Average Error: 39.6 → 0.4
Time: 43.3s
Precision: 64
Internal Precision: 2368
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \frac{\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x\right) \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x\right) - \left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right) \cdot \left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right)}{\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x - \cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x}\]

Error

Bits error versus x

Bits error versus eps

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 39.6

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

  3. Applied diff-cos33.9

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

  4. Applied simplify15.4

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

  5. Taylor expanded around 0 15.4

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

  6. Applied simplify15.4

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

  8. Applied fma-udef15.4

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

  9. Applied sin-sum0.4

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

  11. Applied flip-+0.4

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

Runtime

Time bar (total: 43.3s)Debug logProfile

herbie shell --seed 2018199 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))