2sin (example 3.3)

Average Error: 37.1 → 0.4

Time: 5.9s

Precision: binary64

Math

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

↓

\[\frac{\sin x \cdot \left(-\sin \varepsilon \cdot \sin \varepsilon\right)}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon\]

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	15.2
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 37.1

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

3. Applied sin-sum 21.8

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

4. Taylor expanded around inf 21.8

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

5. Simplified 0.4

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

7. Applied flip-- 0.5

   \[\leadsto \sin x \cdot \color{blue}{\frac{\cos \varepsilon \cdot \cos \varepsilon - 1 \cdot 1}{\cos \varepsilon + 1}} + \cos x \cdot \sin \varepsilon\]

8. Applied associate-*r/ 0.5

   \[\leadsto \color{blue}{\frac{\sin x \cdot \left(\cos \varepsilon \cdot \cos \varepsilon - 1 \cdot 1\right)}{\cos \varepsilon + 1}} + \cos x \cdot \sin \varepsilon\]

9. Simplified 0.5

   \[\leadsto \frac{\color{blue}{\sin x \cdot \left(\cos \varepsilon \cdot \cos \varepsilon - 1\right)}}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon\]
10. Using strategy rm

11. Applied sub-1-cos 0.4

    \[\leadsto \frac{\sin x \cdot \color{blue}{\left(-\sin \varepsilon \cdot \sin \varepsilon\right)}}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon\]

12. Final simplification 0.4

    \[\leadsto \frac{\sin x \cdot \left(-\sin \varepsilon \cdot \sin \varepsilon\right)}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon\]

Reproduce

herbie shell --seed 2020152 
(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)))