Average Error: 39.1 → 0.7
Time: 20.1s
Precision: 64
Internal Precision: 2368
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.7828649927518812 \cdot 10^{-05} \lor \neg \left(\varepsilon \le 0.00010087807500195139\right):\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;(e^{\log_* (1 + \sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right))} - 1)^* \cdot -2\\ \end{array}\]

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 2 regimes

  2. if eps < -1.7828649927518812e-05 or 0.00010087807500195139 < eps

    1. Initial program 30.3

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

    2. Initial simplification30.3

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

    4. Applied cos-sum0.9

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

    if -1.7828649927518812e-05 < eps < 0.00010087807500195139

    1. Initial program 48.5

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

    2. Initial simplification48.5

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

    4. Applied diff-cos36.6

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

    5. Simplified0.5

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

    7. Applied expm1-log1p-u0.5

      \[\leadsto -2 \cdot \color{blue}{(e^{\log_* (1 + \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right))} - 1)^*}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.7828649927518812 \cdot 10^{-05} \lor \neg \left(\varepsilon \le 0.00010087807500195139\right):\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;(e^{\log_* (1 + \sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right))} - 1)^* \cdot -2\\ \end{array}\]

Runtime

Time bar (total: 20.1s)Debug logProfile

herbie shell --seed 2018230 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))