Average Error: 6.8 → 0.3
Time: 1.7m
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[\left(\left(x.re - x.im\right) \cdot x.re\right) \cdot \left(x.re + x.im\right) - x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\]
double f(double x_re, double x_im) {
        double r54500025 = x_re;
        double r54500026 = r54500025 * r54500025;
        double r54500027 = x_im;
        double r54500028 = r54500027 * r54500027;
        double r54500029 = r54500026 - r54500028;
        double r54500030 = r54500029 * r54500025;
        double r54500031 = r54500025 * r54500027;
        double r54500032 = r54500027 * r54500025;
        double r54500033 = r54500031 + r54500032;
        double r54500034 = r54500033 * r54500027;
        double r54500035 = r54500030 - r54500034;
        return r54500035;
}
↓
double f(double x_re, double x_im) {
        double r54500036 = x_re;
        double r54500037 = x_im;
        double r54500038 = r54500036 - r54500037;
        double r54500039 = r54500038 * r54500036;
        double r54500040 = r54500036 + r54500037;
        double r54500041 = r54500039 * r54500040;
        double r54500042 = r54500037 * r54500036;
        double r54500043 = r54500042 + r54500042;
        double r54500044 = r54500037 * r54500043;
        double r54500045 = r54500041 - r54500044;
        return r54500045;
}


\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)

Error

Bits error versus x.re

Bits error versus x.im

Target

Original6.8
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 6.8

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

  2. Taylor expanded around inf 6.7

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

  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. 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 2019101 
(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)))