|
|
/* @(#)s_expm1.c 5.1 93/09/24 */
|
/* @(#)s_expm1.c 5.1 93/09/24 */
|
/*
|
/*
|
* ====================================================
|
* ====================================================
|
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
*
|
*
|
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
* Permission to use, copy, modify, and distribute this
|
* Permission to use, copy, modify, and distribute this
|
* software is freely granted, provided that this notice
|
* software is freely granted, provided that this notice
|
* is preserved.
|
* is preserved.
|
* ====================================================
|
* ====================================================
|
*/
|
*/
|
|
|
/*
|
/*
|
FUNCTION
|
FUNCTION
|
<<expm1>>, <<expm1f>>---exponential minus 1
|
<<expm1>>, <<expm1f>>---exponential minus 1
|
INDEX
|
INDEX
|
expm1
|
expm1
|
INDEX
|
INDEX
|
expm1f
|
expm1f
|
|
|
ANSI_SYNOPSIS
|
ANSI_SYNOPSIS
|
#include <math.h>
|
#include <math.h>
|
double expm1(double <[x]>);
|
double expm1(double <[x]>);
|
float expm1f(float <[x]>);
|
float expm1f(float <[x]>);
|
|
|
TRAD_SYNOPSIS
|
TRAD_SYNOPSIS
|
#include <math.h>
|
#include <math.h>
|
double expm1(<[x]>);
|
double expm1(<[x]>);
|
double <[x]>;
|
double <[x]>;
|
|
|
float expm1f(<[x]>);
|
float expm1f(<[x]>);
|
float <[x]>;
|
float <[x]>;
|
|
|
DESCRIPTION
|
DESCRIPTION
|
<<expm1>> and <<expm1f>> calculate the exponential of <[x]>
|
<<expm1>> and <<expm1f>> calculate the exponential of <[x]>
|
and subtract 1, that is,
|
and subtract 1, that is,
|
@ifinfo
|
@ifinfo
|
e raised to the power <[x]> minus 1 (where e
|
e raised to the power <[x]> minus 1 (where e
|
@end ifinfo
|
@end ifinfo
|
@tex
|
@tex
|
$e^x - 1$ (where $e$
|
$e^x - 1$ (where $e$
|
@end tex
|
@end tex
|
is the base of the natural system of logarithms, approximately
|
is the base of the natural system of logarithms, approximately
|
2.71828). The result is accurate even for small values of
|
2.71828). The result is accurate even for small values of
|
<[x]>, where using <<exp(<[x]>)-1>> would lose many
|
<[x]>, where using <<exp(<[x]>)-1>> would lose many
|
significant digits.
|
significant digits.
|
|
|
RETURNS
|
RETURNS
|
e raised to the power <[x]>, minus 1.
|
e raised to the power <[x]>, minus 1.
|
|
|
PORTABILITY
|
PORTABILITY
|
Neither <<expm1>> nor <<expm1f>> is required by ANSI C or by
|
Neither <<expm1>> nor <<expm1f>> is required by ANSI C or by
|
the System V Interface Definition (Issue 2).
|
the System V Interface Definition (Issue 2).
|
*/
|
*/
|
|
|
/* expm1(x)
|
/* expm1(x)
|
* Returns exp(x)-1, the exponential of x minus 1.
|
* Returns exp(x)-1, the exponential of x minus 1.
|
*
|
*
|
* Method
|
* Method
|
* 1. Argument reduction:
|
* 1. Argument reduction:
|
* Given x, find r and integer k such that
|
* Given x, find r and integer k such that
|
*
|
*
|
* x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
|
* x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
|
*
|
*
|
* Here a correction term c will be computed to compensate
|
* Here a correction term c will be computed to compensate
|
* the error in r when rounded to a floating-point number.
|
* the error in r when rounded to a floating-point number.
|
*
|
*
|
* 2. Approximating expm1(r) by a special rational function on
|
* 2. Approximating expm1(r) by a special rational function on
|
* the interval [0,0.34658]:
|
* the interval [0,0.34658]:
|
* Since
|
* Since
|
* r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
|
* r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
|
* we define R1(r*r) by
|
* we define R1(r*r) by
|
* r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
|
* r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
|
* That is,
|
* That is,
|
* R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
|
* R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
|
* = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
|
* = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
|
* = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
|
* = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
|
* We use a special Reme algorithm on [0,0.347] to generate
|
* We use a special Reme algorithm on [0,0.347] to generate
|
* a polynomial of degree 5 in r*r to approximate R1. The
|
* a polynomial of degree 5 in r*r to approximate R1. The
|
* maximum error of this polynomial approximation is bounded
|
* maximum error of this polynomial approximation is bounded
|
* by 2**-61. In other words,
|
* by 2**-61. In other words,
|
* R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
|
* R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
|
* where Q1 = -1.6666666666666567384E-2,
|
* where Q1 = -1.6666666666666567384E-2,
|
* Q2 = 3.9682539681370365873E-4,
|
* Q2 = 3.9682539681370365873E-4,
|
* Q3 = -9.9206344733435987357E-6,
|
* Q3 = -9.9206344733435987357E-6,
|
* Q4 = 2.5051361420808517002E-7,
|
* Q4 = 2.5051361420808517002E-7,
|
* Q5 = -6.2843505682382617102E-9;
|
* Q5 = -6.2843505682382617102E-9;
|
* (where z=r*r, and the values of Q1 to Q5 are listed below)
|
* (where z=r*r, and the values of Q1 to Q5 are listed below)
|
* with error bounded by
|
* with error bounded by
|
* | 5 | -61
|
* | 5 | -61
|
* | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
|
* | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
|
* | |
|
* | |
|
*
|
*
|
* expm1(r) = exp(r)-1 is then computed by the following
|
* expm1(r) = exp(r)-1 is then computed by the following
|
* specific way which minimize the accumulation rounding error:
|
* specific way which minimize the accumulation rounding error:
|
* 2 3
|
* 2 3
|
* r r [ 3 - (R1 + R1*r/2) ]
|
* r r [ 3 - (R1 + R1*r/2) ]
|
* expm1(r) = r + --- + --- * [--------------------]
|
* expm1(r) = r + --- + --- * [--------------------]
|
* 2 2 [ 6 - r*(3 - R1*r/2) ]
|
* 2 2 [ 6 - r*(3 - R1*r/2) ]
|
*
|
*
|
* To compensate the error in the argument reduction, we use
|
* To compensate the error in the argument reduction, we use
|
* expm1(r+c) = expm1(r) + c + expm1(r)*c
|
* expm1(r+c) = expm1(r) + c + expm1(r)*c
|
* ~ expm1(r) + c + r*c
|
* ~ expm1(r) + c + r*c
|
* Thus c+r*c will be added in as the correction terms for
|
* Thus c+r*c will be added in as the correction terms for
|
* expm1(r+c). Now rearrange the term to avoid optimization
|
* expm1(r+c). Now rearrange the term to avoid optimization
|
* screw up:
|
* screw up:
|
* ( 2 2 )
|
* ( 2 2 )
|
* ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
|
* ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
|
* expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
|
* expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
|
* ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
|
* ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
|
* ( )
|
* ( )
|
*
|
*
|
* = r - E
|
* = r - E
|
* 3. Scale back to obtain expm1(x):
|
* 3. Scale back to obtain expm1(x):
|
* From step 1, we have
|
* From step 1, we have
|
* expm1(x) = either 2^k*[expm1(r)+1] - 1
|
* expm1(x) = either 2^k*[expm1(r)+1] - 1
|
* = or 2^k*[expm1(r) + (1-2^-k)]
|
* = or 2^k*[expm1(r) + (1-2^-k)]
|
* 4. Implementation notes:
|
* 4. Implementation notes:
|
* (A). To save one multiplication, we scale the coefficient Qi
|
* (A). To save one multiplication, we scale the coefficient Qi
|
* to Qi*2^i, and replace z by (x^2)/2.
|
* to Qi*2^i, and replace z by (x^2)/2.
|
* (B). To achieve maximum accuracy, we compute expm1(x) by
|
* (B). To achieve maximum accuracy, we compute expm1(x) by
|
* (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
|
* (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
|
* (ii) if k=0, return r-E
|
* (ii) if k=0, return r-E
|
* (iii) if k=-1, return 0.5*(r-E)-0.5
|
* (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)
|
* (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
|
* else return 1.0+2.0*(r-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)
|
* (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
|
* (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else
|
* (vii) return 2^k(1-((E+2^-k)-r))
|
* (vii) return 2^k(1-((E+2^-k)-r))
|
*
|
*
|
* Special cases:
|
* Special cases:
|
* expm1(INF) is INF, expm1(NaN) is NaN;
|
* expm1(INF) is INF, expm1(NaN) is NaN;
|
* expm1(-INF) is -1, and
|
* expm1(-INF) is -1, and
|
* for finite argument, only expm1(0)=0 is exact.
|
* for finite argument, only expm1(0)=0 is exact.
|
*
|
*
|
* Accuracy:
|
* Accuracy:
|
* according to an error analysis, the error is always less than
|
* according to an error analysis, the error is always less than
|
* 1 ulp (unit in the last place).
|
* 1 ulp (unit in the last place).
|
*
|
*
|
* Misc. info.
|
* Misc. info.
|
* For IEEE double
|
* For IEEE double
|
* if x > 7.09782712893383973096e+02 then expm1(x) overflow
|
* if x > 7.09782712893383973096e+02 then expm1(x) overflow
|
*
|
*
|
* Constants:
|
* Constants:
|
* The hexadecimal values are the intended ones for the following
|
* The hexadecimal values are the intended ones for the following
|
* constants. The decimal values may be used, provided that the
|
* constants. The decimal values may be used, provided that the
|
* compiler will convert from decimal to binary accurately enough
|
* compiler will convert from decimal to binary accurately enough
|
* to produce the hexadecimal values shown.
|
* to produce the hexadecimal values shown.
|
*/
|
*/
|
|
|
#include "fdlibm.h"
|
#include "fdlibm.h"
|
|
|
#ifndef _DOUBLE_IS_32BITS
|
#ifndef _DOUBLE_IS_32BITS
|
|
|
#ifdef __STDC__
|
#ifdef __STDC__
|
static const double
|
static const double
|
#else
|
#else
|
static double
|
static double
|
#endif
|
#endif
|
one = 1.0,
|
one = 1.0,
|
huge = 1.0e+300,
|
huge = 1.0e+300,
|
tiny = 1.0e-300,
|
tiny = 1.0e-300,
|
o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */
|
o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */
|
ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */
|
ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */
|
ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */
|
ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */
|
invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */
|
invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */
|
/* scaled coefficients related to expm1 */
|
/* scaled coefficients related to expm1 */
|
Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */
|
Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */
|
Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
|
Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
|
Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
|
Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
|
Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
|
Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
|
Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
|
Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
|
|
|
#ifdef __STDC__
|
#ifdef __STDC__
|
double expm1(double x)
|
double expm1(double x)
|
#else
|
#else
|
double expm1(x)
|
double expm1(x)
|
double x;
|
double x;
|
#endif
|
#endif
|
{
|
{
|
double y,hi,lo,c,t,e,hxs,hfx,r1;
|
double y,hi,lo,c,t,e,hxs,hfx,r1;
|
__int32_t k,xsb;
|
__int32_t k,xsb;
|
__uint32_t hx;
|
__uint32_t hx;
|
|
|
GET_HIGH_WORD(hx,x);
|
GET_HIGH_WORD(hx,x);
|
xsb = hx&0x80000000; /* sign bit of x */
|
xsb = hx&0x80000000; /* sign bit of x */
|
if(xsb==0) y=x; else y= -x; /* y = |x| */
|
if(xsb==0) y=x; else y= -x; /* y = |x| */
|
hx &= 0x7fffffff; /* high word of |x| */
|
hx &= 0x7fffffff; /* high word of |x| */
|
|
|
/* filter out huge and non-finite argument */
|
/* filter out huge and non-finite argument */
|
if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */
|
if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */
|
if(hx >= 0x40862E42) { /* if |x|>=709.78... */
|
if(hx >= 0x40862E42) { /* if |x|>=709.78... */
|
if(hx>=0x7ff00000) {
|
if(hx>=0x7ff00000) {
|
__uint32_t low;
|
__uint32_t low;
|
GET_LOW_WORD(low,x);
|
GET_LOW_WORD(low,x);
|
if(((hx&0xfffff)|low)!=0)
|
if(((hx&0xfffff)|low)!=0)
|
return x+x; /* NaN */
|
return x+x; /* NaN */
|
else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */
|
else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */
|
}
|
}
|
if(x > o_threshold) return huge*huge; /* overflow */
|
if(x > o_threshold) return huge*huge; /* overflow */
|
}
|
}
|
if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
|
if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
|
if(x+tiny<0.0) /* raise inexact */
|
if(x+tiny<0.0) /* raise inexact */
|
return tiny-one; /* return -1 */
|
return tiny-one; /* return -1 */
|
}
|
}
|
}
|
}
|
|
|
/* argument reduction */
|
/* argument reduction */
|
if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
|
if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
|
if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
|
if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
|
if(xsb==0)
|
if(xsb==0)
|
{hi = x - ln2_hi; lo = ln2_lo; k = 1;}
|
{hi = x - ln2_hi; lo = ln2_lo; k = 1;}
|
else
|
else
|
{hi = x + ln2_hi; lo = -ln2_lo; k = -1;}
|
{hi = x + ln2_hi; lo = -ln2_lo; k = -1;}
|
} else {
|
} else {
|
k = invln2*x+((xsb==0)?0.5:-0.5);
|
k = invln2*x+((xsb==0)?0.5:-0.5);
|
t = k;
|
t = k;
|
hi = x - t*ln2_hi; /* t*ln2_hi is exact here */
|
hi = x - t*ln2_hi; /* t*ln2_hi is exact here */
|
lo = t*ln2_lo;
|
lo = t*ln2_lo;
|
}
|
}
|
x = hi - lo;
|
x = hi - lo;
|
c = (hi-x)-lo;
|
c = (hi-x)-lo;
|
}
|
}
|
else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */
|
else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */
|
t = huge+x; /* return x with inexact flags when x!=0 */
|
t = huge+x; /* return x with inexact flags when x!=0 */
|
return x - (t-(huge+x));
|
return x - (t-(huge+x));
|
}
|
}
|
else k = 0;
|
else k = 0;
|
|
|
/* x is now in primary range */
|
/* x is now in primary range */
|
hfx = 0.5*x;
|
hfx = 0.5*x;
|
hxs = x*hfx;
|
hxs = x*hfx;
|
r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
|
r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
|
t = 3.0-r1*hfx;
|
t = 3.0-r1*hfx;
|
e = hxs*((r1-t)/(6.0 - x*t));
|
e = hxs*((r1-t)/(6.0 - x*t));
|
if(k==0) return x - (x*e-hxs); /* c is 0 */
|
if(k==0) return x - (x*e-hxs); /* c is 0 */
|
else {
|
else {
|
e = (x*(e-c)-c);
|
e = (x*(e-c)-c);
|
e -= hxs;
|
e -= hxs;
|
if(k== -1) return 0.5*(x-e)-0.5;
|
if(k== -1) return 0.5*(x-e)-0.5;
|
if(k==1) {
|
if(k==1) {
|
if(x < -0.25) return -2.0*(e-(x+0.5));
|
if(x < -0.25) return -2.0*(e-(x+0.5));
|
else return one+2.0*(x-e);
|
else return one+2.0*(x-e);
|
}
|
}
|
if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */
|
if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */
|
__uint32_t high;
|
__uint32_t high;
|
y = one-(e-x);
|
y = one-(e-x);
|
GET_HIGH_WORD(high,y);
|
GET_HIGH_WORD(high,y);
|
SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
|
SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
|
return y-one;
|
return y-one;
|
}
|
}
|
t = one;
|
t = one;
|
if(k<20) {
|
if(k<20) {
|
__uint32_t high;
|
__uint32_t high;
|
SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */
|
SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */
|
y = t-(e-x);
|
y = t-(e-x);
|
GET_HIGH_WORD(high,y);
|
GET_HIGH_WORD(high,y);
|
SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
|
SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
|
} else {
|
} else {
|
__uint32_t high;
|
__uint32_t high;
|
SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */
|
SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */
|
y = x-(e+t);
|
y = x-(e+t);
|
y += one;
|
y += one;
|
GET_HIGH_WORD(high,y);
|
GET_HIGH_WORD(high,y);
|
SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
|
SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
|
}
|
}
|
}
|
}
|
return y;
|
return y;
|
}
|
}
|
|
|
#endif /* _DOUBLE_IS_32BITS */
|
#endif /* _DOUBLE_IS_32BITS */
|
|
|
