math.cube on complex, imaginary part

Average Error: 7.1 → 0.2

Time: 23.6s

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 r7246394 = x_re;
        double r7246395 = r7246394 * r7246394;
        double r7246396 = x_im;
        double r7246397 = r7246396 * r7246396;
        double r7246398 = r7246395 - r7246397;
        double r7246399 = r7246398 * r7246396;
        double r7246400 = r7246394 * r7246396;
        double r7246401 = r7246396 * r7246394;
        double r7246402 = r7246400 + r7246401;
        double r7246403 = r7246402 * r7246394;
        double r7246404 = r7246399 + r7246403;
        return r7246404;
}

↓

double f(double x_re, double x_im) {
        double r7246405 = x_im;
        double r7246406 = x_re;
        double r7246407 = r7246405 + r7246406;
        double r7246408 = r7246406 - r7246405;
        double r7246409 = r7246408 * r7246405;
        double r7246410 = r7246406 * r7246405;
        double r7246411 = r7246410 + r7246410;
        double r7246412 = r7246411 * r7246406;
        double r7246413 = fma(r7246407, r7246409, r7246412);
        return r7246413;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.1
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 7.1

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

3. Applied difference-of-squares 7.1

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

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 2019172 +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)))