Average Error: 7.0 → 0.2
Time: 8.3s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[x.re \cdot \left(\left(3 \cdot x.re\right) \cdot x.im\right) - {x.im}^{3}\]
\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
↓
x.re \cdot \left(\left(3 \cdot x.re\right) \cdot x.im\right) - {x.im}^{3}
double f(double x_re, double x_im) {
        double r3246544 = x_re;
        double r3246545 = r3246544 * r3246544;
        double r3246546 = x_im;
        double r3246547 = r3246546 * r3246546;
        double r3246548 = r3246545 - r3246547;
        double r3246549 = r3246548 * r3246546;
        double r3246550 = r3246544 * r3246546;
        double r3246551 = r3246546 * r3246544;
        double r3246552 = r3246550 + r3246551;
        double r3246553 = r3246552 * r3246544;
        double r3246554 = r3246549 + r3246553;
        return r3246554;
}
↓
double f(double x_re, double x_im) {
        double r3246555 = x_re;
        double r3246556 = 3.0;
        double r3246557 = r3246556 * r3246555;
        double r3246558 = x_im;
        double r3246559 = r3246557 * r3246558;
        double r3246560 = r3246555 * r3246559;
        double r3246561 = pow(r3246558, r3246556);
        double r3246562 = r3246560 - r3246561;
        return r3246562;
}

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

Original7.0
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 7.0

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

  2. Simplified7.0

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

  3. Taylor expanded around 0 6.9

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

  4. Simplified0.2

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

  6. Applied pow10.2

    \[\leadsto \left(3 \cdot x.re\right) \cdot \left(x.im \cdot x.re\right) - \left(x.im \cdot x.im\right) \cdot \color{blue}{{x.im}^{1}}\]

  7. Applied pow10.2

    \[\leadsto \left(3 \cdot x.re\right) \cdot \left(x.im \cdot x.re\right) - \left(x.im \cdot \color{blue}{{x.im}^{1}}\right) \cdot {x.im}^{1}\]

  8. Applied pow10.2

    \[\leadsto \left(3 \cdot x.re\right) \cdot \left(x.im \cdot x.re\right) - \left(\color{blue}{{x.im}^{1}} \cdot {x.im}^{1}\right) \cdot {x.im}^{1}\]

  9. Applied pow-prod-up0.2

    \[\leadsto \left(3 \cdot x.re\right) \cdot \left(x.im \cdot x.re\right) - \color{blue}{{x.im}^{\left(1 + 1\right)}} \cdot {x.im}^{1}\]

  10. Applied pow-prod-up0.2

    \[\leadsto \left(3 \cdot x.re\right) \cdot \left(x.im \cdot x.re\right) - \color{blue}{{x.im}^{\left(\left(1 + 1\right) + 1\right)}}\]

  11. Simplified0.2

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

  13. Applied associate-*r*0.2

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

  14. Final simplification0.2

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

Reproduce

herbie shell --seed 2019156 
(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)))