Average Error: 7.5 → 0.2
Time: 3.5s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[{x.re}^{3} - \left(3 \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\]
\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(3 \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)
double f(double x_re, double x_im) {
        double r173957 = x_re;
        double r173958 = r173957 * r173957;
        double r173959 = x_im;
        double r173960 = r173959 * r173959;
        double r173961 = r173958 - r173960;
        double r173962 = r173961 * r173957;
        double r173963 = r173957 * r173959;
        double r173964 = r173959 * r173957;
        double r173965 = r173963 + r173964;
        double r173966 = r173965 * r173959;
        double r173967 = r173962 - r173966;
        return r173967;
}
↓
double f(double x_re, double x_im) {
        double r173968 = x_re;
        double r173969 = 3.0;
        double r173970 = pow(r173968, r173969);
        double r173971 = x_im;
        double r173972 = r173969 * r173971;
        double r173973 = r173968 * r173971;
        double r173974 = r173972 * r173973;
        double r173975 = r173970 - r173974;
        return r173975;
}

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

Original7.5
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 7.5

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

  2. Simplified0.2

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

  4. Applied associate-*r*0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2020033 
(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)))