Average Error: 39.3 → 0.8
Time: 35.4s
Precision: 64
Internal Precision: 2368
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\cos \varepsilon \cdot \cos x - (\left(\sin \varepsilon\right) \cdot \left(\sin x\right) + \left(\cos x\right))_* \le -0.014318772926384227 \lor \neg \left(\cos \varepsilon \cdot \cos x - (\left(\sin \varepsilon\right) \cdot \left(\sin x\right) + \left(\cos x\right))_* \le 0.004408463031335852\right):\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\log_* (1 + (e^{\sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)} - 1)^*) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if (- (* (cos x) (cos eps)) (fma (sin eps) (sin x) (cos x))) < -0.014318772926384227 or 0.004408463031335852 < (- (* (cos x) (cos eps)) (fma (sin eps) (sin x) (cos x)))

    1. Initial program 30.2

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

    3. Applied cos-sum0.6

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

    if -0.014318772926384227 < (- (* (cos x) (cos eps)) (fma (sin eps) (sin x) (cos x))) < 0.004408463031335852

    1. Initial program 48.0

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

    3. Applied diff-cos36.7

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

    4. Applied simplify1.0

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

    6. Applied log1p-expm1-u1.0

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

  4. Applied simplify0.8

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\cos \varepsilon \cdot \cos x - (\left(\sin \varepsilon\right) \cdot \left(\sin x\right) + \left(\cos x\right))_* \le -0.014318772926384227 \lor \neg \left(\cos \varepsilon \cdot \cos x - (\left(\sin \varepsilon\right) \cdot \left(\sin x\right) + \left(\cos x\right))_* \le 0.004408463031335852\right):\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\log_* (1 + (e^{\sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)} - 1)^*) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \end{array}}\]

Runtime

Time bar (total: 35.4s)Debug logProfile

herbie shell --seed '#(1072107073 2127697367 3936270018 2300570620 2134894798 4023771849)' +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))