Average Error: 6.7 → 0.2
Time: 43.9s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\left(\left(x.im + x.re\right) \cdot x.im\right) \cdot \left(x.re - x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\]
\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.im + x.re\right) \cdot x.im\right) \cdot \left(x.re - x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)
double f(double x_re, double x_im) {
        double r54643530 = x_re;
        double r54643531 = r54643530 * r54643530;
        double r54643532 = x_im;
        double r54643533 = r54643532 * r54643532;
        double r54643534 = r54643531 - r54643533;
        double r54643535 = r54643534 * r54643532;
        double r54643536 = r54643530 * r54643532;
        double r54643537 = r54643532 * r54643530;
        double r54643538 = r54643536 + r54643537;
        double r54643539 = r54643538 * r54643530;
        double r54643540 = r54643535 + r54643539;
        return r54643540;
}
↓
double f(double x_re, double x_im) {
        double r54643541 = x_im;
        double r54643542 = x_re;
        double r54643543 = r54643541 + r54643542;
        double r54643544 = r54643543 * r54643541;
        double r54643545 = r54643542 - r54643541;
        double r54643546 = r54643544 * r54643545;
        double r54643547 = r54643542 * r54643541;
        double r54643548 = r54643547 + r54643547;
        double r54643549 = r54643542 * r54643548;
        double r54643550 = r54643546 + r54643549;
        return r54643550;
}

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

Original6.7
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 6.7

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

  2. Taylor expanded around 0 6.6

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

  3. Simplified0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019119 
(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)))