math.cube on complex, imaginary part

Average Error: 6.7 → 0.2

Time: 36.0s

Precision: 64

Math

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

↓

\[\mathsf{fma}\left(x.im + x.re, \left(x.re - x.im\right) \cdot x.im, \left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.re\right)\]

C

double f(double x_re, double x_im) {
        double r4571524 = x_re;
        double r4571525 = r4571524 * r4571524;
        double r4571526 = x_im;
        double r4571527 = r4571526 * r4571526;
        double r4571528 = r4571525 - r4571527;
        double r4571529 = r4571528 * r4571526;
        double r4571530 = r4571524 * r4571526;
        double r4571531 = r4571526 * r4571524;
        double r4571532 = r4571530 + r4571531;
        double r4571533 = r4571532 * r4571524;
        double r4571534 = r4571529 + r4571533;
        return r4571534;
}

↓

double f(double x_re, double x_im) {
        double r4571535 = x_im;
        double r4571536 = x_re;
        double r4571537 = r4571535 + r4571536;
        double r4571538 = r4571536 - r4571535;
        double r4571539 = r4571538 * r4571535;
        double r4571540 = r4571536 * r4571535;
        double r4571541 = r4571540 + r4571540;
        double r4571542 = r4571541 * r4571536;
        double r4571543 = fma(r4571537, r4571539, r4571542);
        return r4571543;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	6.7
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 6.7

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
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 fma-def 0.2

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

7. Final simplification 0.2

   \[\leadsto \mathsf{fma}\left(x.im + x.re, \left(x.re - x.im\right) \cdot x.im, \left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.re\right)\]

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(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)))