Average Error: 40.0 → 1.4
Time: 42.0s
Precision: 64
Internal Precision: 2368
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \log \left(e^{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right)\right) \le -0.049319879375577316:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{if}\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \log \left(e^{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right)\right) \le 0.0001298009043808313:\\ \;\;\;\;\log_* (1 + (e^{-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)} - 1)^*) \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

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)) (log (exp (sin (/ (+ x (+ eps x)) 2)))))) < -0.049319879375577316 or 0.0001298009043808313 < (* -2 (* (sin (/ eps 2)) (log (exp (sin (/ (+ x (+ eps x)) 2))))))

    1. Initial program 31.1

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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 -0.049319879375577316 < (* -2 (* (sin (/ eps 2)) (log (exp (sin (/ (+ x (+ eps x)) 2)))))) < 0.0001298009043808313

    1. Initial program 47.8

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

    3. Applied diff-cos36.5

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

    4. Applied simplify2.1

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

    6. Applied associate-*r*2.1

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

    8. Applied log1p-expm1-u2.1

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

Runtime

Time bar (total: 42.0s)Debug logProfile

herbie shell --seed 2018195 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))