Average Error: 6.8 → 0.3
Time: 46.4s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\left(\left(x.re - x.im\right) \cdot x.im\right) \cdot \left(x.im + x.re\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\]
\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.im + x.re\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)
double f(double x_re, double x_im) {
        double r30892591 = x_re;
        double r30892592 = r30892591 * r30892591;
        double r30892593 = x_im;
        double r30892594 = r30892593 * r30892593;
        double r30892595 = r30892592 - r30892594;
        double r30892596 = r30892595 * r30892593;
        double r30892597 = r30892591 * r30892593;
        double r30892598 = r30892593 * r30892591;
        double r30892599 = r30892597 + r30892598;
        double r30892600 = r30892599 * r30892591;
        double r30892601 = r30892596 + r30892600;
        return r30892601;
}
↓
double f(double x_re, double x_im) {
        double r30892602 = x_re;
        double r30892603 = x_im;
        double r30892604 = r30892602 - r30892603;
        double r30892605 = r30892604 * r30892603;
        double r30892606 = r30892603 + r30892602;
        double r30892607 = r30892605 * r30892606;
        double r30892608 = r30892602 * r30892603;
        double r30892609 = r30892608 + r30892608;
        double r30892610 = r30892602 * r30892609;
        double r30892611 = r30892607 + r30892610;
        return r30892611;
}

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

Original6.8
Target0.3
Herbie0.3
\[\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\]

Derivation

  1. Initial program 6.8

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

  3. Applied difference-of-squares6.8

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

  4. Applied associate-*l*0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"

  :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)))