Average Error: 0.4 → 0.3
Time: 1.1m
Precision: 64
\[\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)\]
↓
\[\left(\left(x.re \cdot \left(\mathsf{qms}\left(\left(\left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\right), \left(x.im + x.im\right), x.im\right)\right)\right)\right)\]
\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)
↓
\left(\left(x.re \cdot \left(\mathsf{qms}\left(\left(\left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\right), \left(x.im + x.im\right), x.im\right)\right)\right)\right)
double f(double x_re, double x_im) {
        double r2616174 = x_re;
        double r2616175 = r2616174 * r2616174;
        double r2616176 = x_im;
        double r2616177 = r2616176 * r2616176;
        double r2616178 = r2616175 - r2616177;
        double r2616179 = r2616178 * r2616174;
        double r2616180 = r2616174 * r2616176;
        double r2616181 = r2616176 * r2616174;
        double r2616182 = r2616180 + r2616181;
        double r2616183 = r2616182 * r2616176;
        double r2616184 = r2616179 - r2616183;
        return r2616184;
}
↓
double f(double x_re, double x_im) {
        double r2616185 = x_re;
        double r2616186 = x_im;
        double r2616187 = r2616185 - r2616186;
        double r2616188 = r2616186 + r2616185;
        double r2616189 = r2616187 * r2616188;
        double r2616190 = /*Error: no posit support in C */;
        double r2616191 = r2616186 + r2616186;
        double r2616192 = /*Error: no posit support in C */;
        double r2616193 = /*Error: no posit support in C */;
        double r2616194 = r2616185 * r2616193;
        double r2616195 = /*Error: no posit support in C */;
        double r2616196 = /*Error: no posit support in C */;
        return r2616196;
}

Error

Bits error versus x.re

Bits error versus x.im

Derivation

  1. Initial program 0.4

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

  3. Applied introduce-quire0.4

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

  4. Simplified0.4

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

  6. Applied introduce-quire0.4

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

  7. Applied insert-quire-fdp-sub0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019151 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  (-.p16 (*.p16 (-.p16 (*.p16 x.re x.re) (*.p16 x.im x.im)) x.re) (*.p16 (+.p16 (*.p16 x.re x.im) (*.p16 x.im x.re)) x.im)))