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

⬇

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.3416378887887987 \cdot 10^{-05}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{if}\;\varepsilon \le 1.1063826297604193 \cdot 10^{-07}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Original	37.4
Comparison	15.3
Herbie	0.5

Derivation

Split input into 2 regimes.
if eps < -2.3416378887887987e-05 or 1.1063826297604193e-07 < eps
1. Initial program 30.2
  \[\sin \left(x + \varepsilon\right) - \sin x\]
2. Using strategy rm
3. Applied sin-sum 0.5
  \[\leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
if -2.3416378887887987e-05 < eps < 1.1063826297604193e-07
1. Initial program 45.0
  \[\sin \left(x + \varepsilon\right) - \sin x\]
2. Using strategy rm
3. Applied diff-sin 45.0
  \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
4. Applied simplify 0.4
  \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)}\]
Recombined 2 regimes into one program.
Removed slow pow expressions

Runtime

Please include this information when filing a bug report:

herbie shell --seed '#(1064555532 179913862 452496668 2441903500 287849034 462453547)'
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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

Error

Target

Derivation

`if eps < -2.3416378887887987e-05 or 1.1063826297604193e-07 < eps`

`if -2.3416378887887987e-05 < eps < 1.1063826297604193e-07`

Runtime