diff --git a/third-party/fdlibm/e_rem_pio2.c b/third-party/fdlibm/e_rem_pio2.c deleted file mode 100644 index d3f793516..000000000 --- a/third-party/fdlibm/e_rem_pio2.c +++ /dev/null @@ -1,156 +0,0 @@ - -/* @(#)e_rem_pio2.c 1.4 95/01/18 */ -/* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. - * - * Developed at SunSoft, a Sun Microsystems, Inc. business. - * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice - * is preserved. - * ==================================================== - * - */ - -/* __ieee754_rem_pio2(x,y) - * - * return the remainder of x rem pi/2 in y[0]+y[1] - * use __kernel_rem_pio2() - */ - -#include "fdlibm.h" - -/* - * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi - */ -static const int two_over_pi[] = { -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 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, -}; - -/* - * invpio2: 53 bits of 2/pi - * 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_3: third 33 bit of pi/2 - * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3) - */ -#define zero 0.00000000000000000000e+00 /* 0x00000000, 0x00000000 */ -#define half 5.00000000000000000000e-01 /* 0x3FE00000, 0x00000000 */ -#define two24 1.67772160000000000000e+07 /* 0x41700000, 0x00000000 */ -#define invpio2 6.36619772367581382433e-01 /* 0x3FE45F30, 0x6DC9C883 */ -#define pio2_1 1.57079632673412561417e+00 /* 0x3FF921FB, 0x54400000 */ -#define pio2_1t 6.07710050650619224932e-11 /* 0x3DD0B461, 0x1A626331 */ -#define pio2_2 6.07710050630396597660e-11 /* 0x3DD0B461, 0x1A600000 */ -#define pio2_2t 2.02226624879595063154e-21 /* 0x3BA3198A, 0x2E037073 */ -#define pio2_3 2.02226624871116645580e-21 /* 0x3BA3198A, 0x2E000000 */ -#define pio2_3t 8.47842766036889956997e-32 /* 0x397B839A, 0x252049C1 */ - -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; - - 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,two_over_pi); - if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;} - return n; -} diff --git a/third-party/fdlibm/include/fdlibm.h b/third-party/fdlibm/include/fdlibm.h index ecf5e7770..680138bd0 100644 --- a/third-party/fdlibm/include/fdlibm.h +++ b/third-party/fdlibm/include/fdlibm.h @@ -81,12 +81,3 @@ extern int finite (double); */ extern double copysign (double, double); extern double scalbn (double, int); - -/* ieee style elementary functions */ -extern int __ieee754_rem_pio2 (double,double*); - -/* fdlibm kernel function */ -extern double __kernel_sin (double,double,int); -extern double __kernel_cos (double,double); -extern double __kernel_tan (double,double,int); -extern int __kernel_rem_pio2 (double*,double*,int,int,int,const int*); diff --git a/third-party/fdlibm/k_cos.c b/third-party/fdlibm/k_cos.c deleted file mode 100644 index ff284107a..000000000 --- a/third-party/fdlibm/k_cos.c +++ /dev/null @@ -1,82 +0,0 @@ - -/* @(#)k_cos.c 1.3 95/01/18 */ -/* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. - * - * Developed at SunSoft, a Sun Microsystems, Inc. business. - * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice - * is preserved. - * ==================================================== - */ - -/* - * __kernel_cos( x, y ) - * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164 - * Input x is assumed to be bounded by ~pi/4 in magnitude. - * 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 - * | | - * - * 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. - */ - -#include "fdlibm.h" - -#define one 1.00000000000000000000e+00 /* 0x3FF00000, 0x00000000 */ -#define C1 4.16666666666666019037e-02 /* 0x3FA55555, 0x5555554C */ -#define C2 -1.38888888888741095749e-03 /* 0xBF56C16C, 0x16C15177 */ -#define C3 2.48015872894767294178e-05 /* 0x3EFA01A0, 0x19CB1590 */ -#define C4 -2.75573143513906633035e-07 /* 0xBE927E4F, 0x809C52AD */ -#define C5 2.08757232129817482790e-09 /* 0x3E21EE9E, 0xBDB4B1C4 */ -#define C6 -1.13596475577881948265e-11 /* 0xBDA8FAE9, 0xBE8838D4 */ - -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)); - } -} diff --git a/third-party/fdlibm/k_rem_pio2.c b/third-party/fdlibm/k_rem_pio2.c deleted file mode 100644 index 9013b49e4..000000000 --- a/third-party/fdlibm/k_rem_pio2.c +++ /dev/null @@ -1,298 +0,0 @@ - -/* @(#)k_rem_pio2.c 1.3 95/01/18 */ -/* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. - * - * Developed at SunSoft, a Sun Microsystems, Inc. business. - * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice - * is preserved. - * ==================================================== - */ - -/* - * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) - * double x[],y[]; int e0,nx,prec; int ipio2[]; - * - * __kernel_rem_pio2 return the last three digits of N with - * y = x - N*pi/2 - * so that |y| < pi/2. - * - * The method is to compute the integer (mod 8) and fraction parts of - * (2/pi)*x without doing the full multiplication. In general we - * skip the part of the product that are known to be a huge integer ( - * more accurately, = 0 mod 8 ). Thus the number of operations are - * independent of the exponent of the input. - * - * (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. - * - * 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: - * - * 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] - * - * 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) - * - * 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)). - * - * External function: - * double scalbn(), floor(); - * - * - * Here is the description of some local variables: - * - * 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. - * - * 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). - * - * 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. - * - * 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. - * - * f[] ipio2[] in floating point - * - * 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. - * - */ - - -/* - * 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 - * to produce the hexadecimal values shown. - */ - -#include "fdlibm.h" - -static const int init_jk[] = {2,3,4,6}; /* initial value for jk */ - -static const double PIo2[] = { - 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */ - 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */ - 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */ - 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */ - 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */ - 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */ - 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */ - 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */ -}; - -#define zero 0.0 -#define one 1.0 -#define two24 1.67772160000000000000e+07 /* 0x41700000, 0x00000000 */ -#define twon24 5.96046447753906250000e-08 /* 0x3E700000, 0x00000000 */ - -int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int *ipio2) -{ - 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; - - /* 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]; - - /* 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; -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; - } - - /* 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); - } - } - - /* 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; - } - } - - /* 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; - } - - /* 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; -} diff --git a/third-party/fdlibm/k_sin.c b/third-party/fdlibm/k_sin.c deleted file mode 100644 index ad11cb3f3..000000000 --- a/third-party/fdlibm/k_sin.c +++ /dev/null @@ -1,64 +0,0 @@ - -/* @(#)k_sin.c 1.3 95/01/18 */ -/* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. - * - * Developed at SunSoft, a Sun Microsystems, Inc. business. - * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice - * is preserved. - * ==================================================== - */ - -/* __kernel_sin( x, y, iy) - * kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854 - * Input x is assumed to be bounded by ~pi/4 in magnitude. - * Input y is the tail of x. - * 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 | - * - * 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)) - */ - -#include "fdlibm.h" - -#define half 5.00000000000000000000e-01 /* 0x3FE00000, 0x00000000 */ -#define S1 -1.66666666666666324348e-01 /* 0xBFC55555, 0x55555549 */ -#define S2 8.33333333332248946124e-03 /* 0x3F811111, 0x1110F8A6 */ -#define S3 -1.98412698298579493134e-04 /* 0xBF2A01A0, 0x19C161D5 */ -#define S4 2.75573137070700676789e-06 /* 0x3EC71DE3, 0x57B1FE7D */ -#define S5 -2.50507602534068634195e-08 /* 0xBE5AE5E6, 0x8A2B9CEB */ -#define S6 1.58969099521155010221e-10 /* 0x3DE5D93A, 0x5ACFD57C */ - -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); -} diff --git a/third-party/fdlibm/k_tan.c b/third-party/fdlibm/k_tan.c deleted file mode 100644 index 3008811d0..000000000 --- a/third-party/fdlibm/k_tan.c +++ /dev/null @@ -1,140 +0,0 @@ - -/* @(#)k_tan.c 1.5 04/04/22 */ -/* - * ==================================================== - * Copyright 2004 Sun Microsystems, Inc. All Rights Reserved. - * - * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice - * is preserved. - * ==================================================== - */ - -/* __kernel_tan( x, y, k ) - * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 - * Input x is assumed to be bounded by ~pi/4 in magnitude. - * Input y is the tail of x. - * 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 - * - * |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)) - * - * 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))) - */ - -#include "fdlibm.h" - -#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 one 1.00000000000000000000e+00 /* 3FF00000, 00000000 */ -#define pio4 7.85398163397448278999e-01 /* 3FE921FB, 54442D18 */ -#define pio4lo 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */ - -double __kernel_tan(double x, double y, int iy) -{ - 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; - - 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); - } -} diff --git a/third-party/fdlibm/s_cos.c b/third-party/fdlibm/s_cos.c deleted file mode 100644 index 3285bc1e7..000000000 --- a/third-party/fdlibm/s_cos.c +++ /dev/null @@ -1,73 +0,0 @@ - -/* @(#)s_cos.c 1.3 95/01/18 */ -/* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. - * - * Developed at SunSoft, a Sun Microsystems, Inc. business. - * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice - * is preserved. - * ==================================================== - */ - -/* 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 - * - * 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 - * - * 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 - * ---------------------------------------------------------- - * - * Special cases: - * Let trig be any of sin, cos, or tan. - * trig(+-INF) is NaN, with signals; - * trig(NaN) is that NaN; - * - * Accuracy: - * TRIG(x) returns trig(x) nearly rounded - */ - -#include "fdlibm.h" - -double cos(double x) -{ - double y[2],z=0.0; - int n, ix; - - /* High word of x. */ - ix = __HI(x); - - /* |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; - - /* 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); - } - } -} diff --git a/third-party/fdlibm/s_sin.c b/third-party/fdlibm/s_sin.c deleted file mode 100644 index 83fa2e558..000000000 --- a/third-party/fdlibm/s_sin.c +++ /dev/null @@ -1,73 +0,0 @@ - -/* @(#)s_sin.c 1.3 95/01/18 */ -/* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. - * - * Developed at SunSoft, a Sun Microsystems, Inc. business. - * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice - * is preserved. - * ==================================================== - */ - -/* 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 - * - * 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 - * - * 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 - * ---------------------------------------------------------- - * - * Special cases: - * Let trig be any of sin, cos, or tan. - * trig(+-INF) is NaN, with signals; - * trig(NaN) is that NaN; - * - * Accuracy: - * TRIG(x) returns trig(x) nearly rounded - */ - -#include "fdlibm.h" - -double sin(double x) -{ - double y[2],z=0.0; - int n, ix; - - /* High word of x. */ - ix = __HI(x); - - /* |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; - - /* 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]); - } - } -} diff --git a/third-party/fdlibm/s_tan.c b/third-party/fdlibm/s_tan.c deleted file mode 100644 index 55fd9f3f3..000000000 --- a/third-party/fdlibm/s_tan.c +++ /dev/null @@ -1,67 +0,0 @@ - -/* @(#)s_tan.c 1.3 95/01/18 */ -/* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. - * - * Developed at SunSoft, a Sun Microsystems, Inc. business. - * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice - * is preserved. - * ==================================================== - */ - -/* tan(x) - * Return tangent function of x. - * - * kernel function: - * __kernel_tan ... tangent function on [-pi/4,pi/4] - * __ieee754_rem_pio2 ... argument reduction routine - * - * 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 - * - * 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 - * ---------------------------------------------------------- - * - * Special cases: - * Let trig be any of sin, cos, or tan. - * trig(+-INF) is NaN, with signals; - * trig(NaN) is that NaN; - * - * Accuracy: - * TRIG(x) returns trig(x) nearly rounded - */ - -#include "fdlibm.h" - -double tan(double x) -{ - double y[2],z=0.0; - int n, ix; - - /* High word of x. */ - ix = __HI(x); - - /* |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 */ - - /* 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 */ - } -} diff --git a/third-party/fdlibm/s_trig.c b/third-party/fdlibm/s_trig.c new file mode 100644 index 000000000..9041dc547 --- /dev/null +++ b/third-party/fdlibm/s_trig.c @@ -0,0 +1,813 @@ + +/* @(#)k_rem_pio2.c 1.3 95/01/18 */ +/* @(#)e_rem_pio2.c 1.4 95/01/18 */ +/* @(#)k_sin.c 1.3 95/01/18 */ +/* @(#)k_cos.c 1.3 95/01/18 */ +/* @(#)k_tan.c 1.5 04/04/22 */ +/* @(#)s_sin.c 1.3 95/01/18 */ +/* @(#)s_cos.c 1.3 95/01/18 */ +/* @(#)s_tan.c 1.3 95/01/18 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include "fdlibm.h" + +#define zero 0.00000000000000000000e+00 /* 0x00000000, 0x00000000 */ +#define half 5.00000000000000000000e-01 /* 0x3FE00000, 0x00000000 */ +#define one 1.00000000000000000000e+00 /* 0x3FF00000, 0x00000000 */ +#define two24 1.67772160000000000000e+07 /* 0x41700000, 0x00000000 */ +#define twon24 5.96046447753906250000e-08 /* 0x3E700000, 0x00000000 */ + +/* + * __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 + * so that |y| < pi/2. + * + * The method is to compute the integer (mod 8) and fraction parts of + * (2/pi)*x without doing the full multiplication. In general we + * skip the part of the product that are known to be a huge integer ( + * more accurately, = 0 mod 8 ). Thus the number of operations are + * independent of the exponent of the input. + * + * (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. + * + * 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: + * + * 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] + * + * 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) + * + * External function: + * 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[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. + * + * jz local integer variable indicating the number of + * terms of ipio2[] used. + * + * 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). + * + * 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. + * + * 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. + * + * f[] ipio2[] in floating point + * + * 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. + * + */ + + +/* + * 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 + * to produce the hexadecimal values shown. + */ + +static const int init_jk[] = {2,3,4,6}; /* initial value for jk */ + +static const double PIo2[] = { + 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */ + 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */ + 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */ + 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */ + 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */ + 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */ + 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */ + 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */ +}; + +/* + * 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 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]; + + /* 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); + + /* 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; + } + + 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; + } + + /* 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); + } + } + + /* 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; + } + } + + /* 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; + } + + /* 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; +} + +/* __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, +}; + +/* + * invpio2: 53 bits of 2/pi + * 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_3: third 33 bit of pi/2 + * 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 */ +#define pio2_1t 6.07710050650619224932e-11 /* 0x3DD0B461, 0x1A626331 */ +#define pio2_2 6.07710050630396597660e-11 /* 0x3DD0B461, 0x1A600000 */ +#define pio2_2t 2.02226624879595063154e-21 /* 0x3BA3198A, 0x2E037073 */ +#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) +{ + 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; +} + +/* __kernel_sin( x, y, iy) + * kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854 + * Input x is assumed to be bounded by ~pi/4 in magnitude. + * Input y is the tail of x. + * 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 | + * + * 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 */ +#define S2 8.33333333332248946124e-03 /* 0x3F811111, 0x1110F8A6 */ +#define S3 -1.98412698298579493134e-04 /* 0xBF2A01A0, 0x19C161D5 */ +#define S4 2.75573137070700676789e-06 /* 0x3EC71DE3, 0x57B1FE7D */ +#define S5 -2.50507602534068634195e-08 /* 0xBE5AE5E6, 0x8A2B9CEB */ +#define S6 1.58969099521155010221e-10 /* 0x3DE5D93A, 0x5ACFD57C */ + +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); +} + +/* + * __kernel_cos( x, y ) + * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164 + * Input x is assumed to be bounded by ~pi/4 in magnitude. + * 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 + * | | + * + * 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 */ +#define C2 -1.38888888888741095749e-03 /* 0xBF56C16C, 0x16C15177 */ +#define C3 2.48015872894767294178e-05 /* 0x3EFA01A0, 0x19CB1590 */ +#define C4 -2.75573143513906633035e-07 /* 0xBE927E4F, 0x809C52AD */ +#define C5 2.08757232129817482790e-09 /* 0x3E21EE9E, 0xBDB4B1C4 */ +#define C6 -1.13596475577881948265e-11 /* 0xBDA8FAE9, 0xBE8838D4 */ + +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)); + } +} + +/* __kernel_tan( x, y, k ) + * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 + * Input x is assumed to be bounded by ~pi/4 in magnitude. + * Input y is the tail of x. + * 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 + * + * |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)) + * + * 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))) + */ + +#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) +{ + 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; + + 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); + } +} + +/* 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 + * + * 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 + * ---------------------------------------------------------- + * + * Special cases: + * Let trig be any of sin, cos, or tan. + * trig(+-INF) is NaN, with signals; + * trig(NaN) is that NaN; + * + * Accuracy: + * 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 + */ +double sin(double x) +{ + double y[2],z=0.0; + int n, ix; + + /* High word of x. */ + ix = __HI(x); + + /* |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; + + /* 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]); + } + } +} + +/* 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 + */ + +double cos(double x) +{ + double y[2],z=0.0; + int n, ix; + + /* High word of x. */ + ix = __HI(x); + + /* |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; + + /* 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); + } + } +} + +/* tan(x) + * Return tangent function of x. + * + * kernel function: + * __kernel_tan ... tangent function on [-pi/4,pi/4] + * __ieee754_rem_pio2 ... argument reduction routine + */ + +double tan(double x) +{ + double y[2],z=0.0; + int n, ix; + + /* High word of x. */ + ix = __HI(x); + + /* |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 */ + + /* 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 */ + } +}