Average Error: 0.4 → 0.3
Time: 2.2m
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
↓
\[\left(\mathsf{qma}\left(\left(\left(x.im \cdot \left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right)\right)\right)\right), \left(x.im \cdot x.re + x.im \cdot x.re\right), x.re\right)\right)\]
\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(x.im \cdot \left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right)\right)\right)\right), \left(x.im \cdot x.re + x.im \cdot x.re\right), x.re\right)\right)
double f(double x_re, double x_im) {
        double r2058182 = x_re;
        double r2058183 = r2058182 * r2058182;
        double r2058184 = x_im;
        double r2058185 = r2058184 * r2058184;
        double r2058186 = r2058183 - r2058185;
        double r2058187 = r2058186 * r2058184;
        double r2058188 = r2058182 * r2058184;
        double r2058189 = r2058184 * r2058182;
        double r2058190 = r2058188 + r2058189;
        double r2058191 = r2058190 * r2058182;
        double r2058192 = r2058187 + r2058191;
        return r2058192;
}
↓
double f(double x_re, double x_im) {
        double r2058193 = x_im;
        double r2058194 = x_re;
        double r2058195 = r2058193 + r2058194;
        double r2058196 = r2058194 - r2058193;
        double r2058197 = r2058195 * r2058196;
        double r2058198 = r2058193 * r2058197;
        double r2058199 = /*Error: no posit support in C */;
        double r2058200 = r2058193 * r2058194;
        double r2058201 = r2058200 + r2058200;
        double r2058202 = /*Error: no posit support in C */;
        double r2058203 = /*Error: no posit support in C */;
        return r2058203;
}

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

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

  7. Applied distribute-rgt-in0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019144 +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)))