Average Error: 7.1 → 0.2
Time: 19.2s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\left(\left(x.re \cdot x.im\right) \cdot 3\right) \cdot x.re - {x.im}^{3}\]
\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(x.re \cdot x.im\right) \cdot 3\right) \cdot x.re - {x.im}^{3}
double f(double x_re, double x_im) {
        double r6431068 = x_re;
        double r6431069 = r6431068 * r6431068;
        double r6431070 = x_im;
        double r6431071 = r6431070 * r6431070;
        double r6431072 = r6431069 - r6431071;
        double r6431073 = r6431072 * r6431070;
        double r6431074 = r6431068 * r6431070;
        double r6431075 = r6431070 * r6431068;
        double r6431076 = r6431074 + r6431075;
        double r6431077 = r6431076 * r6431068;
        double r6431078 = r6431073 + r6431077;
        return r6431078;
}
↓
double f(double x_re, double x_im) {
        double r6431079 = x_re;
        double r6431080 = x_im;
        double r6431081 = r6431079 * r6431080;
        double r6431082 = 3.0;
        double r6431083 = r6431081 * r6431082;
        double r6431084 = r6431083 * r6431079;
        double r6431085 = pow(r6431080, r6431082);
        double r6431086 = r6431084 - r6431085;
        return r6431086;
}

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

Original7.1
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 7.1

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

  2. Taylor expanded around 0 7.0

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

  3. Simplified0.3

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

  5. Applied pow10.3

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

  6. Applied pow10.3

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

  7. Applied pow-prod-up0.3

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

  8. Applied pow10.3

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

  9. Applied pow-prod-up0.2

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

  10. Simplified0.2

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

  12. Applied associate-*r*0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"

  :herbie-target
  (+ (* (* x.re x.im) (* 2.0 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)))