Average Error: 36.8 → 0.6
Time: 56.6s
Precision: 64
Internal precision: 2176
\[\sin \left(x + \varepsilon\right) - \sin x\]
⬇
\[\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.3331382704773037 \cdot 10^{-08}:\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\
\mathbf{if}\;\varepsilon \le 8.172994376936087:\\
\;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)\\
\mathbf{else}:\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\
\end{array}\]
Target
| Original | 36.8 |
|---|
| Comparison | 15.3 |
|---|
| Herbie | 0.6 |
|---|
\[ 2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \]
Derivation
- Split input into 2 regimes.
-
if eps < -1.3331382704773037e-08 or 8.172994376936087 < eps
Initial program 30.1
\[\sin \left(x + \varepsilon\right) - \sin x\]
- Using strategy
rm
Applied sin-sum 0.5
\[\leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
if -1.3331382704773037e-08 < eps < 8.172994376936087
Initial program 43.8
\[\sin \left(x + \varepsilon\right) - \sin x\]
- Using strategy
rm
Applied diff-sin 43.8
\[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
Applied simplify 0.7
\[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)}\]
- Recombined 2 regimes into one program.
- Removed slow pow expressions
Runtime
Please include this information when filing a bug report:
herbie shell --seed '#(1068028399 4028058041 2917032441 2563479541 765645300 1132738916)'
(FPCore (x eps)
:name "2sin (example 3.3)"
:target
(* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))
(- (sin (+ x eps)) (sin x)))
