Error

Target

Derivation

1. Split input into 2 regimes

2. if eps < -7.49725387281966e-09 or 5.925101233597402e-09 < eps

   1. Initial program 29.7

      \[\sin \left(x + \varepsilon\right) - \sin x\]
   2. Using strategy rm

   3. Applied sin-sum0.5

      \[\leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]

   if -7.49725387281966e-09 < eps < 5.925101233597402e-09

   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. Simplified0.2

      \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)}\]
   5. Using strategy rm

   6. Applied associate-*r*0.2

      \[\leadsto \color{blue}{\left(2 \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)}\]
   7. Using strategy rm

   8. Applied expm1-log1p-u0.3

      \[\leadsto \left(2 \cdot \color{blue}{(e^{\log_* (1 + \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right))} - 1)^*}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -7.49725387281966 \cdot 10^{-09}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 5.925101233597402 \cdot 10^{-09}:\\ \;\;\;\;\sin \left(\frac{\varepsilon}{2}\right) \cdot \left((e^{\log_* (1 + \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right))} - 1)^* \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019050 +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)))