Average Error: 39.9 → 0.4
Time: 43.3s
Precision: 64
Internal Precision: 2368
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[-2 \cdot \left(\left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) + \cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

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 39.9

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

  7. Applied sin-sum0.4

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

Runtime

Time bar (total: 43.3s)Debug logProfile

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