* ====================================================
*/
/*
- * 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 */
-
-static const double zero = 0.0;
+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/zero; /* log(+-0)=-inf */
- if (hx<0)
- return (x-x)/zero; /* 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 zero; /* 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;
projects / musl / blobdiff