Average Error: 6.9 → 0.2
Time: 2.6s
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 r174018 = x_re;
        double r174019 = r174018 * r174018;
        double r174020 = x_im;
        double r174021 = r174020 * r174020;
        double r174022 = r174019 - r174021;
        double r174023 = r174022 * r174020;
        double r174024 = r174018 * r174020;
        double r174025 = r174020 * r174018;
        double r174026 = r174024 + r174025;
        double r174027 = r174026 * r174018;
        double r174028 = r174023 + r174027;
        return r174028;
}
↓
double f(double x_re, double x_im) {
        double r174029 = 3.0;
        double r174030 = x_im;
        double r174031 = r174029 * r174030;
        double r174032 = x_re;
        double r174033 = r174031 * r174032;
        double r174034 = r174033 * r174032;
        double r174035 = pow(r174030, r174029);
        double r174036 = r174034 - r174035;
        return r174036;
}

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

Original6.9
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 6.9

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

  2. Simplified6.9

    \[\leadsto \color{blue}{3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right) - {x.im}^{3}}\]
  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 2020064 +o rules:numerics
(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)))