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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.0064995294984633 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{if}\;\varepsilon \le 1.4343600631505567 \cdot 10^{-08}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Original	36.7
Target	14.7
Herbie	0.4

Derivation

Split input into 2 regimes
if eps < -2.0064995294984633e-08 or 1.4343600631505567e-08 < eps
1. Initial program 29.4
  \[\sin \left(x + \varepsilon\right) - \sin x\]
2. Using strategy rm
3. Applied sin-sum0.6
  \[\leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
if -2.0064995294984633e-08 < eps < 1.4343600631505567e-08
1. Initial program 44.4
  \[\sin \left(x + \varepsilon\right) - \sin x\]
2. Using strategy rm
3. Applied diff-sin44.4
  \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
4. Applied simplify0.3
  \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)}\]
Recombined 2 regimes into one program.

Runtime

herbie shell --seed '#(1071246582 2318319007 2683472949 3810440501 3233274817 2724848749)' +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)))

Error

Target

Derivation

`if eps < -2.0064995294984633e-08 or 1.4343600631505567e-08 < eps`

`if -2.0064995294984633e-08 < eps < 1.4343600631505567e-08`

Runtime