math.cube on complex, imaginary part

Average Error: 7.4 → 0.2

Time: 3.8s

Precision: binary64

Math

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

↓

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

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+
  (* (- (* x.re x.re) (* x.im x.im)) x.im)
  (* (+ (* x.re x.im) (* x.im x.re)) x.re)))

↓

(FPCore (x.re x.im)
 :precision binary64
 (+
  (fma
   (+ x.re x.im)
   (* x.im (- x.re x.im))
   (* x.im (fma x.im (- x.im) (* x.im x.im))))
  (* x.re (+ (* x.re x.im) (* x.re x.im)))))

C

double code(double x_46_re, double x_46_im) {
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

↓

double code(double x_46_re, double x_46_im) {
	return fma((x_46_re + x_46_im), (x_46_im * (x_46_re - x_46_im)), (x_46_im * fma(x_46_im, -x_46_im, (x_46_im * x_46_im)))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.4
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 7.4

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

2. Applied egg-rr 0.2

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

3. Final simplification 0.2

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

Reproduce

herbie shell --seed 2022130 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :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)))