Average Error: 39.4 → 1.4
Time: 19.6s
Precision: 64
Internal Precision: 128
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -5.577677896712578 \cdot 10^{-05}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 0.03052679201092073:\\ \;\;\;\;-2 \cdot \left(\left(\left(\left(\sqrt[3]{\sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)}} \cdot \sqrt[3]{\sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)}}\right) \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if eps < -5.577677896712578e-05 or 0.03052679201092073 < eps

    1. Initial program 30.1

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

    3. Applied cos-sum0.8

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

    if -5.577677896712578e-05 < eps < 0.03052679201092073

    1. Initial program 49.1

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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. Simplified0.7

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

    6. Applied add-cube-cbrt1.7

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

    7. Applied associate-*r*1.7

      \[\leadsto -2 \cdot \color{blue}{\left(\left(\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right) \cdot \left(\sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)}\right)\right) \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)}\right)}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt1.8

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

    10. Applied cbrt-prod1.9

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -5.577677896712578 \cdot 10^{-05}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 0.03052679201092073:\\ \;\;\;\;-2 \cdot \left(\left(\left(\left(\sqrt[3]{\sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)}} \cdot \sqrt[3]{\sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)}}\right) \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2019053 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))