math.cube on complex, real part

Average Error: 7.2 → 0.6

Time: 18.4s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r8438733 = x_re;
        double r8438734 = r8438733 * r8438733;
        double r8438735 = x_im;
        double r8438736 = r8438735 * r8438735;
        double r8438737 = r8438734 - r8438736;
        double r8438738 = r8438737 * r8438733;
        double r8438739 = r8438733 * r8438735;
        double r8438740 = r8438735 * r8438733;
        double r8438741 = r8438739 + r8438740;
        double r8438742 = r8438741 * r8438735;
        double r8438743 = r8438738 - r8438742;
        return r8438743;
}

↓

double f(double x_re, double x_im) {
        double r8438744 = x_re;
        double r8438745 = x_im;
        double r8438746 = r8438744 - r8438745;
        double r8438747 = r8438746 * r8438744;
        double r8438748 = r8438745 + r8438744;
        double r8438749 = r8438747 * r8438748;
        double r8438750 = r8438744 * r8438745;
        double r8438751 = r8438750 + r8438750;
        double r8438752 = cbrt(r8438751);
        double r8438753 = r8438752 * r8438752;
        double r8438754 = r8438745 * r8438752;
        double r8438755 = r8438753 * r8438754;
        double r8438756 = r8438749 - r8438755;
        return r8438756;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.2
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 7.2

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

3. Applied difference-of-squares 7.2

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

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 add-cube-cbrt 0.6

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

7. Applied associate-*l* 0.6

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

8. Final simplification 0.6

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

Reproduce

herbie shell --seed 2019165 
(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)))