Average Error: 7.7 → 0.2
Time: 15.2s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[\mathsf{fma}\left(x.im, -3 \cdot \left(x.im \cdot x.re\right), {x.re}^{3}\right)\]
\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im
↓
\mathsf{fma}\left(x.im, -3 \cdot \left(x.im \cdot x.re\right), {x.re}^{3}\right)
double f(double x_re, double x_im) {
        double r226179 = x_re;
        double r226180 = r226179 * r226179;
        double r226181 = x_im;
        double r226182 = r226181 * r226181;
        double r226183 = r226180 - r226182;
        double r226184 = r226183 * r226179;
        double r226185 = r226179 * r226181;
        double r226186 = r226181 * r226179;
        double r226187 = r226185 + r226186;
        double r226188 = r226187 * r226181;
        double r226189 = r226184 - r226188;
        return r226189;
}
↓
double f(double x_re, double x_im) {
        double r226190 = x_im;
        double r226191 = -3.0;
        double r226192 = x_re;
        double r226193 = r226190 * r226192;
        double r226194 = r226191 * r226193;
        double r226195 = 3.0;
        double r226196 = pow(r226192, r226195);
        double r226197 = fma(r226190, r226194, r226196);
        return r226197;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original7.7
Target0.2
Herbie0.2
\[\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right)\]

Derivation

  1. Initial program 7.7

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

  2. Simplified0.2

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3 x.im))))

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