Average Error: 6.9 → 0.2
Time: 52.2s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\mathsf{fma}\left(\left(x.im \cdot \left(x.re + x.im\right)\right), \left(x.re - x.im\right), \left(x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\right)\right)\]
\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.im \cdot \left(x.re + x.im\right)\right), \left(x.re - x.im\right), \left(x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\right)\right)
double f(double x_re, double x_im) {
        double r30223698 = x_re;
        double r30223699 = r30223698 * r30223698;
        double r30223700 = x_im;
        double r30223701 = r30223700 * r30223700;
        double r30223702 = r30223699 - r30223701;
        double r30223703 = r30223702 * r30223700;
        double r30223704 = r30223698 * r30223700;
        double r30223705 = r30223700 * r30223698;
        double r30223706 = r30223704 + r30223705;
        double r30223707 = r30223706 * r30223698;
        double r30223708 = r30223703 + r30223707;
        return r30223708;
}
↓
double f(double x_re, double x_im) {
        double r30223709 = x_im;
        double r30223710 = x_re;
        double r30223711 = r30223710 + r30223709;
        double r30223712 = r30223709 * r30223711;
        double r30223713 = r30223710 - r30223709;
        double r30223714 = r30223709 * r30223710;
        double r30223715 = r30223714 + r30223714;
        double r30223716 = r30223710 * r30223715;
        double r30223717 = fma(r30223712, r30223713, r30223716);
        return r30223717;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original6.9
Target0.2
Herbie0.2
\[\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\]

Derivation

  1. Initial program 6.9

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

  2. Taylor expanded around 0 6.8

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

  3. Simplified0.2

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

  5. Applied fma-def0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019107 +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)))