Average Error: 7.1 → 0.2
Time: 22.8s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\left(\left(x.re + x.im\right) \cdot x.im\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
\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(x.re + x.im\right) \cdot x.im\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re
double f(double x_re, double x_im) {
        double r158625 = x_re;
        double r158626 = r158625 * r158625;
        double r158627 = x_im;
        double r158628 = r158627 * r158627;
        double r158629 = r158626 - r158628;
        double r158630 = r158629 * r158627;
        double r158631 = r158625 * r158627;
        double r158632 = r158627 * r158625;
        double r158633 = r158631 + r158632;
        double r158634 = r158633 * r158625;
        double r158635 = r158630 + r158634;
        return r158635;
}
↓
double f(double x_re, double x_im) {
        double r158636 = x_re;
        double r158637 = x_im;
        double r158638 = r158636 + r158637;
        double r158639 = r158638 * r158637;
        double r158640 = r158636 - r158637;
        double r158641 = r158639 * r158640;
        double r158642 = r158636 * r158637;
        double r158643 = r158637 * r158636;
        double r158644 = r158642 + r158643;
        double r158645 = r158644 * r158636;
        double r158646 = r158641 + r158645;
        return r158646;
}

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
\[\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\]

Derivation

  1. Initial program 7.1

    \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
  2. Using strategy rm

  3. Applied difference-of-squares7.1

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

  4. Applied associate-*l*0.2

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

  5. Taylor expanded around 0 7.0

    \[\leadsto \color{blue}{\left(x.im \cdot {x.re}^{2} - {x.im}^{3}\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]

  6. Simplified0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019325 
(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)))