math.cube on complex, imaginary part

Average Error: 6.8 → 0.2

Time: 40.5s

Precision: 64

Math

\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]

↓

\[\mathsf{fma}\left(\left(x.re - x.im\right) \cdot x.im, x.im + x.re, \left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.re\right)\]

C

double f(double x_re, double x_im) {
        double r6806784 = x_re;
        double r6806785 = r6806784 * r6806784;
        double r6806786 = x_im;
        double r6806787 = r6806786 * r6806786;
        double r6806788 = r6806785 - r6806787;
        double r6806789 = r6806788 * r6806786;
        double r6806790 = r6806784 * r6806786;
        double r6806791 = r6806786 * r6806784;
        double r6806792 = r6806790 + r6806791;
        double r6806793 = r6806792 * r6806784;
        double r6806794 = r6806789 + r6806793;
        return r6806794;
}

↓

double f(double x_re, double x_im) {
        double r6806795 = x_re;
        double r6806796 = x_im;
        double r6806797 = r6806795 - r6806796;
        double r6806798 = r6806797 * r6806796;
        double r6806799 = r6806796 + r6806795;
        double r6806800 = r6806795 * r6806796;
        double r6806801 = r6806800 + r6806800;
        double r6806802 = r6806801 * r6806795;
        double r6806803 = fma(r6806798, r6806799, r6806802);
        return r6806803;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	6.8
Target	0.3
Herbie	0.2

\[\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\]

Derivation

1. Initial program 6.8

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]

2. Taylor expanded around inf 6.7

   \[\leadsto \color{blue}{\left(x.im \cdot {x.re}^{2} - {x.im}^{3}\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]

3. Simplified 0.3

   \[\leadsto \color{blue}{\left(\left(x.re - x.im\right) \cdot x.im\right) \cdot \left(x.re + x.im\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
4. Using strategy rm

5. Applied fma-def 0.2

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x.re - x.im\right) \cdot x.im, x.re + x.im, \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\right)}\]

6. Final simplification 0.2

   \[\leadsto \mathsf{fma}\left(\left(x.re - x.im\right) \cdot x.im, x.im + x.re, \left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.re\right)\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"

  :herbie-target
  (+ (* (* x.re x.im) (* 2 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im)))

  (+ (* (- (* x.re x.re) (* x.im x.im)) x.im) (* (+ (* x.re x.im) (* x.im x.re)) x.re)))