math.cube on complex, imaginary part

Average Error: 6.7 → 0.5

Time: 26.5s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r9701380 = x_re;
        double r9701381 = r9701380 * r9701380;
        double r9701382 = x_im;
        double r9701383 = r9701382 * r9701382;
        double r9701384 = r9701381 - r9701383;
        double r9701385 = r9701384 * r9701382;
        double r9701386 = r9701380 * r9701382;
        double r9701387 = r9701382 * r9701380;
        double r9701388 = r9701386 + r9701387;
        double r9701389 = r9701388 * r9701380;
        double r9701390 = r9701385 + r9701389;
        return r9701390;
}

↓

double f(double x_re, double x_im) {
        double r9701391 = x_re;
        double r9701392 = x_im;
        double r9701393 = r9701391 - r9701392;
        double r9701394 = r9701393 * r9701392;
        double r9701395 = r9701392 + r9701391;
        double r9701396 = r9701394 * r9701395;
        double r9701397 = cbrt(r9701391);
        double r9701398 = r9701397 * r9701397;
        double r9701399 = r9701392 + r9701392;
        double r9701400 = r9701391 * r9701399;
        double r9701401 = r9701398 * r9701400;
        double r9701402 = r9701397 * r9701401;
        double r9701403 = r9701396 + r9701402;
        return r9701403;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	6.7
Target	0.2
Herbie	0.5

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

Derivation

1. Initial program 6.7

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

3. Applied difference-of-squares 6.7

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

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 0.5

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

7. Applied associate-*r* 0.5

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

8. Simplified 0.5

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

9. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019149 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"

  :herbie-target
  (+ (* (* x.re x.im) (* 2 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)))