Average Error: 39.3 → 0.7
Time: 37.6s
Precision: 64
Internal Precision: 2368
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -7.089990623854974 \cdot 10^{-06} \lor \neg \left(\varepsilon \le 0.0057906731298979835\right):\\ \;\;\;\;\cos \varepsilon \cdot \cos x - (\left(\sin \varepsilon\right) \cdot \left(\sin x\right) + \left(\cos x\right))_*\\ \mathbf{else}:\\ \;\;\;\;\left(\log_* (1 + (e^{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} - 1)^*) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot -2\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if eps < -7.089990623854974e-06 or 0.0057906731298979835 < eps

    1. Initial program 29.5

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

    3. Applied cos-sum0.9

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

    4. Applied associate--l-0.9

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

    5. Applied simplify0.9

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

    if -7.089990623854974e-06 < eps < 0.0057906731298979835

    1. Initial program 49.7

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

    3. Applied diff-cos38.3

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

    4. Applied simplify0.5

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

    6. Applied log1p-expm1-u0.6

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

  4. Applied simplify0.7

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\varepsilon \le -7.089990623854974 \cdot 10^{-06} \lor \neg \left(\varepsilon \le 0.0057906731298979835\right):\\ \;\;\;\;\cos \varepsilon \cdot \cos x - (\left(\sin \varepsilon\right) \cdot \left(\sin x\right) + \left(\cos x\right))_*\\ \mathbf{else}:\\ \;\;\;\;\left(\log_* (1 + (e^{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} - 1)^*) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot -2\\ \end{array}}\]

Runtime

Time bar (total: 37.6s)Debug logProfile

herbie shell --seed 2018167 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))