Average Error: 7.6 → 0.2
Time: 4.8s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[{x.re}^{3} - 3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\]
\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im
↓
{x.re}^{3} - 3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)
double f(double x_re, double x_im) {
        double r139919 = x_re;
        double r139920 = r139919 * r139919;
        double r139921 = x_im;
        double r139922 = r139921 * r139921;
        double r139923 = r139920 - r139922;
        double r139924 = r139923 * r139919;
        double r139925 = r139919 * r139921;
        double r139926 = r139921 * r139919;
        double r139927 = r139925 + r139926;
        double r139928 = r139927 * r139921;
        double r139929 = r139924 - r139928;
        return r139929;
}
↓
double f(double x_re, double x_im) {
        double r139930 = x_re;
        double r139931 = 3.0;
        double r139932 = pow(r139930, r139931);
        double r139933 = x_im;
        double r139934 = r139930 * r139933;
        double r139935 = r139933 * r139934;
        double r139936 = r139931 * r139935;
        double r139937 = r139932 - r139936;
        return r139937;
}

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

Original7.6
Target0.2
Herbie0.2
\[\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right)\]

Derivation

  1. Initial program 7.6

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

  2. Simplified0.2

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

  4. Applied add-cube-cbrt0.2

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

  5. Applied associate-*l*0.2

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

  7. Applied associate-*r*0.2

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

  8. Simplified0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019308 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3 x.im))))

  (- (* (- (* x.re x.re) (* x.im x.im)) x.re) (* (+ (* x.re x.im) (* x.im x.re)) x.im)))