2sin (example 3.3)

Average Error: 36.8 → 0.5

Time: 6.2s

Precision: 64

Math

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

↓

\[\mathsf{fma}\left(\sin x, \sqrt[3]{{\left(\cos \varepsilon - 1\right)}^{3}}, \cos x \cdot \sin \varepsilon\right)\]

C

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

↓

double code(double x, double eps) {
	return fma(sin(x), cbrt(pow((cos(eps) - 1.0), 3.0)), (cos(x) * sin(eps)));
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.8
Target	15.7
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 36.8

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

3. Applied sin-sum 20.9

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

4. Taylor expanded around inf 20.9

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

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 add-cbrt-cube 0.5

   \[\leadsto \mathsf{fma}\left(\sin x, \color{blue}{\sqrt[3]{\left(\left(\cos \varepsilon - 1\right) \cdot \left(\cos \varepsilon - 1\right)\right) \cdot \left(\cos \varepsilon - 1\right)}}, \cos x \cdot \sin \varepsilon\right)\]

8. Simplified 0.5

   \[\leadsto \mathsf{fma}\left(\sin x, \sqrt[3]{\color{blue}{{\left(\cos \varepsilon - 1\right)}^{3}}}, \cos x \cdot \sin \varepsilon\right)\]

9. Final simplification 0.5

   \[\leadsto \mathsf{fma}\left(\sin x, \sqrt[3]{{\left(\cos \varepsilon - 1\right)}^{3}}, \cos x \cdot \sin \varepsilon\right)\]

Reproduce

herbie shell --seed 2020102 +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)))