Average Error: 0.4 → 0.3
Time: 19.7s
Precision: 64
\[\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right) + \left(\left(\left(x.re \cdot x.im\right) + \left(x.im \cdot x.re\right)\right) \cdot x.re\right)\]
↓
\[\left(\mathsf{qma}\left(\left(\left(x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\right)\right), \left(\left(x.re \cdot x.im\right) + \left(x.re \cdot x.im\right)\right), x.re\right)\right)\]
\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right) + \left(\left(\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.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\right)\right), \left(\left(x.re \cdot x.im\right) + \left(x.re \cdot x.im\right)\right), x.re\right)\right)
double f(double x_re, double x_im) {
        double r3232183 = x_re;
        double r3232184 = r3232183 * r3232183;
        double r3232185 = x_im;
        double r3232186 = r3232185 * r3232185;
        double r3232187 = r3232184 - r3232186;
        double r3232188 = r3232187 * r3232185;
        double r3232189 = r3232183 * r3232185;
        double r3232190 = r3232185 * r3232183;
        double r3232191 = r3232189 + r3232190;
        double r3232192 = r3232191 * r3232183;
        double r3232193 = r3232188 + r3232192;
        return r3232193;
}
↓
double f(double x_re, double x_im) {
        double r3232194 = x_im;
        double r3232195 = x_re;
        double r3232196 = r3232195 - r3232194;
        double r3232197 = r3232194 + r3232195;
        double r3232198 = r3232196 * r3232197;
        double r3232199 = r3232194 * r3232198;
        double r3232200 = /*Error: no posit support in C */;
        double r3232201 = r3232195 * r3232194;
        double r3232202 = r3232201 + r3232201;
        double r3232203 = /*Error: no posit support in C */;
        double r3232204 = /*Error: no posit support in C */;
        return r3232204;
}

Error

Bits error versus x.re

Bits error versus x.im

Derivation

  1. Initial program 0.4

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

  3. Applied introduce-quire0.4

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

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

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

  5. Simplified0.3

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

  7. Applied distribute-rgt-in0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 0 
(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)))