initial check-in, version 0.5.0 v0.5.0

1 files changed, 131 insertions, 0 deletions

diff --git a/src/math/e_log.c b/src/math/e_log.c
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+
+/* @(#)e_log.c 1.3 95/01/18 */
+/*
+ * ====================================================
+ * 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.
+ * ====================================================
+ */
+
+/* log(x)
+ * Return the logrithm 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"
+
+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 */
+Lg4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
+Lg5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
+Lg6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
+Lg7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
+
+static const double zero   =  0.0;
+
+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);
+
+        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 */
+            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*(0.5-0.33333333333333333*f);
+            if(k==0) return f-R; else {dk=(double)k;
+                     return dk*ln2_hi-((R-dk*ln2_lo)-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)); 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);
+        }
+}