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

⬇

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -9.230164504011904:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{if}\;\varepsilon \le 1183236732512159.2:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \end{array}\]

Derivation

Split input into 2 regimes.
if eps < -9.230164504011904 or 1183236732512159.2 < eps
1. Initial program 30.3
  \[\sin \left(x + \varepsilon\right) - \sin x\]
2. Using strategy rm
3. Applied sin-sum 0.4
  \[\leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
4. Applied associate--l+ 0.4
  \[\leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
if -9.230164504011904 < eps < 1183236732512159.2
1. Initial program 44.1
  \[\sin \left(x + \varepsilon\right) - \sin x\]
2. Using strategy rm
3. Applied diff-sin 44.1
  \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
4. Applied simplify 1.4
  \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)}\]
Recombined 2 regimes into one program.
Removed slow pow expressions

Runtime

Please include this information when filing a bug report:

herbie shell --seed '#(3052192724 3812927732 3686175817 630908657 2373248591 511094450)'
(FPCore (x eps)
  :name "NMSE example 3.3"
  (- (sin (+ x eps)) (sin x)))

Error

Derivation

`if eps < -9.230164504011904 or 1183236732512159.2 < eps`

`if -9.230164504011904 < eps < 1183236732512159.2`

Runtime