math.cube on complex, real part

Average Error: 7.0 → 0.6

Time: 23.5s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r8345417 = x_re;
        double r8345418 = r8345417 * r8345417;
        double r8345419 = x_im;
        double r8345420 = r8345419 * r8345419;
        double r8345421 = r8345418 - r8345420;
        double r8345422 = r8345421 * r8345417;
        double r8345423 = r8345417 * r8345419;
        double r8345424 = r8345419 * r8345417;
        double r8345425 = r8345423 + r8345424;
        double r8345426 = r8345425 * r8345419;
        double r8345427 = r8345422 - r8345426;
        return r8345427;
}

↓

double f(double x_re, double x_im) {
        double r8345428 = x_re;
        double r8345429 = x_im;
        double r8345430 = r8345428 - r8345429;
        double r8345431 = r8345430 * r8345428;
        double r8345432 = r8345429 + r8345428;
        double r8345433 = r8345431 * r8345432;
        double r8345434 = cbrt(r8345429);
        double r8345435 = r8345428 + r8345428;
        double r8345436 = r8345429 * r8345435;
        double r8345437 = r8345434 * r8345436;
        double r8345438 = r8345437 * r8345434;
        double r8345439 = r8345438 * r8345434;
        double r8345440 = r8345433 - r8345439;
        return r8345440;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.0
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 7.0

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
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 add-cube-cbrt 0.6

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

7. Applied associate-*r* 0.6

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

8. Simplified 0.6

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

9. Final simplification 0.6

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

Reproduce

herbie shell --seed 2019170 
(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)))