Average Error: 39.9 → 15.9
Time: 7.8s
Precision: binary64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -6.0153064532234132 \cdot 10^{-7} \lor \neg \left(\varepsilon \le 2.51152766738541848 \cdot 10^{-8}\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \log \left(e^{\sin x \cdot \sin \varepsilon + \cos x}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(-x\right) - \varepsilon \cdot \left(\frac{1}{2} + \varepsilon\right)\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if eps < -6.0153064532234132e-7 or 2.51152766738541848e-8 < eps

    1. Initial program 30.8

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

    3. Applied cos-sum1.1

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

    4. Applied associate--l-1.1

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

    6. Applied add-log-exp1.2

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

    7. Applied add-log-exp1.3

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

    8. Applied sum-log1.3

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

    9. Simplified1.2

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

    if -6.0153064532234132e-7 < eps < 2.51152766738541848e-8

    1. Initial program 49.4

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

    2. Taylor expanded around 0 31.2

      \[\leadsto \color{blue}{-\left(x \cdot \varepsilon + \left(\frac{1}{2} \cdot {\varepsilon}^{2} + {\varepsilon}^{3}\right)\right)}\]

    3. Simplified31.2

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

  4. Final simplification15.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -6.0153064532234132 \cdot 10^{-7} \lor \neg \left(\varepsilon \le 2.51152766738541848 \cdot 10^{-8}\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \log \left(e^{\sin x \cdot \sin \varepsilon + \cos x}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(-x\right) - \varepsilon \cdot \left(\frac{1}{2} + \varepsilon\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020147 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))