Average Error: 39.5 → 0.9
Time: 1.5m
Precision: 64
Internal Precision: 2432
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.00023445401581624484:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\ \mathbf{if}\;\varepsilon \le 71213.23633349378:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 3 regimes

  2. if eps < -0.00023445401581624484

    1. Initial program 30.2

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

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

    if -0.00023445401581624484 < eps < 71213.23633349378

    1. Initial program 49.0

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

    3. Applied diff-cos37.1

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

    4. Applied simplify0.9

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

    if 71213.23633349378 < eps

    1. Initial program 29.4

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

    3. Applied cos-sum0.8

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

Runtime

Time bar (total: 1.5m)Debug log

herbie shell --seed '#(2094665535 3654809497 1731717781 3082199462 566033875 575777438)' 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))