Error

Derivation

1. Split input into 2 regimes

2. if (- (* (cos x) (cos eps)) (fma (sin eps) (sin x) (cos x))) < -0.025003930529898438 or 1.725264261496313e-07 < (- (* (cos x) (cos eps)) (fma (sin eps) (sin x) (cos x)))

   1. Initial program 30.2

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

   3. Applied cos-sum0.6

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

   if -0.025003930529898438 < (- (* (cos x) (cos eps)) (fma (sin eps) (sin x) (cos x))) < 1.725264261496313e-07

   1. Initial program 47.3

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

   3. Applied diff-cos36.4

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]

   4. Applied simplify1.1

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

4. Applied simplify0.9

   \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\cos \varepsilon \cdot \cos x - (\left(\sin \varepsilon\right) \cdot \left(\sin x\right) + \left(\cos x\right))_* \le -0.025003930529898438 \lor \neg \left(\cos \varepsilon \cdot \cos x - (\left(\sin \varepsilon\right) \cdot \left(\sin x\right) + \left(\cos x\right))_* \le 1.725264261496313 \cdot 10^{-07}\right):\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot -2\\ \end{array}}\]

Runtime

Time bar (total: 39.9s)Debug logProfile

herbie shell --seed 2018170 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))