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05b227bdae
With help from Joseph Myers. * sysdeps/ieee754/flt-32/e_j0f.c (__ieee754_y0f): Adjust tinyness cutoff to 2**-13. * sysdeps/ieee754/flt-32/e_j1f.c (__ieee754_y1f): Adjust tinyness cutoff to 2**-25. * sysdeps/ieee754/ldbl-128/e_j0l.c (U0): New constant. ( __ieee754_y0l): Avoid arithmetic underflow when 'x' is very small. * sysdeps/ieee754/ldbl-128/e_j1l.c (__ieee754_y1l): Likewise. * math/libm-test.inc (y0_test): New tests. (y1_test): New tests. * sysdeps/i386/fpu/libm-test-ulps: Update. * sysdeps/x86_64/fpu/libm-test-ulps: Update. * sysdeps/sparc/fpu/libm-test-ulps: Update.
336 lines
10 KiB
C
336 lines
10 KiB
C
/* e_j0f.c -- float version of e_j0.c.
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* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
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*/
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/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunPro, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*/
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#include <math.h>
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#include <math_private.h>
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static float pzerof(float), qzerof(float);
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static const float
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huge = 1e30,
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one = 1.0,
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invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */
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tpi = 6.3661974669e-01, /* 0x3f22f983 */
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/* R0/S0 on [0, 2.00] */
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R02 = 1.5625000000e-02, /* 0x3c800000 */
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R03 = -1.8997929874e-04, /* 0xb947352e */
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R04 = 1.8295404516e-06, /* 0x35f58e88 */
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R05 = -4.6183270541e-09, /* 0xb19eaf3c */
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S01 = 1.5619102865e-02, /* 0x3c7fe744 */
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S02 = 1.1692678527e-04, /* 0x38f53697 */
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S03 = 5.1354652442e-07, /* 0x3509daa6 */
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S04 = 1.1661400734e-09; /* 0x30a045e8 */
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static const float zero = 0.0;
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float
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__ieee754_j0f(float x)
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{
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float z, s,c,ss,cc,r,u,v;
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int32_t hx,ix;
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GET_FLOAT_WORD(hx,x);
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ix = hx&0x7fffffff;
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if(ix>=0x7f800000) return one/(x*x);
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x = fabsf(x);
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if(ix >= 0x40000000) { /* |x| >= 2.0 */
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__sincosf (x, &s, &c);
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ss = s-c;
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cc = s+c;
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if(ix<0x7f000000) { /* make sure x+x not overflow */
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z = -__cosf(x+x);
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if ((s*c)<zero) cc = z/ss;
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else ss = z/cc;
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}
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/*
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* j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
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* y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
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*/
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if(ix>0x48000000) z = (invsqrtpi*cc)/__ieee754_sqrtf(x);
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else {
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u = pzerof(x); v = qzerof(x);
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z = invsqrtpi*(u*cc-v*ss)/__ieee754_sqrtf(x);
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}
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return z;
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}
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if(ix<0x39000000) { /* |x| < 2**-13 */
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math_force_eval(huge+x); /* raise inexact if x != 0 */
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if(ix<0x32000000) return one; /* |x|<2**-27 */
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else return one - (float)0.25*x*x;
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}
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z = x*x;
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r = z*(R02+z*(R03+z*(R04+z*R05)));
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s = one+z*(S01+z*(S02+z*(S03+z*S04)));
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if(ix < 0x3F800000) { /* |x| < 1.00 */
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return one + z*((float)-0.25+(r/s));
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} else {
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u = (float)0.5*x;
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return((one+u)*(one-u)+z*(r/s));
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}
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}
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strong_alias (__ieee754_j0f, __j0f_finite)
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static const float
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u00 = -7.3804296553e-02, /* 0xbd9726b5 */
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u01 = 1.7666645348e-01, /* 0x3e34e80d */
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u02 = -1.3818567619e-02, /* 0xbc626746 */
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u03 = 3.4745343146e-04, /* 0x39b62a69 */
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u04 = -3.8140706238e-06, /* 0xb67ff53c */
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u05 = 1.9559013964e-08, /* 0x32a802ba */
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u06 = -3.9820518410e-11, /* 0xae2f21eb */
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v01 = 1.2730483897e-02, /* 0x3c509385 */
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v02 = 7.6006865129e-05, /* 0x389f65e0 */
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v03 = 2.5915085189e-07, /* 0x348b216c */
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v04 = 4.4111031494e-10; /* 0x2ff280c2 */
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float
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__ieee754_y0f(float x)
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{
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float z, s,c,ss,cc,u,v;
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int32_t hx,ix;
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GET_FLOAT_WORD(hx,x);
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ix = 0x7fffffff&hx;
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/* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0, y0(0) is -inf. */
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if(ix>=0x7f800000) return one/(x+x*x);
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if(ix==0) return -HUGE_VALF+x; /* -inf and overflow exception. */
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if(hx<0) return zero/(zero*x);
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if(ix >= 0x40000000) { /* |x| >= 2.0 */
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/* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
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* where x0 = x-pi/4
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* Better formula:
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* cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
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* = 1/sqrt(2) * (sin(x) + cos(x))
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* sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
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* = 1/sqrt(2) * (sin(x) - cos(x))
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* To avoid cancellation, use
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* sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
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* to compute the worse one.
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*/
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__sincosf (x, &s, &c);
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ss = s-c;
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cc = s+c;
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/*
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* j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
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* y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
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*/
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if(ix<0x7f000000) { /* make sure x+x not overflow */
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z = -__cosf(x+x);
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if ((s*c)<zero) cc = z/ss;
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else ss = z/cc;
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}
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if(ix>0x48000000) z = (invsqrtpi*ss)/__ieee754_sqrtf(x);
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else {
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u = pzerof(x); v = qzerof(x);
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z = invsqrtpi*(u*ss+v*cc)/__ieee754_sqrtf(x);
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}
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return z;
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}
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if(ix<=0x39800000) { /* x < 2**-13 */
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return(u00 + tpi*__ieee754_logf(x));
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}
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z = x*x;
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u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
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v = one+z*(v01+z*(v02+z*(v03+z*v04)));
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return(u/v + tpi*(__ieee754_j0f(x)*__ieee754_logf(x)));
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}
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strong_alias (__ieee754_y0f, __y0f_finite)
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/* The asymptotic expansions of pzero is
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* 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
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* For x >= 2, We approximate pzero by
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* pzero(x) = 1 + (R/S)
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* where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
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* S = 1 + pS0*s^2 + ... + pS4*s^10
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* and
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* | pzero(x)-1-R/S | <= 2 ** ( -60.26)
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*/
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static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
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0.0000000000e+00, /* 0x00000000 */
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-7.0312500000e-02, /* 0xbd900000 */
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-8.0816707611e+00, /* 0xc1014e86 */
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-2.5706311035e+02, /* 0xc3808814 */
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-2.4852163086e+03, /* 0xc51b5376 */
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-5.2530439453e+03, /* 0xc5a4285a */
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};
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static const float pS8[5] = {
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1.1653436279e+02, /* 0x42e91198 */
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3.8337448730e+03, /* 0x456f9beb */
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4.0597855469e+04, /* 0x471e95db */
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1.1675296875e+05, /* 0x47e4087c */
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4.7627726562e+04, /* 0x473a0bba */
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};
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static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
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-1.1412546255e-11, /* 0xad48c58a */
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-7.0312492549e-02, /* 0xbd8fffff */
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-4.1596107483e+00, /* 0xc0851b88 */
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-6.7674766541e+01, /* 0xc287597b */
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-3.3123129272e+02, /* 0xc3a59d9b */
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-3.4643338013e+02, /* 0xc3ad3779 */
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};
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static const float pS5[5] = {
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6.0753936768e+01, /* 0x42730408 */
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1.0512523193e+03, /* 0x44836813 */
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5.9789707031e+03, /* 0x45bad7c4 */
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9.6254453125e+03, /* 0x461665c8 */
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2.4060581055e+03, /* 0x451660ee */
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};
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static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
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-2.5470459075e-09, /* 0xb12f081b */
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-7.0311963558e-02, /* 0xbd8fffb8 */
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-2.4090321064e+00, /* 0xc01a2d95 */
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-2.1965976715e+01, /* 0xc1afba52 */
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-5.8079170227e+01, /* 0xc2685112 */
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-3.1447946548e+01, /* 0xc1fb9565 */
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};
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static const float pS3[5] = {
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3.5856033325e+01, /* 0x420f6c94 */
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3.6151397705e+02, /* 0x43b4c1ca */
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1.1936077881e+03, /* 0x44953373 */
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1.1279968262e+03, /* 0x448cffe6 */
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1.7358093262e+02, /* 0x432d94b8 */
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};
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static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
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-8.8753431271e-08, /* 0xb3be98b7 */
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-7.0303097367e-02, /* 0xbd8ffb12 */
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-1.4507384300e+00, /* 0xbfb9b1cc */
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-7.6356959343e+00, /* 0xc0f4579f */
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-1.1193166733e+01, /* 0xc1331736 */
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-3.2336456776e+00, /* 0xc04ef40d */
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};
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static const float pS2[5] = {
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2.2220300674e+01, /* 0x41b1c32d */
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1.3620678711e+02, /* 0x430834f0 */
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2.7047027588e+02, /* 0x43873c32 */
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1.5387539673e+02, /* 0x4319e01a */
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1.4657617569e+01, /* 0x416a859a */
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};
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static float
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pzerof(float x)
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{
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const float *p,*q;
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float z,r,s;
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int32_t ix;
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GET_FLOAT_WORD(ix,x);
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ix &= 0x7fffffff;
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if(ix>=0x41000000) {p = pR8; q= pS8;}
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else if(ix>=0x40f71c58){p = pR5; q= pS5;}
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else if(ix>=0x4036db68){p = pR3; q= pS3;}
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else if(ix>=0x40000000){p = pR2; q= pS2;}
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z = one/(x*x);
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r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
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s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
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return one+ r/s;
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}
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/* For x >= 8, the asymptotic expansions of qzero is
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* -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
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* We approximate pzero by
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* qzero(x) = s*(-1.25 + (R/S))
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* where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
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* S = 1 + qS0*s^2 + ... + qS5*s^12
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* and
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* | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
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*/
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static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
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0.0000000000e+00, /* 0x00000000 */
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7.3242187500e-02, /* 0x3d960000 */
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1.1768206596e+01, /* 0x413c4a93 */
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5.5767340088e+02, /* 0x440b6b19 */
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8.8591972656e+03, /* 0x460a6cca */
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3.7014625000e+04, /* 0x471096a0 */
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};
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static const float qS8[6] = {
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1.6377603149e+02, /* 0x4323c6aa */
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8.0983447266e+03, /* 0x45fd12c2 */
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1.4253829688e+05, /* 0x480b3293 */
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8.0330925000e+05, /* 0x49441ed4 */
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8.4050156250e+05, /* 0x494d3359 */
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-3.4389928125e+05, /* 0xc8a7eb69 */
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};
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static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
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1.8408595828e-11, /* 0x2da1ec79 */
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7.3242180049e-02, /* 0x3d95ffff */
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5.8356351852e+00, /* 0x40babd86 */
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1.3511157227e+02, /* 0x43071c90 */
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1.0272437744e+03, /* 0x448067cd */
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1.9899779053e+03, /* 0x44f8bf4b */
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};
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static const float qS5[6] = {
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8.2776611328e+01, /* 0x42a58da0 */
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2.0778142090e+03, /* 0x4501dd07 */
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1.8847289062e+04, /* 0x46933e94 */
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5.6751113281e+04, /* 0x475daf1d */
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3.5976753906e+04, /* 0x470c88c1 */
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-5.3543427734e+03, /* 0xc5a752be */
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};
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static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
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4.3774099900e-09, /* 0x3196681b */
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7.3241114616e-02, /* 0x3d95ff70 */
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3.3442313671e+00, /* 0x405607e3 */
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4.2621845245e+01, /* 0x422a7cc5 */
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1.7080809021e+02, /* 0x432acedf */
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1.6673394775e+02, /* 0x4326bbe4 */
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};
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static const float qS3[6] = {
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4.8758872986e+01, /* 0x42430916 */
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7.0968920898e+02, /* 0x44316c1c */
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3.7041481934e+03, /* 0x4567825f */
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6.4604252930e+03, /* 0x45c9e367 */
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2.5163337402e+03, /* 0x451d4557 */
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-1.4924745178e+02, /* 0xc3153f59 */
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};
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static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
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1.5044444979e-07, /* 0x342189db */
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7.3223426938e-02, /* 0x3d95f62a */
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1.9981917143e+00, /* 0x3fffc4bf */
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1.4495602608e+01, /* 0x4167edfd */
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3.1666231155e+01, /* 0x41fd5471 */
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1.6252708435e+01, /* 0x4182058c */
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};
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static const float qS2[6] = {
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3.0365585327e+01, /* 0x41f2ecb8 */
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2.6934811401e+02, /* 0x4386ac8f */
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8.4478375244e+02, /* 0x44533229 */
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8.8293585205e+02, /* 0x445cbbe5 */
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2.1266638184e+02, /* 0x4354aa98 */
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-5.3109550476e+00, /* 0xc0a9f358 */
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};
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static float
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qzerof(float x)
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{
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const float *p,*q;
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float s,r,z;
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int32_t ix;
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GET_FLOAT_WORD(ix,x);
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ix &= 0x7fffffff;
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if(ix>=0x41000000) {p = qR8; q= qS8;}
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else if(ix>=0x40f71c58){p = qR5; q= qS5;}
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else if(ix>=0x4036db68){p = qR3; q= qS3;}
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else if(ix>=0x40000000){p = qR2; q= qS2;}
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z = one/(x*x);
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r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
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s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
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return (-(float).125 + r/s)/x;
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}
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