Average Error: 7.2 → 0.2
Time: 25.6s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[{x.re}^{3} - \left(x.im \cdot 3\right) \cdot \left(x.re \cdot x.im\right)\]
\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(x.im \cdot 3\right) \cdot \left(x.re \cdot x.im\right)
double f(double x_re, double x_im) {
        double r170423 = x_re;
        double r170424 = r170423 * r170423;
        double r170425 = x_im;
        double r170426 = r170425 * r170425;
        double r170427 = r170424 - r170426;
        double r170428 = r170427 * r170423;
        double r170429 = r170423 * r170425;
        double r170430 = r170425 * r170423;
        double r170431 = r170429 + r170430;
        double r170432 = r170431 * r170425;
        double r170433 = r170428 - r170432;
        return r170433;
}
↓
double f(double x_re, double x_im) {
        double r170434 = x_re;
        double r170435 = 3.0;
        double r170436 = pow(r170434, r170435);
        double r170437 = x_im;
        double r170438 = r170437 * r170435;
        double r170439 = r170434 * r170437;
        double r170440 = r170438 * r170439;
        double r170441 = r170436 - r170440;
        return r170441;
}

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

Original7.2
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 7.2

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

  2. Simplified0.2

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

  4. Applied associate-*r*0.2

    \[\leadsto {x.re}^{3} - \color{blue}{\left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.im}\]
  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. Simplified0.2

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

  9. Applied associate-*l*0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019303 
(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)))