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authorUlrich Drepper <drepper@redhat.com>1999-07-14 00:54:57 +0000
committerUlrich Drepper <drepper@redhat.com>1999-07-14 00:54:57 +0000
commitabfbdde177c3a7155070dda1b2cdc8292054cc26 (patch)
treee021306b596381fbf8311d2b7eb294e918ff17c8 /sysdeps/ieee754/dbl-64/e_log.c
parent86421aa57ecfd70963ae66848bd6a6dd3b8e0fe6 (diff)
downloadglibc-abfbdde177c3a7155070dda1b2cdc8292054cc26.tar.gz
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+/* @(#)e_log.c 5.1 93/09/24 */
+/*
+ * ====================================================
+ * 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.
+ * ====================================================
+ */
+/* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25,
+ for performance improvement on pipelined processors.
+*/
+
+#if defined(LIBM_SCCS) && !defined(lint)
+static char rcsid[] = "$NetBSD: e_log.c,v 1.8 1995/05/10 20:45:49 jtc Exp $";
+#endif
+
+/* __ieee754_log(x)
+ * Return the logarithm of x
+ *
+ * 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.
+ */
+
+#include "math.h"
+#include "math_private.h"
+#define half Lg[8]
+#define two Lg[9]
+#ifdef __STDC__
+static const double
+#else
+static double
+#endif
+ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
+ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
+two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
+ Lg[] = {0.0,
+ 6.666666666666735130e-01, /* 3FE55555 55555593 */
+ 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
+ 2.857142874366239149e-01, /* 3FD24924 94229359 */
+ 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
+ 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
+ 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
+ 1.479819860511658591e-01, /* 3FC2F112 DF3E5244 */
+ 0.5,
+ 2.0};
+#ifdef __STDC__
+static const double zero = 0.0;
+#else
+static double zero = 0.0;
+#endif
+
+#ifdef __STDC__
+ double __ieee754_log(double x)
+#else
+ double __ieee754_log(x)
+ double x;
+#endif
+{
+ double hfsq,f,s,z,R,w,dk,t11,t12,t21,t22,w2,zw2;
+#ifdef DO_NOT_USE_THIS
+ double t1,t2;
+#endif
+ int32_t k,hx,i,j;
+ u_int32_t lx;
+
+ EXTRACT_WORDS(hx,lx,x);
+
+ k=0;
+ if (hx < 0x00100000) { /* x < 2**-1022 */
+ if (((hx&0x7fffffff)|lx)==0)
+ return -two54/(x-x); /* log(+-0)=-inf */
+ if (hx<0) return (x-x)/(x-x); /* log(-#) = NaN */
+ k -= 54; x *= two54; /* subnormal number, scale up x */
+ 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);
+ f = x-1.0;
+ if((0x000fffff&(2+hx))<3) { /* |f| < 2**-20 */
+ if(f==zero) {
+ if(k==0) return zero; else {dk=(double)k;
+ return dk*ln2_hi+dk*ln2_lo;}
+ }
+ R = f*f*(half-0.33333333333333333*f);
+ if(k==0) return f-R; else {dk=(double)k;
+ return dk*ln2_hi-((R-dk*ln2_lo)-f);}
+ }
+ s = f/(two+f);
+ dk = (double)k;
+ z = s*s;
+ i = hx-0x6147a;
+ w = z*z;
+ j = 0x6b851-hx;
+#ifdef DO_NOT_USE_THIS
+ t1= w*(Lg2+w*(Lg4+w*Lg6));
+ t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
+ R = t2+t1;
+#else
+ t21 = Lg[5]+w*Lg[7]; w2=w*w;
+ t22 = Lg[1]+w*Lg[3]; zw2=z*w2;
+ t11 = Lg[4]+w*Lg[6];
+ t12 = w*Lg[2];
+ R = t12 + w2*t11 + z*t22 + zw2*t21;
+#endif
+ i |= j;
+ if(i>0) {
+ hfsq=0.5*f*f;
+ if(k==0) return f-(hfsq-s*(hfsq+R)); else
+ return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
+ } else {
+ if(k==0) return f-s*(f-R); else
+ return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
+ }
+}