math.cube on complex, imaginary part

Average Error: 7.1 → 0.6

Time: 56.0s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r10676852 = x_re;
        double r10676853 = r10676852 * r10676852;
        double r10676854 = x_im;
        double r10676855 = r10676854 * r10676854;
        double r10676856 = r10676853 - r10676855;
        double r10676857 = r10676856 * r10676854;
        double r10676858 = r10676852 * r10676854;
        double r10676859 = r10676854 * r10676852;
        double r10676860 = r10676858 + r10676859;
        double r10676861 = r10676860 * r10676852;
        double r10676862 = r10676857 + r10676861;
        return r10676862;
}

↓

double f(double x_re, double x_im) {
        double r10676863 = x_re;
        double r10676864 = x_im;
        double r10676865 = r10676863 - r10676864;
        double r10676866 = r10676865 * r10676864;
        double r10676867 = r10676864 + r10676863;
        double r10676868 = r10676866 * r10676867;
        double r10676869 = r10676863 * r10676864;
        double r10676870 = r10676869 + r10676869;
        double r10676871 = cbrt(r10676870);
        double r10676872 = r10676871 * r10676871;
        double r10676873 = r10676871 * r10676863;
        double r10676874 = r10676872 * r10676873;
        double r10676875 = r10676868 + r10676874;
        return r10676875;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.1
Target	0.2
Herbie	0.6

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

Derivation

1. Initial program 7.1

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

3. Applied difference-of-squares 7.1

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

4. Applied associate-*l* 0.2

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

6. Applied add-cube-cbrt 0.6

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

7. Applied associate-*l* 0.6

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

8. Final simplification 0.6

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

Reproduce

herbie shell --seed 2019168 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"

  :herbie-target
  (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im)))

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