Average Error: 39.9 → 0.9
Time: 18.7s
Precision: 64
Internal Precision: 2368
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right) + \sqrt[3]{\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x} \cdot \left(\sqrt[3]{\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x} \cdot \sqrt[3]{\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x}\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\]

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. Initial simplification39.9

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

  4. Applied diff-cos34.6

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

  5. Simplified15.2

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

  6. Taylor expanded around -inf 15.2

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

  7. Simplified15.1

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

  9. Applied sin-sum0.4

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

  11. Applied add-cube-cbrt0.9

    \[\leadsto \left(\color{blue}{\left(\sqrt[3]{\sin x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right)} \cdot \sqrt[3]{\sin x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right)}\right) \cdot \sqrt[3]{\sin x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right)}} + \cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\]

  12. Final simplification0.9

    \[\leadsto \left(\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right) + \sqrt[3]{\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x} \cdot \left(\sqrt[3]{\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x} \cdot \sqrt[3]{\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x}\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\]

Runtime

Time bar (total: 18.7s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes0.90.90.10.80%
herbie shell --seed 2018296 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))