Average Error: 7.0 → 0.6
Time: 21.2s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\left(\left(x.re - x.im\right) \cdot x.im\right) \cdot \left(x.im + x.re\right) + \sqrt[3]{x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)} \cdot \left(\sqrt[3]{x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)} \cdot \sqrt[3]{x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)}\right)\]
\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
↓
\left(\left(x.re - x.im\right) \cdot x.im\right) \cdot \left(x.im + x.re\right) + \sqrt[3]{x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)} \cdot \left(\sqrt[3]{x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)} \cdot \sqrt[3]{x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)}\right)
double f(double x_re, double x_im) {
        double r2487065 = x_re;
        double r2487066 = r2487065 * r2487065;
        double r2487067 = x_im;
        double r2487068 = r2487067 * r2487067;
        double r2487069 = r2487066 - r2487068;
        double r2487070 = r2487069 * r2487067;
        double r2487071 = r2487065 * r2487067;
        double r2487072 = r2487067 * r2487065;
        double r2487073 = r2487071 + r2487072;
        double r2487074 = r2487073 * r2487065;
        double r2487075 = r2487070 + r2487074;
        return r2487075;
}
↓
double f(double x_re, double x_im) {
        double r2487076 = x_re;
        double r2487077 = x_im;
        double r2487078 = r2487076 - r2487077;
        double r2487079 = r2487078 * r2487077;
        double r2487080 = r2487077 + r2487076;
        double r2487081 = r2487079 * r2487080;
        double r2487082 = r2487076 * r2487077;
        double r2487083 = r2487082 + r2487082;
        double r2487084 = r2487076 * r2487083;
        double r2487085 = cbrt(r2487084);
        double r2487086 = r2487085 * r2487085;
        double r2487087 = r2487085 * r2487086;
        double r2487088 = r2487081 + r2487087;
        return r2487088;
}

Error

Bits error versus x.re

Bits error versus x.im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original7.0
Target0.2
Herbie0.6
\[\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\]

Derivation

  1. Initial program 7.0

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

  2. Taylor expanded around inf 6.9

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

  3. Simplified0.2

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

  5. Applied add-cube-cbrt0.6

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

  6. Final simplification0.6

    \[\leadsto \left(\left(x.re - x.im\right) \cdot x.im\right) \cdot \left(x.im + x.re\right) + \sqrt[3]{x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)} \cdot \left(\sqrt[3]{x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)} \cdot \sqrt[3]{x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)}\right)\]

Reproduce

herbie shell --seed 2019138 
(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)))