Szabolcs Nagy
1d5ba3bb5a
* use unsigned arithmetics on the representation * store arg reduction quotient in unsigned (so n%2 would work like n&1) * use different convention to pass the arg reduction bit to __tan (this argument used to be 1 for even and -1 for odd reduction which meant obscure bithacks, the new n&1 is cleaner) * raise inexact and underflow flags correctly for small x (tanl(x) may still raise spurious underflow for small but normal x) (this exception raising code increases codesize a bit, similar fixes are needed in many other places, it may worth investigating at some point if the inexact and underflow flags are worth raising correctly as this is not strictly required by the standard) * tanf manual reduction optimization is kept for now * tanl code path is cleaned up to follow similar logic to tan and tanf
111 lines
3.9 KiB
C
111 lines
3.9 KiB
C
/* origin: FreeBSD /usr/src/lib/msun/src/k_tan.c */
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/*
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* ====================================================
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* Copyright 2004 Sun Microsystems, Inc. All Rights Reserved.
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*
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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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/* __tan( x, y, k )
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* kernel tan function on ~[-pi/4, pi/4] (except on -0), 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 odd indicates whether tan (if odd = 0) or -1/tan (if odd = 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. Callers must return tan(-0) = -0 without calling here since our
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* odd polynomial is not evaluated in a way that preserves -0.
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* Callers may do the optimization tan(x) ~ x for tiny x.
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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 "libm.h"
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static const double T[] = {
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3.33333333333334091986e-01, /* 3FD55555, 55555563 */
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1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */
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5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */
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2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */
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8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */
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3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */
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1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */
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5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */
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2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */
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7.81794442939557092300e-05, /* 3F147E88, A03792A6 */
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7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */
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-1.85586374855275456654e-05, /* BEF375CB, DB605373 */
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2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */
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},
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pio4 = 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */
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pio4lo = 3.06161699786838301793e-17; /* 3C81A626, 33145C07 */
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double __tan(double x, double y, int odd)
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{
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double_t z, r, v, w, s, a;
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double w0, a0;
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uint32_t hx;
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int big, sign;
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GET_HIGH_WORD(hx,x);
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big = (hx&0x7fffffff) >= 0x3FE59428; /* |x| >= 0.6744 */
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if (big) {
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sign = hx>>31;
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if (sign) {
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x = -x;
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y = -y;
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}
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x = (pio4 - x) + (pio4lo - y);
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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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/*
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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) + s*T[0];
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w = x + r;
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if (big) {
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s = 1 - 2*odd;
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v = s - 2.0 * (x + (r - w*w/(w + s)));
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return sign ? -v : v;
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}
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if (!odd)
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return w;
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/* -1.0/(x+r) has up to 2ulp error, so compute it accurately */
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w0 = w;
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SET_LOW_WORD(w0, 0);
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v = r - (w0 - x); /* w0+v = r+x */
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a0 = a = -1.0 / w;
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SET_LOW_WORD(a0, 0);
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return a0 + a*(1.0 + a0*w0 + a0*v);
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}
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