2sin (example 3.3)

Average Error: 36.8 → 0.4

Time: 8.2s

Precision: binary64

Math

\[\sin \left(x + \varepsilon\right) - \sin x \]

↓

\[\mathsf{fma}\left(\cos x, \sin \varepsilon, -\mathsf{fma}\left(\sin x, 1 - \cos \varepsilon, 0\right)\right) \]

FPCore

(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))

↓

(FPCore (x eps)
 :precision binary64
 (fma (cos x) (sin eps) (- (fma (sin x) (- 1.0 (cos eps)) 0.0))))

C

double code(double x, double eps) {
	return sin((x + eps)) - sin(x);
}

↓

double code(double x, double eps) {
	return fma(cos(x), sin(eps), -fma(sin(x), (1.0 - cos(eps)), 0.0));
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.8
Target	15.4
Herbie	0.4

\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \]

Derivation

1. Initial program 36.8

   \[\sin \left(x + \varepsilon\right) - \sin x \]

2. Applied egg-rr 0.4

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \sin \varepsilon, -\left(\sin x - \sin x \cdot \cos \varepsilon\right)\right)} \]

3. Taylor expanded in x around inf 0.4

   \[\leadsto \mathsf{fma}\left(\cos x, \sin \varepsilon, -\color{blue}{\left(\sin x - \cos \varepsilon \cdot \sin x\right)}\right) \]

4. Simplified 0.4

   \[\leadsto \mathsf{fma}\left(\cos x, \sin \varepsilon, -\color{blue}{\left(1 - \cos \varepsilon\right) \cdot \sin x}\right) \]

5. Applied egg-rr 0.4

   \[\leadsto \mathsf{fma}\left(\cos x, \sin \varepsilon, -\color{blue}{\mathsf{fma}\left(\sin x, 1 - \cos \varepsilon, 0\right)}\right) \]

6. Final simplification 0.4

   \[\leadsto \mathsf{fma}\left(\cos x, \sin \varepsilon, -\mathsf{fma}\left(\sin x, 1 - \cos \varepsilon, 0\right)\right) \]

Reproduce

herbie shell --seed 2022129 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

  (- (sin (+ x eps)) (sin x)))