Average Error: 7.3 → 0.2
Time: 25.4s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[{x.re}^{3} - x.im \cdot \left(\left(x.im \cdot 3\right) \cdot x.re\right)\]
\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} - x.im \cdot \left(\left(x.im \cdot 3\right) \cdot x.re\right)
double f(double x_re, double x_im) {
        double r155447 = x_re;
        double r155448 = r155447 * r155447;
        double r155449 = x_im;
        double r155450 = r155449 * r155449;
        double r155451 = r155448 - r155450;
        double r155452 = r155451 * r155447;
        double r155453 = r155447 * r155449;
        double r155454 = r155449 * r155447;
        double r155455 = r155453 + r155454;
        double r155456 = r155455 * r155449;
        double r155457 = r155452 - r155456;
        return r155457;
}
↓
double f(double x_re, double x_im) {
        double r155458 = x_re;
        double r155459 = 3.0;
        double r155460 = pow(r155458, r155459);
        double r155461 = x_im;
        double r155462 = r155461 * r155459;
        double r155463 = r155462 * r155458;
        double r155464 = r155461 * r155463;
        double r155465 = r155460 - r155464;
        return r155465;
}

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

Original7.3
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 7.3

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

  2. Simplified0.2

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

  4. Applied distribute-lft1-in0.2

    \[\leadsto {x.re}^{3} - x.im \cdot \left(x.im \cdot \color{blue}{\left(\left(2 + 1\right) \cdot x.re\right)}\right)\]

  5. Applied associate-*r*0.2

    \[\leadsto {x.re}^{3} - x.im \cdot \color{blue}{\left(\left(x.im \cdot \left(2 + 1\right)\right) \cdot x.re\right)}\]

  6. Simplified0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019323 
(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)))