Average Error: 7.4 → 0.2
Time: 3.2s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re - {x.im}^{3}\]
\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(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re - {x.im}^{3}
double f(double x_re, double x_im) {
        double r227494 = x_re;
        double r227495 = r227494 * r227494;
        double r227496 = x_im;
        double r227497 = r227496 * r227496;
        double r227498 = r227495 - r227497;
        double r227499 = r227498 * r227496;
        double r227500 = r227494 * r227496;
        double r227501 = r227496 * r227494;
        double r227502 = r227500 + r227501;
        double r227503 = r227502 * r227494;
        double r227504 = r227499 + r227503;
        return r227504;
}
↓
double f(double x_re, double x_im) {
        double r227505 = 3.0;
        double r227506 = x_im;
        double r227507 = x_re;
        double r227508 = r227506 * r227507;
        double r227509 = r227505 * r227508;
        double r227510 = r227509 * r227507;
        double r227511 = pow(r227506, r227505);
        double r227512 = r227510 - r227511;
        return r227512;
}

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

Original7.4
Target0.3
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 7.4

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

  2. Simplified7.3

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

  4. Applied associate-*r*0.2

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

  6. Applied associate-*r*0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2020033 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :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)))