Average Error: 6.9 → 0.2
Time: 52.9s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[\left(\left(x.re - x.im\right) \cdot x.re\right) \cdot \left(x.re + x.im\right) - x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\]
\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
↓
\left(\left(x.re - x.im\right) \cdot x.re\right) \cdot \left(x.re + x.im\right) - x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)
double f(double x_re, double x_im) {
        double r48819834 = x_re;
        double r48819835 = r48819834 * r48819834;
        double r48819836 = x_im;
        double r48819837 = r48819836 * r48819836;
        double r48819838 = r48819835 - r48819837;
        double r48819839 = r48819838 * r48819834;
        double r48819840 = r48819834 * r48819836;
        double r48819841 = r48819836 * r48819834;
        double r48819842 = r48819840 + r48819841;
        double r48819843 = r48819842 * r48819836;
        double r48819844 = r48819839 - r48819843;
        return r48819844;
}
↓
double f(double x_re, double x_im) {
        double r48819845 = x_re;
        double r48819846 = x_im;
        double r48819847 = r48819845 - r48819846;
        double r48819848 = r48819847 * r48819845;
        double r48819849 = r48819845 + r48819846;
        double r48819850 = r48819848 * r48819849;
        double r48819851 = r48819846 * r48819845;
        double r48819852 = r48819851 + r48819851;
        double r48819853 = r48819846 * r48819852;
        double r48819854 = r48819850 - r48819853;
        return r48819854;
}

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

Original6.9
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 6.9

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

  2. Taylor expanded around 0 6.8

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

  3. Simplified0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019104 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"

  :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)))