Average Error: 40.5 → 0.7
Time: 15.5s
Precision: 64
Internal Precision: 2368
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -4.008309058555033 \cdot 10^{-06}:\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 3.6465563790653564 \cdot 10^{-05}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \left(\cos x + \sin \varepsilon \cdot \sin x\right)\\ \end{array}\]

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. Split input into 3 regimes

  2. if eps < -4.008309058555033e-06

    1. Initial program 31.3

      \[\cos \left(x + \varepsilon\right) - \cos x\]

    2. Initial simplification31.3

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

    4. Applied cos-sum1.0

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

    if -4.008309058555033e-06 < eps < 3.6465563790653564e-05

    1. Initial program 49.8

      \[\cos \left(x + \varepsilon\right) - \cos x\]

    2. Initial simplification49.8

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

    4. Applied diff-cos38.0

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

    5. Simplified0.5

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

    if 3.6465563790653564e-05 < eps

    1. Initial program 31.0

      \[\cos \left(x + \varepsilon\right) - \cos x\]

    2. Initial simplification31.0

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

    4. Applied cos-sum0.9

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

    5. Applied associate--l-0.9

      \[\leadsto \color{blue}{\cos \varepsilon \cdot \cos x - \left(\sin \varepsilon \cdot \sin x + \cos x\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -4.008309058555033 \cdot 10^{-06}:\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 3.6465563790653564 \cdot 10^{-05}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \left(\cos x + \sin \varepsilon \cdot \sin x\right)\\ \end{array}\]

Runtime

Time bar (total: 15.5s)Debug logProfile

herbie shell --seed 2018273 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))