math.cube on complex, real part

Average Error: 7.7 → 0.6

Time: 51.1s

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 r5909457 = x_re;
        double r5909458 = r5909457 * r5909457;
        double r5909459 = x_im;
        double r5909460 = r5909459 * r5909459;
        double r5909461 = r5909458 - r5909460;
        double r5909462 = r5909461 * r5909457;
        double r5909463 = r5909457 * r5909459;
        double r5909464 = r5909459 * r5909457;
        double r5909465 = r5909463 + r5909464;
        double r5909466 = r5909465 * r5909459;
        double r5909467 = r5909462 - r5909466;
        return r5909467;
}

↓

double f(double x_re, double x_im) {
        double r5909468 = x_re;
        double r5909469 = x_im;
        double r5909470 = r5909468 - r5909469;
        double r5909471 = r5909470 * r5909468;
        double r5909472 = r5909469 + r5909468;
        double r5909473 = r5909471 * r5909472;
        double r5909474 = r5909468 * r5909469;
        double r5909475 = r5909474 + r5909474;
        double r5909476 = cbrt(r5909475);
        double r5909477 = r5909476 * r5909476;
        double r5909478 = r5909469 * r5909476;
        double r5909479 = r5909477 * r5909478;
        double r5909480 = r5909473 - r5909479;
        return r5909480;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.7
Target	0.2
Herbie	0.6

\[\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3.0 \cdot x.im\right)\]

Derivation

1. Initial program 7.7

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

3. Applied difference-of-squares 7.7

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

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 (* 3.0 x.im))))

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