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This fix for k_tan.c is a copy from fdlibm version 5.3 (see also http://www.netlib.org/fdlibm/readme), adjusted to use the macros available in newlib (SET_LOW_WORD). This fix reduces the ULP error of the value shown in the fdlibm readme (tan(1.7765241907548024E+269)) to 0.45 (thereby reducing the error by 1). This issue only happens for large numbers that get reduced by the range reduction to a value smaller in magnitude than 2^-28, that is also reduced an uneven number of times. This seems rather unlikely given that one ULP is (much) larger than 2^-28 for the values that may cause an issue. Although given the sheer number of values a double can represent, it is still possible that there are more affected values, finding them however will be quite hard, if not impossible. We also took a look at how another library (libm in FreeBSD) handles the issue: In FreeBSD the complete if branch which checks for values smaller than 2^-28 (or rather 2^-27, another change done by FreeBSD) is moved out of the kernel function and into the external function. This means that the value that gets checked for this condition is the unreduced value. Therefore the input value which caused a problem in the fdlibm/newlib kernel tan will run through the full polynomial, including the careful calculation of -1/(x+r). So the difference is really whether r or y is used. r = y + p with p being the result of the polynomial with 1/3*x^3 being the largest (and magnitude defining) value. With x being <2^-27 we therefore know that p is smaller than y (y has to be at least the size of the value of x last mantissa bit divided by 2, which is at least x*2^-51 for doubles) by enough to warrant saying that r ~ y. So we can conclude that the general implementation of this special case is the same, FreeBSD simply has a different philosophy on when to handle especially small numbers.
146 lines
4.2 KiB
C
146 lines
4.2 KiB
C
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/* @(#)k_tan.c 5.1 93/09/24 */
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/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunPro, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*/
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/* __kernel_tan( x, y, k )
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* kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
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* Input x is assumed to be bounded by ~pi/4 in magnitude.
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* Input y is the tail of x.
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* Input k indicates whether tan (if k=1) or
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* -1/tan (if k= -1) is returned.
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*
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* Algorithm
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* 1. Since tan(-x) = -tan(x), we need only to consider positive x.
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* 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
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* 3. tan(x) is approximated by a odd polynomial of degree 27 on
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* [0,0.67434]
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* 3 27
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* tan(x) ~ x + T1*x + ... + T13*x
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* where
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*
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* |tan(x) 2 4 26 | -59.2
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* |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2
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* | x |
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*
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* Note: tan(x+y) = tan(x) + tan'(x)*y
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* ~ tan(x) + (1+x*x)*y
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* Therefore, for better accuracy in computing tan(x+y), let
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* 3 2 2 2 2
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* r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
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* then
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* 3 2
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* tan(x+y) = x + (T1*x + (x *(r+y)+y))
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*
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* 4. For x in [0.67434,pi/4], let y = pi/4 - x, then
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* tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
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* = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
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*/
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#include "fdlibm.h"
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#ifndef _DOUBLE_IS_32BITS
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#ifdef __STDC__
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static const double
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#else
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static double
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#endif
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one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
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pio4 = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
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pio4lo= 3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */
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T[] = {
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3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */
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1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */
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5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */
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2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */
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8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */
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3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */
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1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */
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5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */
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2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */
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7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */
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7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */
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-1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */
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2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */
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};
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#ifdef __STDC__
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double __kernel_tan(double x, double y, int iy)
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#else
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double __kernel_tan(x, y, iy)
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double x,y; int iy;
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#endif
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{
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double z,r,v,w,s;
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__int32_t ix,hx;
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GET_HIGH_WORD(hx,x);
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ix = hx&0x7fffffff; /* high word of |x| */
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if(ix<0x3e300000) { /* x < 2**-28 */
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if((int)x==0) { /* generate inexact */
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__uint32_t low;
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GET_LOW_WORD(low,x);
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if(((ix|low)|(iy+1))==0) return one/fabs(x);
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else {
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if(iy==1)
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return x;
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else {
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double a, t;
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z = w = x + y;
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SET_LOW_WORD(z,0);
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v = y - (z - x);
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t = a = -one / w;
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SET_LOW_WORD(t,0);
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s = one + t * z;
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return t + a * (s + t * v);
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}
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}
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}
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}
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if(ix>=0x3FE59428) { /* |x|>=0.6744 */
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if(hx<0) {x = -x; y = -y;}
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z = pio4-x;
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w = pio4lo-y;
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x = z+w; y = 0.0;
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}
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z = x*x;
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w = z*z;
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/* Break x^5*(T[1]+x^2*T[2]+...) into
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* x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
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* x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
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*/
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r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11]))));
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v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12])))));
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s = z*x;
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r = y + z*(s*(r+v)+y);
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r += T[0]*s;
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w = x+r;
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if(ix>=0x3FE59428) {
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v = (double)iy;
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return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r)));
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}
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if(iy==1) return w;
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else { /* if allow error up to 2 ulp,
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simply return -1.0/(x+r) here */
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/* compute -1.0/(x+r) accurately */
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double a,t;
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z = w;
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SET_LOW_WORD(z,0);
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v = r-(z - x); /* z+v = r+x */
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t = a = -1.0/w; /* a = -1.0/w */
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SET_LOW_WORD(t,0);
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s = 1.0+t*z;
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return t+a*(s+t*v);
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}
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}
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#endif /* defined(_DOUBLE_IS_32BITS) */
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