Average Error: 39.9 → 0.4
Time: 17.2s
Precision: 64
Internal Precision: 2368
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\frac{\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right) \cdot \left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right) - \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x\right) \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x\right)}{\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x - \sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x} \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot -2\right)\]

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-cos34.3

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

  4. Simplified15.0

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

  5. Taylor expanded around -inf 15.0

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

  6. Simplified15.0

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

  8. Applied sin-sum0.4

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

  10. Applied flip-+0.4

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

  11. Final simplification0.4

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

Runtime

Time bar (total: 17.2s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes0.40.40.10.40%
herbie shell --seed 2018297 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))