Average Error: 39.7 → 1.4
Time: 38.8s
Precision: 64
Internal Precision: 2368
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right) \le -6.679132823536548 \cdot 10^{-10}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{if}\;\sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right) \le 0.023608709360561152:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (* (sin (/ (+ eps (+ x x)) 2)) (sin (/ eps 2))) < -6.679132823536548e-10 or 0.023608709360561152 < (* (sin (/ (+ eps (+ x x)) 2)) (sin (/ eps 2)))

    1. Initial program 31.0

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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 -6.679132823536548e-10 < (* (sin (/ (+ eps (+ x x)) 2)) (sin (/ eps 2))) < 0.023608709360561152

    1. Initial program 47.4

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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 simplify2.1

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

Runtime

Time bar (total: 38.8s)Debug logProfile

herbie shell --seed 2020178 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))