Average Error: 39.8 → 1.1
Time: 34.0s
Precision: 64
Internal Precision: 2368
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\cos \varepsilon \cdot \cos x - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \le -0.01804128847162817:\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\ \mathbf{elif}\;\cos \varepsilon \cdot \cos x - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \le 0.03671585948149453:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (- (* (cos x) (cos eps)) (+ (* (sin x) (sin eps)) (cos x))) < -0.01804128847162817

    1. Initial program 21.4

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

    3. Applied cos-sum0.6

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

    4. Applied associate--l-0.6

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

    if -0.01804128847162817 < (- (* (cos x) (cos eps)) (+ (* (sin x) (sin eps)) (cos x))) < 0.03671585948149453

    1. Initial program 48.0

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

    3. Applied diff-cos36.8

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

    4. Simplified1.6

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

    if 0.03671585948149453 < (- (* (cos x) (cos eps)) (+ (* (sin x) (sin eps)) (cos x)))

    1. Initial program 58.4

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

    3. Applied cos-sum0.6

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \varepsilon \cdot \cos x - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \le -0.01804128847162817:\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\ \mathbf{elif}\;\cos \varepsilon \cdot \cos x - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \le 0.03671585948149453:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Runtime

Time bar (total: 34.0s)Debug logProfile

herbie shell --seed 2018216 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))