Average Error: 6.8 → 0.2
Time: 22.6s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\left(\left(x.re \cdot 3\right) \cdot x.im\right) \cdot x.re + x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\]
\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 \cdot 3\right) \cdot x.im\right) \cdot x.re + x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)
double f(double x_re, double x_im) {
        double r7175952 = x_re;
        double r7175953 = r7175952 * r7175952;
        double r7175954 = x_im;
        double r7175955 = r7175954 * r7175954;
        double r7175956 = r7175953 - r7175955;
        double r7175957 = r7175956 * r7175954;
        double r7175958 = r7175952 * r7175954;
        double r7175959 = r7175954 * r7175952;
        double r7175960 = r7175958 + r7175959;
        double r7175961 = r7175960 * r7175952;
        double r7175962 = r7175957 + r7175961;
        return r7175962;
}
↓
double f(double x_re, double x_im) {
        double r7175963 = x_re;
        double r7175964 = 3.0;
        double r7175965 = r7175963 * r7175964;
        double r7175966 = x_im;
        double r7175967 = r7175965 * r7175966;
        double r7175968 = r7175967 * r7175963;
        double r7175969 = -r7175966;
        double r7175970 = r7175966 * r7175969;
        double r7175971 = r7175966 * r7175970;
        double r7175972 = r7175968 + r7175971;
        return r7175972;
}

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

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. Simplified6.8

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

  3. Taylor expanded around 0 6.8

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

  4. Simplified6.8

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

  6. Applied add-cube-cbrt7.5

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

  7. Applied associate-*l*7.5

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

  9. Applied sub-neg7.5

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

  10. Applied distribute-lft-in7.5

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

  11. Applied distribute-rgt-in7.5

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

  12. Simplified0.6

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

  13. Simplified0.2

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

  14. Final simplification0.2

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

Reproduce

herbie shell --seed 2019143 
(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)))