text: /*
   * 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 {
          uint32_t m : 23;
          uint32_t e : 8;
          uint32_t s : 1;
      };
  } float_s_t;
  
  int __signbitf(float f) {
      float_s_t u = {.f = f};
      return u.s;
  }
  
  #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
  
  static const float _M_LN10 = 2.30258509299404; // 0x40135d8e
  float log10f(float x) { return logf(x) / (float)_M_LN10; }
  
  float tanhf(float x) {
      int sign = 0;
      if (x < 0) {
          sign = 1;
          x = -x;
      }
      x = expm1f(-2 * x);
      x = x / (x + 2);
      return sign ? x : -x;
  }
  
  /*****************************************************************************/
  /*****************************************************************************/
  // __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;
  }
  