2sin (example 3.3)

Average Error: 37.3 → 0.5

Time: 5.9s

Precision: 64

Math

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

↓

\[\mathsf{fma}\left(\sin x, \cos \varepsilon - 1, \mathsf{log1p}\left(\mathsf{expm1}\left(\cos x \cdot \sin \varepsilon\right)\right)\right)\]

C

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

↓

double code(double x, double eps) {
	return ((double) fma(((double) sin(x)), ((double) (((double) cos(eps)) - 1.0)), ((double) log1p(((double) expm1(((double) (((double) cos(x)) * ((double) sin(eps))))))))));
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.3
Target	15.2
Herbie	0.5

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

Derivation

1. Initial program 37.3

   \[\sin \left(x + \varepsilon\right) - \sin x\]
2. Using strategy rm

3. Applied sin-sum 22.0

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

4. Taylor expanded around inf 22.0

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

5. Simplified 0.4

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

7. Applied log1p-expm1-u 0.5

   \[\leadsto \mathsf{fma}\left(\sin x, \cos \varepsilon - 1, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos x \cdot \sin \varepsilon\right)\right)}\right)\]

8. Final simplification 0.5

   \[\leadsto \mathsf{fma}\left(\sin x, \cos \varepsilon - 1, \mathsf{log1p}\left(\mathsf{expm1}\left(\cos x \cdot \sin \varepsilon\right)\right)\right)\]

Reproduce

herbie shell --seed 2020121 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))

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