Average Error: 40.1 → 0.7
Time: 18.9s
Precision: 64
Internal Precision: 2368
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.00016202709330682128 \lor \neg \left(\varepsilon \le 0.00011870771635682477\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \log_* (1 + (e^{\sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)} - 1)^*)\right) \cdot -2\\ \end{array}\]

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 < -0.00016202709330682128 or 0.00011870771635682477 < eps

    1. Initial program 30.7

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

    3. Applied cos-sum0.9

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

    if -0.00016202709330682128 < eps < 0.00011870771635682477

    1. Initial program 49.5

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

    3. Applied diff-cos38.0

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

    4. Simplified0.5

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

    6. Applied log1p-expm1-u0.6

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.00016202709330682128 \lor \neg \left(\varepsilon \le 0.00011870771635682477\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \log_* (1 + (e^{\sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)} - 1)^*)\right) \cdot -2\\ \end{array}\]

Runtime

Time bar (total: 18.9s)Debug logProfile

herbie shell --seed 2018221 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))