Average Error: 7.0 → 0.2
Time: 2.0s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[{x.re}^{3} - \left(3 \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\]
\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(3 \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)
double f(double x_re, double x_im) {
        double r260367 = x_re;
        double r260368 = r260367 * r260367;
        double r260369 = x_im;
        double r260370 = r260369 * r260369;
        double r260371 = r260368 - r260370;
        double r260372 = r260371 * r260367;
        double r260373 = r260367 * r260369;
        double r260374 = r260369 * r260367;
        double r260375 = r260373 + r260374;
        double r260376 = r260375 * r260369;
        double r260377 = r260372 - r260376;
        return r260377;
}
↓
double f(double x_re, double x_im) {
        double r260378 = x_re;
        double r260379 = 3.0;
        double r260380 = pow(r260378, r260379);
        double r260381 = x_im;
        double r260382 = r260379 * r260381;
        double r260383 = r260378 * r260381;
        double r260384 = r260382 * r260383;
        double r260385 = r260380 - r260384;
        return r260385;
}

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

Original7.0
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 7.0

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

  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. Final simplification0.2

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

Reproduce

herbie shell --seed 2020018 
(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)))