Average Error: 39.8 → 1.0
Time: 18.8s
Precision: 64
Internal Precision: 128
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -515.6253415996845:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 6294590.97762436:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array}\]

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 3 regimes

  2. if eps < -515.6253415996845

    1. Initial program 31.1

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

    3. Applied cos-sum0.8

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

    if -515.6253415996845 < eps < 6294590.97762436

    1. Initial program 48.6

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

    3. Applied diff-cos37.4

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

    4. Simplified1.2

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

    if 6294590.97762436 < eps

    1. Initial program 30.2

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

    3. Applied cos-sum0.8

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

    4. Applied associate--l-0.8

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -515.6253415996845:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 6294590.97762436:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array}\]

Runtime

Time bar (total: 18.8s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes15.21.00.414.995.7%
herbie shell --seed 2018355 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))