math.cube on complex, real part

Average Error: 7.0 → 0.5

Time: 53.8s

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 r22187561 = x_re;
        double r22187562 = r22187561 * r22187561;
        double r22187563 = x_im;
        double r22187564 = r22187563 * r22187563;
        double r22187565 = r22187562 - r22187564;
        double r22187566 = r22187565 * r22187561;
        double r22187567 = r22187561 * r22187563;
        double r22187568 = r22187563 * r22187561;
        double r22187569 = r22187567 + r22187568;
        double r22187570 = r22187569 * r22187563;
        double r22187571 = r22187566 - r22187570;
        return r22187571;
}

↓

double f(double x_re, double x_im) {
        double r22187572 = x_im;
        double r22187573 = x_re;
        double r22187574 = r22187572 + r22187573;
        double r22187575 = r22187573 - r22187572;
        double r22187576 = r22187575 * r22187573;
        double r22187577 = r22187573 * r22187572;
        double r22187578 = r22187577 + r22187577;
        double r22187579 = r22187572 * r22187578;
        double r22187580 = cbrt(r22187579);
        double r22187581 = cbrt(r22187572);
        double r22187582 = r22187580 * r22187581;
        double r22187583 = cbrt(r22187578);
        double r22187584 = r22187582 * r22187583;
        double r22187585 = -r22187580;
        double r22187586 = r22187584 * r22187585;
        double r22187587 = fma(r22187574, r22187576, r22187586);
        return r22187587;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.0
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 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

   \[\leadsto \color{blue}{(\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \left(-\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right))_*}\]
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 2019112 +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)))