Average Error: 7.4 → 0.2
Time: 27.3s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\mathsf{fma}\left(x.re, \left(3 \cdot x.im\right) \cdot x.re, -{x.im}^{3}\right)\]
\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re
↓
\mathsf{fma}\left(x.re, \left(3 \cdot x.im\right) \cdot x.re, -{x.im}^{3}\right)
double f(double x_re, double x_im) {
        double r162163 = x_re;
        double r162164 = r162163 * r162163;
        double r162165 = x_im;
        double r162166 = r162165 * r162165;
        double r162167 = r162164 - r162166;
        double r162168 = r162167 * r162165;
        double r162169 = r162163 * r162165;
        double r162170 = r162165 * r162163;
        double r162171 = r162169 + r162170;
        double r162172 = r162171 * r162163;
        double r162173 = r162168 + r162172;
        return r162173;
}
↓
double f(double x_re, double x_im) {
        double r162174 = x_re;
        double r162175 = 3.0;
        double r162176 = x_im;
        double r162177 = r162175 * r162176;
        double r162178 = r162177 * r162174;
        double r162179 = pow(r162176, r162175);
        double r162180 = -r162179;
        double r162181 = fma(r162174, r162178, r162180);
        return r162181;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original7.4
Target0.2
Herbie0.2
\[\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\]

Derivation

  1. Initial program 7.4

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

  2. Simplified0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, 3 \cdot \left(x.im \cdot x.re\right), -{x.im}^{3}\right)}\]
  3. Using strategy rm

  4. Applied associate-*r*0.2

    \[\leadsto \mathsf{fma}\left(x.re, \color{blue}{\left(3 \cdot x.im\right) \cdot x.re}, -{x.im}^{3}\right)\]

  5. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(x.re, \left(3 \cdot x.im\right) \cdot x.re, -{x.im}^{3}\right)\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.im) (* 2 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im)))

  (+ (* (- (* x.re x.re) (* x.im x.im)) x.im) (* (+ (* x.re x.im) (* x.im x.re)) x.re)))