| //! A small number of math routines for floats and doubles. |
| //! |
| //! These are adapted from libm, a port of musl libc's libm to Rust. |
| //! libm can be found online [here](https://github.com/rust-lang/libm), |
| //! and is similarly licensed under an Apache2.0/MIT license |
| |
| #![cfg(all(not(feature = "std"), feature = "compact"))] |
| #![doc(hidden)] |
| |
| /* origin: FreeBSD /usr/src/lib/msun/src/e_powf.c */ |
| /* |
| * Conversion to float by Ian Lance Taylor, Cygnus Support, [email protected]. |
| */ |
| /* |
| * ==================================================== |
| * 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. |
| * ==================================================== |
| */ |
| |
| /// # Safety |
| /// |
| /// Safe if `index < array.len()`. |
| macro_rules! i { |
| ($array:ident, $index:expr) => { |
| // SAFETY: safe if `index < array.len()`. |
| unsafe { *$array.get_unchecked($index) } |
| }; |
| } |
| |
| pub fn powf(x: f32, y: f32) -> f32 { |
| const BP: [f32; 2] = [1.0, 1.5]; |
| const DP_H: [f32; 2] = [0.0, 5.84960938e-01]; /* 0x3f15c000 */ |
| const DP_L: [f32; 2] = [0.0, 1.56322085e-06]; /* 0x35d1cfdc */ |
| const TWO24: f32 = 16777216.0; /* 0x4b800000 */ |
| const HUGE: f32 = 1.0e30; |
| const TINY: f32 = 1.0e-30; |
| const L1: f32 = 6.0000002384e-01; /* 0x3f19999a */ |
| const L2: f32 = 4.2857143283e-01; /* 0x3edb6db7 */ |
| const L3: f32 = 3.3333334327e-01; /* 0x3eaaaaab */ |
| const L4: f32 = 2.7272811532e-01; /* 0x3e8ba305 */ |
| const L5: f32 = 2.3066075146e-01; /* 0x3e6c3255 */ |
| const L6: f32 = 2.0697501302e-01; /* 0x3e53f142 */ |
| const P1: f32 = 1.6666667163e-01; /* 0x3e2aaaab */ |
| const P2: f32 = -2.7777778450e-03; /* 0xbb360b61 */ |
| const P3: f32 = 6.6137559770e-05; /* 0x388ab355 */ |
| const P4: f32 = -1.6533901999e-06; /* 0xb5ddea0e */ |
| const P5: f32 = 4.1381369442e-08; /* 0x3331bb4c */ |
| const LG2: f32 = 6.9314718246e-01; /* 0x3f317218 */ |
| const LG2_H: f32 = 6.93145752e-01; /* 0x3f317200 */ |
| const LG2_L: f32 = 1.42860654e-06; /* 0x35bfbe8c */ |
| const OVT: f32 = 4.2995665694e-08; /* -(128-log2(ovfl+.5ulp)) */ |
| const CP: f32 = 9.6179670095e-01; /* 0x3f76384f =2/(3ln2) */ |
| const CP_H: f32 = 9.6191406250e-01; /* 0x3f764000 =12b cp */ |
| const CP_L: f32 = -1.1736857402e-04; /* 0xb8f623c6 =tail of cp_h */ |
| const IVLN2: f32 = 1.4426950216e+00; |
| const IVLN2_H: f32 = 1.4426879883e+00; |
| const IVLN2_L: f32 = 7.0526075433e-06; |
| |
| let mut z: f32; |
| let mut ax: f32; |
| let z_h: f32; |
| let z_l: f32; |
| let mut p_h: f32; |
| let mut p_l: f32; |
| let y1: f32; |
| let mut t1: f32; |
| let t2: f32; |
| let mut r: f32; |
| let s: f32; |
| let mut sn: f32; |
| let mut t: f32; |
| let mut u: f32; |
| let mut v: f32; |
| let mut w: f32; |
| let i: i32; |
| let mut j: i32; |
| let mut k: i32; |
| let mut yisint: i32; |
| let mut n: i32; |
| let hx: i32; |
| let hy: i32; |
| let mut ix: i32; |
| let iy: i32; |
| let mut is: i32; |
| |
| hx = x.to_bits() as i32; |
| hy = y.to_bits() as i32; |
| |
| ix = hx & 0x7fffffff; |
| iy = hy & 0x7fffffff; |
| |
| /* x**0 = 1, even if x is NaN */ |
| if iy == 0 { |
| return 1.0; |
| } |
| |
| /* 1**y = 1, even if y is NaN */ |
| if hx == 0x3f800000 { |
| return 1.0; |
| } |
| |
| /* 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 1.0; |
| } else if ix > 0x3f800000 { |
| /* (|x|>1)**+-inf = inf,0 */ |
| return if hy >= 0 { |
| y |
| } else { |
| 0.0 |
| }; |
| } else { |
| /* (|x|<1)**+-inf = 0,inf */ |
| return if hy >= 0 { |
| 0.0 |
| } else { |
| -y |
| }; |
| } |
| } |
| if iy == 0x3f800000 { |
| /* y is +-1 */ |
| return if hy >= 0 { |
| x |
| } else { |
| 1.0 / x |
| }; |
| } |
| |
| if hy == 0x40000000 { |
| /* y is 2 */ |
| return x * x; |
| } |
| |
| if hy == 0x3f000000 |
| /* y is 0.5 */ |
| && 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 = 1.0 / 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 = 1.0; /* 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 = -1.0; |
| } |
| } |
| |
| /* |y| is HUGE */ |
| if iy > 0x4d000000 { |
| /* if |y| > 2**27 */ |
| /* over/underflow if x is not close to one */ |
| if ix < 0x3f7ffff8 { |
| return if hy < 0 { |
| sn * HUGE * HUGE |
| } else { |
| sn * TINY * TINY |
| }; |
| } |
| |
| if ix > 0x3f800007 { |
| return if hy > 0 { |
| sn * HUGE * HUGE |
| } else { |
| 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) * (0.5 - t * (0.333333333333 - t * 0.25)); |
| u = IVLN2_H * t; /* IVLN2_H has 16 sig. bits */ |
| v = t * IVLN2_L - w * IVLN2; |
| t1 = u + v; |
| is = t1.to_bits() as i32; |
| t1 = f32::from_bits(is as u32 & 0xfffff000); |
| t2 = v - (t1 - u); |
| } else { |
| let mut s2: f32; |
| let mut s_h: f32; |
| let s_l: f32; |
| let mut t_h: f32; |
| let mut t_l: f32; |
| |
| n = 0; |
| /* take care subnormal number */ |
| if ix < 0x00800000 { |
| ax *= TWO24; |
| n -= 24; |
| ix = ax.to_bits() as i32; |
| } |
| 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; |
| } |
| ax = f32::from_bits(ix as u32); |
| |
| /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ |
| u = ax - i!(BP, k as usize); /* bp[0]=1.0, bp[1]=1.5 */ |
| v = 1.0 / (ax + i!(BP, k as usize)); |
| s = u * v; |
| s_h = s; |
| is = s_h.to_bits() as i32; |
| s_h = f32::from_bits(is as u32 & 0xfffff000); |
| /* t_h=ax+bp[k] High */ |
| is = (((ix as u32 >> 1) & 0xfffff000) | 0x20000000) as i32; |
| t_h = f32::from_bits(is as u32 + 0x00400000 + ((k as u32) << 21)); |
| t_l = ax - (t_h - i!(BP, k as usize)); |
| 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 = 3.0 + s2 + r; |
| is = t_h.to_bits() as i32; |
| t_h = f32::from_bits(is as u32 & 0xfffff000); |
| t_l = r - ((t_h - 3.0) - 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; |
| is = p_h.to_bits() as i32; |
| p_h = f32::from_bits(is as u32 & 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 + i!(DP_L, k as usize); |
| /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */ |
| t = n as f32; |
| t1 = ((z_h + z_l) + i!(DP_H, k as usize)) + t; |
| is = t1.to_bits() as i32; |
| t1 = f32::from_bits(is as u32 & 0xfffff000); |
| t2 = z_l - (((t1 - t) - i!(DP_H, k as usize)) - z_h); |
| }; |
| |
| /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ |
| is = y.to_bits() as i32; |
| y1 = f32::from_bits(is as u32 & 0xfffff000); |
| p_l = (y - y1) * t1 + y * t2; |
| p_h = y1 * t1; |
| z = p_l + p_h; |
| j = z.to_bits() as i32; |
| 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 as u32 == 0xc3160000 |
| /* z == -150 */ |
| && 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 */ |
| t = f32::from_bits(n as u32 & !(0x007fffff >> k)); |
| n = ((n & 0x007fffff) | 0x00800000) >> (23 - k); |
| if j < 0 { |
| n = -n; |
| } |
| p_h -= t; |
| } |
| t = p_l + p_h; |
| is = t.to_bits() as i32; |
| t = f32::from_bits(is as u32 & 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 - 2.0) - (w + z * w); |
| z = 1.0 - (r - z); |
| j = z.to_bits() as i32; |
| j += n << 23; |
| if (j >> 23) <= 0 { |
| /* subnormal output */ |
| z = scalbnf(z, n); |
| } else { |
| z = f32::from_bits(j as u32); |
| } |
| sn * z |
| } |
| |
| /* origin: FreeBSD /usr/src/lib/msun/src/e_sqrtf.c */ |
| /* |
| * Conversion to float by Ian Lance Taylor, Cygnus Support, [email protected]. |
| */ |
| /* |
| * ==================================================== |
| * 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. |
| * ==================================================== |
| */ |
| |
| pub fn sqrtf(x: f32) -> f32 { |
| #[cfg(target_feature = "sse")] |
| { |
| // Note: This path is unlikely since LLVM will usually have already |
| // optimized sqrt calls into hardware instructions if sse is available, |
| // but if someone does end up here they'll apprected the speed increase. |
| #[cfg(target_arch = "x86")] |
| use core::arch::x86::*; |
| #[cfg(target_arch = "x86_64")] |
| use core::arch::x86_64::*; |
| // SAFETY: safe, since `_mm_set_ss` takes a 32-bit float, and returns |
| // a 128-bit type with the lowest 32-bits as `x`, `_mm_sqrt_ss` calculates |
| // the sqrt of this 128-bit vector, and `_mm_cvtss_f32` extracts the lower |
| // 32-bits as a 32-bit float. |
| unsafe { |
| let m = _mm_set_ss(x); |
| let m_sqrt = _mm_sqrt_ss(m); |
| _mm_cvtss_f32(m_sqrt) |
| } |
| } |
| #[cfg(not(target_feature = "sse"))] |
| { |
| const TINY: f32 = 1.0e-30; |
| |
| let mut z: f32; |
| let sign: i32 = 0x80000000u32 as i32; |
| let mut ix: i32; |
| let mut s: i32; |
| let mut q: i32; |
| let mut m: i32; |
| let mut t: i32; |
| let mut i: i32; |
| let mut r: u32; |
| |
| ix = x.to_bits() as i32; |
| |
| /* take care of Inf and NaN */ |
| if (ix as u32 & 0x7f800000) == 0x7f800000 { |
| return x * x + x; /* sqrt(NaN)=NaN, sqrt(+inf)=+inf, sqrt(-inf)=sNaN */ |
| } |
| |
| /* take care of zero */ |
| if ix <= 0 { |
| if (ix & !sign) == 0 { |
| return x; /* sqrt(+-0) = +-0 */ |
| } |
| if ix < 0 { |
| return (x - x) / (x - x); /* sqrt(-ve) = sNaN */ |
| } |
| } |
| |
| /* normalize x */ |
| m = ix >> 23; |
| if m == 0 { |
| /* subnormal x */ |
| i = 0; |
| while ix & 0x00800000 == 0 { |
| ix <<= 1; |
| i = i + 1; |
| } |
| m -= i - 1; |
| } |
| m -= 127; /* unbias exponent */ |
| ix = (ix & 0x007fffff) | 0x00800000; |
| if m & 1 == 1 { |
| /* odd m, double x to make it even */ |
| ix += ix; |
| } |
| m >>= 1; /* m = [m/2] */ |
| |
| /* generate sqrt(x) bit by bit */ |
| ix += ix; |
| q = 0; |
| s = 0; |
| r = 0x01000000; /* r = moving bit from right to left */ |
| |
| while r != 0 { |
| t = s + r as i32; |
| if t <= ix { |
| s = t + r as i32; |
| ix -= t; |
| q += r as i32; |
| } |
| ix += ix; |
| r >>= 1; |
| } |
| |
| /* use floating add to find out rounding direction */ |
| if ix != 0 { |
| z = 1.0 - TINY; /* raise inexact flag */ |
| if z >= 1.0 { |
| z = 1.0 + TINY; |
| if z > 1.0 { |
| q += 2; |
| } else { |
| q += q & 1; |
| } |
| } |
| } |
| |
| ix = (q >> 1) + 0x3f000000; |
| ix += m << 23; |
| f32::from_bits(ix as u32) |
| } |
| } |
| |
| /// Absolute value (magnitude) (f32) |
| /// Calculates the absolute value (magnitude) of the argument `x`, |
| /// by direct manipulation of the bit representation of `x`. |
| pub fn fabsf(x: f32) -> f32 { |
| f32::from_bits(x.to_bits() & 0x7fffffff) |
| } |
| |
| pub fn scalbnf(mut x: f32, mut n: i32) -> f32 { |
| let x1p127 = f32::from_bits(0x7f000000); // 0x1p127f === 2 ^ 127 |
| let x1p_126 = f32::from_bits(0x800000); // 0x1p-126f === 2 ^ -126 |
| let x1p24 = f32::from_bits(0x4b800000); // 0x1p24f === 2 ^ 24 |
| |
| if n > 127 { |
| x *= x1p127; |
| n -= 127; |
| if n > 127 { |
| x *= x1p127; |
| n -= 127; |
| if n > 127 { |
| n = 127; |
| } |
| } |
| } else if n < -126 { |
| x *= x1p_126 * x1p24; |
| n += 126 - 24; |
| if n < -126 { |
| x *= x1p_126 * x1p24; |
| n += 126 - 24; |
| if n < -126 { |
| n = -126; |
| } |
| } |
| } |
| x * f32::from_bits(((0x7f + n) as u32) << 23) |
| } |
| |
| /* origin: FreeBSD /usr/src/lib/msun/src/e_pow.c */ |
| /* |
| * ==================================================== |
| * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved. |
| * |
| * Permission to use, copy, modify, and distribute this |
| * software is freely granted, provided that this notice |
| * is preserved. |
| * ==================================================== |
| */ |
| |
| // pow(x,y) return x**y |
| // |
| // n |
| // Method: Let x = 2 * (1+f) |
| // 1. Compute and return log2(x) in two pieces: |
| // log2(x) = w1 + w2, |
| // where w1 has 53-24 = 29 bit trailing zeros. |
| // 2. Perform y*log2(x) = n+y' by simulating muti-precision |
| // arithmetic, where |y'|<=0.5. |
| // 3. Return x**y = 2**n*exp(y'*log2) |
| // |
| // Special cases: |
| // 1. (anything) ** 0 is 1 |
| // 2. 1 ** (anything) is 1 |
| // 3. (anything except 1) ** NAN is NAN |
| // 4. NAN ** (anything except 0) is NAN |
| // 5. +-(|x| > 1) ** +INF is +INF |
| // 6. +-(|x| > 1) ** -INF is +0 |
| // 7. +-(|x| < 1) ** +INF is +0 |
| // 8. +-(|x| < 1) ** -INF is +INF |
| // 9. -1 ** +-INF is 1 |
| // 10. +0 ** (+anything except 0, NAN) is +0 |
| // 11. -0 ** (+anything except 0, NAN, odd integer) is +0 |
| // 12. +0 ** (-anything except 0, NAN) is +INF, raise divbyzero |
| // 13. -0 ** (-anything except 0, NAN, odd integer) is +INF, raise divbyzero |
| // 14. -0 ** (+odd integer) is -0 |
| // 15. -0 ** (-odd integer) is -INF, raise divbyzero |
| // 16. +INF ** (+anything except 0,NAN) is +INF |
| // 17. +INF ** (-anything except 0,NAN) is +0 |
| // 18. -INF ** (+odd integer) is -INF |
| // 19. -INF ** (anything) = -0 ** (-anything), (anything except odd integer) |
| // 20. (anything) ** 1 is (anything) |
| // 21. (anything) ** -1 is 1/(anything) |
| // 22. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) |
| // 23. (-anything except 0 and inf) ** (non-integer) is NAN |
| // |
| // Accuracy: |
| // pow(x,y) returns x**y nearly rounded. In particular |
| // pow(integer,integer) |
| // always returns the correct integer provided it is |
| // representable. |
| // |
| // Constants : |
| // The hexadecimal values are the intended ones for the following |
| // constants. The decimal values may be used, provided that the |
| // compiler will convert from decimal to binary accurately enough |
| // to produce the hexadecimal values shown. |
| |
| pub fn powd(x: f64, y: f64) -> f64 { |
| const BP: [f64; 2] = [1.0, 1.5]; |
| const DP_H: [f64; 2] = [0.0, 5.84962487220764160156e-01]; /* 0x3fe2b803_40000000 */ |
| const DP_L: [f64; 2] = [0.0, 1.35003920212974897128e-08]; /* 0x3E4CFDEB, 0x43CFD006 */ |
| const TWO53: f64 = 9007199254740992.0; /* 0x43400000_00000000 */ |
| const HUGE: f64 = 1.0e300; |
| const TINY: f64 = 1.0e-300; |
| |
| // poly coefs for (3/2)*(log(x)-2s-2/3*s**3: |
| const L1: f64 = 5.99999999999994648725e-01; /* 0x3fe33333_33333303 */ |
| const L2: f64 = 4.28571428578550184252e-01; /* 0x3fdb6db6_db6fabff */ |
| const L3: f64 = 3.33333329818377432918e-01; /* 0x3fd55555_518f264d */ |
| const L4: f64 = 2.72728123808534006489e-01; /* 0x3fd17460_a91d4101 */ |
| const L5: f64 = 2.30660745775561754067e-01; /* 0x3fcd864a_93c9db65 */ |
| const L6: f64 = 2.06975017800338417784e-01; /* 0x3fca7e28_4a454eef */ |
| const P1: f64 = 1.66666666666666019037e-01; /* 0x3fc55555_5555553e */ |
| const P2: f64 = -2.77777777770155933842e-03; /* 0xbf66c16c_16bebd93 */ |
| const P3: f64 = 6.61375632143793436117e-05; /* 0x3f11566a_af25de2c */ |
| const P4: f64 = -1.65339022054652515390e-06; /* 0xbebbbd41_c5d26bf1 */ |
| const P5: f64 = 4.13813679705723846039e-08; /* 0x3e663769_72bea4d0 */ |
| const LG2: f64 = 6.93147180559945286227e-01; /* 0x3fe62e42_fefa39ef */ |
| const LG2_H: f64 = 6.93147182464599609375e-01; /* 0x3fe62e43_00000000 */ |
| const LG2_L: f64 = -1.90465429995776804525e-09; /* 0xbe205c61_0ca86c39 */ |
| const OVT: f64 = 8.0085662595372944372e-017; /* -(1024-log2(ovfl+.5ulp)) */ |
| const CP: f64 = 9.61796693925975554329e-01; /* 0x3feec709_dc3a03fd =2/(3ln2) */ |
| const CP_H: f64 = 9.61796700954437255859e-01; /* 0x3feec709_e0000000 =(float)cp */ |
| const CP_L: f64 = -7.02846165095275826516e-09; /* 0xbe3e2fe0_145b01f5 =tail of cp_h*/ |
| const IVLN2: f64 = 1.44269504088896338700e+00; /* 0x3ff71547_652b82fe =1/ln2 */ |
| const IVLN2_H: f64 = 1.44269502162933349609e+00; /* 0x3ff71547_60000000 =24b 1/ln2*/ |
| const IVLN2_L: f64 = 1.92596299112661746887e-08; /* 0x3e54ae0b_f85ddf44 =1/ln2 tail*/ |
| |
| let t1: f64; |
| let t2: f64; |
| |
| let (hx, lx): (i32, u32) = ((x.to_bits() >> 32) as i32, x.to_bits() as u32); |
| let (hy, ly): (i32, u32) = ((y.to_bits() >> 32) as i32, y.to_bits() as u32); |
| |
| let mut ix: i32 = (hx & 0x7fffffff) as i32; |
| let iy: i32 = (hy & 0x7fffffff) as i32; |
| |
| /* x**0 = 1, even if x is NaN */ |
| if ((iy as u32) | ly) == 0 { |
| return 1.0; |
| } |
| |
| /* 1**y = 1, even if y is NaN */ |
| if hx == 0x3ff00000 && lx == 0 { |
| return 1.0; |
| } |
| |
| /* NaN if either arg is NaN */ |
| if ix > 0x7ff00000 |
| || (ix == 0x7ff00000 && lx != 0) |
| || iy > 0x7ff00000 |
| || (iy == 0x7ff00000 && ly != 0) |
| { |
| 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 |
| */ |
| let mut yisint: i32 = 0; |
| let mut k: i32; |
| let mut j: i32; |
| if hx < 0 { |
| if iy >= 0x43400000 { |
| yisint = 2; /* even integer y */ |
| } else if iy >= 0x3ff00000 { |
| k = (iy >> 20) - 0x3ff; /* exponent */ |
| |
| if k > 20 { |
| j = (ly >> (52 - k)) as i32; |
| |
| if (j << (52 - k)) == (ly as i32) { |
| yisint = 2 - (j & 1); |
| } |
| } else if ly == 0 { |
| j = iy >> (20 - k); |
| |
| if (j << (20 - k)) == iy { |
| yisint = 2 - (j & 1); |
| } |
| } |
| } |
| } |
| |
| if ly == 0 { |
| /* special value of y */ |
| if iy == 0x7ff00000 { |
| /* y is +-inf */ |
| |
| return if ((ix - 0x3ff00000) | (lx as i32)) == 0 { |
| /* (-1)**+-inf is 1 */ |
| 1.0 |
| } else if ix >= 0x3ff00000 { |
| /* (|x|>1)**+-inf = inf,0 */ |
| if hy >= 0 { |
| y |
| } else { |
| 0.0 |
| } |
| } else { |
| /* (|x|<1)**+-inf = 0,inf */ |
| if hy >= 0 { |
| 0.0 |
| } else { |
| -y |
| } |
| }; |
| } |
| |
| if iy == 0x3ff00000 { |
| /* y is +-1 */ |
| return if hy >= 0 { |
| x |
| } else { |
| 1.0 / x |
| }; |
| } |
| |
| if hy == 0x40000000 { |
| /* y is 2 */ |
| return x * x; |
| } |
| |
| if hy == 0x3fe00000 { |
| /* y is 0.5 */ |
| if hx >= 0 { |
| /* x >= +0 */ |
| return sqrtd(x); |
| } |
| } |
| } |
| |
| let mut ax: f64 = fabsd(x); |
| if lx == 0 { |
| /* special value of x */ |
| if ix == 0x7ff00000 || ix == 0 || ix == 0x3ff00000 { |
| /* x is +-0,+-inf,+-1 */ |
| let mut z: f64 = ax; |
| |
| if hy < 0 { |
| /* z = (1/|x|) */ |
| z = 1.0 / z; |
| } |
| |
| if hx < 0 { |
| if ((ix - 0x3ff00000) | 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; |
| } |
| } |
| |
| let mut s: f64 = 1.0; /* 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) */ |
| s = -1.0; |
| } |
| } |
| |
| /* |y| is HUGE */ |
| if iy > 0x41e00000 { |
| /* if |y| > 2**31 */ |
| if iy > 0x43f00000 { |
| /* if |y| > 2**64, must o/uflow */ |
| if ix <= 0x3fefffff { |
| return if hy < 0 { |
| HUGE * HUGE |
| } else { |
| TINY * TINY |
| }; |
| } |
| |
| if ix >= 0x3ff00000 { |
| return if hy > 0 { |
| HUGE * HUGE |
| } else { |
| TINY * TINY |
| }; |
| } |
| } |
| |
| /* over/underflow if x is not close to one */ |
| if ix < 0x3fefffff { |
| return if hy < 0 { |
| s * HUGE * HUGE |
| } else { |
| s * TINY * TINY |
| }; |
| } |
| if ix > 0x3ff00000 { |
| return if hy > 0 { |
| s * HUGE * HUGE |
| } else { |
| s * 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 */ |
| let t: f64 = ax - 1.0; /* t has 20 trailing zeros */ |
| let w: f64 = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25)); |
| let u: f64 = IVLN2_H * t; /* ivln2_h has 21 sig. bits */ |
| let v: f64 = t * IVLN2_L - w * IVLN2; |
| t1 = with_set_low_word(u + v, 0); |
| t2 = v - (t1 - u); |
| } else { |
| // double ss,s2,s_h,s_l,t_h,t_l; |
| let mut n: i32 = 0; |
| |
| if ix < 0x00100000 { |
| /* take care subnormal number */ |
| ax *= TWO53; |
| n -= 53; |
| ix = get_high_word(ax) as i32; |
| } |
| |
| n += (ix >> 20) - 0x3ff; |
| j = ix & 0x000fffff; |
| |
| /* determine interval */ |
| let k: i32; |
| ix = j | 0x3ff00000; /* normalize ix */ |
| if j <= 0x3988E { |
| /* |x|<sqrt(3/2) */ |
| k = 0; |
| } else if j < 0xBB67A { |
| /* |x|<sqrt(3) */ |
| k = 1; |
| } else { |
| k = 0; |
| n += 1; |
| ix -= 0x00100000; |
| } |
| ax = with_set_high_word(ax, ix as u32); |
| |
| /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ |
| let u: f64 = ax - i!(BP, k as usize); /* bp[0]=1.0, bp[1]=1.5 */ |
| let v: f64 = 1.0 / (ax + i!(BP, k as usize)); |
| let ss: f64 = u * v; |
| let s_h = with_set_low_word(ss, 0); |
| |
| /* t_h=ax+bp[k] High */ |
| let t_h: f64 = with_set_high_word( |
| 0.0, |
| ((ix as u32 >> 1) | 0x20000000) + 0x00080000 + ((k as u32) << 18), |
| ); |
| let t_l: f64 = ax - (t_h - i!(BP, k as usize)); |
| let s_l: f64 = v * ((u - s_h * t_h) - s_h * t_l); |
| |
| /* compute log(ax) */ |
| let s2: f64 = ss * ss; |
| let mut r: f64 = s2 * s2 * (L1 + s2 * (L2 + s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6))))); |
| r += s_l * (s_h + ss); |
| let s2: f64 = s_h * s_h; |
| let t_h: f64 = with_set_low_word(3.0 + s2 + r, 0); |
| let t_l: f64 = r - ((t_h - 3.0) - s2); |
| |
| /* u+v = ss*(1+...) */ |
| let u: f64 = s_h * t_h; |
| let v: f64 = s_l * t_h + t_l * ss; |
| |
| /* 2/(3log2)*(ss+...) */ |
| let p_h: f64 = with_set_low_word(u + v, 0); |
| let p_l = v - (p_h - u); |
| let z_h: f64 = CP_H * p_h; /* cp_h+cp_l = 2/(3*log2) */ |
| let z_l: f64 = CP_L * p_h + p_l * CP + i!(DP_L, k as usize); |
| |
| /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */ |
| let t: f64 = n as f64; |
| t1 = with_set_low_word(((z_h + z_l) + i!(DP_H, k as usize)) + t, 0); |
| t2 = z_l - (((t1 - t) - i!(DP_H, k as usize)) - z_h); |
| } |
| |
| /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ |
| let y1: f64 = with_set_low_word(y, 0); |
| let p_l: f64 = (y - y1) * t1 + y * t2; |
| let mut p_h: f64 = y1 * t1; |
| let z: f64 = p_l + p_h; |
| let mut j: i32 = (z.to_bits() >> 32) as i32; |
| let i: i32 = z.to_bits() as i32; |
| // let (j, i): (i32, i32) = ((z.to_bits() >> 32) as i32, z.to_bits() as i32); |
| |
| if j >= 0x40900000 { |
| /* z >= 1024 */ |
| if (j - 0x40900000) | i != 0 { |
| /* if z > 1024 */ |
| return s * HUGE * HUGE; /* overflow */ |
| } |
| |
| if p_l + OVT > z - p_h { |
| return s * HUGE * HUGE; /* overflow */ |
| } |
| } else if (j & 0x7fffffff) >= 0x4090cc00 { |
| /* z <= -1075 */ |
| // FIXME: instead of abs(j) use unsigned j |
| |
| if (((j as u32) - 0xc090cc00) | (i as u32)) != 0 { |
| /* z < -1075 */ |
| return s * TINY * TINY; /* underflow */ |
| } |
| |
| if p_l <= z - p_h { |
| return s * TINY * TINY; /* underflow */ |
| } |
| } |
| |
| /* compute 2**(p_h+p_l) */ |
| let i: i32 = j & (0x7fffffff as i32); |
| k = (i >> 20) - 0x3ff; |
| let mut n: i32 = 0; |
| |
| if i > 0x3fe00000 { |
| /* if |z| > 0.5, set n = [z+0.5] */ |
| n = j + (0x00100000 >> (k + 1)); |
| k = ((n & 0x7fffffff) >> 20) - 0x3ff; /* new k for n */ |
| let t: f64 = with_set_high_word(0.0, (n & !(0x000fffff >> k)) as u32); |
| n = ((n & 0x000fffff) | 0x00100000) >> (20 - k); |
| if j < 0 { |
| n = -n; |
| } |
| p_h -= t; |
| } |
| |
| let t: f64 = with_set_low_word(p_l + p_h, 0); |
| let u: f64 = t * LG2_H; |
| let v: f64 = (p_l - (t - p_h)) * LG2 + t * LG2_L; |
| let mut z: f64 = u + v; |
| let w: f64 = v - (z - u); |
| let t: f64 = z * z; |
| let t1: f64 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5)))); |
| let r: f64 = (z * t1) / (t1 - 2.0) - (w + z * w); |
| z = 1.0 - (r - z); |
| j = get_high_word(z) as i32; |
| j += n << 20; |
| |
| if (j >> 20) <= 0 { |
| /* subnormal output */ |
| z = scalbnd(z, n); |
| } else { |
| z = with_set_high_word(z, j as u32); |
| } |
| |
| s * z |
| } |
| |
| /// Absolute value (magnitude) (f64) |
| /// Calculates the absolute value (magnitude) of the argument `x`, |
| /// by direct manipulation of the bit representation of `x`. |
| pub fn fabsd(x: f64) -> f64 { |
| f64::from_bits(x.to_bits() & (u64::MAX / 2)) |
| } |
| |
| pub fn scalbnd(x: f64, mut n: i32) -> f64 { |
| let x1p1023 = f64::from_bits(0x7fe0000000000000); // 0x1p1023 === 2 ^ 1023 |
| let x1p53 = f64::from_bits(0x4340000000000000); // 0x1p53 === 2 ^ 53 |
| let x1p_1022 = f64::from_bits(0x0010000000000000); // 0x1p-1022 === 2 ^ (-1022) |
| |
| let mut y = x; |
| |
| if n > 1023 { |
| y *= x1p1023; |
| n -= 1023; |
| if n > 1023 { |
| y *= x1p1023; |
| n -= 1023; |
| if n > 1023 { |
| n = 1023; |
| } |
| } |
| } else if n < -1022 { |
| /* make sure final n < -53 to avoid double |
| rounding in the subnormal range */ |
| y *= x1p_1022 * x1p53; |
| n += 1022 - 53; |
| if n < -1022 { |
| y *= x1p_1022 * x1p53; |
| n += 1022 - 53; |
| if n < -1022 { |
| n = -1022; |
| } |
| } |
| } |
| y * f64::from_bits(((0x3ff + n) as u64) << 52) |
| } |
| |
| /* origin: FreeBSD /usr/src/lib/msun/src/e_sqrt.c */ |
| /* |
| * ==================================================== |
| * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
| * |
| * Developed at SunSoft, a Sun Microsystems, Inc. business. |
| * Permission to use, copy, modify, and distribute this |
| * software is freely granted, provided that this notice |
| * is preserved. |
| * ==================================================== |
| */ |
| /* sqrt(x) |
| * Return correctly rounded sqrt. |
| * ------------------------------------------ |
| * | Use the hardware sqrt if you have one | |
| * ------------------------------------------ |
| * Method: |
| * Bit by bit method using integer arithmetic. (Slow, but portable) |
| * 1. Normalization |
| * Scale x to y in [1,4) with even powers of 2: |
| * find an integer k such that 1 <= (y=x*2^(2k)) < 4, then |
| * sqrt(x) = 2^k * sqrt(y) |
| * 2. Bit by bit computation |
| * Let q = sqrt(y) truncated to i bit after binary point (q = 1), |
| * i 0 |
| * i+1 2 |
| * s = 2*q , and y = 2 * ( y - q ). (1) |
| * i i i i |
| * |
| * To compute q from q , one checks whether |
| * i+1 i |
| * |
| * -(i+1) 2 |
| * (q + 2 ) <= y. (2) |
| * i |
| * -(i+1) |
| * If (2) is false, then q = q ; otherwise q = q + 2 . |
| * i+1 i i+1 i |
| * |
| * With some algebraic manipulation, it is not difficult to see |
| * that (2) is equivalent to |
| * -(i+1) |
| * s + 2 <= y (3) |
| * i i |
| * |
| * The advantage of (3) is that s and y can be computed by |
| * i i |
| * the following recurrence formula: |
| * if (3) is false |
| * |
| * s = s , y = y ; (4) |
| * i+1 i i+1 i |
| * |
| * otherwise, |
| * -i -(i+1) |
| * s = s + 2 , y = y - s - 2 (5) |
| * i+1 i i+1 i i |
| * |
| * One may easily use induction to prove (4) and (5). |
| * Note. Since the left hand side of (3) contain only i+2 bits, |
| * it does not necessary to do a full (53-bit) comparison |
| * in (3). |
| * 3. Final rounding |
| * After generating the 53 bits result, we compute one more bit. |
| * Together with the remainder, we can decide whether the |
| * result is exact, bigger than 1/2ulp, or less than 1/2ulp |
| * (it will never equal to 1/2ulp). |
| * The rounding mode can be detected by checking whether |
| * huge + tiny is equal to huge, and whether huge - tiny is |
| * equal to huge for some floating point number "huge" and "tiny". |
| * |
| * Special cases: |
| * sqrt(+-0) = +-0 ... exact |
| * sqrt(inf) = inf |
| * sqrt(-ve) = NaN ... with invalid signal |
| * sqrt(NaN) = NaN ... with invalid signal for signaling NaN |
| */ |
| |
| pub fn sqrtd(x: f64) -> f64 { |
| #[cfg(target_feature = "sse2")] |
| { |
| // Note: This path is unlikely since LLVM will usually have already |
| // optimized sqrt calls into hardware instructions if sse2 is available, |
| // but if someone does end up here they'll apprected the speed increase. |
| #[cfg(target_arch = "x86")] |
| use core::arch::x86::*; |
| #[cfg(target_arch = "x86_64")] |
| use core::arch::x86_64::*; |
| // SAFETY: safe, since `_mm_set_sd` takes a 64-bit float, and returns |
| // a 128-bit type with the lowest 64-bits as `x`, `_mm_sqrt_ss` calculates |
| // the sqrt of this 128-bit vector, and `_mm_cvtss_f64` extracts the lower |
| // 64-bits as a 64-bit float. |
| unsafe { |
| let m = _mm_set_sd(x); |
| let m_sqrt = _mm_sqrt_pd(m); |
| _mm_cvtsd_f64(m_sqrt) |
| } |
| } |
| #[cfg(not(target_feature = "sse2"))] |
| { |
| use core::num::Wrapping; |
| |
| const TINY: f64 = 1.0e-300; |
| |
| let mut z: f64; |
| let sign: Wrapping<u32> = Wrapping(0x80000000); |
| let mut ix0: i32; |
| let mut s0: i32; |
| let mut q: i32; |
| let mut m: i32; |
| let mut t: i32; |
| let mut i: i32; |
| let mut r: Wrapping<u32>; |
| let mut t1: Wrapping<u32>; |
| let mut s1: Wrapping<u32>; |
| let mut ix1: Wrapping<u32>; |
| let mut q1: Wrapping<u32>; |
| |
| ix0 = (x.to_bits() >> 32) as i32; |
| ix1 = Wrapping(x.to_bits() as u32); |
| |
| /* take care of Inf and NaN */ |
| if (ix0 & 0x7ff00000) == 0x7ff00000 { |
| return x * x + x; /* sqrt(NaN)=NaN, sqrt(+inf)=+inf, sqrt(-inf)=sNaN */ |
| } |
| /* take care of zero */ |
| if ix0 <= 0 { |
| if ((ix0 & !(sign.0 as i32)) | ix1.0 as i32) == 0 { |
| return x; /* sqrt(+-0) = +-0 */ |
| } |
| if ix0 < 0 { |
| return (x - x) / (x - x); /* sqrt(-ve) = sNaN */ |
| } |
| } |
| /* normalize x */ |
| m = ix0 >> 20; |
| if m == 0 { |
| /* subnormal x */ |
| while ix0 == 0 { |
| m -= 21; |
| ix0 |= (ix1 >> 11).0 as i32; |
| ix1 <<= 21; |
| } |
| i = 0; |
| while (ix0 & 0x00100000) == 0 { |
| i += 1; |
| ix0 <<= 1; |
| } |
| m -= i - 1; |
| ix0 |= (ix1 >> (32 - i) as usize).0 as i32; |
| ix1 = ix1 << i as usize; |
| } |
| m -= 1023; /* unbias exponent */ |
| ix0 = (ix0 & 0x000fffff) | 0x00100000; |
| if (m & 1) == 1 { |
| /* odd m, double x to make it even */ |
| ix0 += ix0 + ((ix1 & sign) >> 31).0 as i32; |
| ix1 += ix1; |
| } |
| m >>= 1; /* m = [m/2] */ |
| |
| /* generate sqrt(x) bit by bit */ |
| ix0 += ix0 + ((ix1 & sign) >> 31).0 as i32; |
| ix1 += ix1; |
| q = 0; /* [q,q1] = sqrt(x) */ |
| q1 = Wrapping(0); |
| s0 = 0; |
| s1 = Wrapping(0); |
| r = Wrapping(0x00200000); /* r = moving bit from right to left */ |
| |
| while r != Wrapping(0) { |
| t = s0 + r.0 as i32; |
| if t <= ix0 { |
| s0 = t + r.0 as i32; |
| ix0 -= t; |
| q += r.0 as i32; |
| } |
| ix0 += ix0 + ((ix1 & sign) >> 31).0 as i32; |
| ix1 += ix1; |
| r >>= 1; |
| } |
| |
| r = sign; |
| while r != Wrapping(0) { |
| t1 = s1 + r; |
| t = s0; |
| if t < ix0 || (t == ix0 && t1 <= ix1) { |
| s1 = t1 + r; |
| if (t1 & sign) == sign && (s1 & sign) == Wrapping(0) { |
| s0 += 1; |
| } |
| ix0 -= t; |
| if ix1 < t1 { |
| ix0 -= 1; |
| } |
| ix1 -= t1; |
| q1 += r; |
| } |
| ix0 += ix0 + ((ix1 & sign) >> 31).0 as i32; |
| ix1 += ix1; |
| r >>= 1; |
| } |
| |
| /* use floating add to find out rounding direction */ |
| if (ix0 as u32 | ix1.0) != 0 { |
| z = 1.0 - TINY; /* raise inexact flag */ |
| if z >= 1.0 { |
| z = 1.0 + TINY; |
| if q1.0 == 0xffffffff { |
| q1 = Wrapping(0); |
| q += 1; |
| } else if z > 1.0 { |
| if q1.0 == 0xfffffffe { |
| q += 1; |
| } |
| q1 += Wrapping(2); |
| } else { |
| q1 += q1 & Wrapping(1); |
| } |
| } |
| } |
| ix0 = (q >> 1) + 0x3fe00000; |
| ix1 = q1 >> 1; |
| if (q & 1) == 1 { |
| ix1 |= sign; |
| } |
| ix0 += m << 20; |
| f64::from_bits((ix0 as u64) << 32 | ix1.0 as u64) |
| } |
| } |
| |
| #[inline] |
| fn get_high_word(x: f64) -> u32 { |
| (x.to_bits() >> 32) as u32 |
| } |
| |
| #[inline] |
| fn with_set_high_word(f: f64, hi: u32) -> f64 { |
| let mut tmp = f.to_bits(); |
| tmp &= 0x00000000_ffffffff; |
| tmp |= (hi as u64) << 32; |
| f64::from_bits(tmp) |
| } |
| |
| #[inline] |
| fn with_set_low_word(f: f64, lo: u32) -> f64 { |
| let mut tmp = f.to_bits(); |
| tmp &= 0xffffffff_00000000; |
| tmp |= lo as u64; |
| f64::from_bits(tmp) |
| } |
