/* 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;
}
