Average Error: 0.4 → 0.3
Time: 3.0m
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
↓
\[\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
\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)}
↓
\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
double f(double x_re, double x_im) {
        double r1664216 = x_re;
        double r1664217 = r1664216 * r1664216;
        double r1664218 = x_im;
        double r1664219 = r1664218 * r1664218;
        double r1664220 = r1664217 - r1664219;
        double r1664221 = r1664220 * r1664218;
        double r1664222 = r1664216 * r1664218;
        double r1664223 = r1664218 * r1664216;
        double r1664224 = r1664222 + r1664223;
        double r1664225 = r1664224 * r1664216;
        double r1664226 = r1664221 + r1664225;
        return r1664226;
}
↓
double f(double x_re, double x_im) {
        double r1664227 = x_re;
        double r1664228 = x_im;
        double r1664229 = r1664227 + r1664228;
        double r1664230 = r1664227 - r1664228;
        double r1664231 = r1664230 * r1664228;
        double r1664232 = r1664229 * r1664231;
        double r1664233 = r1664227 * r1664228;
        double r1664234 = r1664228 * r1664227;
        double r1664235 = r1664233 + r1664234;
        double r1664236 = r1664235 * r1664227;
        double r1664237 = r1664232 + r1664236;
        return r1664237;
}

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. Using strategy rm

  3. Applied difference-of-squares0.3

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

  4. Applied associate-*l*0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(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)))