Average Error: 7.5 → 0.2
Time: 3.3s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\left(\left(3 \cdot x.im\right) \cdot x.re\right) \cdot x.re - {x.im}^{3}\]
\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
↓
\left(\left(3 \cdot x.im\right) \cdot x.re\right) \cdot x.re - {x.im}^{3}
double f(double x_re, double x_im) {
        double r184293 = x_re;
        double r184294 = r184293 * r184293;
        double r184295 = x_im;
        double r184296 = r184295 * r184295;
        double r184297 = r184294 - r184296;
        double r184298 = r184297 * r184295;
        double r184299 = r184293 * r184295;
        double r184300 = r184295 * r184293;
        double r184301 = r184299 + r184300;
        double r184302 = r184301 * r184293;
        double r184303 = r184298 + r184302;
        return r184303;
}
↓
double f(double x_re, double x_im) {
        double r184304 = 3.0;
        double r184305 = x_im;
        double r184306 = r184304 * r184305;
        double r184307 = x_re;
        double r184308 = r184306 * r184307;
        double r184309 = r184308 * r184307;
        double r184310 = pow(r184305, r184304);
        double r184311 = r184309 - r184310;
        return r184311;
}

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

Original7.5
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 7.5

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

  2. Simplified7.4

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

  6. Applied associate-*r*0.2

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

  8. Applied associate-*r*0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2020089 
(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)))