math.cube on complex, real part

Average Error: 7.0 → 0.6

Time: 17.9s

Precision: 64

Math

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

↓

\[\mathsf{fma}\left(\left(x.im + x.re\right), \left(\left(x.re - x.im\right) \cdot x.re\right), \left(\left(\left(\sqrt[3]{x.im} \cdot \sqrt[3]{x.im}\right) \cdot \left(\sqrt[3]{\left(-x.im\right) \cdot x.re + \left(-x.im\right) \cdot x.re} \cdot \sqrt[3]{\left(-x.im\right) \cdot x.re + \left(-x.im\right) \cdot x.re}\right)\right) \cdot \sqrt[3]{x.im \cdot \left(\left(-x.im\right) \cdot x.re + \left(-x.im\right) \cdot x.re\right)}\right)\right)\]

C

double f(double x_re, double x_im) {
        double r9254738 = x_re;
        double r9254739 = r9254738 * r9254738;
        double r9254740 = x_im;
        double r9254741 = r9254740 * r9254740;
        double r9254742 = r9254739 - r9254741;
        double r9254743 = r9254742 * r9254738;
        double r9254744 = r9254738 * r9254740;
        double r9254745 = r9254740 * r9254738;
        double r9254746 = r9254744 + r9254745;
        double r9254747 = r9254746 * r9254740;
        double r9254748 = r9254743 - r9254747;
        return r9254748;
}

↓

double f(double x_re, double x_im) {
        double r9254749 = x_im;
        double r9254750 = x_re;
        double r9254751 = r9254749 + r9254750;
        double r9254752 = r9254750 - r9254749;
        double r9254753 = r9254752 * r9254750;
        double r9254754 = cbrt(r9254749);
        double r9254755 = r9254754 * r9254754;
        double r9254756 = -r9254749;
        double r9254757 = r9254756 * r9254750;
        double r9254758 = r9254757 + r9254757;
        double r9254759 = cbrt(r9254758);
        double r9254760 = r9254759 * r9254759;
        double r9254761 = r9254755 * r9254760;
        double r9254762 = r9254749 * r9254758;
        double r9254763 = cbrt(r9254762);
        double r9254764 = r9254761 * r9254763;
        double r9254765 = fma(r9254751, r9254753, r9254764);
        return r9254765;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.0
Target	0.2
Herbie	0.6

\[\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right)\]

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. Using strategy rm

3. Applied difference-of-squares 7.0

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

4. Applied associate-*l* 0.2

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

6. Applied fma-neg 0.2

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

7. Simplified 0.2

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

9. Applied add-cube-cbrt 0.6

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

11. Applied cbrt-prod 0.5

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

12. Applied cbrt-prod 0.6

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

13. Applied swap-sqr 0.6

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

14. Final simplification 0.6

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

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(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 x.im))))

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