Error

Derivation

1. Split input into 3 regimes

2. if eps < -388.80618680839126

   1. Initial program 29.4

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

   2. Initial simplification29.4

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

   4. Applied cos-sum0.8

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

   5. Applied associate--l-0.8

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

   if -388.80618680839126 < eps < 6.102937661813991e-07

   1. Initial program 48.8

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

   2. Initial simplification48.8

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

   4. Applied diff-cos37.2

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

   5. Simplified0.8

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

   7. Applied associate-*r*0.8

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

   9. Applied expm1-log1p-u0.9

      \[\leadsto \left(-2 \cdot \color{blue}{(e^{\log_* (1 + \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right))} - 1)^*}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\]

   10. Taylor expanded around -inf 0.8

       \[\leadsto \left(-2 \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(2 \cdot x + \varepsilon\right)\right)}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\]

   11. Simplified0.8

       \[\leadsto \left(-2 \cdot \color{blue}{\sin \left((\frac{1}{2} \cdot \varepsilon + x)_*\right)}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\]

   if 6.102937661813991e-07 < eps

   1. Initial program 30.6

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

   2. Initial simplification30.6

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

   4. Applied cos-sum1.1

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

4. Final simplification0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -388.80618680839126:\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \left(\cos x + \sin \varepsilon \cdot \sin x\right)\\ \mathbf{elif}\;\varepsilon \le 6.102937661813991 \cdot 10^{-07}:\\ \;\;\;\;\left(-2 \cdot \sin \left((\frac{1}{2} \cdot \varepsilon + x)_*\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \end{array}\]

Runtime

Time bar (total: 18.1s)Debug logProfile

herbie shell --seed 2018354 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))