Average Error: 39.9 → 0.4
Time: 50.2s
Precision: 64
Internal Precision: 2368
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[-2 \cdot (e^{\log_* (1 + \sin \left(\frac{\varepsilon}{2}\right) \cdot (\left(\cos x\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) + \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right))_*)} - 1)^*\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Initial program 39.9

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

  3. Applied diff-cos34.6

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

  4. Applied simplify15.2

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

  5. Taylor expanded around 0 15.1

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

  6. Applied simplify15.1

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

  8. Applied fma-udef15.1

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

  9. Applied sin-sum0.4

    \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x + \cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right)}\right)\]
  10. Using strategy rm

  11. Applied expm1-log1p-u0.4

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

  12. Applied simplify0.4

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

Runtime

Time bar (total: 50.2s)Debug logProfile

herbie shell --seed 2018296 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))