Average Error: 7.1 → 0.2
Time: 4.7s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[\mathsf{fma}\left(3, \left(x.re \cdot \left(-x.im\right)\right) \cdot x.im, {x.re}^{3}\right)\]
\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(3, \left(x.re \cdot \left(-x.im\right)\right) \cdot x.im, {x.re}^{3}\right)
double f(double x_re, double x_im) {
        double r204171 = x_re;
        double r204172 = r204171 * r204171;
        double r204173 = x_im;
        double r204174 = r204173 * r204173;
        double r204175 = r204172 - r204174;
        double r204176 = r204175 * r204171;
        double r204177 = r204171 * r204173;
        double r204178 = r204173 * r204171;
        double r204179 = r204177 + r204178;
        double r204180 = r204179 * r204173;
        double r204181 = r204176 - r204180;
        return r204181;
}
↓
double f(double x_re, double x_im) {
        double r204182 = 3.0;
        double r204183 = x_re;
        double r204184 = x_im;
        double r204185 = -r204184;
        double r204186 = r204183 * r204185;
        double r204187 = r204186 * r204184;
        double r204188 = pow(r204183, r204182);
        double r204189 = fma(r204182, r204187, r204188);
        return r204189;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original7.1
Target0.3
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 7.1

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

  2. Simplified7.0

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

  4. Applied distribute-lft-neg-in7.0

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

  5. Applied associate-*r*0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2020036 +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)))