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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.6516291320486477 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{if}\;\varepsilon \le 5.327160090154174 \cdot 10^{-33}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot (e^{\log_* (1 + \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right))} - 1)^*\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \end{array}\]

Original	37.3
Target	14.7
Herbie	0.9

Derivation

Split input into 3 regimes
if eps < -1.6516291320486477e-08
1. Initial program 29.8
  \[\sin \left(x + \varepsilon\right) - \sin x\]
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 -1.6516291320486477e-08 < eps < 5.327160090154174e-33
1. Initial program 45.7
  \[\sin \left(x + \varepsilon\right) - \sin x\]
2. Using strategy rm
3. Applied diff-sin45.7
  \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
4. Applied simplify0.2
  \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)}\]
5. Using strategy rm
6. Applied expm1-log1p-u0.3
  \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{(e^{\log_* (1 + \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right))} - 1)^*}\right)\]
if 5.327160090154174e-33 < eps
1. Initial program 29.7
  \[\sin \left(x + \varepsilon\right) - \sin x\]
2. Using strategy rm
3. Applied sin-sum2.2
  \[\leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
4. Applied associate--l+2.2
  \[\leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
Recombined 3 regimes into one program.

Runtime

herbie shell --seed '#(1070864556 424010669 783715395 1203517814 4070606583 4107618214)' +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 < -1.6516291320486477e-08`

`if -1.6516291320486477e-08 < eps < 5.327160090154174e-33`

`if 5.327160090154174e-33 < eps`

Runtime