math.cube on complex, imaginary part

Average Error: 6.8 → 0.2

Time: 16.7s

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 r4085370 = x_re;
        double r4085371 = r4085370 * r4085370;
        double r4085372 = x_im;
        double r4085373 = r4085372 * r4085372;
        double r4085374 = r4085371 - r4085373;
        double r4085375 = r4085374 * r4085372;
        double r4085376 = r4085370 * r4085372;
        double r4085377 = r4085372 * r4085370;
        double r4085378 = r4085376 + r4085377;
        double r4085379 = r4085378 * r4085370;
        double r4085380 = r4085375 + r4085379;
        return r4085380;
}

↓

double f(double x_re, double x_im) {
        double r4085381 = x_re;
        double r4085382 = x_im;
        double r4085383 = r4085381 - r4085382;
        double r4085384 = r4085383 * r4085382;
        double r4085385 = r4085382 + r4085381;
        double r4085386 = r4085384 * r4085385;
        double r4085387 = r4085381 * r4085382;
        double r4085388 = r4085387 + r4085387;
        double r4085389 = r4085381 * r4085388;
        double r4085390 = r4085386 + r4085389;
        return r4085390;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	6.8
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 6.8

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

3. Applied difference-of-squares 6.8

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

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 2019154 
(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)))