Average Error: 0.4 → 0.3
Time: 45.4s
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
↓
\[\left(\mathsf{qma}\left(\left(\left(\left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.im + x.re\right)\right)\right), \left(x.re \cdot x.im + x.re \cdot x.im\right), x.re\right)\right)\]
\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)}
↓
\left(\mathsf{qma}\left(\left(\left(\left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.im + x.re\right)\right)\right), \left(x.re \cdot x.im + x.re \cdot x.im\right), x.re\right)\right)
double f(double x_re, double x_im) {
        double r1994250 = x_re;
        double r1994251 = r1994250 * r1994250;
        double r1994252 = x_im;
        double r1994253 = r1994252 * r1994252;
        double r1994254 = r1994251 - r1994253;
        double r1994255 = r1994254 * r1994252;
        double r1994256 = r1994250 * r1994252;
        double r1994257 = r1994252 * r1994250;
        double r1994258 = r1994256 + r1994257;
        double r1994259 = r1994258 * r1994250;
        double r1994260 = r1994255 + r1994259;
        return r1994260;
}
↓
double f(double x_re, double x_im) {
        double r1994261 = x_im;
        double r1994262 = x_re;
        double r1994263 = r1994262 - r1994261;
        double r1994264 = r1994261 * r1994263;
        double r1994265 = r1994261 + r1994262;
        double r1994266 = r1994264 * r1994265;
        double r1994267 = /*Error: no posit support in C */;
        double r1994268 = r1994262 * r1994261;
        double r1994269 = r1994268 + r1994268;
        double r1994270 = /*Error: no posit support in C */;
        double r1994271 = /*Error: no posit support in C */;
        return r1994271;
}

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 introduce-quire0.4

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

  4. Applied insert-quire-fdp-add0.3

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

  5. Simplified0.3

    \[\leadsto \color{blue}{\left(\mathsf{qma}\left(\left(\left(\left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(\frac{x.im}{x.re}\right)\right)\right), \left(x.re \cdot \left(\frac{x.im}{x.im}\right)\right), x.re\right)\right)}\]
  6. Using strategy rm

  7. Applied distribute-lft-in0.3

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

  8. Final simplification0.3

    \[\leadsto \left(\mathsf{qma}\left(\left(\left(\left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.im + x.re\right)\right)\right), \left(x.re \cdot x.im + x.re \cdot x.im\right), x.re\right)\right)\]

Reproduce

herbie shell --seed 2019152 
(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)))