math.cube on complex, real part

Average Error: 6.9 → 0.5

Time: 54.2s

Precision: 64

Math

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

↓

\[(\left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \left(\left(\left(\sqrt[3]{x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)} \cdot \sqrt[3]{x.im}\right) \cdot \sqrt[3]{x.re \cdot x.im + x.re \cdot x.im}\right) \cdot \left(-\sqrt[3]{x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)}\right)\right))_*\]

C

double f(double x_re, double x_im) {
        double r30648921 = x_re;
        double r30648922 = r30648921 * r30648921;
        double r30648923 = x_im;
        double r30648924 = r30648923 * r30648923;
        double r30648925 = r30648922 - r30648924;
        double r30648926 = r30648925 * r30648921;
        double r30648927 = r30648921 * r30648923;
        double r30648928 = r30648923 * r30648921;
        double r30648929 = r30648927 + r30648928;
        double r30648930 = r30648929 * r30648923;
        double r30648931 = r30648926 - r30648930;
        return r30648931;
}

↓

double f(double x_re, double x_im) {
        double r30648932 = x_im;
        double r30648933 = x_re;
        double r30648934 = r30648932 + r30648933;
        double r30648935 = r30648933 - r30648932;
        double r30648936 = r30648935 * r30648933;
        double r30648937 = r30648933 * r30648932;
        double r30648938 = r30648937 + r30648937;
        double r30648939 = r30648932 * r30648938;
        double r30648940 = cbrt(r30648939);
        double r30648941 = cbrt(r30648932);
        double r30648942 = r30648940 * r30648941;
        double r30648943 = cbrt(r30648938);
        double r30648944 = r30648942 * r30648943;
        double r30648945 = -r30648940;
        double r30648946 = r30648944 * r30648945;
        double r30648947 = fma(r30648934, r30648936, r30648946);
        return r30648947;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	6.9
Target	0.2
Herbie	0.5

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

Derivation

1. Initial program 6.9

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
2. Using strategy rm

3. Applied difference-of-squares 6.9

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

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

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

8. Applied add-cube-cbrt 0.6

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

10. Applied cbrt-prod 0.5

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

11. Applied associate-*l* 0.5

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

12. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019107 +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)))