math.cube on complex, imaginary part

Average Error: 7.5 → 0.2

Time: 18.7s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r12952482 = x_re;
        double r12952483 = r12952482 * r12952482;
        double r12952484 = x_im;
        double r12952485 = r12952484 * r12952484;
        double r12952486 = r12952483 - r12952485;
        double r12952487 = r12952486 * r12952484;
        double r12952488 = r12952482 * r12952484;
        double r12952489 = r12952484 * r12952482;
        double r12952490 = r12952488 + r12952489;
        double r12952491 = r12952490 * r12952482;
        double r12952492 = r12952487 + r12952491;
        return r12952492;
}

↓

double f(double x_re, double x_im) {
        double r12952493 = x_re;
        double r12952494 = x_im;
        double r12952495 = r12952493 + r12952494;
        double r12952496 = r12952493 - r12952494;
        double r12952497 = r12952496 * r12952494;
        double r12952498 = r12952494 * r12952493;
        double r12952499 = r12952498 + r12952498;
        double r12952500 = r12952499 * r12952493;
        double r12952501 = fma(r12952495, r12952497, r12952500);
        return r12952501;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.5
Target	0.3
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 7.5

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

3. Applied difference-of-squares 7.5

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

4. Applied associate-*l* 0.3

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

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"

  :herbie-target
  (+ (* (* x.re x.im) (* 2.0 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)))