Average Error: 6.8 → 0.2
Time: 13.9s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\mathsf{fma}\left(x.im \cdot \left(x.re + x.im\right), x.re - x.im, x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\right)\]
\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(x.im \cdot \left(x.re + x.im\right), x.re - x.im, x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\right)
double f(double x_re, double x_im) {
        double r3973087 = x_re;
        double r3973088 = r3973087 * r3973087;
        double r3973089 = x_im;
        double r3973090 = r3973089 * r3973089;
        double r3973091 = r3973088 - r3973090;
        double r3973092 = r3973091 * r3973089;
        double r3973093 = r3973087 * r3973089;
        double r3973094 = r3973089 * r3973087;
        double r3973095 = r3973093 + r3973094;
        double r3973096 = r3973095 * r3973087;
        double r3973097 = r3973092 + r3973096;
        return r3973097;
}
↓
double f(double x_re, double x_im) {
        double r3973098 = x_im;
        double r3973099 = x_re;
        double r3973100 = r3973099 + r3973098;
        double r3973101 = r3973098 * r3973100;
        double r3973102 = r3973099 - r3973098;
        double r3973103 = r3973098 * r3973099;
        double r3973104 = r3973103 + r3973103;
        double r3973105 = r3973099 * r3973104;
        double r3973106 = fma(r3973101, r3973102, r3973105);
        return r3973106;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original6.8
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 6.8

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

  2. Taylor expanded around 0 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. 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

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019154 +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)))