diff options
Diffstat (limited to 'libm/s_expm1.c')
| -rw-r--r-- | libm/s_expm1.c | 48 |
1 files changed, 24 insertions, 24 deletions
diff --git a/libm/s_expm1.c b/libm/s_expm1.c index 5e7e9d84f..aee70835b 100644 --- a/libm/s_expm1.c +++ b/libm/s_expm1.c @@ -5,7 +5,7 @@ * * Developed at SunPro, a Sun Microsystems, Inc. business. * 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. * ==================================================== */ @@ -21,9 +21,9 @@ static char rcsid[] = "$NetBSD: s_expm1.c,v 1.8 1995/05/10 20:47:09 jtc Exp $"; * 1. Argument reduction: * Given x, find r and integer k such that * - * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658 + * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658 * - * Here a correction term c will be computed to compensate + * Here a correction term c will be computed to compensate * the error in r when rounded to a floating-point number. * * 2. Approximating expm1(r) by a special rational function on @@ -36,9 +36,9 @@ static char rcsid[] = "$NetBSD: s_expm1.c,v 1.8 1995/05/10 20:47:09 jtc Exp $"; * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r) * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r)) * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ... - * We use a special Reme algorithm on [0,0.347] to generate - * a polynomial of degree 5 in r*r to approximate R1. The - * maximum error of this polynomial approximation is bounded + * We use a special Reme algorithm on [0,0.347] to generate + * a polynomial of degree 5 in r*r to approximate R1. The + * maximum error of this polynomial approximation is bounded * by 2**-61. In other words, * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5 * where Q1 = -1.6666666666666567384E-2, @@ -49,28 +49,28 @@ static char rcsid[] = "$NetBSD: s_expm1.c,v 1.8 1995/05/10 20:47:09 jtc Exp $"; * (where z=r*r, and the values of Q1 to Q5 are listed below) * with error bounded by * | 5 | -61 - * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2 + * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2 * | | - * - * expm1(r) = exp(r)-1 is then computed by the following - * specific way which minimize the accumulation rounding error: + * + * expm1(r) = exp(r)-1 is then computed by the following + * specific way which minimize the accumulation rounding error: * 2 3 * r r [ 3 - (R1 + R1*r/2) ] * expm1(r) = r + --- + --- * [--------------------] * 2 2 [ 6 - r*(3 - R1*r/2) ] - * + * * To compensate the error in the argument reduction, we use - * expm1(r+c) = expm1(r) + c + expm1(r)*c - * ~ expm1(r) + c + r*c + * expm1(r+c) = expm1(r) + c + expm1(r)*c + * ~ expm1(r) + c + r*c * Thus c+r*c will be added in as the correction terms for - * expm1(r+c). Now rearrange the term to avoid optimization + * expm1(r+c). Now rearrange the term to avoid optimization * screw up: * ( 2 2 ) * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r ) * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- ) * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 ) * ( ) - * + * * = r - E * 3. Scale back to obtain expm1(x): * From step 1, we have @@ -87,7 +87,7 @@ static char rcsid[] = "$NetBSD: s_expm1.c,v 1.8 1995/05/10 20:47:09 jtc Exp $"; * else return 1.0+2.0*(r-E); * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1) * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else - * (vii) return 2^k(1-((E+2^-k)-r)) + * (vii) return 2^k(1-((E+2^-k)-r)) * * Special cases: * expm1(INF) is INF, expm1(NaN) is NaN; @@ -99,12 +99,12 @@ static char rcsid[] = "$NetBSD: s_expm1.c,v 1.8 1995/05/10 20:47:09 jtc Exp $"; * 1 ulp (unit in the last place). * * Misc. info. - * For IEEE double + * 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 + * 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. */ @@ -153,7 +153,7 @@ Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */ if(hx>=0x7ff00000) { u_int32_t low; GET_LOW_WORD(low,x); - if(((hx&0xfffff)|low)!=0) + if(((hx&0xfffff)|low)!=0) return x+x; /* NaN */ else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */ } @@ -166,7 +166,7 @@ Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */ } /* 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(xsb==0) {hi = x - ln2_hi; lo = ln2_lo; k = 1;} @@ -180,10 +180,10 @@ Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */ } 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)); + return x - (t-(huge+x)); } else k = 0; @@ -198,7 +198,7 @@ Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */ e = (x*(e-c)-c); e -= hxs; 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)); else return one+2.0*(x-e); } |