Average Error: 6.9 → 0.2
Time: 2.1m
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 r56104493 = x_re;
        double r56104494 = r56104493 * r56104493;
        double r56104495 = x_im;
        double r56104496 = r56104495 * r56104495;
        double r56104497 = r56104494 - r56104496;
        double r56104498 = r56104497 * r56104495;
        double r56104499 = r56104493 * r56104495;
        double r56104500 = r56104495 * r56104493;
        double r56104501 = r56104499 + r56104500;
        double r56104502 = r56104501 * r56104493;
        double r56104503 = r56104498 + r56104502;
        return r56104503;
}
↓
double f(double x_re, double x_im) {
        double r56104504 = x_im;
        double r56104505 = x_re;
        double r56104506 = r56104504 + r56104505;
        double r56104507 = r56104506 * r56104504;
        double r56104508 = r56104505 - r56104504;
        double r56104509 = r56104507 * r56104508;
        double r56104510 = r56104505 * r56104504;
        double r56104511 = r56104510 + r56104510;
        double r56104512 = r56104505 * r56104511;
        double r56104513 = r56104509 + r56104512;
        return r56104513;
}

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
\[\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\]

Derivation

  1. Initial program 6.9

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

  2. Taylor expanded around 0 6.9

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

  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 2019125 
(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)))