Average Error: 30.1 → 10.1
Time: 26.4s
Precision: 64
Internal Precision: 128
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -3.479208942241992 \cdot 10^{-122} \lor \neg \left(\varepsilon \le 2.899050323811387 \cdot 10^{-36}\right):\\ \;\;\;\;(\left(\cos \varepsilon\right) \cdot \left(\cos x\right) + \left(-(\left(\sin \varepsilon\right) \cdot \left(\sin x\right) + \left(\cos x\right))_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if eps < -3.479208942241992e-122 or 2.899050323811387e-36 < eps

    1. Initial program 30.4

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

    2. Initial simplification30.4

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

    4. Applied cos-sum6.5

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

    5. Taylor expanded around -inf 6.5

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

    6. Simplified6.5

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

    8. Applied fma-neg6.5

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

    if -3.479208942241992e-122 < eps < 2.899050323811387e-36

    1. Initial program 29.5

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

    2. Initial simplification29.5

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

    4. Applied diff-cos37.4

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

    5. Simplified16.1

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -3.479208942241992 \cdot 10^{-122} \lor \neg \left(\varepsilon \le 2.899050323811387 \cdot 10^{-36}\right):\\ \;\;\;\;(\left(\cos \varepsilon\right) \cdot \left(\cos x\right) + \left(-(\left(\sin \varepsilon\right) \cdot \left(\sin x\right) + \left(\cos x\right))_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)\\ \end{array}\]

Runtime

Time bar (total: 26.4s)Debug logProfile

herbie shell --seed 2018277 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))