TonBERRY/micropython
3
0
Fork 0
Code Pull Requests Activity

lib: Add lib and libm, moving current files from stmhal.

Browse Source 
Top-level lib directory is for standard C libraries that we want to
provide our own versions of (for efficiency and stand-alone reasons).
It currently has libm in it for math functions.

Also add atanf and atan2f, which addresses issue #837.
This commit is contained in:
Damien George
2014-08-29 22:42:26 +01:00
parent 17ae2395c2
commit 8707ea3421
6 changed files with 250 additions and 57 deletions
Download Patch File Download Diff File Expand all files Collapse all files

89
lib/libm/atan2f.c Normal file
View File

@@ -0,0 +1,89 @@
/*****************************************************************************/
/*****************************************************************************/
// atan2f from musl-0.9.15
/*****************************************************************************/
/*****************************************************************************/
/* origin: FreeBSD /usr/src/lib/msun/src/e_atan2f.c */
/*
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
*/
/*
* ====================================================
* 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.
* ====================================================
*/
#include "libm.h"
static const float
pi = 3.1415927410e+00, /* 0x40490fdb */
pi_lo = -8.7422776573e-08; /* 0xb3bbbd2e */
float atan2f(float y, float x)
{
float z;
uint32_t m,ix,iy;
if (isnan(x) || isnan(y))
return x+y;
GET_FLOAT_WORD(ix, x);
GET_FLOAT_WORD(iy, y);
if (ix == 0x3f800000) /* x=1.0 */
return atanf(y);
m = ((iy>>31)&1) | ((ix>>30)&2); /* 2*sign(x)+sign(y) */
ix &= 0x7fffffff;
iy &= 0x7fffffff;
/* when y = 0 */
if (iy == 0) {
switch (m) {
case 0:
case 1: return y; /* atan(+-0,+anything)=+-0 */
case 2: return pi; /* atan(+0,-anything) = pi */
case 3: return -pi; /* atan(-0,-anything) =-pi */
}
}
/* when x = 0 */
if (ix == 0)
return m&1 ? -pi/2 : pi/2;
/* when x is INF */
if (ix == 0x7f800000) {
if (iy == 0x7f800000) {
switch (m) {
case 0: return pi/4; /* atan(+INF,+INF) */
case 1: return -pi/4; /* atan(-INF,+INF) */
case 2: return 3*pi/4; /*atan(+INF,-INF)*/
case 3: return -3*pi/4; /*atan(-INF,-INF)*/
}
} else {
switch (m) {
case 0: return 0.0f; /* atan(+...,+INF) */
case 1: return -0.0f; /* atan(-...,+INF) */
case 2: return pi; /* atan(+...,-INF) */
case 3: return -pi; /* atan(-...,-INF) */
}
}
}
/* |y/x| > 0x1p26 */
if (ix+(26<<23) < iy || iy == 0x7f800000)
return m&1 ? -pi/2 : pi/2;
/* z = atan(|y/x|) with correct underflow */
if ((m&2) && iy+(26<<23) < ix) /*|y/x| < 0x1p-26, x < 0 */
z = 0.0;
else
z = atanf(fabsf(y/x));
switch (m) {
case 0: return z; /* atan(+,+) */
case 1: return -z; /* atan(-,+) */
case 2: return pi - (z-pi_lo); /* atan(+,-) */
default: /* case 3 */
return (z-pi_lo) - pi; /* atan(-,-) */
}
}

100
lib/libm/atanf.c Normal file
View File

@@ -0,0 +1,100 @@
/*****************************************************************************/
/*****************************************************************************/
// atanf from musl-0.9.15
/*****************************************************************************/
/*****************************************************************************/
/* origin: FreeBSD /usr/src/lib/msun/src/s_atanf.c */
/*
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
*/
/*
* ====================================================
* 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.
* ====================================================
*/
#include "libm.h"
static const float atanhi[] = {
4.6364760399e-01, /* atan(0.5)hi 0x3eed6338 */
7.8539812565e-01, /* atan(1.0)hi 0x3f490fda */
9.8279368877e-01, /* atan(1.5)hi 0x3f7b985e */
1.5707962513e+00, /* atan(inf)hi 0x3fc90fda */
};
static const float atanlo[] = {
5.0121582440e-09, /* atan(0.5)lo 0x31ac3769 */
3.7748947079e-08, /* atan(1.0)lo 0x33222168 */
3.4473217170e-08, /* atan(1.5)lo 0x33140fb4 */
7.5497894159e-08, /* atan(inf)lo 0x33a22168 */
};
static const float aT[] = {
3.3333328366e-01,
-1.9999158382e-01,
1.4253635705e-01,
-1.0648017377e-01,
6.1687607318e-02,
};
float atanf(float x)
{
float_t w,s1,s2,z;
uint32_t ix,sign;
int id;
GET_FLOAT_WORD(ix, x);
sign = ix>>31;
ix &= 0x7fffffff;
if (ix >= 0x4c800000) { /* if |x| >= 2**26 */
if (isnan(x))
return x;
z = atanhi[3] + 0x1p-120f;
return sign ? -z : z;
}
if (ix < 0x3ee00000) { /* |x| < 0.4375 */
if (ix < 0x39800000) { /* |x| < 2**-12 */
if (ix < 0x00800000)
/* raise underflow for subnormal x */
FORCE_EVAL(x*x);
return x;
}
id = -1;
} else {
x = fabsf(x);
if (ix < 0x3f980000) { /* |x| < 1.1875 */
if (ix < 0x3f300000) { /* 7/16 <= |x| < 11/16 */
id = 0;
x = (2.0f*x - 1.0f)/(2.0f + x);
} else { /* 11/16 <= |x| < 19/16 */
id = 1;
x = (x - 1.0f)/(x + 1.0f);
}
} else {
if (ix < 0x401c0000) { /* |x| < 2.4375 */
id = 2;
x = (x - 1.5f)/(1.0f + 1.5f*x);
} else { /* 2.4375 <= |x| < 2**26 */
id = 3;
x = -1.0f/x;
}
}
}
/* end of argument reduction */
z = x*x;
w = z*z;
/* break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly */
s1 = z*(aT[0]+w*(aT[2]+w*aT[4]));
s2 = w*(aT[1]+w*aT[3]);
if (id < 0)
return x - x*(s1+s2);
z = atanhi[id] - ((x*(s1+s2) - atanlo[id]) - x);
return sign ? -z : z;
}

52
lib/libm/libm.h Normal file
View File

@@ -0,0 +1,52 @@
/*****************************************************************************/
/*****************************************************************************/
// portions extracted from musl-0.9.15 libm.h
/*****************************************************************************/
/*****************************************************************************/
/* origin: FreeBSD /usr/src/lib/msun/src/math_private.h */
/*
* ====================================================
* 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.
* ====================================================
*/
#include <stdint.h>
#include <math.h>
#define FORCE_EVAL(x) do { \
if (sizeof(x) == sizeof(float)) { \
volatile float __x; \
__x = (x); \
(void)__x; \
} else if (sizeof(x) == sizeof(double)) { \
volatile double __x; \
__x = (x); \
(void)__x; \
} else { \
volatile long double __x; \
__x = (x); \
(void)__x; \
} \
} while(0)
/* Get a 32 bit int from a float. */
#define GET_FLOAT_WORD(w,d) \
do { \
union {float f; uint32_t i;} __u; \
__u.f = (d); \
(w) = __u.i; \
} while (0)
/* Set a float from a 32 bit int. */
#define SET_FLOAT_WORD(d,w) \
do { \
union {float f; uint32_t i;} __u; \
__u.i = (w); \
(d) = __u.f; \
} while (0)

886
lib/libm/math.c Normal file
View File

@@ -0,0 +1,886 @@
/*
* This file is part of the Micro Python project, http://micropython.org/
*
* The MIT License (MIT)
*
* Copyright (c) 2013, 2014 Damien P. George
*
* Permission is hereby granted, free of charge, to any person obtaining a copy
* of this software and associated documentation files (the "Software"), to deal
* in the Software without restriction, including without limitation the rights
* to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
* copies of the Software, and to permit persons to whom the Software is
* furnished to do so, subject to the following conditions:
*
* The above copyright notice and this permission notice shall be included in
* all copies or substantial portions of the Software.
*
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
* IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
* FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
* AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
* LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
* OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
* THE SOFTWARE.
*/
#include "libm.h"
typedef float float_t;
typedef union {
float f;
struct {
uint64_t m : 23;
uint64_t e : 8;
uint64_t s : 1;
};
} float_s_t;
typedef union {
double d;
struct {
uint64_t m : 52;
uint64_t e : 11;
uint64_t s : 1;
};
} double_s_t;
double __attribute__((pcs("aapcs"))) __aeabi_i2d(int32_t x) {
return (float)x;
}
double __attribute__((pcs("aapcs"))) __aeabi_f2d(float x) {
float_s_t fx={0};
double_s_t dx={0};
fx.f = x;
dx.s = (fx.s);
dx.e = (fx.e-127+1023) & 0x7FF;
dx.m = fx.m;
dx.m <<=(52-23); // left justify
return dx.d;
}
float __attribute__((pcs("aapcs"))) __aeabi_d2f(double x) {
float_s_t fx={0};
double_s_t dx={0};
dx.d = x;
fx.s = (dx.s);
fx.e = (dx.e-1023+127) & 0xFF;
fx.m = (dx.m>>(52-23)); // right justify
return fx.f;
}
double __aeabi_dmul(double x , double y) {
return 0.0;
}
float sqrtf(float x) {
asm volatile (
"vsqrt.f32 %[r], %[x]\n"
: [r] "=t" (x)
: [x] "t" (x));
return x;
}
#ifndef NDEBUG
float copysignf(float x, float y) {
float_s_t fx={.f = x};
float_s_t fy={.f = y};
// copy sign bit;
fx.s = fy.s;
return fx.f;
}
#endif
// some compilers define log2f in terms of logf
#ifdef log2f
#undef log2f
#endif
// some compilers have _M_LN2 defined in math.h, some don't
#ifndef _M_LN2
#define _M_LN2 (0.69314718055994530942)
#endif
float log2f(float x) { return logf(x) / (float)_M_LN2; }
static const float _M_LN10 = 2.30258509299404; // 0x40135d8e
float log10f(float x) { return logf(x) / (float)_M_LN10; }
float tanhf(float x) { return sinhf(x) / coshf(x); }
// TODO we need import these functions from some library (eg musl or newlib)
float acoshf(float x) { return 0.0; }
float asinhf(float x) { return 0.0; }
float atanhf(float x) { return 0.0; }
float acosf(float x) { return 0.0; }
float asinf(float x) { return 0.0; }
float fmodf(float x, float y) { return 0.0; }
float tgammaf(float x) { return 0.0; }
float lgammaf(float x) { return 0.0; }
float erff(float x) { return 0.0; }
float erfcf(float x) { return 0.0; }
float modff(float x, float *y) { return 0.0; }
float frexpf(float x, int *exp) { return 0.0; }
float ldexpf(float x, int exp) { return 0.0; }
/*****************************************************************************/
/*****************************************************************************/
// __fpclassifyf from musl-0.9.15
/*****************************************************************************/
/*****************************************************************************/
int __fpclassifyf(float x)
{
union {float f; uint32_t i;} u = {x};
int e = u.i>>23 & 0xff;
if (!e) return u.i<<1 ? FP_SUBNORMAL : FP_ZERO;
if (e==0xff) return u.i<<9 ? FP_NAN : FP_INFINITE;
return FP_NORMAL;
}
/*****************************************************************************/
/*****************************************************************************/
// scalbnf from musl-0.9.15
/*****************************************************************************/
/*****************************************************************************/
float scalbnf(float x, int n)
{
union {float f; uint32_t i;} u;
float_t y = x;
if (n > 127) {
y *= 0x1p127f;
n -= 127;
if (n > 127) {
y *= 0x1p127f;
n -= 127;
if (n > 127)
n = 127;
}
} else if (n < -126) {
y *= 0x1p-126f;
n += 126;
if (n < -126) {
y *= 0x1p-126f;
n += 126;
if (n < -126)
n = -126;
}
}
u.i = (uint32_t)(0x7f+n)<<23;
x = y * u.f;
return x;
}
/*****************************************************************************/
/*****************************************************************************/
// powf from musl-0.9.15
/*****************************************************************************/
/*****************************************************************************/
/* origin: FreeBSD /usr/src/lib/msun/src/e_powf.c */
/*
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
*/
/*
* ====================================================
* 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.
* ====================================================
*/
static const float
bp[] = {1.0, 1.5,},
dp_h[] = { 0.0, 5.84960938e-01,}, /* 0x3f15c000 */
dp_l[] = { 0.0, 1.56322085e-06,}, /* 0x35d1cfdc */
two24 = 16777216.0, /* 0x4b800000 */
huge = 1.0e30,
tiny = 1.0e-30,
/* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
L1 = 6.0000002384e-01, /* 0x3f19999a */
L2 = 4.2857143283e-01, /* 0x3edb6db7 */
L3 = 3.3333334327e-01, /* 0x3eaaaaab */
L4 = 2.7272811532e-01, /* 0x3e8ba305 */
L5 = 2.3066075146e-01, /* 0x3e6c3255 */
L6 = 2.0697501302e-01, /* 0x3e53f142 */
P1 = 1.6666667163e-01, /* 0x3e2aaaab */
P2 = -2.7777778450e-03, /* 0xbb360b61 */
P3 = 6.6137559770e-05, /* 0x388ab355 */
P4 = -1.6533901999e-06, /* 0xb5ddea0e */
P5 = 4.1381369442e-08, /* 0x3331bb4c */
lg2 = 6.9314718246e-01, /* 0x3f317218 */
lg2_h = 6.93145752e-01, /* 0x3f317200 */
lg2_l = 1.42860654e-06, /* 0x35bfbe8c */
ovt = 4.2995665694e-08, /* -(128-log2(ovfl+.5ulp)) */
cp = 9.6179670095e-01, /* 0x3f76384f =2/(3ln2) */
cp_h = 9.6191406250e-01, /* 0x3f764000 =12b cp */
cp_l = -1.1736857402e-04, /* 0xb8f623c6 =tail of cp_h */
ivln2 = 1.4426950216e+00, /* 0x3fb8aa3b =1/ln2 */
ivln2_h = 1.4426879883e+00, /* 0x3fb8aa00 =16b 1/ln2*/
ivln2_l = 7.0526075433e-06; /* 0x36eca570 =1/ln2 tail*/
float powf(float x, float y)
{
float z,ax,z_h,z_l,p_h,p_l;
float y1,t1,t2,r,s,sn,t,u,v,w;
int32_t i,j,k,yisint,n;
int32_t hx,hy,ix,iy,is;
GET_FLOAT_WORD(hx, x);
GET_FLOAT_WORD(hy, y);
ix = hx & 0x7fffffff;
iy = hy & 0x7fffffff;
/* x**0 = 1, even if x is NaN */
if (iy == 0)
return 1.0f;
/* 1**y = 1, even if y is NaN */
if (hx == 0x3f800000)
return 1.0f;
/* NaN if either arg is NaN */
if (ix > 0x7f800000 || iy > 0x7f800000)
return x + y;
/* determine if y is an odd int when x < 0
* yisint = 0 ... y is not an integer
* yisint = 1 ... y is an odd int
* yisint = 2 ... y is an even int
*/
yisint = 0;
if (hx < 0) {
if (iy >= 0x4b800000)
yisint = 2; /* even integer y */
else if (iy >= 0x3f800000) {
k = (iy>>23) - 0x7f; /* exponent */
j = iy>>(23-k);
if ((j<<(23-k)) == iy)
yisint = 2 - (j & 1);
}
}
/* special value of y */
if (iy == 0x7f800000) { /* y is +-inf */
if (ix == 0x3f800000) /* (-1)**+-inf is 1 */
return 1.0f;
else if (ix > 0x3f800000) /* (|x|>1)**+-inf = inf,0 */
return hy >= 0 ? y : 0.0f;
else if (ix != 0) /* (|x|<1)**+-inf = 0,inf if x!=0 */
return hy >= 0 ? 0.0f: -y;
}
if (iy == 0x3f800000) /* y is +-1 */
return hy >= 0 ? x : 1.0f/x;
if (hy == 0x40000000) /* y is 2 */
return x*x;
if (hy == 0x3f000000) { /* y is 0.5 */
if (hx >= 0) /* x >= +0 */
return sqrtf(x);
}
ax = fabsf(x);
/* special value of x */
if (ix == 0x7f800000 || ix == 0 || ix == 0x3f800000) { /* x is +-0,+-inf,+-1 */
z = ax;
if (hy < 0) /* z = (1/|x|) */
z = 1.0f/z;
if (hx < 0) {
if (((ix-0x3f800000)|yisint) == 0) {
z = (z-z)/(z-z); /* (-1)**non-int is NaN */
} else if (yisint == 1)
z = -z; /* (x<0)**odd = -(|x|**odd) */
}
return z;
}
sn = 1.0f; /* sign of result */
if (hx < 0) {
if (yisint == 0) /* (x<0)**(non-int) is NaN */
return (x-x)/(x-x);
if (yisint == 1) /* (x<0)**(odd int) */
sn = -1.0f;
}
/* |y| is huge */
if (iy > 0x4d000000) { /* if |y| > 2**27 */
/* over/underflow if x is not close to one */
if (ix < 0x3f7ffff8)
return hy < 0 ? sn*huge*huge : sn*tiny*tiny;
if (ix > 0x3f800007)
return hy > 0 ? sn*huge*huge : sn*tiny*tiny;
/* now |1-x| is tiny <= 2**-20, suffice to compute
log(x) by x-x^2/2+x^3/3-x^4/4 */
t = ax - 1; /* t has 20 trailing zeros */
w = (t*t)*(0.5f - t*(0.333333333333f - t*0.25f));
u = ivln2_h*t; /* ivln2_h has 16 sig. bits */
v = t*ivln2_l - w*ivln2;
t1 = u + v;
GET_FLOAT_WORD(is, t1);
SET_FLOAT_WORD(t1, is & 0xfffff000);
t2 = v - (t1-u);
} else {
float s2,s_h,s_l,t_h,t_l;
n = 0;
/* take care subnormal number */
if (ix < 0x00800000) {
ax *= two24;
n -= 24;
GET_FLOAT_WORD(ix, ax);
}
n += ((ix)>>23) - 0x7f;
j = ix & 0x007fffff;
/* determine interval */
ix = j | 0x3f800000; /* normalize ix */
if (j <= 0x1cc471) /* |x|<sqrt(3/2) */
k = 0;
else if (j < 0x5db3d7) /* |x|<sqrt(3) */
k = 1;
else {
k = 0;
n += 1;
ix -= 0x00800000;
}
SET_FLOAT_WORD(ax, ix);
/* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
u = ax - bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
v = 1.0f/(ax+bp[k]);
s = u*v;
s_h = s;
GET_FLOAT_WORD(is, s_h);
SET_FLOAT_WORD(s_h, is & 0xfffff000);
/* t_h=ax+bp[k] High */
is = ((ix>>1) & 0xfffff000) | 0x20000000;
SET_FLOAT_WORD(t_h, is + 0x00400000 + (k<<21));
t_l = ax - (t_h - bp[k]);
s_l = v*((u - s_h*t_h) - s_h*t_l);
/* compute log(ax) */
s2 = s*s;
r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
r += s_l*(s_h+s);
s2 = s_h*s_h;
t_h = 3.0f + s2 + r;
GET_FLOAT_WORD(is, t_h);
SET_FLOAT_WORD(t_h, is & 0xfffff000);
t_l = r - ((t_h - 3.0f) - s2);
/* u+v = s*(1+...) */
u = s_h*t_h;
v = s_l*t_h + t_l*s;
/* 2/(3log2)*(s+...) */
p_h = u + v;
GET_FLOAT_WORD(is, p_h);
SET_FLOAT_WORD(p_h, is & 0xfffff000);
p_l = v - (p_h - u);
z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
z_l = cp_l*p_h + p_l*cp+dp_l[k];
/* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
t = (float)n;
t1 = (((z_h + z_l) + dp_h[k]) + t);
GET_FLOAT_WORD(is, t1);
SET_FLOAT_WORD(t1, is & 0xfffff000);
t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
}
/* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
GET_FLOAT_WORD(is, y);
SET_FLOAT_WORD(y1, is & 0xfffff000);
p_l = (y-y1)*t1 + y*t2;
p_h = y1*t1;
z = p_l + p_h;
GET_FLOAT_WORD(j, z);
if (j > 0x43000000) /* if z > 128 */
return sn*huge*huge; /* overflow */
else if (j == 0x43000000) { /* if z == 128 */
if (p_l + ovt > z - p_h)
return sn*huge*huge; /* overflow */
} else if ((j&0x7fffffff) > 0x43160000) /* z < -150 */ // FIXME: check should be (uint32_t)j > 0xc3160000
return sn*tiny*tiny; /* underflow */
else if (j == 0xc3160000) { /* z == -150 */
if (p_l <= z-p_h)
return sn*tiny*tiny; /* underflow */
}
/*
* compute 2**(p_h+p_l)
*/
i = j & 0x7fffffff;
k = (i>>23) - 0x7f;
n = 0;
if (i > 0x3f000000) { /* if |z| > 0.5, set n = [z+0.5] */
n = j + (0x00800000>>(k+1));
k = ((n&0x7fffffff)>>23) - 0x7f; /* new k for n */
SET_FLOAT_WORD(t, n & ~(0x007fffff>>k));
n = ((n&0x007fffff)|0x00800000)>>(23-k);
if (j < 0)
n = -n;
p_h -= t;
}
t = p_l + p_h;
GET_FLOAT_WORD(is, t);
SET_FLOAT_WORD(t, is & 0xffff8000);
u = t*lg2_h;
v = (p_l-(t-p_h))*lg2 + t*lg2_l;
z = u + v;
w = v - (z - u);
t = z*z;
t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
r = (z*t1)/(t1-2.0f) - (w+z*w);
z = 1.0f - (r - z);
GET_FLOAT_WORD(j, z);
j += n<<23;
if ((j>>23) <= 0) /* subnormal output */
z = scalbnf(z, n);
else
SET_FLOAT_WORD(z, j);
return sn*z;
}
/*****************************************************************************/
/*****************************************************************************/
// expf from musl-0.9.15
/*****************************************************************************/
/*****************************************************************************/
/* origin: FreeBSD /usr/src/lib/msun/src/e_expf.c */
/*
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
*/
/*
* ====================================================
* 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.
* ====================================================
*/
static const float
half[2] = {0.5,-0.5},
ln2hi = 6.9314575195e-1f, /* 0x3f317200 */
ln2lo = 1.4286067653e-6f, /* 0x35bfbe8e */
invln2 = 1.4426950216e+0f, /* 0x3fb8aa3b */
/*
* Domain [-0.34568, 0.34568], range ~[-4.278e-9, 4.447e-9]:
* |x*(exp(x)+1)/(exp(x)-1) - p(x)| < 2**-27.74
*/
expf_P1 = 1.6666625440e-1f, /* 0xaaaa8f.0p-26 */
expf_P2 = -2.7667332906e-3f; /* -0xb55215.0p-32 */
float expf(float x)
{
float_t hi, lo, c, xx, y;
int k, sign;
uint32_t hx;
GET_FLOAT_WORD(hx, x);
sign = hx >> 31; /* sign bit of x */
hx &= 0x7fffffff; /* high word of |x| */
/* special cases */
if (hx >= 0x42aeac50) { /* if |x| >= -87.33655f or NaN */
if (hx >= 0x42b17218 && !sign) { /* x >= 88.722839f */
/* overflow */
x *= 0x1p127f;
return x;
}
if (sign) {
/* underflow */
FORCE_EVAL(-0x1p-149f/x);
if (hx >= 0x42cff1b5) /* x <= -103.972084f */
return 0;
}
}
/* argument reduction */
if (hx > 0x3eb17218) { /* if |x| > 0.5 ln2 */
if (hx > 0x3f851592) /* if |x| > 1.5 ln2 */
k = invln2*x + half[sign];
else
k = 1 - sign - sign;
hi = x - k*ln2hi; /* k*ln2hi is exact here */
lo = k*ln2lo;
x = hi - lo;
} else if (hx > 0x39000000) { /* |x| > 2**-14 */
k = 0;
hi = x;
lo = 0;
} else {
/* raise inexact */
FORCE_EVAL(0x1p127f + x);
return 1 + x;
}
/* x is now in primary range */
xx = x*x;
c = x - xx*(expf_P1+xx*expf_P2);
y = 1 + (x*c/(2-c) - lo + hi);
if (k == 0)
return y;
return scalbnf(y, k);
}
/*****************************************************************************/
/*****************************************************************************/
// expm1f from musl-0.9.15
/*****************************************************************************/
/*****************************************************************************/
/* origin: FreeBSD /usr/src/lib/msun/src/s_expm1f.c */
/*
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
*/
/*
* ====================================================
* 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.
* ====================================================
*/
static const float
o_threshold = 8.8721679688e+01, /* 0x42b17180 */
ln2_hi = 6.9313812256e-01, /* 0x3f317180 */
ln2_lo = 9.0580006145e-06, /* 0x3717f7d1 */
//invln2 = 1.4426950216e+00, /* 0x3fb8aa3b */
/*
* Domain [-0.34568, 0.34568], range ~[-6.694e-10, 6.696e-10]:
* |6 / x * (1 + 2 * (1 / (exp(x) - 1) - 1 / x)) - q(x)| < 2**-30.04
* Scaled coefficients: Qn_here = 2**n * Qn_for_q (see s_expm1.c):
*/
Q1 = -3.3333212137e-2, /* -0x888868.0p-28 */
Q2 = 1.5807170421e-3; /* 0xcf3010.0p-33 */
float expm1f(float x)
{
float_t y,hi,lo,c,t,e,hxs,hfx,r1,twopk;
union {float f; uint32_t i;} u = {x};
uint32_t hx = u.i & 0x7fffffff;
int k, sign = u.i >> 31;
/* filter out huge and non-finite argument */
if (hx >= 0x4195b844) { /* if |x|>=27*ln2 */
if (hx > 0x7f800000) /* NaN */
return x;
if (sign)
return -1;
if (x > o_threshold) {
x *= 0x1p127f;
return x;
}
}
/* argument reduction */
if (hx > 0x3eb17218) { /* if |x| > 0.5 ln2 */
if (hx < 0x3F851592) { /* 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 ? -0.5f : 0.5f);
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 < 0x33000000) { /* when |x|<2**-25, return x */
if (hx < 0x00800000)
FORCE_EVAL(x*x);
return x;
} else
k = 0;
/* x is now in primary range */
hfx = 0.5f*x;
hxs = x*hfx;
r1 = 1.0f+hxs*(Q1+hxs*Q2);
t = 3.0f - r1*hfx;
e = hxs*((r1-t)/(6.0f - 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 0.5f*(x-e) - 0.5f;
if (k == 1) {
if (x < -0.25f)
return -2.0f*(e-(x+0.5f));
return 1.0f + 2.0f*(x-e);
}
u.i = (0x7f+k)<<23; /* 2^k */
twopk = u.f;
if (k < 0 || k > 56) { /* suffice to return exp(x)-1 */
y = x - e + 1.0f;
if (k == 128)
y = y*2.0f*0x1p127f;
else
y = y*twopk;
return y - 1.0f;
}
u.i = (0x7f-k)<<23; /* 2^-k */
if (k < 23)
y = (x-e+(1-u.f))*twopk;
else
y = (x-(e+u.f)+1)*twopk;
return y;
}
/*****************************************************************************/
/*****************************************************************************/
// __expo2f from musl-0.9.15
/*****************************************************************************/
/*****************************************************************************/
/* k is such that k*ln2 has minimal relative error and x - kln2 > log(FLT_MIN) */
static const int k = 235;
static const float kln2 = 0x1.45c778p+7f;
/* expf(x)/2 for x >= log(FLT_MAX), slightly better than 0.5f*expf(x/2)*expf(x/2) */
float __expo2f(float x)
{
float scale;
/* note that k is odd and scale*scale overflows */
SET_FLOAT_WORD(scale, (uint32_t)(0x7f + k/2) << 23);
/* exp(x - k ln2) * 2**(k-1) */
return expf(x - kln2) * scale * scale;
}
/*****************************************************************************/
/*****************************************************************************/
// logf from musl-0.9.15
/*****************************************************************************/
/*****************************************************************************/
/* origin: FreeBSD /usr/src/lib/msun/src/e_logf.c */
/*
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
*/
/*
* ====================================================
* 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.
* ====================================================
*/
static const float
/* |(log(1+s)-log(1-s))/s - Lg(s)| < 2**-34.24 (~[-4.95e-11, 4.97e-11]). */
Lg1 = 0xaaaaaa.0p-24, /* 0.66666662693 */
Lg2 = 0xccce13.0p-25, /* 0.40000972152 */
Lg3 = 0x91e9ee.0p-25, /* 0.28498786688 */
Lg4 = 0xf89e26.0p-26; /* 0.24279078841 */
float logf(float x)
{
union {float f; uint32_t i;} u = {x};
float_t hfsq,f,s,z,R,w,t1,t2,dk;
uint32_t ix;
int k;
ix = u.i;
k = 0;
if (ix < 0x00800000 || ix>>31) { /* x < 2**-126 */
if (ix<<1 == 0)
return -1/(x*x); /* log(+-0)=-inf */
if (ix>>31)
return (x-x)/0.0f; /* log(-#) = NaN */
/* subnormal number, scale up x */
k -= 25;
x *= 0x1p25f;
u.f = x;
ix = u.i;
} else if (ix >= 0x7f800000) {
return x;
} else if (ix == 0x3f800000)
return 0;
/* reduce x into [sqrt(2)/2, sqrt(2)] */
ix += 0x3f800000 - 0x3f3504f3;
k += (int)(ix>>23) - 0x7f;
ix = (ix&0x007fffff) + 0x3f3504f3;
u.i = ix;
x = u.f;
f = x - 1.0f;
s = f/(2.0f + f);
z = s*s;
w = z*z;
t1= w*(Lg2+w*Lg4);
t2= z*(Lg1+w*Lg3);
R = t2 + t1;
hfsq = 0.5f*f*f;
dk = k;
return s*(hfsq+R) + dk*ln2_lo - hfsq + f + dk*ln2_hi;
}
/*****************************************************************************/
/*****************************************************************************/
// coshf from musl-0.9.15
/*****************************************************************************/
/*****************************************************************************/
float coshf(float x)
{
union {float f; uint32_t i;} u = {.f = x};
uint32_t w;
float t;
/* |x| */
u.i &= 0x7fffffff;
x = u.f;
w = u.i;
/* |x| < log(2) */
if (w < 0x3f317217) {
if (w < 0x3f800000 - (12<<23)) {
FORCE_EVAL(x + 0x1p120f);
return 1;
}
t = expm1f(x);
return 1 + t*t/(2*(1+t));
}
/* |x| < log(FLT_MAX) */
if (w < 0x42b17217) {
t = expf(x);
return 0.5f*(t + 1/t);
}
/* |x| > log(FLT_MAX) or nan */
t = __expo2f(x);
return t;
}
/*****************************************************************************/
/*****************************************************************************/
// sinhf from musl-0.9.15
/*****************************************************************************/
/*****************************************************************************/
float sinhf(float x)
{
union {float f; uint32_t i;} u = {.f = x};
uint32_t w;
float t, h, absx;
h = 0.5;
if (u.i >> 31)
h = -h;
/* |x| */
u.i &= 0x7fffffff;
absx = u.f;
w = u.i;
/* |x| < log(FLT_MAX) */
if (w < 0x42b17217) {
t = expm1f(absx);
if (w < 0x3f800000) {
if (w < 0x3f800000 - (12<<23))
return x;
return h*(2*t - t*t/(t+1));
}
return h*(t + t/(t+1));
}
/* |x| > logf(FLT_MAX) or nan */
t = 2*h*__expo2f(absx);
return t;
}
/*****************************************************************************/
/*****************************************************************************/
// ceilf, floorf and truncf from musl-0.9.15
/*****************************************************************************/
/*****************************************************************************/
float ceilf(float x)
{
union {float f; uint32_t i;} u = {x};
int e = (int)(u.i >> 23 & 0xff) - 0x7f;
uint32_t m;
if (e >= 23)
return x;
if (e >= 0) {
m = 0x007fffff >> e;
if ((u.i & m) == 0)
return x;
FORCE_EVAL(x + 0x1p120f);
if (u.i >> 31 == 0)
u.i += m;
u.i &= ~m;
} else {
FORCE_EVAL(x + 0x1p120f);
if (u.i >> 31)
u.f = -0.0;
else if (u.i << 1)
u.f = 1.0;
}
return u.f;
}
float floorf(float x)
{
union {float f; uint32_t i;} u = {x};
int e = (int)(u.i >> 23 & 0xff) - 0x7f;
uint32_t m;
if (e >= 23)
return x;
if (e >= 0) {
m = 0x007fffff >> e;
if ((u.i & m) == 0)
return x;
FORCE_EVAL(x + 0x1p120f);
if (u.i >> 31)
u.i += m;
u.i &= ~m;
} else {
FORCE_EVAL(x + 0x1p120f);
if (u.i >> 31 == 0)
u.i = 0;
else if (u.i << 1)
u.f = -1.0;
}
return u.f;
}
float truncf(float x)
{
union {float f; uint32_t i;} u = {x};
int e = (int)(u.i >> 23 & 0xff) - 0x7f + 9;
uint32_t m;
if (e >= 23 + 9)
return x;
if (e < 9)
e = 1;
m = -1U >> e;
if ((u.i & m) == 0)
return x;
FORCE_EVAL(x + 0x1p120f);
u.i &= ~m;
return u.f;
}

627
lib/libm/mathsincos.c Normal file
View File

@@ -0,0 +1,627 @@
/*
* This file is part of the Micro Python project, http://micropython.org/
*
* These math functions are taken from newlib 2.1.0, the newlib/libm directory.
* Appropriate copyright headers are reproduced below.
*
* The MIT License (MIT)
*
* Copyright (c) 2013, 2014 Damien P. George
*
* Permission is hereby granted, free of charge, to any person obtaining a copy
* of this software and associated documentation files (the "Software"), to deal
* in the Software without restriction, including without limitation the rights
* to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
* copies of the Software, and to permit persons to whom the Software is
* furnished to do so, subject to the following conditions:
*
* The above copyright notice and this permission notice shall be included in
* all copies or substantial portions of the Software.
*
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
* IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
* FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
* AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
* LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
* OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
* THE SOFTWARE.
*/
#include <stdint.h>
#include <math.h>
/* @(#)fdlibm.h 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.
* ====================================================
*/
#define FLT_UWORD_IS_FINITE(x) ((x)<0x7f800000L)
/* A union which permits us to convert between a float and a 32 bit
int. */
typedef union
{
float value;
__uint32_t word;
} ieee_float_shape_type;
/* Get a 32 bit int from a float. */
#define GET_FLOAT_WORD(i,d) \
do { \
ieee_float_shape_type gf_u; \
gf_u.value = (d); \
(i) = gf_u.word; \
} while (0)
/* Set a float from a 32 bit int. */
#define SET_FLOAT_WORD(d,i) \
do { \
ieee_float_shape_type sf_u; \
sf_u.word = (i); \
(d) = sf_u.value; \
} while (0)
/* kf_rem_pio2.c -- float version of k_rem_pio2.c
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
*/
/*
* ====================================================
* 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.
* ====================================================
*/
/* In the float version, the input parameter x contains 8 bit
integers, not 24 bit integers. 113 bit precision is not supported. */
#ifdef __STDC__
static const int init_jk[] = {4,7,9}; /* initial value for jk */
#else
static int init_jk[] = {4,7,9};
#endif
#ifdef __STDC__
static const float PIo2[] = {
#else
static float PIo2[] = {
#endif
1.5703125000e+00, /* 0x3fc90000 */
4.5776367188e-04, /* 0x39f00000 */
2.5987625122e-05, /* 0x37da0000 */
7.5437128544e-08, /* 0x33a20000 */
6.0026650317e-11, /* 0x2e840000 */
7.3896444519e-13, /* 0x2b500000 */
5.3845816694e-15, /* 0x27c20000 */
5.6378512969e-18, /* 0x22d00000 */
8.3009228831e-20, /* 0x1fc40000 */
3.2756352257e-22, /* 0x1bc60000 */
6.3331015649e-25, /* 0x17440000 */
};
static const float
zero = 0.0000000000e+00, /* 0x00000000 */
one = 1.0000000000e+00, /* 0x3f800000 */
two8 = 2.5600000000e+02, /* 0x43800000 */
twon8 = 3.9062500000e-03; /* 0x3b800000 */
int __kernel_rem_pio2f(float *x, float *y, int e0, int nx, int prec, const __int32_t *ipio2)
{
__int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
float z,fw,f[20],fq[20],q[20];
/* initialize jk*/
jk = init_jk[prec];
jp = jk;
/* determine jx,jv,q0, note that 3>q0 */
jx = nx-1;
jv = (e0-3)/8; if(jv<0) jv=0;
q0 = e0-8*(jv+1);
/* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
j = jv-jx; m = jx+jk;
for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (float) ipio2[j];
/* compute q[0],q[1],...q[jk] */
for (i=0;i<=jk;i++) {
for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;
}
jz = jk;
recompute:
/* distill q[] into iq[] reversingly */
for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
fw = (float)((__int32_t)(twon8* z));
iq[i] = (__int32_t)(z-two8*fw);
z = q[j-1]+fw;
}
/* compute n */
z = scalbnf(z,(int)q0); /* actual value of z */
z -= (float)8.0*floorf(z*(float)0.125); /* trim off integer >= 8 */
n = (__int32_t) z;
z -= (float)n;
ih = 0;
if(q0>0) { /* need iq[jz-1] to determine n */
i = (iq[jz-1]>>(8-q0)); n += i;
iq[jz-1] -= i<<(8-q0);
ih = iq[jz-1]>>(7-q0);
}
else if(q0==0) ih = iq[jz-1]>>8;
else if(z>=(float)0.5) ih=2;
if(ih>0) { /* q > 0.5 */
n += 1; carry = 0;
for(i=0;i<jz ;i++) { /* compute 1-q */
j = iq[i];
if(carry==0) {
if(j!=0) {
carry = 1; iq[i] = 0x100- j;
}
} else iq[i] = 0xff - j;
}
if(q0>0) { /* rare case: chance is 1 in 12 */
switch(q0) {
case 1:
iq[jz-1] &= 0x7f; break;
case 2:
iq[jz-1] &= 0x3f; break;
}
}
if(ih==2) {
z = one - z;
if(carry!=0) z -= scalbnf(one,(int)q0);
}
}
/* check if recomputation is needed */
if(z==zero) {
j = 0;
for (i=jz-1;i>=jk;i--) j |= iq[i];
if(j==0) { /* need recomputation */
for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */
for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */
f[jx+i] = (float) ipio2[jv+i];
for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
q[i] = fw;
}
jz += k;
goto recompute;
}
}
/* chop off zero terms */
if(z==(float)0.0) {
jz -= 1; q0 -= 8;
while(iq[jz]==0) { jz--; q0-=8;}
} else { /* break z into 8-bit if necessary */
z = scalbnf(z,-(int)q0);
if(z>=two8) {
fw = (float)((__int32_t)(twon8*z));
iq[jz] = (__int32_t)(z-two8*fw);
jz += 1; q0 += 8;
iq[jz] = (__int32_t) fw;
} else iq[jz] = (__int32_t) z ;
}
/* convert integer "bit" chunk to floating-point value */
fw = scalbnf(one,(int)q0);
for(i=jz;i>=0;i--) {
q[i] = fw*(float)iq[i]; fw*=twon8;
}
/* compute PIo2[0,...,jp]*q[jz,...,0] */
for(i=jz;i>=0;i--) {
for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
fq[jz-i] = fw;
}
/* compress fq[] into y[] */
switch(prec) {
case 0:
fw = 0.0;
for (i=jz;i>=0;i--) fw += fq[i];
y[0] = (ih==0)? fw: -fw;
break;
case 1:
case 2:
fw = 0.0;
for (i=jz;i>=0;i--) fw += fq[i];
y[0] = (ih==0)? fw: -fw;
fw = fq[0]-fw;
for (i=1;i<=jz;i++) fw += fq[i];
y[1] = (ih==0)? fw: -fw;
break;
case 3: /* painful */
for (i=jz;i>0;i--) {
fw = fq[i-1]+fq[i];
fq[i] += fq[i-1]-fw;
fq[i-1] = fw;
}
for (i=jz;i>1;i--) {
fw = fq[i-1]+fq[i];
fq[i] += fq[i-1]-fw;
fq[i-1] = fw;
}
for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
if(ih==0) {
y[0] = fq[0]; y[1] = fq[1]; y[2] = fw;
} else {
y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
}
}
return n&7;
}
/* ef_rem_pio2.c -- float version of e_rem_pio2.c
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
*/
/*
* ====================================================
* 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.
* ====================================================
*
*/
/* __ieee754_rem_pio2f(x,y)
*
* return the remainder of x rem pi/2 in y[0]+y[1]
* use __kernel_rem_pio2f()
*/
/*
* Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
*/
static const __int32_t two_over_pi[] = {
0xA2, 0xF9, 0x83, 0x6E, 0x4E, 0x44, 0x15, 0x29, 0xFC,
0x27, 0x57, 0xD1, 0xF5, 0x34, 0xDD, 0xC0, 0xDB, 0x62,
0x95, 0x99, 0x3C, 0x43, 0x90, 0x41, 0xFE, 0x51, 0x63,
0xAB, 0xDE, 0xBB, 0xC5, 0x61, 0xB7, 0x24, 0x6E, 0x3A,
0x42, 0x4D, 0xD2, 0xE0, 0x06, 0x49, 0x2E, 0xEA, 0x09,
0xD1, 0x92, 0x1C, 0xFE, 0x1D, 0xEB, 0x1C, 0xB1, 0x29,
0xA7, 0x3E, 0xE8, 0x82, 0x35, 0xF5, 0x2E, 0xBB, 0x44,
0x84, 0xE9, 0x9C, 0x70, 0x26, 0xB4, 0x5F, 0x7E, 0x41,
0x39, 0x91, 0xD6, 0x39, 0x83, 0x53, 0x39, 0xF4, 0x9C,
0x84, 0x5F, 0x8B, 0xBD, 0xF9, 0x28, 0x3B, 0x1F, 0xF8,
0x97, 0xFF, 0xDE, 0x05, 0x98, 0x0F, 0xEF, 0x2F, 0x11,
0x8B, 0x5A, 0x0A, 0x6D, 0x1F, 0x6D, 0x36, 0x7E, 0xCF,
0x27, 0xCB, 0x09, 0xB7, 0x4F, 0x46, 0x3F, 0x66, 0x9E,
0x5F, 0xEA, 0x2D, 0x75, 0x27, 0xBA, 0xC7, 0xEB, 0xE5,
0xF1, 0x7B, 0x3D, 0x07, 0x39, 0xF7, 0x8A, 0x52, 0x92,
0xEA, 0x6B, 0xFB, 0x5F, 0xB1, 0x1F, 0x8D, 0x5D, 0x08,
0x56, 0x03, 0x30, 0x46, 0xFC, 0x7B, 0x6B, 0xAB, 0xF0,
0xCF, 0xBC, 0x20, 0x9A, 0xF4, 0x36, 0x1D, 0xA9, 0xE3,
0x91, 0x61, 0x5E, 0xE6, 0x1B, 0x08, 0x65, 0x99, 0x85,
0x5F, 0x14, 0xA0, 0x68, 0x40, 0x8D, 0xFF, 0xD8, 0x80,
0x4D, 0x73, 0x27, 0x31, 0x06, 0x06, 0x15, 0x56, 0xCA,
0x73, 0xA8, 0xC9, 0x60, 0xE2, 0x7B, 0xC0, 0x8C, 0x6B,
};
/* This array is like the one in e_rem_pio2.c, but the numbers are
single precision and the last 8 bits are forced to 0. */
static const __int32_t npio2_hw[] = {
0x3fc90f00, 0x40490f00, 0x4096cb00, 0x40c90f00, 0x40fb5300, 0x4116cb00,
0x412fed00, 0x41490f00, 0x41623100, 0x417b5300, 0x418a3a00, 0x4196cb00,
0x41a35c00, 0x41afed00, 0x41bc7e00, 0x41c90f00, 0x41d5a000, 0x41e23100,
0x41eec200, 0x41fb5300, 0x4203f200, 0x420a3a00, 0x42108300, 0x4216cb00,
0x421d1400, 0x42235c00, 0x4229a500, 0x422fed00, 0x42363600, 0x423c7e00,
0x4242c700, 0x42490f00
};
/*
* invpio2: 24 bits of 2/pi
* pio2_1: first 17 bit of pi/2
* pio2_1t: pi/2 - pio2_1
* pio2_2: second 17 bit of pi/2
* pio2_2t: pi/2 - (pio2_1+pio2_2)
* pio2_3: third 17 bit of pi/2
* pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
*/
static const float
half = 5.0000000000e-01, /* 0x3f000000 */
invpio2 = 6.3661980629e-01, /* 0x3f22f984 */
pio2_1 = 1.5707855225e+00, /* 0x3fc90f80 */
pio2_1t = 1.0804334124e-05, /* 0x37354443 */
pio2_2 = 1.0804273188e-05, /* 0x37354400 */
pio2_2t = 6.0770999344e-11, /* 0x2e85a308 */
pio2_3 = 6.0770943833e-11, /* 0x2e85a300 */
pio2_3t = 6.1232342629e-17; /* 0x248d3132 */
__int32_t __ieee754_rem_pio2f(float x, float *y)
{
float z,w,t,r,fn;
float tx[3];
__int32_t i,j,n,ix,hx;
int e0,nx;
GET_FLOAT_WORD(hx,x);
ix = hx&0x7fffffff;
if(ix<=0x3f490fd8) /* |x| ~<= pi/4 , no need for reduction */
{y[0] = x; y[1] = 0; return 0;}
if(ix<0x4016cbe4) { /* |x| < 3pi/4, special case with n=+-1 */
if(hx>0) {
z = x - pio2_1;
if((ix&0xfffffff0)!=0x3fc90fd0) { /* 24+24 bit pi OK */
y[0] = z - pio2_1t;
y[1] = (z-y[0])-pio2_1t;
} else { /* near pi/2, use 24+24+24 bit pi */
z -= pio2_2;
y[0] = z - pio2_2t;
y[1] = (z-y[0])-pio2_2t;
}
return 1;
} else { /* negative x */
z = x + pio2_1;
if((ix&0xfffffff0)!=0x3fc90fd0) { /* 24+24 bit pi OK */
y[0] = z + pio2_1t;
y[1] = (z-y[0])+pio2_1t;
} else { /* near pi/2, use 24+24+24 bit pi */
z += pio2_2;
y[0] = z + pio2_2t;
y[1] = (z-y[0])+pio2_2t;
}
return -1;
}
}
if(ix<=0x43490f80) { /* |x| ~<= 2^7*(pi/2), medium size */
t = fabsf(x);
n = (__int32_t) (t*invpio2+half);
fn = (float)n;
r = t-fn*pio2_1;
w = fn*pio2_1t; /* 1st round good to 40 bit */
if(n<32&&(ix&0xffffff00)!=npio2_hw[n-1]) {
y[0] = r-w; /* quick check no cancellation */
} else {
__uint32_t high;
j = ix>>23;
y[0] = r-w;
GET_FLOAT_WORD(high,y[0]);
i = j-((high>>23)&0xff);
if(i>8) { /* 2nd iteration needed, good to 57 */
t = r;
w = fn*pio2_2;
r = t-w;
w = fn*pio2_2t-((t-r)-w);
y[0] = r-w;
GET_FLOAT_WORD(high,y[0]);
i = j-((high>>23)&0xff);
if(i>25) { /* 3rd iteration need, 74 bits acc */
t = r; /* will cover all possible cases */
w = fn*pio2_3;
r = t-w;
w = fn*pio2_3t-((t-r)-w);
y[0] = r-w;
}
}
}
y[1] = (r-y[0])-w;
if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
else return n;
}
/*
* all other (large) arguments
*/
if(!FLT_UWORD_IS_FINITE(ix)) {
y[0]=y[1]=x-x; return 0;
}
/* set z = scalbn(|x|,ilogb(x)-7) */
e0 = (int)((ix>>23)-134); /* e0 = ilogb(z)-7; */
SET_FLOAT_WORD(z, ix - ((__int32_t)e0<<23));
for(i=0;i<2;i++) {
tx[i] = (float)((__int32_t)(z));
z = (z-tx[i])*two8;
}
tx[2] = z;
nx = 3;
while(tx[nx-1]==zero) nx--; /* skip zero term */
n = __kernel_rem_pio2f(tx,y,e0,nx,2,two_over_pi);
if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
return n;
}
/* kf_sin.c -- float version of k_sin.c
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
*/
/*
* ====================================================
* 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.
* ====================================================
*/
static const float
S1 = -1.6666667163e-01, /* 0xbe2aaaab */
S2 = 8.3333337680e-03, /* 0x3c088889 */
S3 = -1.9841270114e-04, /* 0xb9500d01 */
S4 = 2.7557314297e-06, /* 0x3638ef1b */
S5 = -2.5050759689e-08, /* 0xb2d72f34 */
S6 = 1.5896910177e-10; /* 0x2f2ec9d3 */
float __kernel_sinf(float x, float y, int iy) /* iy=0 if y is zero */
{
float z,r,v;
__int32_t ix;
GET_FLOAT_WORD(ix,x);
ix &= 0x7fffffff; /* high word of x */
if(ix<0x32000000) /* |x| < 2**-27 */
{if((int)x==0) return x;} /* generate inexact */
z = x*x;
v = z*x;
r = S2+z*(S3+z*(S4+z*(S5+z*S6)));
if(iy==0) return x+v*(S1+z*r);
else return x-((z*(half*y-v*r)-y)-v*S1);
}
/* kf_cos.c -- float version of k_cos.c
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
*/
/*
* ====================================================
* 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.
* ====================================================
*/
static const float
C1 = 4.1666667908e-02, /* 0x3d2aaaab */
C2 = -1.3888889225e-03, /* 0xbab60b61 */
C3 = 2.4801587642e-05, /* 0x37d00d01 */
C4 = -2.7557314297e-07, /* 0xb493f27c */
C5 = 2.0875723372e-09, /* 0x310f74f6 */
C6 = -1.1359647598e-11; /* 0xad47d74e */
float __kernel_cosf(float x, float y)
{
float a,hz,z,r,qx;
__int32_t ix;
GET_FLOAT_WORD(ix,x);
ix &= 0x7fffffff; /* ix = |x|'s high word*/
if(ix<0x32000000) { /* if x < 2**27 */
if(((int)x)==0) return one; /* generate inexact */
}
z = x*x;
r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6)))));
if(ix < 0x3e99999a) /* if |x| < 0.3 */
return one - ((float)0.5*z - (z*r - x*y));
else {
if(ix > 0x3f480000) { /* x > 0.78125 */
qx = (float)0.28125;
} else {
SET_FLOAT_WORD(qx,ix-0x01000000); /* x/4 */
}
hz = (float)0.5*z-qx;
a = one-qx;
return a - (hz - (z*r-x*y));
}
}
/* sf_sin.c -- float version of s_sin.c.
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
*/
/*
* ====================================================
* 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.
* ====================================================
*/
float sinf(float x)
{
float y[2],z=0.0;
__int32_t n,ix;
GET_FLOAT_WORD(ix,x);
/* |x| ~< pi/4 */
ix &= 0x7fffffff;
if(ix <= 0x3f490fd8) return __kernel_sinf(x,z,0);
/* sin(Inf or NaN) is NaN */
else if (!FLT_UWORD_IS_FINITE(ix)) return x-x;
/* argument reduction needed */
else {
n = __ieee754_rem_pio2f(x,y);
switch(n&3) {
case 0: return __kernel_sinf(y[0],y[1],1);
case 1: return __kernel_cosf(y[0],y[1]);
case 2: return -__kernel_sinf(y[0],y[1],1);
default:
return -__kernel_cosf(y[0],y[1]);
}
}
}
/* sf_cos.c -- float version of s_cos.c.
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
*/
/*
* ====================================================
* 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.
* ====================================================
*/
float cosf(float x)
{
float y[2],z=0.0;
__int32_t n,ix;
GET_FLOAT_WORD(ix,x);
/* |x| ~< pi/4 */
ix &= 0x7fffffff;
if(ix <= 0x3f490fd8) return __kernel_cosf(x,z);
/* cos(Inf or NaN) is NaN */
else if (!FLT_UWORD_IS_FINITE(ix)) return x-x;
/* argument reduction needed */
else {
n = __ieee754_rem_pio2f(x,y);
switch(n&3) {
case 0: return __kernel_cosf(y[0],y[1]);
case 1: return -__kernel_sinf(y[0],y[1],1);
case 2: return -__kernel_cosf(y[0],y[1]);
default:
return __kernel_sinf(y[0],y[1],1);
}
}
}