Average Error: 7.1 → 0.7
Time: 19.5s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \sqrt[3]{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right)} \cdot \left(\left(\sqrt[3]{x.re + x.im} \cdot \sqrt[3]{\left(x.re - x.im\right) \cdot x.im}\right) \cdot \sqrt[3]{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right)}\right)\]
\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(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \sqrt[3]{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right)} \cdot \left(\left(\sqrt[3]{x.re + x.im} \cdot \sqrt[3]{\left(x.re - x.im\right) \cdot x.im}\right) \cdot \sqrt[3]{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right)}\right)
double f(double x_re, double x_im) {
        double r9121613 = x_re;
        double r9121614 = r9121613 * r9121613;
        double r9121615 = x_im;
        double r9121616 = r9121615 * r9121615;
        double r9121617 = r9121614 - r9121616;
        double r9121618 = r9121617 * r9121615;
        double r9121619 = r9121613 * r9121615;
        double r9121620 = r9121615 * r9121613;
        double r9121621 = r9121619 + r9121620;
        double r9121622 = r9121621 * r9121613;
        double r9121623 = r9121618 + r9121622;
        return r9121623;
}
↓
double f(double x_re, double x_im) {
        double r9121624 = x_im;
        double r9121625 = x_re;
        double r9121626 = r9121624 * r9121625;
        double r9121627 = r9121626 + r9121626;
        double r9121628 = r9121627 * r9121625;
        double r9121629 = r9121625 + r9121624;
        double r9121630 = r9121625 - r9121624;
        double r9121631 = r9121630 * r9121624;
        double r9121632 = r9121629 * r9121631;
        double r9121633 = cbrt(r9121632);
        double r9121634 = cbrt(r9121629);
        double r9121635 = cbrt(r9121631);
        double r9121636 = r9121634 * r9121635;
        double r9121637 = r9121636 * r9121633;
        double r9121638 = r9121633 * r9121637;
        double r9121639 = r9121628 + r9121638;
        return r9121639;
}

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

Original7.1
Target0.2
Herbie0.7
\[\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\]

Derivation

  1. Initial program 7.1

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

  3. Applied difference-of-squares7.1

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

  4. Applied associate-*l*0.2

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

  6. Applied add-cube-cbrt0.7

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

  8. Applied cbrt-prod0.7

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

  10. Applied add-cbrt-cube0.7

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

  11. Simplified0.7

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

  12. Final simplification0.7

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

Reproduce

herbie shell --seed 2019172 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"

  :herbie-target
  (+ (* (* x.re x.im) (* 2.0 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)))