Average Error: 39.9 → 0.8
Time: 19.4s
Precision: 64
Internal Precision: 128
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -51.35704505414333 \lor \neg \left(\varepsilon \le 5.70303659793194 \cdot 10^{-05}\right):\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(-2 \cdot (e^{\log_* (1 + \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right))} - 1)^*\right)\\ \end{array}\]

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 < -51.35704505414333 or 5.70303659793194e-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

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

    if -51.35704505414333 < eps < 5.70303659793194e-05

    1. Initial program 49.0

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

    2. Initial simplification49.0

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

    4. Applied diff-cos37.7

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

    5. Simplified0.7

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

    7. Applied associate-*r*0.7

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

    9. Applied expm1-log1p-u0.8

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -51.35704505414333 \lor \neg \left(\varepsilon \le 5.70303659793194 \cdot 10^{-05}\right):\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(-2 \cdot (e^{\log_* (1 + \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right))} - 1)^*\right)\\ \end{array}\]

Runtime

Time bar (total: 19.4s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes15.60.80.415.297.2%
herbie shell --seed 2018353 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))