+++ /dev/null
-/* origin: FreeBSD /usr/src/lib/msun/src/k_log.h */
-/*
- * ====================================================
- * 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.
- * ====================================================
- */
-/*
- * __log1p(f):
- * Return log(1+f) - f for 1+f in ~[sqrt(2)/2, sqrt(2)].
- *
- * The following describes the overall strategy for computing
- * logarithms in base e. The argument reduction and adding the final
- * term of the polynomial are done by the caller for increased accuracy
- * when different bases are used.
- *
- * Method :
- * 1. Argument Reduction: find k and f such that
- * x = 2^k * (1+f),
- * where sqrt(2)/2 < 1+f < sqrt(2) .
- *
- * 2. Approximation of log(1+f).
- * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
- * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
- * = 2s + s*R
- * We use a special Reme algorithm on [0,0.1716] to generate
- * a polynomial of degree 14 to approximate R The maximum error
- * of this polynomial approximation is bounded by 2**-58.45. In
- * other words,
- * 2 4 6 8 10 12 14
- * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
- * (the values of Lg1 to Lg7 are listed in the program)
- * and
- * | 2 14 | -58.45
- * | Lg1*s +...+Lg7*s - R(z) | <= 2
- * | |
- * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
- * In order to guarantee error in log below 1ulp, we compute log
- * by
- * log(1+f) = f - s*(f - R) (if f is not too large)
- * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
- *
- * 3. Finally, log(x) = k*ln2 + log(1+f).
- * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
- * Here ln2 is split into two floating point number:
- * ln2_hi + ln2_lo,
- * where n*ln2_hi is always exact for |n| < 2000.
- *
- * Special cases:
- * log(x) is NaN with signal if x < 0 (including -INF) ;
- * log(+INF) is +INF; log(0) is -INF with signal;
- * log(NaN) is that NaN with no signal.
- *
- * Accuracy:
- * according to an error analysis, the error is always less than
- * 1 ulp (unit in the last place).
- *
- * 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 double
-Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
-Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
-Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
-Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
-Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
-Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
-Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
-
-/*
- * We always inline __log1p(), since doing so produces a
- * substantial performance improvement (~40% on amd64).
- */
-static inline double __log1p(double f)
-{
- double_t hfsq,s,z,R,w,t1,t2;
-
- s = f/(2.0+f);
- z = s*s;
- w = z*z;
- t1= w*(Lg2+w*(Lg4+w*Lg6));
- t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
- R = t2+t1;
- hfsq = 0.5*f*f;
- return s*(hfsq+R);
-}
+++ /dev/null
-/* origin: FreeBSD /usr/src/lib/msun/src/k_logf.h */
-/*
- * ====================================================
- * 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.
- * ====================================================
- */
-/*
- * See comments in __log1p.h.
- */
-
-/* |(log(1+s)-log(1-s))/s - Lg(s)| < 2**-34.24 (~[-4.95e-11, 4.97e-11]). */
-static const float
-Lg1 = 0xaaaaaa.0p-24, /* 0.66666662693 */
-Lg2 = 0xccce13.0p-25, /* 0.40000972152 */
-Lg3 = 0x91e9ee.0p-25, /* 0.28498786688 */
-Lg4 = 0xf89e26.0p-26; /* 0.24279078841 */
-
-static inline float __log1pf(float f)
-{
- float_t hfsq,s,z,R,w,t1,t2;
-
- s = f/(2.0f + f);
- z = s*s;
- w = z*z;
- t1 = w*(Lg2+w*Lg4);
- t2 = z*(Lg1+w*Lg3);
- R = t2+t1;
- hfsq = 0.5f * f * f;
- return s*(hfsq+R);
-}
* ====================================================
*/
/* log(x)
- * Return the logrithm of x
+ * Return the logarithm of x
*
* Method :
* 1. Argument Reduction: find k and f such that
* to produce the hexadecimal values shown.
*/
-#include "libm.h"
+#include <math.h>
+#include <stdint.h>
static const double
ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
-two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
double log(double x)
{
- double hfsq,f,s,z,R,w,t1,t2,dk;
- int32_t k,hx,i,j;
- uint32_t lx;
-
- EXTRACT_WORDS(hx, lx, x);
+ union {double f; uint64_t i;} u = {x};
+ double_t hfsq,f,s,z,R,w,t1,t2,dk;
+ uint32_t hx;
+ int k;
+ hx = u.i>>32;
k = 0;
- if (hx < 0x00100000) { /* x < 2**-1022 */
- if (((hx&0x7fffffff)|lx) == 0)
- return -two54/0.0; /* log(+-0)=-inf */
- if (hx < 0)
- return (x-x)/0.0; /* log(-#) = NaN */
- /* subnormal number, scale up x */
+ if (hx < 0x00100000 || hx>>31) {
+ if (u.i<<1 == 0)
+ return -1/(x*x); /* log(+-0)=-inf */
+ if (hx>>31)
+ return (x-x)/0.0; /* log(-#) = NaN */
+ /* subnormal number, scale x up */
k -= 54;
- x *= two54;
- GET_HIGH_WORD(hx,x);
- }
- if (hx >= 0x7ff00000)
- return x+x;
- k += (hx>>20) - 1023;
- hx &= 0x000fffff;
- i = (hx+0x95f64)&0x100000;
- SET_HIGH_WORD(x, hx|(i^0x3ff00000)); /* normalize x or x/2 */
- k += i>>20;
+ x *= 0x1p54;
+ u.f = x;
+ hx = u.i>>32;
+ } else if (hx >= 0x7ff00000) {
+ return x;
+ } else if (hx == 0x3ff00000 && u.i<<32 == 0)
+ return 0;
+
+ /* reduce x into [sqrt(2)/2, sqrt(2)] */
+ hx += 0x3ff00000 - 0x3fe6a09e;
+ k += (int)(hx>>20) - 0x3ff;
+ hx = (hx&0x000fffff) + 0x3fe6a09e;
+ u.i = (uint64_t)hx<<32 | (u.i&0xffffffff);
+ x = u.f;
+
f = x - 1.0;
- if ((0x000fffff&(2+hx)) < 3) { /* -2**-20 <= f < 2**-20 */
- if (f == 0.0) {
- if (k == 0) {
- return 0.0;
- }
- dk = (double)k;
- return dk*ln2_hi + dk*ln2_lo;
- }
- R = f*f*(0.5-0.33333333333333333*f);
- if (k == 0)
- return f - R;
- dk = (double)k;
- return dk*ln2_hi - ((R-dk*ln2_lo)-f);
- }
+ hfsq = 0.5*f*f;
s = f/(2.0+f);
- dk = (double)k;
z = s*s;
- i = hx - 0x6147a;
w = z*z;
- j = 0x6b851 - hx;
t1 = w*(Lg2+w*(Lg4+w*Lg6));
t2 = z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
- i |= j;
R = t2 + t1;
- if (i > 0) {
- hfsq = 0.5*f*f;
- if (k == 0)
- return f - (hfsq-s*(hfsq+R));
- return dk*ln2_hi - ((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
- } else {
- if (k == 0)
- return f - s*(f-R);
- return dk*ln2_hi - ((s*(f-R)-dk*ln2_lo)-f);
- }
+ dk = k;
+ return s*(hfsq+R) + dk*ln2_lo - hfsq + f + dk*ln2_hi;
}
* ====================================================
*/
/*
- * Return the base 10 logarithm of x. See e_log.c and k_log.h for most
- * comments.
+ * Return the base 10 logarithm of x. See log.c for most comments.
*
- * log10(x) = (f - 0.5*f*f + k_log1p(f)) / ln10 + k * log10(2)
- * in not-quite-routine extra precision.
+ * Reduce x to 2^k (1+f) and calculate r = log(1+f) - f + f*f/2
+ * as in log.c, then combine and scale in extra precision:
+ * log10(x) = (f - f*f/2 + r)/log(10) + k*log10(2)
*/
-#include "libm.h"
-#include "__log1p.h"
+#include <math.h>
+#include <stdint.h>
static const double
-two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */
ivln10hi = 4.34294481878168880939e-01, /* 0x3fdbcb7b, 0x15200000 */
ivln10lo = 2.50829467116452752298e-11, /* 0x3dbb9438, 0xca9aadd5 */
log10_2hi = 3.01029995663611771306e-01, /* 0x3FD34413, 0x509F6000 */
-log10_2lo = 3.69423907715893078616e-13; /* 0x3D59FEF3, 0x11F12B36 */
+log10_2lo = 3.69423907715893078616e-13, /* 0x3D59FEF3, 0x11F12B36 */
+Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
+Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
+Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
+Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
+Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
+Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
+Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
double log10(double x)
{
- double f,hfsq,hi,lo,r,val_hi,val_lo,w,y,y2;
- int32_t i,k,hx;
- uint32_t lx;
-
- EXTRACT_WORDS(hx, lx, x);
+ union {double f; uint64_t i;} u = {x};
+ double_t hfsq,f,s,z,R,w,t1,t2,dk,y,hi,lo,val_hi,val_lo;
+ uint32_t hx;
+ int k;
+ hx = u.i>>32;
k = 0;
- if (hx < 0x00100000) { /* x < 2**-1022 */
- if (((hx&0x7fffffff)|lx) == 0)
- return -two54/0.0; /* log(+-0)=-inf */
- if (hx<0)
- return (x-x)/0.0; /* log(-#) = NaN */
- /* subnormal number, scale up x */
+ if (hx < 0x00100000 || hx>>31) {
+ if (u.i<<1 == 0)
+ return -1/(x*x); /* log(+-0)=-inf */
+ if (hx>>31)
+ return (x-x)/0.0; /* log(-#) = NaN */
+ /* subnormal number, scale x up */
k -= 54;
- x *= two54;
- GET_HIGH_WORD(hx, x);
- }
- if (hx >= 0x7ff00000)
- return x+x;
- if (hx == 0x3ff00000 && lx == 0)
- return 0.0; /* log(1) = +0 */
- k += (hx>>20) - 1023;
- hx &= 0x000fffff;
- i = (hx+0x95f64)&0x100000;
- SET_HIGH_WORD(x, hx|(i^0x3ff00000)); /* normalize x or x/2 */
- k += i>>20;
- y = (double)k;
+ x *= 0x1p54;
+ u.f = x;
+ hx = u.i>>32;
+ } else if (hx >= 0x7ff00000) {
+ return x;
+ } else if (hx == 0x3ff00000 && u.i<<32 == 0)
+ return 0;
+
+ /* reduce x into [sqrt(2)/2, sqrt(2)] */
+ hx += 0x3ff00000 - 0x3fe6a09e;
+ k += (int)(hx>>20) - 0x3ff;
+ hx = (hx&0x000fffff) + 0x3fe6a09e;
+ u.i = (uint64_t)hx<<32 | (u.i&0xffffffff);
+ x = u.f;
+
f = x - 1.0;
hfsq = 0.5*f*f;
- r = __log1p(f);
+ s = f/(2.0+f);
+ z = s*s;
+ w = z*z;
+ t1 = w*(Lg2+w*(Lg4+w*Lg6));
+ t2 = z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
+ R = t2 + t1;
/* See log2.c for details. */
+ /* hi+lo = f - hfsq + s*(hfsq+R) ~ log(1+f) */
hi = f - hfsq;
- SET_LOW_WORD(hi, 0);
- lo = (f - hi) - hfsq + r;
+ u.f = hi;
+ u.i &= (uint64_t)-1<<32;
+ hi = u.f;
+ lo = f - hi - hfsq + s*(hfsq+R);
+
+ /* val_hi+val_lo ~ log10(1+f) + k*log10(2) */
val_hi = hi*ivln10hi;
- y2 = y*log10_2hi;
- val_lo = y*log10_2lo + (lo+hi)*ivln10lo + lo*ivln10hi;
+ dk = k;
+ y = dk*log10_2hi;
+ val_lo = dk*log10_2lo + (lo+hi)*ivln10lo + lo*ivln10hi;
/*
- * Extra precision in for adding y*log10_2hi is not strictly needed
+ * Extra precision in for adding y is not strictly needed
* since there is no very large cancellation near x = sqrt(2) or
* x = 1/sqrt(2), but we do it anyway since it costs little on CPUs
* with some parallelism and it reduces the error for many args.
*/
- w = y2 + val_hi;
- val_lo += (y2 - w) + val_hi;
+ w = y + val_hi;
+ val_lo += (y - w) + val_hi;
val_hi = w;
return val_lo + val_hi;
* See comments in log10.c.
*/
-#include "libm.h"
-#include "__log1pf.h"
+#include <math.h>
+#include <stdint.h>
static const float
-two25 = 3.3554432000e+07, /* 0x4c000000 */
ivln10hi = 4.3432617188e-01, /* 0x3ede6000 */
ivln10lo = -3.1689971365e-05, /* 0xb804ead9 */
log10_2hi = 3.0102920532e-01, /* 0x3e9a2080 */
-log10_2lo = 7.9034151668e-07; /* 0x355427db */
+log10_2lo = 7.9034151668e-07, /* 0x355427db */
+/* |(log(1+s)-log(1-s))/s - Lg(s)| < 2**-34.24 (~[-4.95e-11, 4.97e-11]). */
+Lg1 = 0xaaaaaa.0p-24, /* 0.66666662693 */
+Lg2 = 0xccce13.0p-25, /* 0.40000972152 */
+Lg3 = 0x91e9ee.0p-25, /* 0.28498786688 */
+Lg4 = 0xf89e26.0p-26; /* 0.24279078841 */
float log10f(float x)
{
- float f,hfsq,hi,lo,r,y;
- int32_t i,k,hx;
-
- GET_FLOAT_WORD(hx, x);
+ union {float f; uint32_t i;} u = {x};
+ float_t hfsq,f,s,z,R,w,t1,t2,dk,hi,lo;
+ uint32_t ix;
+ int k;
+ ix = u.i;
k = 0;
- if (hx < 0x00800000) { /* x < 2**-126 */
- if ((hx&0x7fffffff) == 0)
- return -two25/0.0f; /* log(+-0)=-inf */
- if (hx < 0)
- return (x-x)/0.0f; /* log(-#) = NaN */
+ if (ix < 0x00800000 || ix>>31) { /* x < 2**-126 */
+ if (ix<<1 == 0)
+ return -1/(x*x); /* log(+-0)=-inf */
+ if (ix>>31)
+ return (x-x)/0.0f; /* log(-#) = NaN */
/* subnormal number, scale up x */
k -= 25;
- x *= two25;
- GET_FLOAT_WORD(hx, x);
- }
- if (hx >= 0x7f800000)
- return x+x;
- if (hx == 0x3f800000)
- return 0.0f; /* log(1) = +0 */
- k += (hx>>23) - 127;
- hx &= 0x007fffff;
- i = (hx+(0x4afb0d))&0x800000;
- SET_FLOAT_WORD(x, hx|(i^0x3f800000)); /* normalize x or x/2 */
- k += i>>23;
- y = (float)k;
+ x *= 0x1p25f;
+ u.f = x;
+ ix = u.i;
+ } else if (ix >= 0x7f800000) {
+ return x;
+ } else if (ix == 0x3f800000)
+ return 0;
+
+ /* reduce x into [sqrt(2)/2, sqrt(2)] */
+ ix += 0x3f800000 - 0x3f3504f3;
+ k += (int)(ix>>23) - 0x7f;
+ ix = (ix&0x007fffff) + 0x3f3504f3;
+ u.i = ix;
+ x = u.f;
+
f = x - 1.0f;
- hfsq = 0.5f * f * f;
- r = __log1pf(f);
+ s = f/(2.0f + f);
+ z = s*s;
+ w = z*z;
+ t1= w*(Lg2+w*Lg4);
+ t2= z*(Lg1+w*Lg3);
+ R = t2 + t1;
+ hfsq = 0.5f*f*f;
-// FIXME
-// /* See log2f.c and log2.c for details. */
-// if (sizeof(float_t) > sizeof(float))
-// return (r - hfsq + f) * ((float_t)ivln10lo + ivln10hi) +
-// y * ((float_t)log10_2lo + log10_2hi);
hi = f - hfsq;
- GET_FLOAT_WORD(hx, hi);
- SET_FLOAT_WORD(hi, hx&0xfffff000);
- lo = (f - hi) - hfsq + r;
- return y*log10_2lo + (lo+hi)*ivln10lo + lo*ivln10hi +
- hi*ivln10hi + y*log10_2hi;
+ u.f = hi;
+ u.i &= 0xfffff000;
+ hi = u.f;
+ lo = f - hi - hfsq + s*(hfsq+R);
+ dk = k;
+ return dk*log10_2lo + (lo+hi)*ivln10lo + lo*ivln10hi + hi*ivln10hi + dk*log10_2hi;
}
long double log10l(long double x)
{
- long double y;
- volatile long double z;
+ long double y, z;
int e;
if (isnan(x))
return x;
if(x <= 0.0) {
if(x == 0.0)
- return -1.0 / (x - x);
- return (x - x) / (x - x);
+ return -1.0 / (x*x);
+ return (x - x) / 0.0;
}
if (x == INFINITY)
return INFINITY;
* ====================================================
*/
/* double log1p(double x)
+ * Return the natural logarithm of 1+x.
*
* Method :
* 1. Argument Reduction: find k and f such that
* and add back the correction term c/u.
* (Note: when x > 2**53, one can simply return log(x))
*
- * 2. Approximation of log1p(f).
- * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
- * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
- * = 2s + s*R
- * We use a special Reme algorithm on [0,0.1716] to generate
- * a polynomial of degree 14 to approximate R The maximum error
- * of this polynomial approximation is bounded by 2**-58.45. In
- * other words,
- * 2 4 6 8 10 12 14
- * R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s
- * (the values of Lp1 to Lp7 are listed in the program)
- * and
- * | 2 14 | -58.45
- * | Lp1*s +...+Lp7*s - R(z) | <= 2
- * | |
- * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
- * In order to guarantee error in log below 1ulp, we compute log
- * by
- * log1p(f) = f - (hfsq - s*(hfsq+R)).
+ * 2. Approximation of log(1+f): See log.c
*
- * 3. Finally, log1p(x) = k*ln2 + log1p(f).
- * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
- * Here ln2 is split into two floating point number:
- * ln2_hi + ln2_lo,
- * where n*ln2_hi is always exact for |n| < 2000.
+ * 3. Finally, log1p(x) = k*ln2 + log(1+f) + c/u. See log.c
*
* Special cases:
* log1p(x) is NaN with signal if x < -1 (including -INF) ;
static const double
ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
-two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
-Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
-Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
-Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
-Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
-Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
-Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
-Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
+Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
+Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
+Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
+Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
+Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
+Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
+Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
double log1p(double x)
{
- double hfsq,f,c,s,z,R,u;
- int32_t k,hx,hu,ax;
-
- GET_HIGH_WORD(hx, x);
- ax = hx & 0x7fffffff;
+ union {double f; uint64_t i;} u = {x};
+ double_t hfsq,f,c,s,z,R,w,t1,t2,dk;
+ uint32_t hx,hu;
+ int k;
+ hx = u.i>>32;
k = 1;
- if (hx < 0x3FDA827A) { /* 1+x < sqrt(2)+ */
- if (ax >= 0x3ff00000) { /* x <= -1.0 */
- if (x == -1.0)
- return -two54/0.0; /* log1p(-1)=+inf */
- return (x-x)/(x-x); /* log1p(x<-1)=NaN */
+ if (hx < 0x3fda827a || hx>>31) { /* 1+x < sqrt(2)+ */
+ if (hx >= 0xbff00000) { /* x <= -1.0 */
+ if (x == -1)
+ return x/0.0; /* log1p(-1) = -inf */
+ return (x-x)/0.0; /* log1p(x<-1) = NaN */
}
- if (ax < 0x3e200000) { /* |x| < 2**-29 */
- /* if 0x1p-1022 <= |x| < 0x1p-54, avoid raising underflow */
- if (ax < 0x3c900000 && ax >= 0x00100000)
- return x;
-#if FLT_EVAL_METHOD != 0
- FORCE_EVAL((float)x);
-#endif
- return x - x*x*0.5;
+ if (hx<<1 < 0x3ca00000<<1) { /* |x| < 2**-53 */
+ /* underflow if subnormal */
+ if ((hx&0x7ff00000) == 0)
+ FORCE_EVAL((float)x);
+ return x;
}
- if (hx > 0 || hx <= (int32_t)0xbfd2bec4) { /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
+ if (hx <= 0xbfd2bec4) { /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
k = 0;
+ c = 0;
f = x;
- hu = 1;
}
- }
- if (hx >= 0x7ff00000)
- return x+x;
- if (k != 0) {
- if (hx < 0x43400000) {
- u = 1 + x;
- GET_HIGH_WORD(hu, u);
- k = (hu>>20) - 1023;
- c = k > 0 ? 1.0-(u-x) : x-(u-1.0); /* correction term */
- c /= u;
- } else {
- u = x;
- GET_HIGH_WORD(hu,u);
- k = (hu>>20) - 1023;
+ } else if (hx >= 0x7ff00000)
+ return x;
+ if (k) {
+ u.f = 1 + x;
+ hu = u.i>>32;
+ hu += 0x3ff00000 - 0x3fe6a09e;
+ k = (int)(hu>>20) - 0x3ff;
+ /* correction term ~ log(1+x)-log(u), avoid underflow in c/u */
+ if (k < 54) {
+ c = k >= 2 ? 1-(u.f-x) : x-(u.f-1);
+ c /= u.f;
+ } else
c = 0;
- }
- hu &= 0x000fffff;
- /*
- * The approximation to sqrt(2) used in thresholds is not
- * critical. However, the ones used above must give less
- * strict bounds than the one here so that the k==0 case is
- * never reached from here, since here we have committed to
- * using the correction term but don't use it if k==0.
- */
- if (hu < 0x6a09e) { /* u ~< sqrt(2) */
- SET_HIGH_WORD(u, hu|0x3ff00000); /* normalize u */
- } else {
- k += 1;
- SET_HIGH_WORD(u, hu|0x3fe00000); /* normalize u/2 */
- hu = (0x00100000-hu)>>2;
- }
- f = u - 1.0;
+ /* reduce u into [sqrt(2)/2, sqrt(2)] */
+ hu = (hu&0x000fffff) + 0x3fe6a09e;
+ u.i = (uint64_t)hu<<32 | (u.i&0xffffffff);
+ f = u.f - 1;
}
hfsq = 0.5*f*f;
- if (hu == 0) { /* |f| < 2**-20 */
- if (f == 0.0) {
- if(k == 0)
- return 0.0;
- c += k*ln2_lo;
- return k*ln2_hi + c;
- }
- R = hfsq*(1.0 - 0.66666666666666666*f);
- if (k == 0)
- return f - R;
- return k*ln2_hi - ((R-(k*ln2_lo+c))-f);
- }
s = f/(2.0+f);
z = s*s;
- R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))));
- if (k == 0)
- return f - (hfsq-s*(hfsq+R));
- return k*ln2_hi - ((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
+ w = z*z;
+ t1 = w*(Lg2+w*(Lg4+w*Lg6));
+ t2 = z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
+ R = t2 + t1;
+ dk = k;
+ return s*(hfsq+R) + (dk*ln2_lo+c) - hfsq + f + dk*ln2_hi;
}
/* origin: FreeBSD /usr/src/lib/msun/src/s_log1pf.c */
-/*
- * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
- */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
static const float
ln2_hi = 6.9313812256e-01, /* 0x3f317180 */
ln2_lo = 9.0580006145e-06, /* 0x3717f7d1 */
-two25 = 3.355443200e+07, /* 0x4c000000 */
-Lp1 = 6.6666668653e-01, /* 3F2AAAAB */
-Lp2 = 4.0000000596e-01, /* 3ECCCCCD */
-Lp3 = 2.8571429849e-01, /* 3E924925 */
-Lp4 = 2.2222198546e-01, /* 3E638E29 */
-Lp5 = 1.8183572590e-01, /* 3E3A3325 */
-Lp6 = 1.5313838422e-01, /* 3E1CD04F */
-Lp7 = 1.4798198640e-01; /* 3E178897 */
+/* |(log(1+s)-log(1-s))/s - Lg(s)| < 2**-34.24 (~[-4.95e-11, 4.97e-11]). */
+Lg1 = 0xaaaaaa.0p-24, /* 0.66666662693 */
+Lg2 = 0xccce13.0p-25, /* 0.40000972152 */
+Lg3 = 0x91e9ee.0p-25, /* 0.28498786688 */
+Lg4 = 0xf89e26.0p-26; /* 0.24279078841 */
float log1pf(float x)
{
- float hfsq,f,c,s,z,R,u;
- int32_t k,hx,hu,ax;
-
- GET_FLOAT_WORD(hx, x);
- ax = hx & 0x7fffffff;
+ union {float f; uint32_t i;} u = {x};
+ float_t hfsq,f,c,s,z,R,w,t1,t2,dk;
+ uint32_t ix,iu;
+ int k;
+ ix = u.i;
k = 1;
- if (hx < 0x3ed413d0) { /* 1+x < sqrt(2)+ */
- if (ax >= 0x3f800000) { /* x <= -1.0 */
- if (x == -1.0f)
- return -two25/0.0f; /* log1p(-1)=+inf */
- return (x-x)/(x-x); /* log1p(x<-1)=NaN */
+ if (ix < 0x3ed413d0 || ix>>31) { /* 1+x < sqrt(2)+ */
+ if (ix >= 0xbf800000) { /* x <= -1.0 */
+ if (x == -1)
+ return x/0.0f; /* log1p(-1)=+inf */
+ return (x-x)/0.0f; /* log1p(x<-1)=NaN */
}
- if (ax < 0x38000000) { /* |x| < 2**-15 */
- /* if 0x1p-126 <= |x| < 0x1p-24, avoid raising underflow */
- if (ax < 0x33800000 && ax >= 0x00800000)
- return x;
-#if FLT_EVAL_METHOD != 0
- FORCE_EVAL(x*x);
-#endif
- return x - x*x*0.5f;
+ if (ix<<1 < 0x33800000<<1) { /* |x| < 2**-24 */
+ /* underflow if subnormal */
+ if ((ix&0x7f800000) == 0)
+ FORCE_EVAL(x*x);
+ return x;
}
- if (hx > 0 || hx <= (int32_t)0xbe95f619) { /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
+ if (ix <= 0xbe95f619) { /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
k = 0;
+ c = 0;
f = x;
- hu = 1;
}
- }
- if (hx >= 0x7f800000)
- return x+x;
- if (k != 0) {
- if (hx < 0x5a000000) {
- u = 1 + x;
- GET_FLOAT_WORD(hu, u);
- k = (hu>>23) - 127;
- /* correction term */
- c = k > 0 ? 1.0f-(u-x) : x-(u-1.0f);
- c /= u;
- } else {
- u = x;
- GET_FLOAT_WORD(hu,u);
- k = (hu>>23) - 127;
+ } else if (ix >= 0x7f800000)
+ return x;
+ if (k) {
+ u.f = 1 + x;
+ iu = u.i;
+ iu += 0x3f800000 - 0x3f3504f3;
+ k = (int)(iu>>23) - 0x7f;
+ /* correction term ~ log(1+x)-log(u), avoid underflow in c/u */
+ if (k < 25) {
+ c = k >= 2 ? 1-(u.f-x) : x-(u.f-1);
+ c /= u.f;
+ } else
c = 0;
- }
- hu &= 0x007fffff;
- /*
- * The approximation to sqrt(2) used in thresholds is not
- * critical. However, the ones used above must give less
- * strict bounds than the one here so that the k==0 case is
- * never reached from here, since here we have committed to
- * using the correction term but don't use it if k==0.
- */
- if (hu < 0x3504f4) { /* u < sqrt(2) */
- SET_FLOAT_WORD(u, hu|0x3f800000); /* normalize u */
- } else {
- k += 1;
- SET_FLOAT_WORD(u, hu|0x3f000000); /* normalize u/2 */
- hu = (0x00800000-hu)>>2;
- }
- f = u - 1.0f;
- }
- hfsq = 0.5f * f * f;
- if (hu == 0) { /* |f| < 2**-20 */
- if (f == 0.0f) {
- if (k == 0)
- return 0.0f;
- c += k*ln2_lo;
- return k*ln2_hi+c;
- }
- R = hfsq*(1.0f - 0.66666666666666666f * f);
- if (k == 0)
- return f - R;
- return k*ln2_hi - ((R-(k*ln2_lo+c))-f);
+ /* reduce u into [sqrt(2)/2, sqrt(2)] */
+ iu = (iu&0x007fffff) + 0x3f3504f3;
+ u.i = iu;
+ f = u.f - 1;
}
s = f/(2.0f + f);
z = s*s;
- R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))));
- if (k == 0)
- return f - (hfsq-s*(hfsq+R));
- return k*ln2_hi - ((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
+ w = z*z;
+ t1= w*(Lg2+w*Lg4);
+ t2= z*(Lg1+w*Lg3);
+ R = t2 + t1;
+ hfsq = 0.5f*f*f;
+ dk = k;
+ return s*(hfsq+R) + (dk*ln2_lo+c) - hfsq + f + dk*ln2_hi;
}
/* Test for domain errors. */
if (x <= 0.0) {
if (x == 0.0)
- return -1/x; /* -inf with divbyzero */
+ return -1/(x*x); /* -inf with divbyzero */
return 0/0.0f; /* nan with invalid */
}
* ====================================================
*/
/*
- * Return the base 2 logarithm of x. See log.c and __log1p.h for most
- * comments.
+ * Return the base 2 logarithm of x. See log.c for most comments.
*
- * This reduces x to {k, 1+f} exactly as in e_log.c, then calls the kernel,
- * then does the combining and scaling steps
- * log2(x) = (f - 0.5*f*f + k_log1p(f)) / ln2 + k
- * in not-quite-routine extra precision.
+ * Reduce x to 2^k (1+f) and calculate r = log(1+f) - f + f*f/2
+ * as in log.c, then combine and scale in extra precision:
+ * log2(x) = (f - f*f/2 + r)/log(2) + k
*/
-#include "libm.h"
-#include "__log1p.h"
+#include <math.h>
+#include <stdint.h>
static const double
-two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */
ivln2hi = 1.44269504072144627571e+00, /* 0x3ff71547, 0x65200000 */
-ivln2lo = 1.67517131648865118353e-10; /* 0x3de705fc, 0x2eefa200 */
+ivln2lo = 1.67517131648865118353e-10, /* 0x3de705fc, 0x2eefa200 */
+Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
+Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
+Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
+Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
+Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
+Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
+Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
double log2(double x)
{
- double f,hfsq,hi,lo,r,val_hi,val_lo,w,y;
- int32_t i,k,hx;
- uint32_t lx;
-
- EXTRACT_WORDS(hx, lx, x);
+ union {double f; uint64_t i;} u = {x};
+ double_t hfsq,f,s,z,R,w,t1,t2,y,hi,lo,val_hi,val_lo;
+ uint32_t hx;
+ int k;
+ hx = u.i>>32;
k = 0;
- if (hx < 0x00100000) { /* x < 2**-1022 */
- if (((hx&0x7fffffff)|lx) == 0)
- return -two54/0.0; /* log(+-0)=-inf */
- if (hx < 0)
- return (x-x)/0.0; /* log(-#) = NaN */
- /* subnormal number, scale up x */
+ if (hx < 0x00100000 || hx>>31) {
+ if (u.i<<1 == 0)
+ return -1/(x*x); /* log(+-0)=-inf */
+ if (hx>>31)
+ return (x-x)/0.0; /* log(-#) = NaN */
+ /* subnormal number, scale x up */
k -= 54;
- x *= two54;
- GET_HIGH_WORD(hx, x);
- }
- if (hx >= 0x7ff00000)
- return x+x;
- if (hx == 0x3ff00000 && lx == 0)
- return 0.0; /* log(1) = +0 */
- k += (hx>>20) - 1023;
- hx &= 0x000fffff;
- i = (hx+0x95f64) & 0x100000;
- SET_HIGH_WORD(x, hx|(i^0x3ff00000)); /* normalize x or x/2 */
- k += i>>20;
- y = (double)k;
+ x *= 0x1p54;
+ u.f = x;
+ hx = u.i>>32;
+ } else if (hx >= 0x7ff00000) {
+ return x;
+ } else if (hx == 0x3ff00000 && u.i<<32 == 0)
+ return 0;
+
+ /* reduce x into [sqrt(2)/2, sqrt(2)] */
+ hx += 0x3ff00000 - 0x3fe6a09e;
+ k += (int)(hx>>20) - 0x3ff;
+ hx = (hx&0x000fffff) + 0x3fe6a09e;
+ u.i = (uint64_t)hx<<32 | (u.i&0xffffffff);
+ x = u.f;
+
f = x - 1.0;
hfsq = 0.5*f*f;
- r = __log1p(f);
+ s = f/(2.0+f);
+ z = s*s;
+ w = z*z;
+ t1 = w*(Lg2+w*(Lg4+w*Lg6));
+ t2 = z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
+ R = t2 + t1;
/*
* f-hfsq must (for args near 1) be evaluated in extra precision
* The multi-precision calculations for the multiplications are
* routine.
*/
+
+ /* hi+lo = f - hfsq + s*(hfsq+R) ~ log(1+f) */
hi = f - hfsq;
- SET_LOW_WORD(hi, 0);
- lo = (f - hi) - hfsq + r;
+ u.f = hi;
+ u.i &= (uint64_t)-1<<32;
+ hi = u.f;
+ lo = f - hi - hfsq + s*(hfsq+R);
+
val_hi = hi*ivln2hi;
val_lo = (lo+hi)*ivln2lo + lo*ivln2hi;
/* spadd(val_hi, val_lo, y), except for not using double_t: */
+ y = k;
w = y + val_hi;
val_lo += (y - w) + val_hi;
val_hi = w;
* See comments in log2.c.
*/
-#include "libm.h"
-#include "__log1pf.h"
+#include <math.h>
+#include <stdint.h>
static const float
-two25 = 3.3554432000e+07, /* 0x4c000000 */
ivln2hi = 1.4428710938e+00, /* 0x3fb8b000 */
-ivln2lo = -1.7605285393e-04; /* 0xb9389ad4 */
+ivln2lo = -1.7605285393e-04, /* 0xb9389ad4 */
+/* |(log(1+s)-log(1-s))/s - Lg(s)| < 2**-34.24 (~[-4.95e-11, 4.97e-11]). */
+Lg1 = 0xaaaaaa.0p-24, /* 0.66666662693 */
+Lg2 = 0xccce13.0p-25, /* 0.40000972152 */
+Lg3 = 0x91e9ee.0p-25, /* 0.28498786688 */
+Lg4 = 0xf89e26.0p-26; /* 0.24279078841 */
float log2f(float x)
{
- float f,hfsq,hi,lo,r,y;
- int32_t i,k,hx;
-
- GET_FLOAT_WORD(hx, x);
+ union {float f; uint32_t i;} u = {x};
+ float_t hfsq,f,s,z,R,w,t1,t2,hi,lo;
+ uint32_t ix;
+ int k;
+ ix = u.i;
k = 0;
- if (hx < 0x00800000) { /* x < 2**-126 */
- if ((hx&0x7fffffff) == 0)
- return -two25/0.0f; /* log(+-0)=-inf */
- if (hx < 0)
- return (x-x)/0.0f; /* log(-#) = NaN */
+ if (ix < 0x00800000 || ix>>31) { /* x < 2**-126 */
+ if (ix<<1 == 0)
+ return -1/(x*x); /* log(+-0)=-inf */
+ if (ix>>31)
+ return (x-x)/0.0f; /* log(-#) = NaN */
/* subnormal number, scale up x */
k -= 25;
- x *= two25;
- GET_FLOAT_WORD(hx, x);
- }
- if (hx >= 0x7f800000)
- return x+x;
- if (hx == 0x3f800000)
- return 0.0f; /* log(1) = +0 */
- k += (hx>>23) - 127;
- hx &= 0x007fffff;
- i = (hx+(0x4afb0d))&0x800000;
- SET_FLOAT_WORD(x, hx|(i^0x3f800000)); /* normalize x or x/2 */
- k += i>>23;
- y = (float)k;
- f = x - 1.0f;
- hfsq = 0.5f * f * f;
- r = __log1pf(f);
+ x *= 0x1p25f;
+ u.f = x;
+ ix = u.i;
+ } else if (ix >= 0x7f800000) {
+ return x;
+ } else if (ix == 0x3f800000)
+ return 0;
- /*
- * We no longer need to avoid falling into the multi-precision
- * calculations due to compiler bugs breaking Dekker's theorem.
- * Keep avoiding this as an optimization. See log2.c for more
- * details (some details are here only because the optimization
- * is not yet available in double precision).
- *
- * Another compiler bug turned up. With gcc on i386,
- * (ivln2lo + ivln2hi) would be evaluated in float precision
- * despite runtime evaluations using double precision. So we
- * must cast one of its terms to float_t. This makes the whole
- * expression have type float_t, so return is forced to waste
- * time clobbering its extra precision.
- */
-// FIXME
-// if (sizeof(float_t) > sizeof(float))
-// return (r - hfsq + f) * ((float_t)ivln2lo + ivln2hi) + y;
+ /* reduce x into [sqrt(2)/2, sqrt(2)] */
+ ix += 0x3f800000 - 0x3f3504f3;
+ k += (int)(ix>>23) - 0x7f;
+ ix = (ix&0x007fffff) + 0x3f3504f3;
+ u.i = ix;
+ x = u.f;
+
+ f = x - 1.0f;
+ s = f/(2.0f + f);
+ z = s*s;
+ w = z*z;
+ t1= w*(Lg2+w*Lg4);
+ t2= z*(Lg1+w*Lg3);
+ R = t2 + t1;
+ hfsq = 0.5f*f*f;
hi = f - hfsq;
- GET_FLOAT_WORD(hx,hi);
- SET_FLOAT_WORD(hi,hx&0xfffff000);
- lo = (f - hi) - hfsq + r;
- return (lo+hi)*ivln2lo + lo*ivln2hi + hi*ivln2hi + y;
+ u.f = hi;
+ u.i &= 0xfffff000;
+ hi = u.f;
+ lo = f - hi - hfsq + s*(hfsq+R);
+ return (lo+hi)*ivln2lo + lo*ivln2hi + hi*ivln2hi + k;
}
return x;
if (x <= 0.0) {
if (x == 0.0)
- return -1/(x+0); /* -inf with divbyzero */
+ return -1/(x*x); /* -inf with divbyzero */
return 0/0.0f; /* nan with invalid */
}
* ====================================================
*/
-#include "libm.h"
+#include <math.h>
+#include <stdint.h>
static const float
ln2_hi = 6.9313812256e-01, /* 0x3f317180 */
ln2_lo = 9.0580006145e-06, /* 0x3717f7d1 */
-two25 = 3.355443200e+07, /* 0x4c000000 */
/* |(log(1+s)-log(1-s))/s - Lg(s)| < 2**-34.24 (~[-4.95e-11, 4.97e-11]). */
Lg1 = 0xaaaaaa.0p-24, /* 0.66666662693 */
Lg2 = 0xccce13.0p-25, /* 0.40000972152 */
float logf(float x)
{
- float hfsq,f,s,z,R,w,t1,t2,dk;
- int32_t k,ix,i,j;
-
- GET_FLOAT_WORD(ix, x);
+ union {float f; uint32_t i;} u = {x};
+ float_t hfsq,f,s,z,R,w,t1,t2,dk;
+ uint32_t ix;
+ int k;
+ ix = u.i;
k = 0;
- if (ix < 0x00800000) { /* x < 2**-126 */
- if ((ix & 0x7fffffff) == 0)
- return -two25/0.0f; /* log(+-0)=-inf */
- if (ix < 0)
- return (x-x)/0.0f; /* log(-#) = NaN */
+ if (ix < 0x00800000 || ix>>31) { /* x < 2**-126 */
+ if (ix<<1 == 0)
+ return -1/(x*x); /* log(+-0)=-inf */
+ if (ix>>31)
+ return (x-x)/0.0f; /* log(-#) = NaN */
/* subnormal number, scale up x */
k -= 25;
- x *= two25;
- GET_FLOAT_WORD(ix, x);
- }
- if (ix >= 0x7f800000)
- return x+x;
- k += (ix>>23) - 127;
- ix &= 0x007fffff;
- i = (ix + (0x95f64<<3)) & 0x800000;
- SET_FLOAT_WORD(x, ix|(i^0x3f800000)); /* normalize x or x/2 */
- k += i>>23;
+ x *= 0x1p25f;
+ u.f = x;
+ ix = u.i;
+ } else if (ix >= 0x7f800000) {
+ return x;
+ } else if (ix == 0x3f800000)
+ return 0;
+
+ /* reduce x into [sqrt(2)/2, sqrt(2)] */
+ ix += 0x3f800000 - 0x3f3504f3;
+ k += (int)(ix>>23) - 0x7f;
+ ix = (ix&0x007fffff) + 0x3f3504f3;
+ u.i = ix;
+ x = u.f;
+
f = x - 1.0f;
- if ((0x007fffff & (0x8000 + ix)) < 0xc000) { /* -2**-9 <= f < 2**-9 */
- if (f == 0.0f) {
- if (k == 0)
- return 0.0f;
- dk = (float)k;
- return dk*ln2_hi + dk*ln2_lo;
- }
- R = f*f*(0.5f - 0.33333333333333333f*f);
- if (k == 0)
- return f-R;
- dk = (float)k;
- return dk*ln2_hi - ((R-dk*ln2_lo)-f);
- }
s = f/(2.0f + f);
- dk = (float)k;
z = s*s;
- i = ix-(0x6147a<<3);
w = z*z;
- j = (0x6b851<<3)-ix;
t1= w*(Lg2+w*Lg4);
t2= z*(Lg1+w*Lg3);
- i |= j;
R = t2 + t1;
- if (i > 0) {
- hfsq = 0.5f * f * f;
- if (k == 0)
- return f - (hfsq-s*(hfsq+R));
- return dk*ln2_hi - ((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
- } else {
- if (k == 0)
- return f - s*(f-R);
- return dk*ln2_hi - ((s*(f-R)-dk*ln2_lo)-f);
- }
+ hfsq = 0.5f*f*f;
+ dk = k;
+ return s*(hfsq+R) + dk*ln2_lo - hfsq + f + dk*ln2_hi;
}
*
* log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x).
*
- * Otherwise, setting z = 2(x-1)/x+1),
+ * Otherwise, setting z = 2(x-1)/(x+1),
*
- * log(x) = z + z**3 P(z)/Q(z).
+ * log(x) = log(1+z/2) - log(1-z/2) = z + z**3 P(z)/Q(z).
*
*
* ACCURACY:
return x;
if (x <= 0.0) {
if (x == 0.0)
- return -1/(x+0); /* -inf with divbyzero */
+ return -1/(x*x); /* -inf with divbyzero */
return 0/0.0f; /* nan with invalid */
}
x = frexpl(x, &e);
/* logarithm using log(x) = z + z**3 P(z)/Q(z),
- * where z = 2(x-1)/x+1)
+ * where z = 2(x-1)/(x+1)
*/
if (e > 2 || e < -2) {
if (x < SQRTH) { /* 2(2x-1)/(2x+1) */
projects / oweals / musl.git / commitdiff