Average Error: 6.8 → 0.2
Time: 1.2m
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[(\left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\right))_*\]
double f(double x_re, double x_im) {
        double r42004049 = x_re;
        double r42004050 = r42004049 * r42004049;
        double r42004051 = x_im;
        double r42004052 = r42004051 * r42004051;
        double r42004053 = r42004050 - r42004052;
        double r42004054 = r42004053 * r42004051;
        double r42004055 = r42004049 * r42004051;
        double r42004056 = r42004051 * r42004049;
        double r42004057 = r42004055 + r42004056;
        double r42004058 = r42004057 * r42004049;
        double r42004059 = r42004054 + r42004058;
        return r42004059;
}
↓
double f(double x_re, double x_im) {
        double r42004060 = x_im;
        double r42004061 = x_re;
        double r42004062 = r42004061 + r42004060;
        double r42004063 = r42004060 * r42004062;
        double r42004064 = r42004061 - r42004060;
        double r42004065 = r42004060 * r42004061;
        double r42004066 = r42004065 + r42004065;
        double r42004067 = r42004061 * r42004066;
        double r42004068 = fma(r42004063, r42004064, r42004067);
        return r42004068;
}


\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(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\right))_*

Error

Bits error versus x.re

Bits error versus x.im

Target

Original6.8
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 6.8

    \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
  2. Using strategy rm

  3. Applied difference-of-squares6.8

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

  4. Applied associate-*l*0.2

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

  5. Taylor expanded around inf 6.7

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

  6. Simplified0.2

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

  8. Applied fma-def0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"

  :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)))