Average Error: 6.9 → 0.2
Time: 52.7s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[(\left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\right))_*\]
\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 r33281230 = x_re;
        double r33281231 = r33281230 * r33281230;
        double r33281232 = x_im;
        double r33281233 = r33281232 * r33281232;
        double r33281234 = r33281231 - r33281233;
        double r33281235 = r33281234 * r33281232;
        double r33281236 = r33281230 * r33281232;
        double r33281237 = r33281232 * r33281230;
        double r33281238 = r33281236 + r33281237;
        double r33281239 = r33281238 * r33281230;
        double r33281240 = r33281235 + r33281239;
        return r33281240;
}
↓
double f(double x_re, double x_im) {
        double r33281241 = x_im;
        double r33281242 = x_re;
        double r33281243 = r33281242 + r33281241;
        double r33281244 = r33281241 * r33281243;
        double r33281245 = r33281242 - r33281241;
        double r33281246 = r33281241 * r33281242;
        double r33281247 = r33281246 + r33281246;
        double r33281248 = r33281242 * r33281247;
        double r33281249 = fma(r33281244, r33281245, r33281248);
        return r33281249;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original6.9
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 6.9

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

  2. Taylor expanded around inf 6.8

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

  3. Simplified0.2

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

  5. Applied fma-def0.2

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

  6. 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 2019107 +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)))