cbrt.rs - source
Description: Source of the Rust file `/Users/erlendbasso/.cargo/registry/src/github.com-1ecc6299db9ec823/libm-0.2.6/src/math/cbrt.rs`.
Keywords: rust, rustlang, rust-lang

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    /* origin: FreeBSD /usr/src/lib/msun/src/s_cbrt.c */
    /*
     * ====================================================
     * 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.
     * ====================================================
     *
     * Optimized by Bruce D. Evans.
     */
    /* cbrt(x)
     * Return cube root of x
     */

    use core::f64;

    const B1: u32 = 715094163; /* B1 = (1023-1023/3-0.03306235651)*2**20 */
    const B2: u32 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */

    /* |1/cbrt(x) - p(x)| < 2**-23.5 (~[-7.93e-8, 7.929e-8]). */
    const P0: f64 = 1.87595182427177009643; /* 0x3ffe03e6, 0x0f61e692 */
    const P1: f64 = -1.88497979543377169875; /* 0xbffe28e0, 0x92f02420 */
    const P2: f64 = 1.621429720105354466140; /* 0x3ff9f160, 0x4a49d6c2 */
    const P3: f64 = -0.758397934778766047437; /* 0xbfe844cb, 0xbee751d9 */
    const P4: f64 = 0.145996192886612446982; /* 0x3fc2b000, 0xd4e4edd7 */

    // Cube root (f64)
    ///
    /// Computes the cube root of the argument.
    #[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)]
    pub fn cbrt(x: f64) -> f64 {
        let x1p54 = f64::from_bits(0x4350000000000000); // 0x1p54 === 2 ^ 54

        let mut ui: u64 = x.to_bits();
        let mut r: f64;
        let s: f64;
        let mut t: f64;
        let w: f64;
        let mut hx: u32 = (ui >> 32) as u32 & 0x7fffffff;

        if hx >= 0x7ff00000 {
            /* cbrt(NaN,INF) is itself */
            return x + x;
        }

        /*
         * Rough cbrt to 5 bits:
         *    cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3)
         * where e is integral and >= 0, m is real and in [0, 1), and "/" and
         * "%" are integer division and modulus with rounding towards minus
         * infinity.  The RHS is always >= the LHS and has a maximum relative
         * error of about 1 in 16.  Adding a bias of -0.03306235651 to the
         * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE
         * floating point representation, for finite positive normal values,
         * ordinary integer divison of the value in bits magically gives
         * almost exactly the RHS of the above provided we first subtract the
         * exponent bias (1023 for doubles) and later add it back.  We do the
         * subtraction virtually to keep e >= 0 so that ordinary integer
         * division rounds towards minus infinity; this is also efficient.
         */
        if hx < 0x00100000 {
            /* zero or subnormal? */
            ui = (x * x1p54).to_bits();
            hx = (ui >> 32) as u32 & 0x7fffffff;
            if hx == 0 {
                return x; /* cbrt(0) is itself */
            }
            hx = hx / 3 + B2;
        } else {
            hx = hx / 3 + B1;
        }
        ui &= 1 << 63;
        ui |= (hx as u64) << 32;
        t = f64::from_bits(ui);

        /*
         * New cbrt to 23 bits:
         *    cbrt(x) = t*cbrt(x/t**3) ~= t*P(t**3/x)
         * where P(r) is a polynomial of degree 4 that approximates 1/cbrt(r)
         * to within 2**-23.5 when |r - 1| < 1/10.  The rough approximation
         * has produced t such than |t/cbrt(x) - 1| ~< 1/32, and cubing this
         * gives us bounds for r = t**3/x.
         *
         * Try to optimize for parallel evaluation as in __tanf.c.
         */
        r = (t * t) * (t / x);
        t = t * ((P0 + r * (P1 + r * P2)) + ((r * r) * r) * (P3 + r * P4));

        /*
         * Round t away from zero to 23 bits (sloppily except for ensuring that
         * the result is larger in magnitude than cbrt(x) but not much more than
         * 2 23-bit ulps larger).  With rounding towards zero, the error bound
         * would be ~5/6 instead of ~4/6.  With a maximum error of 2 23-bit ulps
         * in the rounded t, the infinite-precision error in the Newton
         * approximation barely affects third digit in the final error
         * 0.667; the error in the rounded t can be up to about 3 23-bit ulps
         * before the final error is larger than 0.667 ulps.
         */
        ui = t.to_bits();
        ui = (ui + 0x80000000) & 0xffffffffc0000000;
        t = f64::from_bits(ui);

        /* one step Newton iteration to 53 bits with error < 0.667 ulps */
        s = t * t; /* t*t is exact */
        r = x / s; /* error <= 0.5 ulps; |r| < |t| */
        w = t + t; /* t+t is exact */
        r = (r - t) / (w + r); /* r-t is exact; w+r ~= 3*t */
        t = t + t * r; /* error <= 0.5 + 0.5/3 + epsilon */
        t
    }