Average Error: 36.5 → 0.4

Time: 32.1s

Precision: 64

Internal Precision: 2368

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

↓

\[(\left(\cos x\right) \cdot \left(\sin \varepsilon\right) + \left(\sin x \cdot \cos \varepsilon - \sin x\right))_*\]

Error

The X axis uses an exponential scale — Bits error versus `x`

Target

Original	36.5
Target	15.1
Herbie	0.4

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

Derivation

Initial program 36.5
\[\sin \left(x + \varepsilon\right) - \sin x\]
Initial simplification36.5
\[\leadsto \sin \left(\varepsilon + x\right) - \sin x\]
Using strategy rm
Applied sin-sum21.2
\[\leadsto \color{blue}{\left(\sin \varepsilon \cdot \cos x + \cos \varepsilon \cdot \sin x\right)} - \sin x\]
Applied associate--l+0.4
\[\leadsto \color{blue}{\sin \varepsilon \cdot \cos x + \left(\cos \varepsilon \cdot \sin x - \sin x\right)}\]
Taylor expanded around -inf 21.2
\[\leadsto \color{blue}{\left(\cos x \cdot \sin \varepsilon + \sin x \cdot \cos \varepsilon\right) - \sin x}\]
Simplified0.4
\[\leadsto \color{blue}{(\left(\cos x\right) \cdot \left(\sin \varepsilon\right) + \left(\sin x \cdot \cos \varepsilon - \sin x\right))_*}\]
Final simplification0.4
\[\leadsto (\left(\cos x\right) \cdot \left(\sin \varepsilon\right) + \left(\sin x \cdot \cos \varepsilon - \sin x\right))_*\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	0.4	0.4	0.2	0.2	0%

herbie shell --seed 2018295 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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