[lmi-commits] [lmi] master c4e7bd93 01/22: Import expm1() and log1p() from fdlibm

[Greg Chicares]

branch: master
commit c4e7bd939218737db2c8f669ae501256ec937449
Author: Gregory W. Chicares <gchicares@sbcglobal.net>
Commit: Gregory W. Chicares <gchicares@sbcglobal.net>

    Import expm1() and log1p() from fdlibm
    
    These are pristine copies of netlib.org sources:
    
    wget -O fdlibm_expm1.c.txt 
"http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=fdlibm%2Fs_expm1.c";
    
    wget -O fdlibm_log1p.c.txt 
"http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=fdlibm%2Fs_log1p.c";
    
    The '.txt' suffix lets them elude lmi's concinnity checks for now.
---
 fdlibm_expm1.c.txt | 431 +++++++++++++++++++++++++++++++++++++++++++++++++++++
 fdlibm_log1p.c.txt | 381 ++++++++++++++++++++++++++++++++++++++++++++++
 2 files changed, 812 insertions(+)

diff --git a/fdlibm_expm1.c.txt b/fdlibm_expm1.c.txt
new file mode 100644
index 00000000..6f0bd778
--- /dev/null
+++ b/fdlibm_expm1.c.txt
@@ -0,0 +1,431 @@
+
+/* @(#)fdlibm.h 1.5 04/04/22 */
+/*
+ * ====================================================
+ * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
+ *
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice 
+ * is preserved.
+ * ====================================================
+ */
+
+/* Sometimes it's necessary to define __LITTLE_ENDIAN explicitly
+   but these catch some common cases. */
+
+#if defined(i386) || defined(i486) || \
+       defined(intel) || defined(x86) || defined(i86pc) || \
+       defined(__alpha) || defined(__osf__)
+#define __LITTLE_ENDIAN
+#endif
+
+#ifdef __LITTLE_ENDIAN
+#define __HI(x) *(1+(int*)&x)
+#define __LO(x) *(int*)&x
+#define __HIp(x) *(1+(int*)x)
+#define __LOp(x) *(int*)x
+#else
+#define __HI(x) *(int*)&x
+#define __LO(x) *(1+(int*)&x)
+#define __HIp(x) *(int*)x
+#define __LOp(x) *(1+(int*)x)
+#endif
+
+#ifdef __STDC__
+#define        __P(p)  p
+#else
+#define        __P(p)  ()
+#endif
+
+/*
+ * ANSI/POSIX
+ */
+
+extern int signgam;
+
+#define        MAXFLOAT        ((float)3.40282346638528860e+38)
+
+enum fdversion {fdlibm_ieee = -1, fdlibm_svid, fdlibm_xopen, fdlibm_posix};
+
+#define _LIB_VERSION_TYPE enum fdversion
+#define _LIB_VERSION _fdlib_version  
+
+/* if global variable _LIB_VERSION is not desirable, one may 
+ * change the following to be a constant by: 
+ *     #define _LIB_VERSION_TYPE const enum version
+ * In that case, after one initializes the value _LIB_VERSION (see
+ * s_lib_version.c) during compile time, it cannot be modified
+ * in the middle of a program
+ */ 
+extern  _LIB_VERSION_TYPE  _LIB_VERSION;
+
+#define _IEEE_  fdlibm_ieee
+#define _SVID_  fdlibm_svid
+#define _XOPEN_ fdlibm_xopen
+#define _POSIX_ fdlibm_posix
+
+struct exception {
+       int type;
+       char *name;
+       double arg1;
+       double arg2;
+       double retval;
+};
+
+#define        HUGE            MAXFLOAT
+
+/* 
+ * set X_TLOSS = pi*2**52, which is possibly defined in <values.h>
+ * (one may replace the following line by "#include <values.h>")
+ */
+
+#define X_TLOSS                1.41484755040568800000e+16 
+
+#define        DOMAIN          1
+#define        SING            2
+#define        OVERFLOW        3
+#define        UNDERFLOW       4
+#define        TLOSS           5
+#define        PLOSS           6
+
+/*
+ * ANSI/POSIX
+ */
+extern double acos __P((double));
+extern double asin __P((double));
+extern double atan __P((double));
+extern double atan2 __P((double, double));
+extern double cos __P((double));
+extern double sin __P((double));
+extern double tan __P((double));
+
+extern double cosh __P((double));
+extern double sinh __P((double));
+extern double tanh __P((double));
+
+extern double exp __P((double));
+extern double frexp __P((double, int *));
+extern double ldexp __P((double, int));
+extern double log __P((double));
+extern double log10 __P((double));
+extern double modf __P((double, double *));
+
+extern double pow __P((double, double));
+extern double sqrt __P((double));
+
+extern double ceil __P((double));
+extern double fabs __P((double));
+extern double floor __P((double));
+extern double fmod __P((double, double));
+
+extern double erf __P((double));
+extern double erfc __P((double));
+extern double gamma __P((double));
+extern double hypot __P((double, double));
+extern int isnan __P((double));
+extern int finite __P((double));
+extern double j0 __P((double));
+extern double j1 __P((double));
+extern double jn __P((int, double));
+extern double lgamma __P((double));
+extern double y0 __P((double));
+extern double y1 __P((double));
+extern double yn __P((int, double));
+
+extern double acosh __P((double));
+extern double asinh __P((double));
+extern double atanh __P((double));
+extern double cbrt __P((double));
+extern double logb __P((double));
+extern double nextafter __P((double, double));
+extern double remainder __P((double, double));
+#ifdef _SCALB_INT
+extern double scalb __P((double, int));
+#else
+extern double scalb __P((double, double));
+#endif
+
+extern int matherr __P((struct exception *));
+
+/*
+ * IEEE Test Vector
+ */
+extern double significand __P((double));
+
+/*
+ * Functions callable from C, intended to support IEEE arithmetic.
+ */
+extern double copysign __P((double, double));
+extern int ilogb __P((double));
+extern double rint __P((double));
+extern double scalbn __P((double, int));
+
+/*
+ * BSD math library entry points
+ */
+extern double expm1 __P((double));
+extern double log1p __P((double));
+
+/*
+ * Reentrant version of gamma & lgamma; passes signgam back by reference
+ * as the second argument; user must allocate space for signgam.
+ */
+#ifdef _REENTRANT
+extern double gamma_r __P((double, int *));
+extern double lgamma_r __P((double, int *));
+#endif /* _REENTRANT */
+
+/* ieee style elementary functions */
+extern double __ieee754_sqrt __P((double));                    
+extern double __ieee754_acos __P((double));                    
+extern double __ieee754_acosh __P((double));                   
+extern double __ieee754_log __P((double));                     
+extern double __ieee754_atanh __P((double));                   
+extern double __ieee754_asin __P((double));                    
+extern double __ieee754_atan2 __P((double,double));                    
+extern double __ieee754_exp __P((double));
+extern double __ieee754_cosh __P((double));
+extern double __ieee754_fmod __P((double,double));
+extern double __ieee754_pow __P((double,double));
+extern double __ieee754_lgamma_r __P((double,int *));
+extern double __ieee754_gamma_r __P((double,int *));
+extern double __ieee754_lgamma __P((double));
+extern double __ieee754_gamma __P((double));
+extern double __ieee754_log10 __P((double));
+extern double __ieee754_sinh __P((double));
+extern double __ieee754_hypot __P((double,double));
+extern double __ieee754_j0 __P((double));
+extern double __ieee754_j1 __P((double));
+extern double __ieee754_y0 __P((double));
+extern double __ieee754_y1 __P((double));
+extern double __ieee754_jn __P((int,double));
+extern double __ieee754_yn __P((int,double));
+extern double __ieee754_remainder __P((double,double));
+extern int    __ieee754_rem_pio2 __P((double,double*));
+#ifdef _SCALB_INT
+extern double __ieee754_scalb __P((double,int));
+#else
+extern double __ieee754_scalb __P((double,double));
+#endif
+
+/* fdlibm kernel function */
+extern double __kernel_standard __P((double,double,int));      
+extern double __kernel_sin __P((double,double,int));
+extern double __kernel_cos __P((double,double));
+extern double __kernel_tan __P((double,double,int));
+extern int    __kernel_rem_pio2 __P((double*,double*,int,int,int,const int*));
+
+/* @(#)s_expm1.c 1.5 04/04/22 */
+/*
+ * ====================================================
+ * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
+ *
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice 
+ * is preserved.
+ * ====================================================
+ */
+
+/* expm1(x)
+ * Returns exp(x)-1, the exponential of x minus 1.
+ *
+ * Method
+ *   1. Argument reduction:
+ *     Given x, find r and integer k such that
+ *
+ *               x = k*ln2 + r,  |r| <= 0.5*ln2 ~ 0.34658  
+ *
+ *      Here a correction term c will be computed to compensate 
+ *     the error in r when rounded to a floating-point number.
+ *
+ *   2. Approximating expm1(r) by a special rational function on
+ *     the interval [0,0.34658]:
+ *     Since
+ *         r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
+ *     we define R1(r*r) by
+ *         r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
+ *     That is,
+ *         R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
+ *                  = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
+ *                  = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
+ *      We use a special Remes algorithm on [0,0.347] to generate 
+ *     a polynomial of degree 5 in r*r to approximate R1. The 
+ *     maximum error of this polynomial approximation is bounded 
+ *     by 2**-61. In other words,
+ *         R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
+ *     where   Q1  =  -1.6666666666666567384E-2,
+ *             Q2  =   3.9682539681370365873E-4,
+ *             Q3  =  -9.9206344733435987357E-6,
+ *             Q4  =   2.5051361420808517002E-7,
+ *             Q5  =  -6.2843505682382617102E-9;
+ *     (where z=r*r, and the values of Q1 to Q5 are listed below)
+ *     with error bounded by
+ *         |                  5           |     -61
+ *         | 1.0+Q1*z+...+Q5*z   -  R1(z) | <= 2 
+ *         |                              |
+ *     
+ *     expm1(r) = exp(r)-1 is then computed by the following 
+ *     specific way which minimize the accumulation rounding error: 
+ *                            2     3
+ *                           r     r    [ 3 - (R1 + R1*r/2)  ]
+ *           expm1(r) = r + --- + --- * [--------------------]
+ *                           2     2    [ 6 - r*(3 - R1*r/2) ]
+ *     
+ *     To compensate the error in the argument reduction, we use
+ *             expm1(r+c) = expm1(r) + c + expm1(r)*c 
+ *                        ~ expm1(r) + c + r*c 
+ *     Thus c+r*c will be added in as the correction terms for
+ *     expm1(r+c). Now rearrange the term to avoid optimization 
+ *     screw up:
+ *                     (      2                                    2 )
+ *                     ({  ( r    [ R1 -  (3 - R1*r/2) ]  )  }    r  )
+ *      expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
+ *                     ({  ( 2    [ 6 - r*(3 - R1*r/2) ]  )  }    2  )
+ *                      (                                             )
+ *     
+ *                = r - E
+ *   3. Scale back to obtain expm1(x):
+ *     From step 1, we have
+ *        expm1(x) = either 2^k*[expm1(r)+1] - 1
+ *                 = or     2^k*[expm1(r) + (1-2^-k)]
+ *   4. Implementation notes:
+ *     (A). To save one multiplication, we scale the coefficient Qi
+ *          to Qi*2^i, and replace z by (x^2)/2.
+ *     (B). To achieve maximum accuracy, we compute expm1(x) by
+ *       (i)   if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
+ *       (ii)  if k=0, return r-E
+ *       (iii) if k=-1, return 0.5*(r-E)-0.5
+ *        (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
+ *                    else          return  1.0+2.0*(r-E);
+ *       (v)   if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
+ *       (vi)  if k <= 20, return 2^k((1-2^-k)-(E-r)), else
+ *       (vii) return 2^k(1-((E+2^-k)-r)) 
+ *
+ * Special cases:
+ *     expm1(INF) is INF, expm1(NaN) is NaN;
+ *     expm1(-INF) is -1, and
+ *     for finite argument, only expm1(0)=0 is exact.
+ *
+ * Accuracy:
+ *     according to an error analysis, the error is always less than
+ *     1 ulp (unit in the last place).
+ *
+ * Misc. info.
+ *     For IEEE double 
+ *         if x >  7.09782712893383973096e+02 then expm1(x) overflow
+ *
+ * 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 "fdlibm.h"
+
+#ifdef __STDC__
+static const double
+#else
+static double
+#endif
+one            = 1.0,
+huge           = 1.0e+300,
+tiny           = 1.0e-300,
+o_threshold    = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */
+ln2_hi         = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */
+ln2_lo         = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */
+invln2         = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */
+       /* scaled coefficients related to expm1 */
+Q1  =  -3.33333333333331316428e-02, /* BFA11111 111110F4 */
+Q2  =   1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
+Q3  =  -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
+Q4  =   4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
+Q5  =  -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
+
+#ifdef __STDC__
+       double expm1(double x)
+#else
+       double expm1(x)
+       double x;
+#endif
+{
+       double y,hi,lo,c,t,e,hxs,hfx,r1;
+       int k,xsb;
+       unsigned hx;
+
+       hx  = __HI(x);  /* high word of x */
+       xsb = hx&0x80000000;            /* sign bit of x */
+       if(xsb==0) y=x; else y= -x;     /* y = |x| */
+       hx &= 0x7fffffff;               /* high word of |x| */
+
+    /* filter out huge and non-finite argument */
+       if(hx >= 0x4043687A) {                  /* if |x|>=56*ln2 */
+           if(hx >= 0x40862E42) {              /* if |x|>=709.78... */
+                if(hx>=0x7ff00000) {
+                   if(((hx&0xfffff)|__LO(x))!=0) 
+                        return x+x;     /* NaN */
+                   else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */
+               }
+               if(x > o_threshold) return huge*huge; /* overflow */
+           }
+           if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
+               if(x+tiny<0.0)          /* raise inexact */
+               return tiny-one;        /* return -1 */
+           }
+       }
+
+    /* argument reduction */
+       if(hx > 0x3fd62e42) {           /* if  |x| > 0.5 ln2 */ 
+           if(hx < 0x3FF0A2B2) {       /* and |x| < 1.5 ln2 */
+               if(xsb==0)
+                   {hi = x - ln2_hi; lo =  ln2_lo;  k =  1;}
+               else
+                   {hi = x + ln2_hi; lo = -ln2_lo;  k = -1;}
+           } else {
+               k  = invln2*x+((xsb==0)?0.5:-0.5);
+               t  = k;
+               hi = x - t*ln2_hi;      /* t*ln2_hi is exact here */
+               lo = t*ln2_lo;
+           }
+           x  = hi - lo;
+           c  = (hi-x)-lo;
+       } 
+       else if(hx < 0x3c900000) {      /* when |x|<2**-54, return x */
+           t = huge+x; /* return x with inexact flags when x!=0 */
+           return x - (t-(huge+x));    
+       }
+       else k = 0;
+
+    /* x is now in primary range */
+       hfx = 0.5*x;
+       hxs = x*hfx;
+       r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
+       t  = 3.0-r1*hfx;
+       e  = hxs*((r1-t)/(6.0 - x*t));
+       if(k==0) return x - (x*e-hxs);          /* c is 0 */
+       else {
+           e  = (x*(e-c)-c);
+           e -= hxs;
+           if(k== -1) return 0.5*(x-e)-0.5;
+           if(k==1) 
+               if(x < -0.25) return -2.0*(e-(x+0.5));
+               else          return  one+2.0*(x-e);
+           if (k <= -2 || k>56) {   /* suffice to return exp(x)-1 */
+               y = one-(e-x);
+               __HI(y) += (k<<20);     /* add k to y's exponent */
+               return y-one;
+           }
+           t = one;
+           if(k<20) {
+               __HI(t) = 0x3ff00000 - (0x200000>>k);  /* t=1-2^-k */
+               y = t-(e-x);
+               __HI(y) += (k<<20);     /* add k to y's exponent */
+          } else {
+               __HI(t)  = ((0x3ff-k)<<20);     /* 2^-k */
+               y = x-(e+t);
+               y += one;
+               __HI(y) += (k<<20);     /* add k to y's exponent */
+           }
+       }
+       return y;
+}
diff --git a/fdlibm_log1p.c.txt b/fdlibm_log1p.c.txt
new file mode 100644
index 00000000..75b3d564
--- /dev/null
+++ b/fdlibm_log1p.c.txt
@@ -0,0 +1,381 @@
+
+/* @(#)fdlibm.h 1.5 04/04/22 */
+/*
+ * ====================================================
+ * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
+ *
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice 
+ * is preserved.
+ * ====================================================
+ */
+
+/* Sometimes it's necessary to define __LITTLE_ENDIAN explicitly
+   but these catch some common cases. */
+
+#if defined(i386) || defined(i486) || \
+       defined(intel) || defined(x86) || defined(i86pc) || \
+       defined(__alpha) || defined(__osf__)
+#define __LITTLE_ENDIAN
+#endif
+
+#ifdef __LITTLE_ENDIAN
+#define __HI(x) *(1+(int*)&x)
+#define __LO(x) *(int*)&x
+#define __HIp(x) *(1+(int*)x)
+#define __LOp(x) *(int*)x
+#else
+#define __HI(x) *(int*)&x
+#define __LO(x) *(1+(int*)&x)
+#define __HIp(x) *(int*)x
+#define __LOp(x) *(1+(int*)x)
+#endif
+
+#ifdef __STDC__
+#define        __P(p)  p
+#else
+#define        __P(p)  ()
+#endif
+
+/*
+ * ANSI/POSIX
+ */
+
+extern int signgam;
+
+#define        MAXFLOAT        ((float)3.40282346638528860e+38)
+
+enum fdversion {fdlibm_ieee = -1, fdlibm_svid, fdlibm_xopen, fdlibm_posix};
+
+#define _LIB_VERSION_TYPE enum fdversion
+#define _LIB_VERSION _fdlib_version  
+
+/* if global variable _LIB_VERSION is not desirable, one may 
+ * change the following to be a constant by: 
+ *     #define _LIB_VERSION_TYPE const enum version
+ * In that case, after one initializes the value _LIB_VERSION (see
+ * s_lib_version.c) during compile time, it cannot be modified
+ * in the middle of a program
+ */ 
+extern  _LIB_VERSION_TYPE  _LIB_VERSION;
+
+#define _IEEE_  fdlibm_ieee
+#define _SVID_  fdlibm_svid
+#define _XOPEN_ fdlibm_xopen
+#define _POSIX_ fdlibm_posix
+
+struct exception {
+       int type;
+       char *name;
+       double arg1;
+       double arg2;
+       double retval;
+};
+
+#define        HUGE            MAXFLOAT
+
+/* 
+ * set X_TLOSS = pi*2**52, which is possibly defined in <values.h>
+ * (one may replace the following line by "#include <values.h>")
+ */
+
+#define X_TLOSS                1.41484755040568800000e+16 
+
+#define        DOMAIN          1
+#define        SING            2
+#define        OVERFLOW        3
+#define        UNDERFLOW       4
+#define        TLOSS           5
+#define        PLOSS           6
+
+/*
+ * ANSI/POSIX
+ */
+extern double acos __P((double));
+extern double asin __P((double));
+extern double atan __P((double));
+extern double atan2 __P((double, double));
+extern double cos __P((double));
+extern double sin __P((double));
+extern double tan __P((double));
+
+extern double cosh __P((double));
+extern double sinh __P((double));
+extern double tanh __P((double));
+
+extern double exp __P((double));
+extern double frexp __P((double, int *));
+extern double ldexp __P((double, int));
+extern double log __P((double));
+extern double log10 __P((double));
+extern double modf __P((double, double *));
+
+extern double pow __P((double, double));
+extern double sqrt __P((double));
+
+extern double ceil __P((double));
+extern double fabs __P((double));
+extern double floor __P((double));
+extern double fmod __P((double, double));
+
+extern double erf __P((double));
+extern double erfc __P((double));
+extern double gamma __P((double));
+extern double hypot __P((double, double));
+extern int isnan __P((double));
+extern int finite __P((double));
+extern double j0 __P((double));
+extern double j1 __P((double));
+extern double jn __P((int, double));
+extern double lgamma __P((double));
+extern double y0 __P((double));
+extern double y1 __P((double));
+extern double yn __P((int, double));
+
+extern double acosh __P((double));
+extern double asinh __P((double));
+extern double atanh __P((double));
+extern double cbrt __P((double));
+extern double logb __P((double));
+extern double nextafter __P((double, double));
+extern double remainder __P((double, double));
+#ifdef _SCALB_INT
+extern double scalb __P((double, int));
+#else
+extern double scalb __P((double, double));
+#endif
+
+extern int matherr __P((struct exception *));
+
+/*
+ * IEEE Test Vector
+ */
+extern double significand __P((double));
+
+/*
+ * Functions callable from C, intended to support IEEE arithmetic.
+ */
+extern double copysign __P((double, double));
+extern int ilogb __P((double));
+extern double rint __P((double));
+extern double scalbn __P((double, int));
+
+/*
+ * BSD math library entry points
+ */
+extern double expm1 __P((double));
+extern double log1p __P((double));
+
+/*
+ * Reentrant version of gamma & lgamma; passes signgam back by reference
+ * as the second argument; user must allocate space for signgam.
+ */
+#ifdef _REENTRANT
+extern double gamma_r __P((double, int *));
+extern double lgamma_r __P((double, int *));
+#endif /* _REENTRANT */
+
+/* ieee style elementary functions */
+extern double __ieee754_sqrt __P((double));                    
+extern double __ieee754_acos __P((double));                    
+extern double __ieee754_acosh __P((double));                   
+extern double __ieee754_log __P((double));                     
+extern double __ieee754_atanh __P((double));                   
+extern double __ieee754_asin __P((double));                    
+extern double __ieee754_atan2 __P((double,double));                    
+extern double __ieee754_exp __P((double));
+extern double __ieee754_cosh __P((double));
+extern double __ieee754_fmod __P((double,double));
+extern double __ieee754_pow __P((double,double));
+extern double __ieee754_lgamma_r __P((double,int *));
+extern double __ieee754_gamma_r __P((double,int *));
+extern double __ieee754_lgamma __P((double));
+extern double __ieee754_gamma __P((double));
+extern double __ieee754_log10 __P((double));
+extern double __ieee754_sinh __P((double));
+extern double __ieee754_hypot __P((double,double));
+extern double __ieee754_j0 __P((double));
+extern double __ieee754_j1 __P((double));
+extern double __ieee754_y0 __P((double));
+extern double __ieee754_y1 __P((double));
+extern double __ieee754_jn __P((int,double));
+extern double __ieee754_yn __P((int,double));
+extern double __ieee754_remainder __P((double,double));
+extern int    __ieee754_rem_pio2 __P((double,double*));
+#ifdef _SCALB_INT
+extern double __ieee754_scalb __P((double,int));
+#else
+extern double __ieee754_scalb __P((double,double));
+#endif
+
+/* fdlibm kernel function */
+extern double __kernel_standard __P((double,double,int));      
+extern double __kernel_sin __P((double,double,int));
+extern double __kernel_cos __P((double,double));
+extern double __kernel_tan __P((double,double,int));
+extern int    __kernel_rem_pio2 __P((double*,double*,int,int,int,const int*));
+
+/* @(#)s_log1p.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.
+ * ====================================================
+ */
+
+/* double log1p(double x)
+ *
+ * Method :                  
+ *   1. Argument Reduction: find k and f such that 
+ *                     1+x = 2^k * (1+f), 
+ *        where  sqrt(2)/2 < 1+f < sqrt(2) .
+ *
+ *      Note. If k=0, then f=x is exact. However, if k!=0, then f
+ *     may not be representable exactly. In that case, a correction
+ *     term is need. Let u=1+x rounded. Let c = (1+x)-u, then
+ *     log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
+ *     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)).
+ *     
+ *     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.
+ *
+ * Special cases:
+ *     log1p(x) is NaN with signal if x < -1 (including -INF) ; 
+ *     log1p(+INF) is +INF; log1p(-1) is -INF with signal;
+ *     log1p(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.
+ *
+ * Note: Assuming log() return accurate answer, the following
+ *      algorithm can be used to compute log1p(x) to within a few ULP:
+ *     
+ *             u = 1+x;
+ *             if(u==1.0) return x ; else
+ *                        return log(u)*(x/(u-1.0));
+ *
+ *      See HP-15C Advanced Functions Handbook, p.193.
+ */
+
+#include "fdlibm.h"
+
+#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 */
+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 */
+
+static double zero = 0.0;
+
+#ifdef __STDC__
+       double log1p(double x)
+#else
+       double log1p(x)
+       double x;
+#endif
+{
+       double hfsq,f,c,s,z,R,u;
+       int k,hx,hu,ax;
+
+       hx = __HI(x);           /* high word of x */
+       ax = hx&0x7fffffff;
+
+       k = 1;
+       if (hx < 0x3FDA827A) {                  /* x < 0.41422  */
+           if(ax>=0x3ff00000) {                /* x <= -1.0 */
+               if(x==-1.0) return -two54/zero; /* log1p(-1)=+inf */
+               else return (x-x)/(x-x);        /* log1p(x<-1)=NaN */
+           }
+           if(ax<0x3e200000) {                 /* |x| < 2**-29 */
+               if(two54+x>zero                 /* raise inexact */
+                   &&ax<0x3c900000)            /* |x| < 2**-54 */
+                   return x;
+               else
+                   return x - x*x*0.5;
+           }
+           if(hx>0||hx<=((int)0xbfd2bec3)) {
+               k=0;f=x;hu=1;}  /* -0.2929<x<0.41422 */
+       } 
+       if (hx >= 0x7ff00000) return x+x;
+       if(k!=0) {
+           if(hx<0x43400000) {
+               u  = 1.0+x; 
+               hu = __HI(u);           /* high word of u */
+               k  = (hu>>20)-1023;
+               c  = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */
+               c /= u;
+           } else {
+               u  = x;
+               hu = __HI(u);           /* high word of u */
+               k  = (hu>>20)-1023;
+               c  = 0;
+           }
+           hu &= 0x000fffff;
+           if(hu<0x6a09e) {
+               __HI(u) = hu|0x3ff00000;        /* normalize u */
+           } else {
+               k += 1; 
+               __HI(u) = hu|0x3fe00000;        /* normalize u/2 */
+               hu = (0x00100000-hu)>>2;
+           }
+           f = u-1.0;
+       }
+       hfsq=0.5*f*f;
+       if(hu==0) {     /* |f| < 2**-20 */
+           if(f==zero) if(k==0) return zero;  
+                       else {c += k*ln2_lo; return k*ln2_hi+c;}
+           R = hfsq*(1.0-0.66666666666666666*f);
+           if(k==0) return f-R; else
+                    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)); else
+                return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
+}