Average Error: 7.3 → 0.2
Time: 3.2s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[\left(\left(-3 \cdot x.re\right) \cdot x.im\right) \cdot x.im + {x.re}^{3}\]
\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
↓
\left(\left(-3 \cdot x.re\right) \cdot x.im\right) \cdot x.im + {x.re}^{3}
double f(double x_re, double x_im) {
        double r218754 = x_re;
        double r218755 = r218754 * r218754;
        double r218756 = x_im;
        double r218757 = r218756 * r218756;
        double r218758 = r218755 - r218757;
        double r218759 = r218758 * r218754;
        double r218760 = r218754 * r218756;
        double r218761 = r218756 * r218754;
        double r218762 = r218760 + r218761;
        double r218763 = r218762 * r218756;
        double r218764 = r218759 - r218763;
        return r218764;
}
↓
double f(double x_re, double x_im) {
        double r218765 = -3.0;
        double r218766 = x_re;
        double r218767 = r218765 * r218766;
        double r218768 = x_im;
        double r218769 = r218767 * r218768;
        double r218770 = r218769 * r218768;
        double r218771 = 3.0;
        double r218772 = pow(r218766, r218771);
        double r218773 = r218770 + r218772;
        return r218773;
}

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. Simplified7.3

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

  4. Applied distribute-lft-neg-in7.3

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

  5. Applied associate-*r*0.2

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

  7. Applied fma-udef0.2

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

  8. Simplified0.2

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

  10. Applied associate-*r*0.2

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

  12. Applied associate-*r*0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(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)))