Add a "fork" of musl as //fusl.

fusl/src/math/expm1.c

+/* origin: FreeBSD /usr/src/lib/msun/src/s_expm1.c */
+/*
+ * ====================================================
+ * 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.
+ * ====================================================
+ */
+/* 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 Remez 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;
+ *              z   =  r*r,
+ *      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 "libm.h"
+
+static const double
+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 Q's: Qn_here = 2**n * Qn_above, for R(2*z) where z = hxs = x*x/2: */
+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 */
+
+double expm1(double x)
+{
+        double_t y,hi,lo,c,t,e,hxs,hfx,r1,twopk;
+        union {double f; uint64_t i;} u = {x};
+        uint32_t hx = u.i>>32 & 0x7fffffff;
+        int k, sign = u.i>>63;
+
+        /* filter out huge and non-finite argument */
+        if (hx >= 0x4043687A) {  /* if |x|>=56*ln2 */
+                if (isnan(x))
+                        return x;
+                if (sign)
+                        return -1;
+                if (x > o_threshold) {
+                        x *= 0x1p1023;
+                        return x;
+                }
+        }
+
+        /* argument reduction */
+        if (hx > 0x3fd62e42) {  /* if  |x| > 0.5 ln2 */
+                if (hx < 0x3FF0A2B2) {  /* and |x| < 1.5 ln2 */
+                        if (!sign) {
+                                hi = x - ln2_hi;
+                                lo = ln2_lo;
+                                k =  1;
+                        } else {
+                                hi = x + ln2_hi;
+                                lo = -ln2_lo;
+                                k = -1;
+                        }
+                } else {
+                        k  = invln2*x + (sign ? -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) {  /* |x| < 2**-54, return x */
+                if (hx < 0x00100000)
+                        FORCE_EVAL((float)x);
+                return x;
+        } else
+                k = 0;
+
+        /* x is now in primary range */
+        hfx = 0.5*x;
+        hxs = x*hfx;
+        r1 = 1.0+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)   /* c is 0 */
+                return x - (x*e-hxs);
+        e  = x*(e-c) - c;
+        e -= hxs;
+        /* exp(x) ~ 2^k (x_reduced - e + 1) */
+        if (k == -1)
+                return 0.5*(x-e) - 0.5;
+        if (k == 1) {
+                if (x < -0.25)
+                        return -2.0*(e-(x+0.5));
+                return 1.0+2.0*(x-e);
+        }
+        u.i = (uint64_t)(0x3ff + k)<<52;  /* 2^k */
+        twopk = u.f;
+        if (k < 0 || k > 56) {  /* suffice to return exp(x)-1 */
+                y = x - e + 1.0;
+                if (k == 1024)
+                        y = y*2.0*0x1p1023;
+                else
+                        y = y*twopk;
+                return y - 1.0;
+        }
+        u.i = (uint64_t)(0x3ff - k)<<52;  /* 2^-k */
+        if (k < 20)
+                y = (x-e+(1-u.f))*twopk;
+        else
+                y = (x-(e+u.f)+1)*twopk;
+        return y;
+}