Error

Derivation

1. Split input into 3 regimes

2. if (- (* (cos x) (cos eps)) (fma (sin eps) (sin x) (cos x))) < -0.005648658341913538

   1. Initial program 20.6

      \[\cos \left(x + \varepsilon\right) - \cos x\]
   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 -0.005648658341913538 < (- (* (cos x) (cos eps)) (fma (sin eps) (sin x) (cos x))) < 0.10149915156676655

   1. Initial program 48.7

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

   3. Applied diff-cos37.8

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

   4. Applied simplify2.2

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

   if 0.10149915156676655 < (- (* (cos x) (cos eps)) (fma (sin eps) (sin x) (cos x)))

   1. Initial program 58.6

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

   3. Applied cos-sum0.6

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

   4. Applied associate--l-0.6

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

   5. Applied simplify0.6

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

Runtime

Time bar (total: 25.7s)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))