Average Error: 7.0 → 0.2
Time: 2.0s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[{x.re}^{3} - \left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.im\]
\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 \left(x.im \cdot x.re\right)\right) \cdot x.im
double f(double x_re, double x_im) {
        double r177046 = x_re;
        double r177047 = r177046 * r177046;
        double r177048 = x_im;
        double r177049 = r177048 * r177048;
        double r177050 = r177047 - r177049;
        double r177051 = r177050 * r177046;
        double r177052 = r177046 * r177048;
        double r177053 = r177048 * r177046;
        double r177054 = r177052 + r177053;
        double r177055 = r177054 * r177048;
        double r177056 = r177051 - r177055;
        return r177056;
}
↓
double f(double x_re, double x_im) {
        double r177057 = x_re;
        double r177058 = 3.0;
        double r177059 = pow(r177057, r177058);
        double r177060 = x_im;
        double r177061 = r177060 * r177057;
        double r177062 = r177058 * r177061;
        double r177063 = r177062 * r177060;
        double r177064 = r177059 - r177063;
        return r177064;
}

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. Using strategy rm

  6. Applied associate-*r*0.2

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

  8. Applied associate-*l*0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2020056 
(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)))