Average Error: 7.1 → 0.2
Time: 12.7s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[{x.re}^{3} - \left(\left(3 \cdot x.im\right) \cdot x.re\right) \cdot x.im\]
\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
↓
{x.re}^{3} - \left(\left(3 \cdot x.im\right) \cdot x.re\right) \cdot x.im
double f(double x_re, double x_im) {
        double r128171 = x_re;
        double r128172 = r128171 * r128171;
        double r128173 = x_im;
        double r128174 = r128173 * r128173;
        double r128175 = r128172 - r128174;
        double r128176 = r128175 * r128171;
        double r128177 = r128171 * r128173;
        double r128178 = r128173 * r128171;
        double r128179 = r128177 + r128178;
        double r128180 = r128179 * r128173;
        double r128181 = r128176 - r128180;
        return r128181;
}
↓
double f(double x_re, double x_im) {
        double r128182 = x_re;
        double r128183 = 3.0;
        double r128184 = pow(r128182, r128183);
        double r128185 = x_im;
        double r128186 = r128183 * r128185;
        double r128187 = r128186 * r128182;
        double r128188 = r128187 * r128185;
        double r128189 = r128184 - r128188;
        return r128189;
}

Error

Bits error versus x.re

Bits error versus x.im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original7.1
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 7.1

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

  2. Simplified0.2

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

  4. Applied associate-*r*0.2

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

  6. Applied associate-*r*0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019291 
(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)))