From e0cf66b4210359f1caa92f9d98f0fa6e0667d556 Mon Sep 17 00:00:00 2001 From: Natalia Portillo Date: Fri, 9 Dec 2016 03:26:17 +0000 Subject: [PATCH] Use precomputed trigonometry for GNU-EFI. --- e_rem_pio2.c | 163 -------------------------- k_cos.c | 83 -------------- k_rem_pio2.c | 302 ------------------------------------------------- k_sin.c | 66 ----------- math_private.h | 167 --------------------------- s_copysign.c | 29 ----- s_cos.c | 73 ------------ s_fabs.c | 27 ----- s_floor.c | 68 ----------- s_scalbn.c | 54 --------- s_sin.c | 73 ------------ 11 files changed, 1105 deletions(-) delete mode 100644 e_rem_pio2.c delete mode 100644 k_cos.c delete mode 100644 k_rem_pio2.c delete mode 100644 k_sin.c delete mode 100644 math_private.h delete mode 100644 s_copysign.c delete mode 100644 s_cos.c delete mode 100644 s_fabs.c delete mode 100644 s_floor.c delete mode 100644 s_scalbn.c delete mode 100644 s_sin.c diff --git a/e_rem_pio2.c b/e_rem_pio2.c deleted file mode 100644 index 9dd3754..0000000 --- a/e_rem_pio2.c +++ /dev/null @@ -1,163 +0,0 @@ -/* @(#)e_rem_pio2.c 5.1 93/09/24 */ -/* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. - * - * Developed at SunPro, 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 "math_private.h" - -/* - * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi - */ -static const __INT32_TYPE__ 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 __INT32_TYPE__ 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) - */ - -static const double -zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ -half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ -two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ -invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */ -pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */ -pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */ -pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */ -pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */ -pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */ -pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */ - -__INT32_TYPE__ -__ieee754_rem_pio2(double x, double *y) -{ - double z,w,t,r,fn; - double tx[3]; - __INT32_TYPE__ e0,i,j,nx,n,ix,hx; - __UINT32_TYPE__ low; - - z = 0; - GET_HIGH_WORD(hx,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 = (__INT32_TYPE__) (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 { - __UINT32_TYPE__ high; - j = ix>>20; - y[0] = r-w; - GET_HIGH_WORD(high,y[0]); - i = j-((high>>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; - GET_HIGH_WORD(high,y[0]); - i = j-((high>>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) */ - GET_LOW_WORD(low,x); - SET_LOW_WORD(z,low); - e0 = (ix>>20)-1046; /* e0 = ilogb(z)-23; */ - SET_HIGH_WORD(z, ix - ((__INT32_TYPE__)(e0<<20))); - for(i=0;i<2;i++) { - tx[i] = (double)((__INT32_TYPE__)(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; -} \ No newline at end of file diff --git a/k_cos.c b/k_cos.c deleted file mode 100644 index 159bb1a..0000000 --- a/k_cos.c +++ /dev/null @@ -1,83 +0,0 @@ -/* @(#)k_cos.c 5.1 93/09/24 */ -/* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. - * - * Developed at SunPro, 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 "math_private.h" - -static const double -one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ -C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */ -C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */ -C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */ -C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */ -C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */ -C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */ - -double -__kernel_cos(double x, double y) -{ - double a,hz,z,r,qx; - __UINT32_TYPE__ ix; - GET_HIGH_WORD(ix,x); - ix &= 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 { - INSERT_WORDS(qx,ix-0x00200000,0); /* x/4 */ - } - hz = 0.5*z-qx; - a = one-qx; - return a - (hz - (z*r-x*y)); - } -} \ No newline at end of file diff --git a/k_rem_pio2.c b/k_rem_pio2.c deleted file mode 100644 index 159b2a0..0000000 --- a/k_rem_pio2.c +++ /dev/null @@ -1,302 +0,0 @@ -/* @(#)k_rem_pio2.c 5.1 93/09/24 */ -/* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. - * - * Developed at SunPro, 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[] output 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 "math.h" -#include "math_private.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 */ -}; - -static const double -zero = 0.0, -one = 1.0, -two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ -twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */ - -int -__kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const __INT32_TYPE__ *ipio2) -{ - __INT32_TYPE__ 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)((__INT32_TYPE__)(twon24* z)); - iq[i] = (__INT32_TYPE__)(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 = (__INT32_TYPE__) 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)((__INT32_TYPE__)(twon24*z)); - iq[jz] = (__INT32_TYPE__)(z-two24*fw); - jz += 1; q0 += 24; - iq[jz] = (__INT32_TYPE__) fw; - } else iq[jz] = (__INT32_TYPE__) 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; -} \ No newline at end of file diff --git a/k_sin.c b/k_sin.c deleted file mode 100644 index 510d426..0000000 --- a/k_sin.c +++ /dev/null @@ -1,66 +0,0 @@ -/* @(#)k_sin.c 5.1 93/09/24 */ -/* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. - * - * Developed at SunPro, 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 "math_private.h" - -static const double -half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ -S1 = -1.66666666666666324348e-01, /* 0xBFC55555, 0x55555549 */ -S2 = 8.33333333332248946124e-03, /* 0x3F811111, 0x1110F8A6 */ -S3 = -1.98412698298579493134e-04, /* 0xBF2A01A0, 0x19C161D5 */ -S4 = 2.75573137070700676789e-06, /* 0x3EC71DE3, 0x57B1FE7D */ -S5 = -2.50507602534068634195e-08, /* 0xBE5AE5E6, 0x8A2B9CEB */ -S6 = 1.58969099521155010221e-10; /* 0x3DE5D93A, 0x5ACFD57C */ - -double -__kernel_sin(double x, double y, int iy) -{ - double z,r,v; - __UINT32_TYPE__ ix; - GET_HIGH_WORD(ix,x); - ix &= 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); -} \ No newline at end of file diff --git a/math_private.h b/math_private.h deleted file mode 100644 index 3703b5e..0000000 --- a/math_private.h +++ /dev/null @@ -1,167 +0,0 @@ -/* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. - * - * Developed at SunPro, a Sun Microsystems, Inc. business. - * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice - * is preserved. - * ==================================================== - */ - -/* - * from: @(#)fdlibm.h 5.1 93/09/24 - * $NetBSD: math_private.h,v 1.12 2005/07/21 12:55:58 christos Exp $ - */ - -#ifndef _MATH_PRIVATE_H_ -#define _MATH_PRIVATE_H_ - -/* The original fdlibm code used statements like: - n0 = ((*(int*)&one)>>29)^1; * index of high word * - ix0 = *(n0+(int*)&x); * high word of x * - ix1 = *((1-n0)+(int*)&x); * low word of x * - to dig two 32 bit words out of the 64 bit IEEE floating point - value. That is non-ANSI, and, moreover, the gcc instruction - scheduler gets it wrong. We instead use the following macros. - Unlike the original code, we determine the endianness at compile - time, not at run time; I don't see much benefit to selecting - endianness at run time. */ - -/* A union which permits us to convert between a double and two 32 bit - ints. */ - -/* - * The ARM ports are little endian except for the FPA word order which is - * big endian. - */ - -#if (__BYTE_ORDER__ == __ORDER_BIG_ENDIAN__) || (defined(__arm__) && !defined(__VFP_FP__)) - -typedef union -{ - double value; - struct - { - unsigned int msw; - unsigned int lsw; - } parts; -} ieee_double_shape_type; - -#endif - -#if (__BYTE_ORDER__ == __ORDER_LITTLE_ENDIAN__) && !(defined(__arm__) && !defined(__VFP_FP__)) - -typedef union -{ - double value; - struct - { - unsigned int lsw; - unsigned int msw; - } parts; -} ieee_double_shape_type; - -#endif - -/* Get two 32 bit ints from a double. */ - -#define EXTRACT_WORDS(ix0,ix1,d) \ -do { \ - ieee_double_shape_type ew_u; \ - ew_u.value = (d); \ - (ix0) = ew_u.parts.msw; \ - (ix1) = ew_u.parts.lsw; \ -} while (0) - -/* Get the more significant 32 bit int from a double. */ - -#define GET_HIGH_WORD(i,d) \ -do { \ - ieee_double_shape_type gh_u; \ - gh_u.value = (d); \ - (i) = gh_u.parts.msw; \ -} while (0) - -/* Get the less significant 32 bit int from a double. */ - -#define GET_LOW_WORD(i,d) \ -do { \ - ieee_double_shape_type gl_u; \ - gl_u.value = (d); \ - (i) = gl_u.parts.lsw; \ -} while (0) - -/* Set a double from two 32 bit ints. */ - -#define INSERT_WORDS(d,ix0,ix1) \ -do { \ - ieee_double_shape_type iw_u; \ - iw_u.parts.msw = (ix0); \ - iw_u.parts.lsw = (ix1); \ - (d) = iw_u.value; \ -} while (0) - -/* Set the more significant 32 bits of a double from an int. */ - -#define SET_HIGH_WORD(d,v) \ -do { \ - ieee_double_shape_type sh_u; \ - sh_u.value = (d); \ - sh_u.parts.msw = (v); \ - (d) = sh_u.value; \ -} while (0) - -/* Set the less significant 32 bits of a double from an int. */ - -#define SET_LOW_WORD(d,v) \ -do { \ - ieee_double_shape_type sl_u; \ - sl_u.value = (d); \ - sl_u.parts.lsw = (v); \ - (d) = sl_u.value; \ -} while (0) - -/** Compute the value of the cosine of Arg, measured in radians. - @param[in] Arg The value to compute the cosine of. - @return The computed value of the cosine of Arg. -**/ -double cos(double Arg); - -/** Compute the value of the sine of Arg. - @param[in] Arg The value to compute the sine of. - @return The computed value of the sine of Arg. -**/ -double sin(double Arg); - -/** Compute the absolute value of Arg. - @param[in] Arg The value to compute the absolute value of. - @return The absolute value of Arg. -**/ -double fabs(double Arg); - -/** Compute the largest integer value not greater than Arg. - @param[in] Arg The value to compute the floor of. - @return The largest integer value not greater than Arg, expressed as a floating-point number. -**/ -double floor(double); - -/* 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 int __kernel_rem_pio2 (double*,double*,int,int,int,const int*); - - -/**@{ - C99, Posix, or NetBSD functions that are not part of the C95 specification. -**/ -/* - * Functions callable from C, intended to support IEEE arithmetic. - */ -double scalbn(double, int); -double copysign(double, double); - -#endif /* _MATH_PRIVATE_H_ */ \ No newline at end of file diff --git a/s_copysign.c b/s_copysign.c deleted file mode 100644 index 9650223..0000000 --- a/s_copysign.c +++ /dev/null @@ -1,29 +0,0 @@ -/* @(#)s_copysign.c 5.1 93/09/24 */ -/* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. - * - * Developed at SunPro, a Sun Microsystems, Inc. business. - * Permission to use, copy, modify, and distribute this - * 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 - * with the sign bit of y. - */ - -#include "math_private.h" - -double -copysign(double x, double y) -{ - __UINT32_TYPE__ hx,hy; - GET_HIGH_WORD(hx,x); - GET_HIGH_WORD(hy,y); - SET_HIGH_WORD(x,(hx&0x7fffffff)|(hy&0x80000000)); - return x; -} \ No newline at end of file diff --git a/s_cos.c b/s_cos.c deleted file mode 100644 index ee09e1b..0000000 --- a/s_cos.c +++ /dev/null @@ -1,73 +0,0 @@ -/* @(#)s_cos.c 5.1 93/09/24 */ -/* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. - * - * Developed at SunPro, 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 "math_private.h" - -double -cos(double x) -{ - double y[2],z=0.0; - __UINT32_TYPE__ n, ix; - - /* High word of x. */ - GET_HIGH_WORD(ix,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); - } - } -} \ No newline at end of file diff --git a/s_fabs.c b/s_fabs.c deleted file mode 100644 index b3dbbd8..0000000 --- a/s_fabs.c +++ /dev/null @@ -1,27 +0,0 @@ - -/* @(#)s_fabs.c 5.1 93/09/24 */ -/* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. - * - * Developed at SunPro, a Sun Microsystems, Inc. business. - * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice - * is preserved. - * ==================================================== - */ - -/* - * fabs(x) returns the absolute value of x. - */ - -#include "math_private.h" - -double -fabs(double x) -{ - __UINT32_TYPE__ high; - GET_HIGH_WORD(high,x); - SET_HIGH_WORD(x,high&0x7fffffff); - return x; -} \ No newline at end of file diff --git a/s_floor.c b/s_floor.c deleted file mode 100644 index 6c9d747..0000000 --- a/s_floor.c +++ /dev/null @@ -1,68 +0,0 @@ -/* @(#)s_floor.c 5.1 93/09/24 */ -/* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. - * - * Developed at SunPro, a Sun Microsystems, Inc. business. - * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice - * is preserved. - * ==================================================== - */ - -/* - * floor(x) - * Return x rounded toward -inf to integral value - * Method: - * Bit twiddling. - * Exception: - * Inexact flag raised if x not equal to floor(x). - */ - -#include "math_private.h" - -static const double huge = 1.0e300; - -double -floor(double x) -{ - __INT32_TYPE__ i0,i1,j0; - __UINT32_TYPE__ i,j; - EXTRACT_WORDS(i0,i1,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 = ((__UINT32_TYPE__)(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((__INT32_TYPE__)j>20; /* extract exponent */ - if (k==0) { /* 0 or subnormal x */ - if ((lx|(hx&0x7fffffff))==0) return x; /* +-0 */ - x *= two54; - GET_HIGH_WORD(hx,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 */ - {SET_HIGH_WORD(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 */ - SET_HIGH_WORD(x,(hx&0x800fffff)|(k<<20)); - return x*twom54; -} \ No newline at end of file diff --git a/s_sin.c b/s_sin.c deleted file mode 100644 index 5f0b4a1..0000000 --- a/s_sin.c +++ /dev/null @@ -1,73 +0,0 @@ -/* @(#)s_sin.c 5.1 93/09/24 */ -/* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. - * - * Developed at SunPro, 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 "math_private.h" - -double -sin(double x) -{ - double y[2],z=0.0; - __UINT32_TYPE__ n, ix; - - /* High word of x. */ - GET_HIGH_WORD(ix,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]); - } - } -} \ No newline at end of file