0,0 → 1,272 |
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/* @(#)s_expm1.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. |
* ==================================================== |
*/ |
|
/* |
FUNCTION |
<<expm1>>, <<expm1f>>---exponential minus 1 |
INDEX |
expm1 |
INDEX |
expm1f |
|
ANSI_SYNOPSIS |
#include <math.h> |
double expm1(double <[x]>); |
float expm1f(float <[x]>); |
|
TRAD_SYNOPSIS |
#include <math.h> |
double expm1(<[x]>); |
double <[x]>; |
|
float expm1f(<[x]>); |
float <[x]>; |
|
DESCRIPTION |
<<expm1>> and <<expm1f>> calculate the exponential of <[x]> |
and subtract 1, that is, |
@ifnottex |
e raised to the power <[x]> minus 1 (where e |
@end ifnottex |
@tex |
$e^x - 1$ (where $e$ |
@end tex |
is the base of the natural system of logarithms, approximately |
2.71828). The result is accurate even for small values of |
<[x]>, where using <<exp(<[x]>)-1>> would lose many |
significant digits. |
|
RETURNS |
e raised to the power <[x]>, minus 1. |
|
PORTABILITY |
Neither <<expm1>> nor <<expm1f>> is required by ANSI C or by |
the System V Interface Definition (Issue 2). |
*/ |
|
/* 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 Reme 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" |
|
#ifndef _DOUBLE_IS_32BITS |
|
#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; |
__int32_t k,xsb; |
__uint32_t hx; |
|
GET_HIGH_WORD(hx,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) { |
__uint32_t low; |
GET_LOW_WORD(low,x); |
if(((hx&0xfffff)|low)!=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 */ |
__uint32_t high; |
y = one-(e-x); |
GET_HIGH_WORD(high,y); |
SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */ |
return y-one; |
} |
t = one; |
if(k<20) { |
__uint32_t high; |
SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */ |
y = t-(e-x); |
GET_HIGH_WORD(high,y); |
SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */ |
} else { |
__uint32_t high; |
SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */ |
y = x-(e+t); |
y += one; |
GET_HIGH_WORD(high,y); |
SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */ |
} |
} |
return y; |
} |
|
#endif /* _DOUBLE_IS_32BITS */ |