Average Error: 7.0 → 0.2
Time: 17.0s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[x.re \cdot \left(\left(x.im \cdot 3\right) \cdot x.re\right) - {x.im}^{3}\]
\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(x.im \cdot 3\right) \cdot x.re\right) - {x.im}^{3}
double f(double x_re, double x_im) {
        double r8328421 = x_re;
        double r8328422 = r8328421 * r8328421;
        double r8328423 = x_im;
        double r8328424 = r8328423 * r8328423;
        double r8328425 = r8328422 - r8328424;
        double r8328426 = r8328425 * r8328423;
        double r8328427 = r8328421 * r8328423;
        double r8328428 = r8328423 * r8328421;
        double r8328429 = r8328427 + r8328428;
        double r8328430 = r8328429 * r8328421;
        double r8328431 = r8328426 + r8328430;
        return r8328431;
}
↓
double f(double x_re, double x_im) {
        double r8328432 = x_re;
        double r8328433 = x_im;
        double r8328434 = 3.0;
        double r8328435 = r8328433 * r8328434;
        double r8328436 = r8328435 * r8328432;
        double r8328437 = r8328432 * r8328436;
        double r8328438 = pow(r8328433, r8328434);
        double r8328439 = r8328437 - r8328438;
        return r8328439;
}

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
Target0.3
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 7.0

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

  2. Taylor expanded around -inf 7.0

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

  3. Simplified0.3

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

  4. Taylor expanded around -inf 7.0

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

  5. Simplified0.3

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

  7. Applied pow20.3

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

  8. Applied pow10.3

    \[\leadsto \left(\left(x.im \cdot 3\right) \cdot x.re\right) \cdot x.re - \color{blue}{{x.im}^{1}} \cdot {x.im}^{2}\]

  9. Applied pow-prod-up0.2

    \[\leadsto \left(\left(x.im \cdot 3\right) \cdot x.re\right) \cdot x.re - \color{blue}{{x.im}^{\left(1 + 2\right)}}\]

  10. Simplified0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(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)))