Average Error: 39.2 → 0.8
Time: 27.4s
Precision: 64
Internal Precision: 2368
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot \left(\sin \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \le -0.0015008851931660302:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;-2 \cdot \left(\sin \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \le 3.522092692045019 \cdot 10^{-07}:\\ \;\;\;\;-2 \cdot \left(\log_* (1 + (e^{\sin \left(\frac{\left(\varepsilon + x\right) + x}{2}\right)} - 1)^*) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (* -2 (* (sin (/ eps 2)) (sin (/ (+ x (+ eps x)) 2)))) < -0.0015008851931660302 or 3.522092692045019e-07 < (* -2 (* (sin (/ eps 2)) (sin (/ (+ x (+ eps x)) 2))))

    1. Initial program 30.3

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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 -0.0015008851931660302 < (* -2 (* (sin (/ eps 2)) (sin (/ (+ x (+ eps x)) 2)))) < 3.522092692045019e-07

    1. Initial program 48.4

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

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

    6. Applied log1p-expm1-u0.7

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\log_* (1 + (e^{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} - 1)^*)}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot \left(\sin \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \le -0.0015008851931660302:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;-2 \cdot \left(\sin \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \le 3.522092692045019 \cdot 10^{-07}:\\ \;\;\;\;-2 \cdot \left(\log_* (1 + (e^{\sin \left(\frac{\left(\varepsilon + x\right) + x}{2}\right)} - 1)^*) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Runtime

Time bar (total: 27.4s)Debug logProfile

herbie shell --seed 2018215 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))