Average Error: 0.4 → 0.4
Time: 10.7s
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
↓
\[x.im \cdot \left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right) + x.re \cdot \left(x.re + x.re\right)\right)\]
\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}
↓
x.im \cdot \left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right) + x.re \cdot \left(x.re + x.re\right)\right)
double f(double x_re, double x_im) {
        double r1358185 = x_re;
        double r1358186 = r1358185 * r1358185;
        double r1358187 = x_im;
        double r1358188 = r1358187 * r1358187;
        double r1358189 = r1358186 - r1358188;
        double r1358190 = r1358189 * r1358187;
        double r1358191 = r1358185 * r1358187;
        double r1358192 = r1358187 * r1358185;
        double r1358193 = r1358191 + r1358192;
        double r1358194 = r1358193 * r1358185;
        double r1358195 = r1358190 + r1358194;
        return r1358195;
}
↓
double f(double x_re, double x_im) {
        double r1358196 = x_im;
        double r1358197 = x_re;
        double r1358198 = r1358196 + r1358197;
        double r1358199 = r1358197 - r1358196;
        double r1358200 = r1358198 * r1358199;
        double r1358201 = r1358197 + r1358197;
        double r1358202 = r1358197 * r1358201;
        double r1358203 = r1358200 + r1358202;
        double r1358204 = r1358196 * r1358203;
        return r1358204;
}

Error

Bits error versus x.re

Bits error versus x.im

Derivation

  1. Initial program 0.4

    \[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]

  2. Simplified0.4

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

  3. Final simplification0.4

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

Reproduce

herbie shell --seed 2019121 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  (+.p16 (*.p16 (-.p16 (*.p16 x.re x.re) (*.p16 x.im x.im)) x.im) (*.p16 (+.p16 (*.p16 x.re x.im) (*.p16 x.im x.re)) x.re)))