Average Error: 7.3 → 0.2
Time: 23.7s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[{x.re}^{3} - \left(\left(x.im \cdot 3\right) \cdot x.re\right) \cdot x.im\]
\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
↓
{x.re}^{3} - \left(\left(x.im \cdot 3\right) \cdot x.re\right) \cdot x.im
double f(double x_re, double x_im) {
        double r97468 = x_re;
        double r97469 = r97468 * r97468;
        double r97470 = x_im;
        double r97471 = r97470 * r97470;
        double r97472 = r97469 - r97471;
        double r97473 = r97472 * r97468;
        double r97474 = r97468 * r97470;
        double r97475 = r97470 * r97468;
        double r97476 = r97474 + r97475;
        double r97477 = r97476 * r97470;
        double r97478 = r97473 - r97477;
        return r97478;
}
↓
double f(double x_re, double x_im) {
        double r97479 = x_re;
        double r97480 = 3.0;
        double r97481 = pow(r97479, r97480);
        double r97482 = x_im;
        double r97483 = r97482 * r97480;
        double r97484 = r97483 * r97479;
        double r97485 = r97484 * r97482;
        double r97486 = r97481 - r97485;
        return r97486;
}

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

Original7.3
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 7.3

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

  2. Simplified0.2

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

  4. Applied associate-*r*0.2

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

  6. Applied associate-*r*0.2

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

  7. Simplified0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019323 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :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)))