math.cube on complex, real part

Average Error: 7.4 → 0.2

Time: 3.2s

Precision: binary64

Math

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

↓

\[\mathsf{fma}\left(-3, x.im \cdot \left(x.re \cdot x.im\right), {x.re}^{3}\right) \]

FPCore

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

↓

(FPCore (x.re x.im)
 :precision binary64
 (fma -3.0 (* x.im (* x.re x.im)) (pow x.re 3.0)))

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_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
}

↓

double code(double x_46_re, double x_46_im) {
	return fma(-3.0, (x_46_im * (x_46_re * x_46_im)), pow(x_46_re, 3.0));
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.4
Target	0.2
Herbie	0.2

\[\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right) \]

Derivation

1. Initial program 7.4

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

2. Simplified 7.4

   \[\leadsto \color{blue}{x.re \cdot \mathsf{fma}\left(x.re, x.re, \left(x.im \cdot x.im\right) \cdot -3\right)} \]

3. Taylor expanded in x.re around 0 7.3

   \[\leadsto \color{blue}{{x.re}^{3} - 3 \cdot \left(x.re \cdot {x.im}^{2}\right)} \]

4. Simplified 7.3

   \[\leadsto \color{blue}{\mathsf{fma}\left(-3, x.re \cdot \left(x.im \cdot x.im\right), {x.re}^{3}\right)} \]

5. Applied associate-*r*_binary64 0.2

   \[\leadsto \mathsf{fma}\left(-3, \color{blue}{\left(x.re \cdot x.im\right) \cdot x.im}, {x.re}^{3}\right) \]

6. Final simplification 0.2

   \[\leadsto \mathsf{fma}\left(-3, x.im \cdot \left(x.re \cdot x.im\right), {x.re}^{3}\right) \]

Reproduce

herbie shell --seed 2022125 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im))))

  (- (* (- (* x.re x.re) (* x.im x.im)) x.re) (* (+ (* x.re x.im) (* x.im x.re)) x.im)))