Average Error: 7.0 → 0.5
Time: 17.7s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\sqrt[3]{x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)} \cdot \left(\sqrt[3]{x.re \cdot x.im + x.re \cdot x.im} \cdot \left(\sqrt[3]{x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)} \cdot \sqrt[3]{x.re}\right)\right) + \left(\left(x.re - x.im\right) \cdot x.im\right) \cdot \left(x.im + x.re\right)\]
\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
↓
\sqrt[3]{x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)} \cdot \left(\sqrt[3]{x.re \cdot x.im + x.re \cdot x.im} \cdot \left(\sqrt[3]{x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)} \cdot \sqrt[3]{x.re}\right)\right) + \left(\left(x.re - x.im\right) \cdot x.im\right) \cdot \left(x.im + x.re\right)
double f(double x_re, double x_im) {
        double r5753159 = x_re;
        double r5753160 = r5753159 * r5753159;
        double r5753161 = x_im;
        double r5753162 = r5753161 * r5753161;
        double r5753163 = r5753160 - r5753162;
        double r5753164 = r5753163 * r5753161;
        double r5753165 = r5753159 * r5753161;
        double r5753166 = r5753161 * r5753159;
        double r5753167 = r5753165 + r5753166;
        double r5753168 = r5753167 * r5753159;
        double r5753169 = r5753164 + r5753168;
        return r5753169;
}
↓
double f(double x_re, double x_im) {
        double r5753170 = x_re;
        double r5753171 = x_im;
        double r5753172 = r5753170 * r5753171;
        double r5753173 = r5753172 + r5753172;
        double r5753174 = r5753170 * r5753173;
        double r5753175 = cbrt(r5753174);
        double r5753176 = cbrt(r5753173);
        double r5753177 = cbrt(r5753170);
        double r5753178 = r5753175 * r5753177;
        double r5753179 = r5753176 * r5753178;
        double r5753180 = r5753175 * r5753179;
        double r5753181 = r5753170 - r5753171;
        double r5753182 = r5753181 * r5753171;
        double r5753183 = r5753171 + r5753170;
        double r5753184 = r5753182 * r5753183;
        double r5753185 = r5753180 + r5753184;
        return r5753185;
}

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
Target0.3
Herbie0.5
\[\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\]

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. Using strategy rm

  3. Applied difference-of-squares7.0

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

  4. Applied associate-*l*0.3

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

  6. Applied add-cube-cbrt0.6

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

  8. Applied cbrt-prod0.5

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

  9. Applied associate-*l*0.5

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

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2019133 
(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)))