq[jz] = (int) (z - two24 * fw); + jz += 1; + q0 += 24; + iq[jz] = (int) fw; + } + else + { + iq[jz] = (int) z; + } + } - /* convert integer "bit" chunk to floating-point value */ - fw = scalbn(one,q0); - for(i=jz;i>=0;i--) { - q[i] = fw*(double)iq[i]; fw*=twon24; - } + /* convert integer "bit" chunk to floating-point value */ + fw = scalbn (one, q0); + for (i = jz; i >= 0; i--) + { + q[i] = fw * (double) iq[i]; + fw *= twon24; + } - /* compute PIo2[0,...,jp]*q[jz,...,0] */ - for(i=jz;i>=0;i--) { - for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k]; - fq[jz-i] = fw; - } + /* compute PIo2[0, ..., jp] * q[jz, ..., 0] */ + for (i = jz; i >= 0; i--) + { + for (fw = 0.0, k = 0; k <= jp && k <= jz - i; k++) + { + fw += PIo2[k] * q[i + k]; + } + fq[jz - i] = fw; + } - /* compress fq[] into y[] */ - switch(prec) { - case 0: - fw = 0.0; - for (i=jz;i>=0;i--) fw += fq[i]; - y[0] = (ih==0)? fw: -fw; - break; - case 1: - case 2: - fw = 0.0; - for (i=jz;i>=0;i--) fw += fq[i]; - y[0] = (ih==0)? fw: -fw; - fw = fq[0]-fw; - for (i=1;i<=jz;i++) fw += fq[i]; - y[1] = (ih==0)? fw: -fw; - break; - case 3: /* painful */ - for (i=jz;i>0;i--) { - fw = fq[i-1]+fq[i]; - fq[i] += fq[i-1]-fw; - fq[i-1] = fw; - } - for (i=jz;i>1;i--) { - fw = fq[i-1]+fq[i]; - fq[i] += fq[i-1]-fw; - fq[i-1] = fw; - } - for (fw=0.0,i=jz;i>=2;i--) fw += fq[i]; - if(ih==0) { - y[0] = fq[0]; y[1] = fq[1]; y[2] = fw; - } else { - y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw; - } - } - return n&7; -} + /* compress fq[] into y[] */ + switch (prec) + { + case 0: + { + fw = 0.0; + for (i = jz; i >= 0; i--) + { + fw += fq[i]; + } + y[0] = (ih == 0) ? fw : -fw; + break; + } + case 1: + case 2: + { + fw = 0.0; + for (i = jz; i >= 0; i--) + { + fw += fq[i]; + } + y[0] = (ih == 0) ? fw : -fw; + fw = fq[0] - fw; + for (i = 1; i <= jz; i++) + { + fw += fq[i]; + } + y[1] = (ih == 0) ? fw : -fw; + break; + } + case 3: /* painful */ + { + for (i = jz; i > 0; i--) + { + fw = fq[i - 1] + fq[i]; + fq[i] += fq[i - 1] - fw; + fq[i - 1] = fw; + } + for (i = jz; i > 1; i--) + { + fw = fq[i - 1] + fq[i]; + fq[i] += fq[i - 1] - fw; + fq[i - 1] = fw; + } + for (fw = 0.0, i = jz; i >= 2; i--) + { + fw += fq[i]; + } + if (ih == 0) + { + y[0] = fq[0]; + y[1] = fq[1]; + y[2] = fw; + } + else + { + y[0] = -fq[0]; + y[1] = -fq[1]; + y[2] = -fw; + } + } + } + return n & 7; +} /* __kernel_rem_pio2 */ /* __ieee754_rem_pio2(x,y) - * * return the remainder of x rem pi/2 in y[0]+y[1] * use __kernel_rem_pio2() */ -static const int npio2_hw[] = { -0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C, -0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C, -0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A, -0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C, -0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB, -0x404858EB, 0x404921FB, +static const int npio2_hw[] = +{ + 0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C, + 0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C, + 0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A, + 0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C, + 0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB, + 0x404858EB, 0x404921FB, }; /* @@ -342,9 +446,9 @@ static const int npio2_hw[] = { * pio2_1: first 33 bit of pi/2 * pio2_1t: pi/2 - pio2_1 * pio2_2: second 33 bit of pi/2 - * pio2_2t: pi/2 - (pio2_1+pio2_2) + * pio2_2t: pi/2 - (pio2_1 + pio2_2) * pio2_3: third 33 bit of pi/2 - * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3) + * pio2_3t: pi/2 - (pio2_1 + pio2_2 + pio2_3) */ #define invpio2 6.36619772367581382433e-01 /* 0x3FE45F30, 0x6DC9C883 */ #define pio2_1 1.57079632673412561417e+00 /* 0x3FF921FB, 0x54400000 */ @@ -354,94 +458,134 @@ static const int npio2_hw[] = { #define pio2_3 2.02226624871116645580e-21 /* 0x3BA3198A, 0x2E000000 */ #define pio2_3t 8.47842766036889956997e-32 /* 0x397B839A, 0x252049C1 */ -static int __ieee754_rem_pio2(double x, double *y) +static int +__ieee754_rem_pio2 (double x, double *y) { - double z,w,t,r,fn; - double tx[3]; - int e0,i,j,nx,n,ix,hx; + double z, w, t, r, fn; + double tx[3]; + int e0, i, j, nx, n, ix, hx; - hx = __HI(x); /* high word of x */ - ix = hx&0x7fffffff; - if(ix<=0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */ - {y[0] = x; y[1] = 0; return 0;} - if(ix<0x4002d97c) { /* |x| < 3pi/4, special case with n=+-1 */ - if(hx>0) { - z = x - pio2_1; - if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */ - y[0] = z - pio2_1t; - y[1] = (z-y[0])-pio2_1t; - } else { /* near pi/2, use 33+33+53 bit pi */ - z -= pio2_2; - y[0] = z - pio2_2t; - y[1] = (z-y[0])-pio2_2t; - } - return 1; - } else { /* negative x */ - z = x + pio2_1; - if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */ - y[0] = z + pio2_1t; - y[1] = (z-y[0])+pio2_1t; - } else { /* near pi/2, use 33+33+53 bit pi */ - z += pio2_2; - y[0] = z + pio2_2t; - y[1] = (z-y[0])+pio2_2t; - } - return -1; - } - } - if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */ - t = fabs(x); - n = (int) (t*invpio2+half); - fn = (double)n; - r = t-fn*pio2_1; - w = fn*pio2_1t; /* 1st round good to 85 bit */ - if(n<32&&ix!=npio2_hw[n-1]) { - y[0] = r-w; /* quick check no cancellation */ - } else { - j = ix>>20; - y[0] = r-w; - i = j-(((__HI(y[0]))>>20)&0x7ff); - if(i>16) { /* 2nd iteration needed, good to 118 */ - t = r; - w = fn*pio2_2; - r = t-w; - w = fn*pio2_2t-((t-r)-w); - y[0] = r-w; - i = j-(((__HI(y[0]))>>20)&0x7ff); - if(i>49) { /* 3rd iteration need, 151 bits acc */ - t = r; /* will cover all possible cases */ - w = fn*pio2_3; - r = t-w; - w = fn*pio2_3t-((t-r)-w); - y[0] = r-w; - } - } - } - y[1] = (r-y[0])-w; - if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;} - else return n; - } - /* - * all other (large) arguments - */ - if(ix>=0x7ff00000) { /* x is inf or NaN */ - y[0]=y[1]=x-x; return 0; - } - /* set z = scalbn(|x|,ilogb(x)-23) */ - __LO(z) = __LO(x); - e0 = (ix>>20)-1046; /* e0 = ilogb(z)-23; */ - __HI(z) = ix - (e0<<20); - for(i=0;i<2;i++) { - tx[i] = (double)((int)(z)); - z = (z-tx[i])*two24; - } - tx[2] = z; - nx = 3; - while(tx[nx-1]==zero) nx--; /* skip zero term */ - n = __kernel_rem_pio2(tx,y,e0,nx,2); - if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;} - return n; -} + hx = __HI (x); /* high word of x */ + ix = hx & 0x7fffffff; + if (ix <= 0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */ + { + y[0] = x; + y[1] = 0; + return 0; + } + if (ix < 0x4002d97c) /* |x| < 3pi/4, special case with n = +-1 */ + { + if (hx > 0) + { + z = x - pio2_1; + if (ix != 0x3ff921fb) /* 33 + 53 bit pi is good enough */ + { + y[0] = z - pio2_1t; + y[1] = (z - y[0]) - pio2_1t; + } + else /* near pi/2, use 33 + 33 + 53 bit pi */ + { + z -= pio2_2; + y[0] = z - pio2_2t; + y[1] = (z - y[0]) - pio2_2t; + } + return 1; + } + else /* negative x */ + { + z = x + pio2_1; + if (ix != 0x3ff921fb) /* 33 + 53 bit pi is good enough */ + { + y[0] = z + pio2_1t; + y[1] = (z - y[0]) + pio2_1t; + } + else /* near pi/2, use 33 + 33 + 53 bit pi */ + { + z += pio2_2; + y[0] = z + pio2_2t; + y[1] = (z - y[0]) + pio2_2t; + } + return -1; + } + } + if (ix <= 0x413921fb) /* |x| ~<= 2^19 * (pi/2), medium size */ + { + t = fabs (x); + n = (int) (t * invpio2 + half); + fn = (double) n; + r = t - fn * pio2_1; + w = fn * pio2_1t; /* 1st round good to 85 bit */ + if (n < 32 && ix != npio2_hw[n - 1]) + { + y[0] = r - w; /* quick check no cancellation */ + } + else + { + j = ix >> 20; + y[0] = r - w; + i = j - (((__HI (y[0])) >> 20) & 0x7ff); + if (i > 16) /* 2nd iteration needed, good to 118 */ + { + t = r; + w = fn * pio2_2; + r = t - w; + w = fn * pio2_2t - ((t - r) - w); + y[0] = r - w; + i = j - (((__HI (y[0])) >> 20) & 0x7ff); + if (i > 49) /* 3rd iteration need, 151 bits acc, will cover all possible cases */ + { + t = r; + w = fn * pio2_3; + r = t - w; + w = fn * pio2_3t - ((t - r) - w); + y[0] = r - w; + } + } + } + y[1] = (r - y[0]) - w; + if (hx < 0) + { + y[0] = -y[0]; + y[1] = -y[1]; + return -n; + } + else + { + return n; + } + } + /* + * all other (large) arguments + */ + if (ix >= 0x7ff00000) /* x is inf or NaN */ + { + y[0] = y[1] = x - x; + return 0; + } + /* set z = scalbn(|x|, ilogb(x) - 23) */ + __LO (z) = __LO (x); + e0 = (ix >> 20) - 1046; /* e0 = ilogb(z) - 23; */ + __HI (z) = ix - (e0 << 20); + for (i = 0; i < 2; i++) + { + tx[i] = (double) ((int) (z)); + z = (z - tx[i]) * two24; + } + tx[2] = z; + nx = 3; + while (tx[nx - 1] == zero) /* skip zero term */ + { + nx--; + } + n = __kernel_rem_pio2 (tx, y, e0, nx, 2); + if (hx < 0) + { + y[0] = -y[0]; + y[1] = -y[1]; + return -n; + } + return n; +} /* __ieee754_rem_pio2 */ /* __kernel_sin( x, y, iy) * kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854 @@ -450,25 +594,25 @@ static int __ieee754_rem_pio2(double x, double *y) * Input iy indicates whether y is 0. (if iy=0, y assume to be 0). * * Algorithm - * 1. Since sin(-x) = -sin(x), we need only to consider positive x. - * 2. if x < 2^-27 (hx<0x3e400000 0), return x with inexact if x!=0. - * 3. sin(x) is approximated by a polynomial of degree 13 on - * [0,pi/4] - * 3 13 - * sin(x) ~ x + S1*x + ... + S6*x - * where - * - * |sin(x) 2 4 6 8 10 12 | -58 - * |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x +S6*x )| <= 2 - * | x | + * 1. Since sin(-x) = -sin(x), we need only to consider positive x. + * 2. if x < 2^-27 (hx<0x3e400000 0), return x with inexact if x!=0. + * 3. sin(x) is approximated by a polynomial of degree 13 on + * [0,pi/4] + * 3 13 + * sin(x) ~ x + S1*x + ... + S6*x + * where * - * 4. sin(x+y) = sin(x) + sin'(x')*y - * ~ sin(x) + (1-x*x/2)*y - * For better accuracy, let - * 3 2 2 2 2 - * r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6)))) - * then 3 2 - * sin(x) = x + (S1*x + (x *(r-y/2)+y)) + * |sin(x) 2 4 6 8 10 12 | -58 + * |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x +S6*x )| <= 2 + * | x | + * + * 4. sin(x+y) = sin(x) + sin'(x')*y + * ~ sin(x) + (1-x*x/2)*y + * For better accuracy, let + * 3 2 2 2 2 + * r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6)))) + * then 3 2 + * sin(x) = x + (S1*x + (x *(r-y/2)+y)) */ #define S1 -1.66666666666666324348e-01 /* 0xBFC55555, 0x55555549 */ @@ -478,19 +622,32 @@ static int __ieee754_rem_pio2(double x, double *y) #define S5 -2.50507602534068634195e-08 /* 0xBE5AE5E6, 0x8A2B9CEB */ #define S6 1.58969099521155010221e-10 /* 0x3DE5D93A, 0x5ACFD57C */ -static double __kernel_sin(double x, double y, int iy) +static double +__kernel_sin (double x, double y, int iy) { - double z,r,v; - int ix; - ix = __HI(x)&0x7fffffff; /* high word of x */ - if(ix<0x3e400000) /* |x| < 2**-27 */ - {if((int)x==0) return x;} /* generate inexact */ - z = x*x; - v = z*x; - r = S2+z*(S3+z*(S4+z*(S5+z*S6))); - if(iy==0) return x+v*(S1+z*r); - else return x-((z*(half*y-v*r)-y)-v*S1); -} + double z, r, v; + int ix; + + ix = __HI (x) & 0x7fffffff; /* high word of x */ + if (ix < 0x3e400000) /* |x| < 2**-27 */ + { + if ((int) x == 0) + { + return x; /* generate inexact */ + } + } + z = x * x; + v = z * x; + r = S2 + z * (S3 + z * (S4 + z * (S5 + z * S6))); + if (iy == 0) + { + return x + v * (S1 + z * r); + } + else + { + return x - ((z * (half * y - v * r) - y) - v * S1); + } +} /* __kernel_sin */ /* * __kernel_cos( x, y ) @@ -499,32 +656,32 @@ static double __kernel_sin(double x, double y, int iy) * Input y is the tail of x. * * Algorithm - * 1. Since cos(-x) = cos(x), we need only to consider positive x. - * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0. - * 3. cos(x) is approximated by a polynomial of degree 14 on - * [0,pi/4] - * 4 14 - * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x - * where the remez error is - * - * | 2 4 6 8 10 12 14 | -58 - * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2 - * | | + * 1. Since cos(-x) = cos(x), we need only to consider positive x. + * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0. + * 3. cos(x) is approximated by a polynomial of degree 14 on + * [0,pi/4] + * 4 14 + * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x + * where the remez error is * - * 4 6 8 10 12 14 - * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then - * cos(x) = 1 - x*x/2 + r - * since cos(x+y) ~ cos(x) - sin(x)*y - * ~ cos(x) - x*y, - * a correction term is necessary in cos(x) and hence - * cos(x+y) = 1 - (x*x/2 - (r - x*y)) - * For better accuracy when x > 0.3, let qx = |x|/4 with - * the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125. - * Then - * cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)). - * Note that 1-qx and (x*x/2-qx) is EXACT here, and the - * magnitude of the latter is at least a quarter of x*x/2, - * thus, reducing the rounding error in the subtraction. + * | 2 4 6 8 10 12 14 | -58 + * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2 + * | | + * + * 4 6 8 10 12 14 + * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then + * cos(x) = 1 - x*x/2 + r + * since cos(x+y) ~ cos(x) - sin(x)*y + * ~ cos(x) - x*y, + * a correction term is necessary in cos(x) and hence + * cos(x+y) = 1 - (x*x/2 - (r - x*y)) + * For better accuracy when x > 0.3, let qx = |x|/4 with + * the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125. + * Then + * cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)). + * Note that 1-qx and (x*x/2-qx) is EXACT here, and the + * magnitude of the latter is at least a quarter of x*x/2, + * thus, reducing the rounding error in the subtraction. */ #define C1 4.16666666666666019037e-02 /* 0x3FA55555, 0x5555554C */ @@ -534,30 +691,42 @@ static double __kernel_sin(double x, double y, int iy) #define C5 2.08757232129817482790e-09 /* 0x3E21EE9E, 0xBDB4B1C4 */ #define C6 -1.13596475577881948265e-11 /* 0xBDA8FAE9, 0xBE8838D4 */ -static double __kernel_cos(double x, double y) +static double +__kernel_cos (double x, double y) { - double a,hz,z,r,qx; - int ix; - ix = __HI(x)&0x7fffffff; /* ix = |x|'s high word*/ - if(ix<0x3e400000) { /* if x < 2**27 */ - if(((int)x)==0) return one; /* generate inexact */ - } - z = x*x; - r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6))))); - if(ix < 0x3FD33333) /* if |x| < 0.3 */ - return one - (0.5*z - (z*r - x*y)); - else { - if(ix > 0x3fe90000) { /* x > 0.78125 */ - qx = 0.28125; - } else { - __HI(qx) = ix-0x00200000; /* x/4 */ - __LO(qx) = 0; - } - hz = 0.5*z-qx; - a = one-qx; - return a - (hz - (z*r-x*y)); - } -} + double a, hz, z, r, qx; + int ix; + + ix = __HI (x) & 0x7fffffff; /* ix = |x|'s high word */ + if (ix < 0x3e400000) /* if x < 2**27 */ + { + if (((int) x) == 0) + { + return one; /* generate inexact */ + } + } + z = x * x; + r = z * (C1 + z * (C2 + z * (C3 + z * (C4 + z * (C5 + z * C6))))); + if (ix < 0x3FD33333) /* if |x| < 0.3 */ + { + return one - (0.5 * z - (z * r - x * y)); + } + else + { + if (ix > 0x3fe90000) /* x > 0.78125 */ + { + qx = 0.28125; + } + else + { + __HI (qx) = ix - 0x00200000; /* x / 4 */ + __LO (qx) = 0; + } + hz = 0.5 * z - qx; + a = one - qx; + return a - (hz - (z * r - x * y)); + } +} /* __kernel_cos */ /* __kernel_tan( x, y, k ) * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 @@ -566,137 +735,150 @@ static double __kernel_cos(double x, double y) * Input k indicates whether tan (if k = 1) or -1/tan (if k = -1) is returned. * * Algorithm - * 1. Since tan(-x) = -tan(x), we need only to consider positive x. - * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0. - * 3. tan(x) is approximated by a odd polynomial of degree 27 on - * [0,0.67434] - * 3 27 - * tan(x) ~ x + T1*x + ... + T13*x - * where + * 1. Since tan(-x) = -tan(x), we need only to consider positive x. + * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0. + * 3. tan(x) is approximated by a odd polynomial of degree 27 on + * [0,0.67434] + * 3 27 + * tan(x) ~ x + T1*x + ... + T13*x + * where * - * |tan(x) 2 4 26 | -59.2 - * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 - * | x | + * |tan(x) 2 4 26 | -59.2 + * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 + * | x | * - * Note: tan(x+y) = tan(x) + tan'(x)*y - * ~ tan(x) + (1+x*x)*y - * Therefore, for better accuracy in computing tan(x+y), let - * 3 2 2 2 2 - * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) - * then - * 3 2 - * tan(x+y) = x + (T1*x + (x *(r+y)+y)) + * Note: tan(x+y) = tan(x) + tan'(x)*y + * ~ tan(x) + (1+x*x)*y + * Therefore, for better accuracy in computing tan(x+y), let + * 3 2 2 2 2 + * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) + * then + * 3 2 + * tan(x+y) = x + (T1*x + (x *(r+y)+y)) * * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then - * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) - * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) + * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) + * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) */ -#define T0 3.33333333333334091986e-01 /* 3FD55555, 55555563 */ -#define T1 1.33333333333201242699e-01 /* 3FC11111, 1110FE7A */ -#define T2 5.39682539762260521377e-02 /* 3FABA1BA, 1BB341FE */ -#define T3 2.18694882948595424599e-02 /* 3F9664F4, 8406D637 */ -#define T4 8.86323982359930005737e-03 /* 3F8226E3, E96E8493 */ -#define T5 3.59207910759131235356e-03 /* 3F6D6D22, C9560328 */ -#define T6 1.45620945432529025516e-03 /* 3F57DBC8, FEE08315 */ -#define T7 5.88041240820264096874e-04 /* 3F4344D8, F2F26501 */ -#define T8 2.46463134818469906812e-04 /* 3F3026F7, 1A8D1068 */ -#define T9 7.81794442939557092300e-05 /* 3F147E88, A03792A6 */ -#define T10 7.14072491382608190305e-05 /* 3F12B80F, 32F0A7E9 */ -#define T11 -1.85586374855275456654e-05 /* BEF375CB, DB605373 */ -#define T12 2.59073051863633712884e-05 /* 3EFB2A70, 74BF7AD4 */ -#define pio4 7.85398163397448278999e-01 /* 3FE921FB, 54442D18 */ -#define pio4lo 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */ +#define T0 3.33333333333334091986e-01 /* 3FD55555, 55555563 */ +#define T1 1.33333333333201242699e-01 /* 3FC11111, 1110FE7A */ +#define T2 5.39682539762260521377e-02 /* 3FABA1BA, 1BB341FE */ +#define T3 2.18694882948595424599e-02 /* 3F9664F4, 8406D637 */ +#define T4 8.86323982359930005737e-03 /* 3F8226E3, E96E8493 */ +#define T5 3.59207910759131235356e-03 /* 3F6D6D22, C9560328 */ +#define T6 1.45620945432529025516e-03 /* 3F57DBC8, FEE08315 */ +#define T7 5.88041240820264096874e-04 /* 3F4344D8, F2F26501 */ +#define T8 2.46463134818469906812e-04 /* 3F3026F7, 1A8D1068 */ +#define T9 7.81794442939557092300e-05 /* 3F147E88, A03792A6 */ +#define T10 7.14072491382608190305e-05 /* 3F12B80F, 32F0A7E9 */ +#define T11 -1.85586374855275456654e-05 /* BEF375CB, DB605373 */ +#define T12 2.59073051863633712884e-05 /* 3EFB2A70, 74BF7AD4 */ +#define pio4 7.85398163397448278999e-01 /* 3FE921FB, 54442D18 */ +#define pio4lo 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */ -static double __kernel_tan(double x, double y, int iy) +static double +__kernel_tan (double x, double y, int iy) { - double z, r, v, w, s; - int ix, hx; + double z, r, v, w, s; + int ix, hx; - hx = __HI(x); /* high word of x */ - ix = hx & 0x7fffffff; /* high word of |x| */ - if (ix < 0x3e300000) { /* x < 2**-28 */ - if ((int) x == 0) { /* generate inexact */ - if (((ix | __LO(x)) | (iy + 1)) == 0) - return one / fabs(x); - else { - if (iy == 1) - return x; - else { /* compute -1 / (x+y) carefully */ - double a, t; + hx = __HI (x); /* high word of x */ + ix = hx & 0x7fffffff; /* high word of |x| */ + if (ix < 0x3e300000) /* x < 2**-28 */ + { + if ((int) x == 0) /* generate inexact */ + { + if (((ix | __LO (x)) | (iy + 1)) == 0) + { + return one / fabs (x); + } + else + { + if (iy == 1) + { + return x; + } + else /* compute -1 / (x + y) carefully */ + { + double a, t; - z = w = x + y; - __LO(z) = 0; - v = y - (z - x); - t = a = -one / w; - __LO(t) = 0; - s = one + t * z; - return t + a * (s + t * v); - } - } - } - } - if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */ - if (hx < 0) { - x = -x; - y = -y; - } - z = pio4 - x; - w = pio4lo - y; - x = z + w; - y = 0.0; - } - z = x * x; - w = z * z; - /* - * Break x^5*(T[1]+x^2*T[2]+...) into - * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + - * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) - */ - r = T1 + w * (T3 + w * (T5 + w * (T7 + w * (T9 + - w * T11)))); - v = z * (T2 + w * (T4 + w * (T6 + w * (T8 + w * (T10 + - w * T12))))); - s = z * x; - r = y + z * (s * (r + v) + y); - r += T0 * s; - w = x + r; - if (ix >= 0x3FE59428) { - v = (double) iy; - return (double) (1 - ((hx >> 30) & 2)) * - (v - 2.0 * (x - (w * w / (w + v) - r))); - } - if (iy == 1) - return w; - else { - /* - * if allow error up to 2 ulp, simply return - * -1.0 / (x+r) here - */ - /* compute -1.0 / (x+r) accurately */ - double a, t; - z = w; - __LO(z) = 0; - v = r - (z - x); /* z+v = r+x */ - t = a = -1.0 / w; /* a = -1.0/w */ - __LO(t) = 0; - s = 1.0 + t * z; - return t + a * (s + t * v); - } -} + z = w = x + y; + __LO (z) = 0; + v = y - (z - x); + t = a = -one / w; + __LO (t) = 0; + s = one + t * z; + return t + a * (s + t * v); + } + } + } + } + if (ix >= 0x3FE59428) /* |x| >= 0.6744 */ + { + if (hx < 0) + { + x = -x; + y = -y; + } + z = pio4 - x; + w = pio4lo - y; + x = z + w; + y = 0.0; + } + z = x * x; + w = z * z; + /* + * Break x^5 * (T[1] + x^2 * T[2] + ...) into + * x^5 (T[1] + x^4 * T[3] + ... + x^20 * T[11]) + + * x^5 (x^2 * (T[2] + x^4 * T[4] + ... + x^22 * [T12])) + */ + r = T1 + w * (T3 + w * (T5 + w * (T7 + w * (T9 + w * T11)))); + v = z * (T2 + w * (T4 + w * (T6 + w * (T8 + w * (T10 + w * T12))))); + s = z * x; + r = y + z * (s * (r + v) + y); + r += T0 * s; + w = x + r; + if (ix >= 0x3FE59428) + { + v = (double) iy; + return (double) (1 - ((hx >> 30) & 2)) * (v - 2.0 * (x - (w * w / (w + v) - r))); + } + if (iy == 1) + { + return w; + } + else + { + /* + * if allow error up to 2 ulp, simply return + * -1.0 / (x + r) here + */ + /* compute -1.0 / (x + r) accurately */ + double a, t; -/* Method. + z = w; + __LO (z) = 0; + v = r - (z - x); /* z + v = r + x */ + t = a = -1.0 / w; /* a = -1.0 / w */ + __LO (t) = 0; + s = 1.0 + t * z; + return t + a * (s + t * v); + } +} /* __kernel_tan */ + +/* Method: * Let S,C and T denote the sin, cos and tan respectively on - * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 - * in [-pi/4 , +pi/4], and let n = k mod 4. - * We have + * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 + * in [-pi/4 , +pi/4], and let n = k mod 4. + * We have * * n sin(x) cos(x) tan(x) * ---------------------------------------------------------- - * 0 S C T - * 1 C -S -1/T - * 2 -S -C T - * 3 -C S -1/T + * 0 S C T + * 1 C -S -1/T + * 2 -S -C T + * 3 -C S -1/T * ---------------------------------------------------------- * * Special cases: @@ -705,109 +887,156 @@ static double __kernel_tan(double x, double y, int iy) * trig(NaN) is that NaN; * * Accuracy: - * TRIG(x) returns trig(x) nearly rounded + * TRIG(x) returns trig(x) nearly rounded */ /* sin(x) * Return sine function of x. * * kernel function: - * __kernel_sin ... sine function on [-pi/4,pi/4] - * __kernel_cos ... cose function on [-pi/4,pi/4] - * __ieee754_rem_pio2 ... argument reduction routine + * __kernel_sin ... sine function on [-pi/4,pi/4] + * __kernel_cos ... cose function on [-pi/4,pi/4] + * __ieee754_rem_pio2 ... argument reduction routine */ -double sin(double x) +double +sin (double x) { - double y[2],z=0.0; - int n, ix; + double y[2], z = 0.0; + int n, ix; - /* High word of x. */ - ix = __HI(x); + /* High word of x. */ + ix = __HI (x); - /* |x| ~< pi/4 */ - ix &= 0x7fffffff; - if(ix <= 0x3fe921fb) return __kernel_sin(x,z,0); + /* |x| ~< pi/4 */ + ix &= 0x7fffffff; + if (ix <= 0x3fe921fb) + { + return __kernel_sin (x, z, 0); + } - /* sin(Inf or NaN) is NaN */ - else if (ix>=0x7ff00000) return x-x; + /* sin(Inf or NaN) is NaN */ + else if (ix >= 0x7ff00000) + { + return x - x; + } - /* argument reduction needed */ - else { - n = __ieee754_rem_pio2(x,y); - switch(n&3) { - case 0: return __kernel_sin(y[0],y[1],1); - case 1: return __kernel_cos(y[0],y[1]); - case 2: return -__kernel_sin(y[0],y[1],1); - default: - return -__kernel_cos(y[0],y[1]); - } - } -} + /* argument reduction needed */ + else + { + n = __ieee754_rem_pio2 (x, y); + switch (n & 3) + { + case 0: + { + return __kernel_sin (y[0], y[1], 1); + } + case 1: + { + return __kernel_cos (y[0], y[1]); + } + case 2: + { + return -__kernel_sin (y[0], y[1], 1); + } + default: + { + return -__kernel_cos (y[0], y[1]); + } + } + } +} /* sin */ /* cos(x) * Return cosine function of x. * * kernel function: - * __kernel_sin ... sine function on [-pi/4,pi/4] - * __kernel_cos ... cosine function on [-pi/4,pi/4] - * __ieee754_rem_pio2 ... argument reduction routine + * __kernel_sin ... sine function on [-pi/4,pi/4] + * __kernel_cos ... cosine function on [-pi/4,pi/4] + * __ieee754_rem_pio2 ... argument reduction routine */ -double cos(double x) +double +cos (double x) { - double y[2],z=0.0; - int n, ix; + double y[2], z = 0.0; + int n, ix; - /* High word of x. */ - ix = __HI(x); + /* High word of x. */ + ix = __HI (x); - /* |x| ~< pi/4 */ - ix &= 0x7fffffff; - if(ix <= 0x3fe921fb) return __kernel_cos(x,z); + /* |x| ~< pi/4 */ + ix &= 0x7fffffff; + if (ix <= 0x3fe921fb) + { + return __kernel_cos (x, z); + } - /* cos(Inf or NaN) is NaN */ - else if (ix>=0x7ff00000) return x-x; + /* cos(Inf or NaN) is NaN */ + else if (ix >= 0x7ff00000) + { + return x - x; + } - /* argument reduction needed */ - else { - n = __ieee754_rem_pio2(x,y); - switch(n&3) { - case 0: return __kernel_cos(y[0],y[1]); - case 1: return -__kernel_sin(y[0],y[1],1); - case 2: return -__kernel_cos(y[0],y[1]); - default: - return __kernel_sin(y[0],y[1],1); - } - } -} + /* argument reduction needed */ + else + { + n = __ieee754_rem_pio2 (x, y); + switch (n & 3) + { + case 0: + { + return __kernel_cos (y[0], y[1]); + } + case 1: + { + return -__kernel_sin (y[0], y[1], 1); + } + case 2: + { + return -__kernel_cos (y[0], y[1]); + } + default: + { + return __kernel_sin (y[0], y[1], 1); + } + } + } +} /* cos */ /* tan(x) * Return tangent function of x. * * kernel function: - * __kernel_tan ... tangent function on [-pi/4,pi/4] - * __ieee754_rem_pio2 ... argument reduction routine + * __kernel_tan ... tangent function on [-pi/4,pi/4] + * __ieee754_rem_pio2 ... argument reduction routine */ -double tan(double x) +double +tan (double x) { - double y[2],z=0.0; - int n, ix; + double y[2], z = 0.0; + int n, ix; - /* High word of x. */ - ix = __HI(x); + /* High word of x. */ + ix = __HI (x); - /* |x| ~< pi/4 */ - ix &= 0x7fffffff; - if(ix <= 0x3fe921fb) return __kernel_tan(x,z,1); + /* |x| ~< pi/4 */ + ix &= 0x7fffffff; + if (ix <= 0x3fe921fb) + { + return __kernel_tan (x, z, 1); + } - /* tan(Inf or NaN) is NaN */ - else if (ix>=0x7ff00000) return x-x; /* NaN */ + /* tan(Inf or NaN) is NaN */ + else if (ix >= 0x7ff00000) + { + return x - x; /* NaN */ + } - /* argument reduction needed */ - else { - n = __ieee754_rem_pio2(x,y); - return __kernel_tan(y[0],y[1],1-((n&1)<<1)); /* 1 -- n even - -1 -- n odd */ - } -} + /* argument reduction needed */ + else + { + n = __ieee754_rem_pio2 (x, y); + return __kernel_tan (y[0], y[1], 1 - ((n & 1) << 1)); /* 1 -- n even, -1 -- n odd */ + } +} /* tan */