Average Error: 7.6 → 0.2
Time: 10.4s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[x.im \cdot \left(\left(x.im \cdot x.re\right) \cdot -3\right) + {x.re}^{3}\]
\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
↓
x.im \cdot \left(\left(x.im \cdot x.re\right) \cdot -3\right) + {x.re}^{3}
double f(double x_re, double x_im) {
        double r100689 = x_re;
        double r100690 = r100689 * r100689;
        double r100691 = x_im;
        double r100692 = r100691 * r100691;
        double r100693 = r100690 - r100692;
        double r100694 = r100693 * r100689;
        double r100695 = r100689 * r100691;
        double r100696 = r100691 * r100689;
        double r100697 = r100695 + r100696;
        double r100698 = r100697 * r100691;
        double r100699 = r100694 - r100698;
        return r100699;
}
↓
double f(double x_re, double x_im) {
        double r100700 = x_im;
        double r100701 = x_re;
        double r100702 = r100700 * r100701;
        double r100703 = -3.0;
        double r100704 = r100702 * r100703;
        double r100705 = r100700 * r100704;
        double r100706 = 3.0;
        double r100707 = pow(r100701, r100706);
        double r100708 = r100705 + r100707;
        return r100708;
}

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

Original7.6
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 7.6

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

  2. Simplified7.6

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

  4. Applied pow17.6

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

  5. Applied pow17.6

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

  6. Applied pow-prod-down7.6

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

  7. Simplified7.6

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

  9. Applied associate-*r*7.5

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

  10. Simplified0.2

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

  12. Applied associate-*l*0.2

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

  13. Simplified0.2

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

  15. Applied associate-*l*0.2

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

  16. Simplified0.2

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

  17. Final simplification0.2

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

Reproduce

herbie shell --seed 2019196 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"

  :herbie-target
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im))))

  (- (* (- (* x.re x.re) (* x.im x.im)) x.re) (* (+ (* x.re x.im) (* x.im x.re)) x.im)))