Average Error: 7.0 → 0.3
Time: 1.1m
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[\left(\left(x.re + x.im\right) \cdot x.re\right) \cdot \left(x.re - x.im\right) - x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\]
\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
↓
\left(\left(x.re + x.im\right) \cdot x.re\right) \cdot \left(x.re - x.im\right) - x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)
double f(double x_re, double x_im) {
        double r31893970 = x_re;
        double r31893971 = r31893970 * r31893970;
        double r31893972 = x_im;
        double r31893973 = r31893972 * r31893972;
        double r31893974 = r31893971 - r31893973;
        double r31893975 = r31893974 * r31893970;
        double r31893976 = r31893970 * r31893972;
        double r31893977 = r31893972 * r31893970;
        double r31893978 = r31893976 + r31893977;
        double r31893979 = r31893978 * r31893972;
        double r31893980 = r31893975 - r31893979;
        return r31893980;
}
↓
double f(double x_re, double x_im) {
        double r31893981 = x_re;
        double r31893982 = x_im;
        double r31893983 = r31893981 + r31893982;
        double r31893984 = r31893983 * r31893981;
        double r31893985 = r31893981 - r31893982;
        double r31893986 = r31893984 * r31893985;
        double r31893987 = r31893982 * r31893981;
        double r31893988 = r31893987 + r31893987;
        double r31893989 = r31893982 * r31893988;
        double r31893990 = r31893986 - r31893989;
        return r31893990;
}

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
Herbie0.3
\[\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right)\]

Derivation

  1. Initial program 7.0

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

  2. Taylor expanded around -inf 6.9

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

  3. Simplified0.3

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

  5. Applied associate-*r*0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019121 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"

  :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)))