math.cube on complex, imaginary part

Average Error: 7.0 → 0.2

Time: 15.0s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r4274412 = x_re;
        double r4274413 = r4274412 * r4274412;
        double r4274414 = x_im;
        double r4274415 = r4274414 * r4274414;
        double r4274416 = r4274413 - r4274415;
        double r4274417 = r4274416 * r4274414;
        double r4274418 = r4274412 * r4274414;
        double r4274419 = r4274414 * r4274412;
        double r4274420 = r4274418 + r4274419;
        double r4274421 = r4274420 * r4274412;
        double r4274422 = r4274417 + r4274421;
        return r4274422;
}

↓

double f(double x_re, double x_im) {
        double r4274423 = x_re;
        double r4274424 = x_im;
        double r4274425 = r4274423 - r4274424;
        double r4274426 = r4274425 * r4274424;
        double r4274427 = r4274424 + r4274423;
        double r4274428 = r4274426 * r4274427;
        double r4274429 = r4274423 * r4274424;
        double r4274430 = r4274429 + r4274429;
        double r4274431 = r4274423 * r4274430;
        double r4274432 = r4274428 + r4274431;
        return r4274432;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.0
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 7.0

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

3. Applied difference-of-squares 7.0

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

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. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019153 
(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)))