Average Error: 39.5 → 0.8
Time: 50.9s
Precision: 64
Internal Precision: 2368
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.4194207403981303 \cdot 10^{-05} \lor \neg \left(\varepsilon \le 2.6713101620310196 \cdot 10^{-06}\right):\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if eps < -1.4194207403981303e-05 or 2.6713101620310196e-06 < eps

    1. Initial program 30.6

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

    3. Applied cos-sum1.0

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

    if -1.4194207403981303e-05 < eps < 2.6713101620310196e-06

    1. Initial program 48.8

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

    3. Applied diff-cos37.1

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

    4. Applied simplify0.6

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

  4. Applied simplify0.8

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.4194207403981303 \cdot 10^{-05} \lor \neg \left(\varepsilon \le 2.6713101620310196 \cdot 10^{-06}\right):\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \end{array}}\]

Runtime

Time bar (total: 50.9s)Debug logProfile

herbie shell --seed 2018193 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))