Average Error: 7.6 → 0.2
Time: 3.0s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\left(\left(3 \cdot x.im\right) \cdot x.re\right) \cdot x.re - {x.im}^{3}\]
\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(\left(3 \cdot x.im\right) \cdot x.re\right) \cdot x.re - {x.im}^{3}
double f(double x_re, double x_im) {
        double r380467 = x_re;
        double r380468 = r380467 * r380467;
        double r380469 = x_im;
        double r380470 = r380469 * r380469;
        double r380471 = r380468 - r380470;
        double r380472 = r380471 * r380469;
        double r380473 = r380467 * r380469;
        double r380474 = r380469 * r380467;
        double r380475 = r380473 + r380474;
        double r380476 = r380475 * r380467;
        double r380477 = r380472 + r380476;
        return r380477;
}
↓
double f(double x_re, double x_im) {
        double r380478 = 3.0;
        double r380479 = x_im;
        double r380480 = r380478 * r380479;
        double r380481 = x_re;
        double r380482 = r380480 * r380481;
        double r380483 = r380482 * r380481;
        double r380484 = pow(r380479, r380478);
        double r380485 = r380483 - r380484;
        return r380485;
}

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

Derivation

  1. Initial program 7.6

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

  2. Simplified7.5

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

  8. Applied associate-*r*0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2020034 
(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)))