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

⬇

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.5947871865450548 \cdot 10^{-13}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{if}\;\varepsilon \le 4.6574913990588864 \cdot 10^{-35}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Original	37.0
Comparison	15.1
Herbie	1.0

Derivation

Split input into 2 regimes.
if eps < -1.5947871865450548e-13 or 4.6574913990588864e-35 < eps
1. Initial program 29.8
  \[\sin \left(x + \varepsilon\right) - \sin x\]
2. Using strategy rm
3. Applied sin-sum 1.7
  \[\leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
if -1.5947871865450548e-13 < eps < 4.6574913990588864e-35
1. Initial program 45.3
  \[\sin \left(x + \varepsilon\right) - \sin x\]
2. Using strategy rm
3. Applied diff-sin 45.3
  \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
4. Applied simplify 0.2
  \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)}\]
Recombined 2 regimes into one program.
Removed slow pow expressions

Runtime

Please include this information when filing a bug report:

herbie --seed '#(83214553 2015247279 762247982 3765665317 878970787 861607559)'
(FPCore (x eps)
  :name "NMSE example 3.3"

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

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

Error

Target

Derivation

`if eps < -1.5947871865450548e-13 or 4.6574913990588864e-35 < eps`

`if -1.5947871865450548e-13 < eps < 4.6574913990588864e-35`

Runtime