Average Error: 39.2 → 0.7
Time: 19.8s
Precision: 64
Internal Precision: 2368
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -3.447773278636287 \cdot 10^{-06} \lor \neg \left(\varepsilon \le 0.11700375221913029\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\log_* (1 + (e^{\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot -2\right) \cdot \sin \left((x \cdot 2 + \varepsilon)_* \cdot \frac{1}{2}\right)} - 1)^*)\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if eps < -3.447773278636287e-06 or 0.11700375221913029 < eps

    1. Initial program 30.4

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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 -3.447773278636287e-06 < eps < 0.11700375221913029

    1. Initial program 48.7

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

    3. Applied diff-cos37.3

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

    4. Simplified0.5

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

    5. Taylor expanded around inf 0.5

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

    6. Simplified0.5

      \[\leadsto \color{blue}{\sin \left((x \cdot 2 + \varepsilon)_* \cdot \frac{1}{2}\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)}\]
    7. Using strategy rm

    8. Applied log1p-expm1-u0.5

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -3.447773278636287 \cdot 10^{-06} \lor \neg \left(\varepsilon \le 0.11700375221913029\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\log_* (1 + (e^{\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot -2\right) \cdot \sin \left((x \cdot 2 + \varepsilon)_* \cdot \frac{1}{2}\right)} - 1)^*)\\ \end{array}\]

Runtime

Time bar (total: 19.8s)Debug logProfile

herbie shell --seed 2018227 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))