Average Error: 7.3 → 0.2
Time: 11.5s
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 r107965 = x_re;
        double r107966 = r107965 * r107965;
        double r107967 = x_im;
        double r107968 = r107967 * r107967;
        double r107969 = r107966 - r107968;
        double r107970 = r107969 * r107967;
        double r107971 = r107965 * r107967;
        double r107972 = r107967 * r107965;
        double r107973 = r107971 + r107972;
        double r107974 = r107973 * r107965;
        double r107975 = r107970 + r107974;
        return r107975;
}
↓
double f(double x_re, double x_im) {
        double r107976 = 3.0;
        double r107977 = x_im;
        double r107978 = r107976 * r107977;
        double r107979 = x_re;
        double r107980 = r107978 * r107979;
        double r107981 = r107980 * r107979;
        double r107982 = pow(r107977, r107976);
        double r107983 = r107981 - r107982;
        return r107983;
}

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

Original7.3
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 7.3

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

  2. Simplified0.2

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

  4. Applied associate-*r*0.2

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

  6. Applied associate-*r*0.2

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

  7. 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 2019195 +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)))