Average Error: 39.8 → 0.8
Time: 1.7m
Precision: 64
Ground Truth: 128
\[\cos \left(x + \varepsilon\right) - \cos x\]
⬇
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.5947871865450548 \cdot 10^{-13}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{if}\;\varepsilon \le 0.00054274381733396:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes.

  2. if eps < -1.5947871865450548e-13 or 0.00054274381733396 < eps

    1. Initial program 30.5

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

    3. Applied cos-sum 1.3

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

    if -1.5947871865450548e-13 < eps < 0.00054274381733396

    1. Initial program 49.5

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

    3. Applied diff-cos 37.7

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

    4. Applied simplify 0.3

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

Runtime

Total time: 1.7m Debug log

Please include this information when filing a bug report:

herbie --seed '#(2970406553 2426707880 2544292409 178877700 3142095965 2263235033)'
(FPCore (x eps)
  :name "NMSE problem 3.3.5"
  (- (cos (+ x eps)) (cos x)))