math.cube on complex, imaginary part

Average Error: 6.8 → 0.2

Time: 17.0s

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 r3345416 = x_re;
        double r3345417 = r3345416 * r3345416;
        double r3345418 = x_im;
        double r3345419 = r3345418 * r3345418;
        double r3345420 = r3345417 - r3345419;
        double r3345421 = r3345420 * r3345418;
        double r3345422 = r3345416 * r3345418;
        double r3345423 = r3345418 * r3345416;
        double r3345424 = r3345422 + r3345423;
        double r3345425 = r3345424 * r3345416;
        double r3345426 = r3345421 + r3345425;
        return r3345426;
}

↓

double f(double x_re, double x_im) {
        double r3345427 = x_im;
        double r3345428 = x_re;
        double r3345429 = r3345427 + r3345428;
        double r3345430 = r3345428 - r3345427;
        double r3345431 = r3345430 * r3345427;
        double r3345432 = r3345428 * r3345427;
        double r3345433 = r3345432 + r3345432;
        double r3345434 = r3345433 * r3345428;
        double r3345435 = fma(r3345429, r3345431, r3345434);
        return r3345435;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	6.8
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 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* 0.3

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

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

Reproduce

herbie shell --seed 2019151 +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)))