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math: long double inverse trigonometric cleanup (acosl, asinl, atanl, atan2l)
* added ld128 support from freebsd fdlibm (untested) * using new ldshape union instead of IEEEl2bits * inexact status flag is not supported
This commit is contained in:
Szabolcs Nagy
parent
c2a0dfea62
commit
bcd797a5ba
@ -1,7 +1,7 @@
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#include <float.h>
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#include "__invtrigl.h"
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#if LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
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static const long double
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pS0 = 1.66666666666666666631e-01L,
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pS1 = -4.16313987993683104320e-01L,
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@ -28,4 +28,36 @@ long double __invtrigl_R(long double z)
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q = 1.0+z*(qS1+z*(qS2+z*(qS3+z*(qS4+z*qS5))));
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return p/q;
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}
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#elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
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static const long double
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pS0 = 1.66666666666666666666666666666700314e-01L,
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pS1 = -7.32816946414566252574527475428622708e-01L,
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pS2 = 1.34215708714992334609030036562143589e+00L,
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pS3 = -1.32483151677116409805070261790752040e+00L,
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pS4 = 7.61206183613632558824485341162121989e-01L,
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pS5 = -2.56165783329023486777386833928147375e-01L,
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pS6 = 4.80718586374448793411019434585413855e-02L,
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pS7 = -4.42523267167024279410230886239774718e-03L,
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pS8 = 1.44551535183911458253205638280410064e-04L,
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pS9 = -2.10558957916600254061591040482706179e-07L,
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qS1 = -4.84690167848739751544716485245697428e+00L,
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qS2 = 9.96619113536172610135016921140206980e+00L,
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qS3 = -1.13177895428973036660836798461641458e+01L,
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qS4 = 7.74004374389488266169304117714658761e+00L,
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qS5 = -3.25871986053534084709023539900339905e+00L,
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qS6 = 8.27830318881232209752469022352928864e-01L,
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qS7 = -1.18768052702942805423330715206348004e-01L,
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qS8 = 8.32600764660522313269101537926539470e-03L,
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qS9 = -1.99407384882605586705979504567947007e-04L;
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const long double pio2_hi = 1.57079632679489661923132169163975140L;
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const long double pio2_lo = 4.33590506506189051239852201302167613e-35L;
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long double __invtrigl_R(long double z)
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{
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long double p, q;
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p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*(pS5+z*(pS6+z*(pS7+z*(pS8+z*pS9)))))))));
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q = 1.0+z*(qS1+z*(qS2+z*(qS3+z*(qS4+z*(qS5+z*(pS6+z*(pS7+z*(pS8+z*pS9))))))));
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return p/q;
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}
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#endif
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@ -1,10 +1,6 @@
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#include <float.h>
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#if LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
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/* shared by acosl, asinl and atan2l */
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#define pio2_hi __pio2_hi
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#define pio2_lo __pio2_lo
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extern const long double pio2_hi, pio2_lo;
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long double __invtrigl_R(long double z);
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#endif
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@ -23,46 +23,45 @@ long double acosl(long double x)
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}
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#elif (LDBL_MANT_DIG == 64 || LDBL_MANT_DIG == 113) && LDBL_MAX_EXP == 16384
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#include "__invtrigl.h"
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#if LDBL_MANT_DIG == 64
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#define CLEARBOTTOM(u) (u.i.m &= -1ULL << 32)
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#elif LDBL_MANT_DIG == 113
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#define CLEARBOTTOM(u) (u.i.lo = 0)
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#endif
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long double acosl(long double x)
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{
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union IEEEl2bits u;
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long double z, w, s, c, df;
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int16_t expsign, expt;
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u.e = x;
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expsign = u.xbits.expsign;
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expt = expsign & 0x7fff;
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union ldshape u = {x};
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long double z, s, c, f;
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uint16_t e = u.i.se & 0x7fff;
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/* |x| >= 1 or nan */
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if (expt >= 0x3fff) {
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if (expt == 0x3fff &&
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((u.bits.manh & ~LDBL_NBIT) | u.bits.manl) == 0) {
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if (expsign > 0)
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return 0; /* acos(1) = 0 */
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return 2*pio2_hi + 0x1p-120f; /* acos(-1)= pi */
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}
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return 0/(x-x); /* acos(|x|>1) is NaN */
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if (e >= 0x3fff) {
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if (x == 1)
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return 0;
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if (x == -1)
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return 2*pio2_hi + 0x1p-120f;
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return 0/(x-x);
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}
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/* |x| < 0.5 */
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if (expt < 0x3fff - 1) {
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if (expt < 0x3fff - 65)
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return pio2_hi + 0x1p-120f; /* x < 0x1p-65: acosl(x)=pi/2 */
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return pio2_hi - (x - (pio2_lo - x * __invtrigl_R(x*x)));
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if (e < 0x3fff - 1) {
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if (e < 0x3fff - LDBL_MANT_DIG - 1)
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return pio2_hi + 0x1p-120f;
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return pio2_hi - (__invtrigl_R(x*x)*x - pio2_lo + x);
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}
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/* x < -0.5 */
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if (expsign < 0) {
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z = (1.0 + x) * 0.5;
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if (u.i.se >> 15) {
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z = (1 + x)*0.5;
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s = sqrtl(z);
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w = __invtrigl_R(z) * s - pio2_lo;
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return 2*(pio2_hi - (s + w));
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return 2*(pio2_hi - (__invtrigl_R(z)*s - pio2_lo + s));
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}
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/* x > 0.5 */
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z = (1.0 - x) * 0.5;
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z = (1 - x)*0.5;
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s = sqrtl(z);
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u.e = s;
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u.bits.manl = 0;
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df = u.e;
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c = (z - df * df) / (s + df);
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w = __invtrigl_R(z) * s + c;
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return 2*(df + w);
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u.f = s;
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CLEARBOTTOM(u);
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f = u.f;
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c = (z - f*f)/(s + f);
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return 2*(__invtrigl_R(z)*s + c + f);
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}
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#endif
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@ -23,27 +23,29 @@ long double asinl(long double x)
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}
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#elif (LDBL_MANT_DIG == 64 || LDBL_MANT_DIG == 113) && LDBL_MAX_EXP == 16384
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#include "__invtrigl.h"
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/* 0.95 */
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#define THRESH ((0xe666666666666666ULL>>(64-(LDBL_MANH_SIZE-1)))|LDBL_NBIT)
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#if LDBL_MANT_DIG == 64
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#define CLOSETO1(u) (u.i.m>>56 >= 0xf7)
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#define CLEARBOTTOM(u) (u.i.m &= -1ULL << 32)
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#elif LDBL_MANT_DIG == 113
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#define CLOSETO1(u) (u.i.top >= 0xee00)
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#define CLEARBOTTOM(u) (u.i.lo = 0)
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#endif
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long double asinl(long double x)
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{
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union IEEEl2bits u;
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long double z,r,s;
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uint16_t expsign, expt;
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union ldshape u = {x};
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long double z, r, s;
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uint16_t e = u.i.se & 0x7fff;
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int sign = u.i.se >> 15;
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u.e = x;
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expsign = u.xbits.expsign;
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expt = expsign & 0x7fff;
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if (expt >= 0x3fff) { /* |x| >= 1 or nan */
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if (expt == 0x3fff &&
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((u.bits.manh&~LDBL_NBIT)|u.bits.manl) == 0)
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/* asin(+-1)=+-pi/2 with inexact */
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if (e >= 0x3fff) { /* |x| >= 1 or nan */
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/* asin(+-1)=+-pi/2 with inexact */
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if (x == 1 || x == -1)
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return x*pio2_hi + 0x1p-120f;
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return 0/(x-x);
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}
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if (expt < 0x3fff - 1) { /* |x| < 0.5 */
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if (expt < 0x3fff - 32) { /* |x|<0x1p-32, asinl(x)=x */
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if (e < 0x3fff - 1) { /* |x| < 0.5 */
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if (e < 0x3fff - (LDBL_MANT_DIG+1)/2) {
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/* return x with inexact if x!=0 */
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FORCE_EVAL(x + 0x1p120f);
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return x;
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@ -54,18 +56,16 @@ long double asinl(long double x)
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z = (1.0 - fabsl(x))*0.5;
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s = sqrtl(z);
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r = __invtrigl_R(z);
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if (u.bits.manh >= THRESH) { /* if |x| is close to 1 */
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if (CLOSETO1(u)) {
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x = pio2_hi - (2*(s+s*r)-pio2_lo);
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} else {
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long double f, c;
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u.e = s;
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u.bits.manl = 0;
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f = u.e;
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c = (z-f*f)/(s+f);
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u.f = s;
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CLEARBOTTOM(u);
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f = u.f;
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c = (z - f*f)/(s + f);
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x = 0.5*pio2_hi-(2*s*r - (pio2_lo-2*c) - (0.5*pio2_hi-2*f));
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}
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if (expsign>>15)
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return -x;
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return x;
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return sign ? -x : x;
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}
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#endif
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@ -27,56 +27,50 @@ long double atan2l(long double y, long double x)
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long double atan2l(long double y, long double x)
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{
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union IEEEl2bits ux, uy;
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union ldshape ux, uy;
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long double z;
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int m;
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uint16_t exptx, expsignx, expty, expsigny;
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int m, ex, ey;
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if (isnan(x) || isnan(y))
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return x+y;
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if (x == 1)
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return atanl(y);
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uy.e = y;
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expsigny = uy.xbits.expsign;
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expty = expsigny & 0x7fff;
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ux.e = x;
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expsignx = ux.xbits.expsign;
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exptx = expsignx & 0x7fff;
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m = ((expsigny>>15)&1) | ((expsignx>>14)&2); /* 2*sign(x)+sign(y) */
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ux.f = x;
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uy.f = y;
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ex = ux.i.se & 0x7fff;
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ey = uy.i.se & 0x7fff;
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m = 2*(ux.i.se>>15) | uy.i.se>>15;
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if (y == 0) {
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switch(m) {
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case 0:
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case 1: return y; /* atan(+-0,+anything)=+-0 */
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case 2: return 2*pio2_hi+0x1p-120f; /* atan(+0,-anything) = pi */
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case 3: return -2*pio2_hi-0x1p-120f; /* atan(-0,-anything) =-pi */
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case 2: return 2*pio2_hi; /* atan(+0,-anything) = pi */
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case 3: return -2*pio2_hi; /* atan(-0,-anything) =-pi */
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}
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}
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if (x == 0)
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return m&1 ? -pio2_hi-0x1p-120f : pio2_hi+0x1p-120f;
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/* when x is INF */
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if (exptx == 0x7fff) {
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if (expty == 0x7fff) {
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return m&1 ? -pio2_hi : pio2_hi;
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if (ex == 0x7fff) {
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if (ey == 0x7fff) {
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switch(m) {
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case 0: return pio2_hi*0.5+0x1p-120f; /* atan(+INF,+INF) */
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case 1: return -pio2_hi*0.5-0x1p-120f; /* atan(-INF,+INF) */
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case 2: return 1.5*pio2_hi+0x1p-120f; /* atan(+INF,-INF) */
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case 3: return -1.5*pio2_hi-0x1p-120f; /* atan(-INF,-INF) */
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case 0: return pio2_hi/2; /* atan(+INF,+INF) */
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case 1: return -pio2_hi/2; /* atan(-INF,+INF) */
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case 2: return 1.5*pio2_hi; /* atan(+INF,-INF) */
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case 3: return -1.5*pio2_hi; /* atan(-INF,-INF) */
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}
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} else {
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switch(m) {
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case 0: return 0.0; /* atan(+...,+INF) */
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case 1: return -0.0; /* atan(-...,+INF) */
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case 2: return 2*pio2_hi+0x1p-120f; /* atan(+...,-INF) */
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case 3: return -2*pio2_hi-0x1p-120f; /* atan(-...,-INF) */
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case 2: return 2*pio2_hi; /* atan(+...,-INF) */
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case 3: return -2*pio2_hi; /* atan(-...,-INF) */
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}
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}
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}
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/* when y is INF */
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if (exptx+120 < expty || expty == 0x7fff)
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return m&1 ? -pio2_hi-0x1p-120f : pio2_hi+0x1p-120f;
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if ((m&2) && expty+120 < exptx) /* |y/x| tiny, x<0 */
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if (ex+120 < ey || ey == 0x7fff)
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return m&1 ? -pio2_hi : pio2_hi;
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/* z = atan(|y/x|) without spurious underflow */
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if ((m&2) && ey+120 < ex) /* |y/x| < 0x1p-120, x<0 */
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z = 0.0;
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else
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z = atanl(fabsl(y/x));
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@ -23,6 +23,9 @@ long double atanl(long double x)
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}
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#elif (LDBL_MANT_DIG == 64 || LDBL_MANT_DIG == 113) && LDBL_MAX_EXP == 16384
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#if LDBL_MANT_DIG == 64
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#define EXPMAN(u) ((u.i.se & 0x7fff)<<8 | (u.i.m>>55 & 0xff))
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static const long double atanhi[] = {
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4.63647609000806116202e-01L,
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7.85398163397448309628e-01L,
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@ -64,32 +67,85 @@ static long double T_odd(long double x)
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return aT[1] + x * (aT[3] + x * (aT[5] + x * (aT[7] +
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x * (aT[9] + x * aT[11]))));
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}
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#elif LDBL_MANT_DIG == 113
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#define EXPMAN(u) ((u.i.se & 0x7fff)<<8 | u.i.top>>8)
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const long double atanhi[] = {
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4.63647609000806116214256231461214397e-01L,
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7.85398163397448309615660845819875699e-01L,
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9.82793723247329067985710611014666038e-01L,
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1.57079632679489661923132169163975140e+00L,
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};
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const long double atanlo[] = {
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4.89509642257333492668618435220297706e-36L,
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2.16795253253094525619926100651083806e-35L,
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-2.31288434538183565909319952098066272e-35L,
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4.33590506506189051239852201302167613e-35L,
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};
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const long double aT[] = {
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3.33333333333333333333333333333333125e-01L,
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-1.99999999999999999999999999999180430e-01L,
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1.42857142857142857142857142125269827e-01L,
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-1.11111111111111111111110834490810169e-01L,
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9.09090909090909090908522355708623681e-02L,
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-7.69230769230769230696553844935357021e-02L,
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6.66666666666666660390096773046256096e-02L,
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-5.88235294117646671706582985209643694e-02L,
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5.26315789473666478515847092020327506e-02L,
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-4.76190476189855517021024424991436144e-02L,
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4.34782608678695085948531993458097026e-02L,
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-3.99999999632663469330634215991142368e-02L,
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3.70370363987423702891250829918659723e-02L,
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-3.44827496515048090726669907612335954e-02L,
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3.22579620681420149871973710852268528e-02L,
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-3.03020767654269261041647570626778067e-02L,
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2.85641979882534783223403715930946138e-02L,
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-2.69824879726738568189929461383741323e-02L,
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2.54194698498808542954187110873675769e-02L,
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-2.35083879708189059926183138130183215e-02L,
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2.04832358998165364349957325067131428e-02L,
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-1.54489555488544397858507248612362957e-02L,
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8.64492360989278761493037861575248038e-03L,
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-2.58521121597609872727919154569765469e-03L,
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};
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static long double T_even(long double x)
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{
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return (aT[0] + x * (aT[2] + x * (aT[4] + x * (aT[6] + x * (aT[8] +
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x * (aT[10] + x * (aT[12] + x * (aT[14] + x * (aT[16] +
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x * (aT[18] + x * (aT[20] + x * aT[22])))))))))));
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}
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static long double T_odd(long double x)
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{
|
||||
return (aT[1] + x * (aT[3] + x * (aT[5] + x * (aT[7] + x * (aT[9] +
|
||||
x * (aT[11] + x * (aT[13] + x * (aT[15] + x * (aT[17] +
|
||||
x * (aT[19] + x * (aT[21] + x * aT[23])))))))))));
|
||||
}
|
||||
#endif
|
||||
|
||||
long double atanl(long double x)
|
||||
{
|
||||
union IEEEl2bits u;
|
||||
long double w,s1,s2,z;
|
||||
union ldshape u = {x};
|
||||
long double w, s1, s2, z;
|
||||
int id;
|
||||
uint16_t expsign, expt;
|
||||
uint32_t expman;
|
||||
unsigned e = u.i.se & 0x7fff;
|
||||
unsigned sign = u.i.se >> 15;
|
||||
unsigned expman;
|
||||
|
||||
u.e = x;
|
||||
expsign = u.xbits.expsign;
|
||||
expt = expsign & 0x7fff;
|
||||
if (expt >= 0x3fff + 65) { /* if |x| is large, atan(x)~=pi/2 */
|
||||
if (expt == 0x7fff &&
|
||||
((u.bits.manh&~LDBL_NBIT)|u.bits.manl)!=0) /* NaN */
|
||||
return x+x;
|
||||
z = atanhi[3] + 0x1p-120f;
|
||||
return expsign>>15 ? -z : z;
|
||||
if (e >= 0x3fff + LDBL_MANT_DIG + 1) { /* if |x| is large, atan(x)~=pi/2 */
|
||||
if (isnan(x))
|
||||
return x;
|
||||
return sign ? -atanhi[3] : atanhi[3];
|
||||
}
|
||||
/* Extract the exponent and the first few bits of the mantissa. */
|
||||
/* XXX There should be a more convenient way to do this. */
|
||||
expman = (expt << 8) | ((u.bits.manh >> (LDBL_MANH_SIZE - 9)) & 0xff);
|
||||
expman = EXPMAN(u);
|
||||
if (expman < ((0x3fff - 2) << 8) + 0xc0) { /* |x| < 0.4375 */
|
||||
if (expt < 0x3fff - 32) { /* if |x| is small, atanl(x)~=x */
|
||||
if (e < 0x3fff - (LDBL_MANT_DIG+1)/2) { /* if |x| is small, atanl(x)~=x */
|
||||
/* raise underflow if subnormal */
|
||||
if (expt == 0)
|
||||
if (e == 0)
|
||||
FORCE_EVAL((float)x);
|
||||
return x;
|
||||
}
|
||||
@ -108,7 +164,7 @@ long double atanl(long double x)
|
||||
if (expman < ((0x3fff + 1) << 8) + 0x38) { /* |x| < 2.4375 */
|
||||
id = 2;
|
||||
x = (x-1.5)/(1.0+1.5*x);
|
||||
} else { /* 2.4375 <= |x| < 2^ATAN_CONST */
|
||||
} else { /* 2.4375 <= |x| */
|
||||
id = 3;
|
||||
x = -1.0/x;
|
||||
}
|
||||
@ -123,6 +179,6 @@ long double atanl(long double x)
|
||||
if (id < 0)
|
||||
return x - x*(s1+s2);
|
||||
z = atanhi[id] - ((x*(s1+s2) - atanlo[id]) - x);
|
||||
return expsign>>15 ? -z : z;
|
||||
return sign ? -z : z;
|
||||
}
|
||||
#endif
|
||||
|
||||
Loading…
Reference in New Issue
Block a user