Average Error: 7.7 → 0.2
Time: 3.8s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[3 \cdot \left(\left(x.im \cdot x.re\right) \cdot x.re\right) - {x.im}^{3}\]
\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
↓
3 \cdot \left(\left(x.im \cdot x.re\right) \cdot x.re\right) - {x.im}^{3}
double f(double x_re, double x_im) {
        double r191232 = x_re;
        double r191233 = r191232 * r191232;
        double r191234 = x_im;
        double r191235 = r191234 * r191234;
        double r191236 = r191233 - r191235;
        double r191237 = r191236 * r191234;
        double r191238 = r191232 * r191234;
        double r191239 = r191234 * r191232;
        double r191240 = r191238 + r191239;
        double r191241 = r191240 * r191232;
        double r191242 = r191237 + r191241;
        return r191242;
}
↓
double f(double x_re, double x_im) {
        double r191243 = 3.0;
        double r191244 = x_im;
        double r191245 = x_re;
        double r191246 = r191244 * r191245;
        double r191247 = r191246 * r191245;
        double r191248 = r191243 * r191247;
        double r191249 = pow(r191244, r191243);
        double r191250 = r191248 - r191249;
        return r191250;
}

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

Original7.7
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 7.7

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

  2. Simplified7.7

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

  4. Applied associate-*r*0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2020036 +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)))