Average Error: 39.7 → 0.7
Time: 26.0s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.00010087453472037896:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \le 3.441463654237003 \cdot 10^{-05}:\\ \;\;\;\;\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(-2 \cdot \log_* (1 + (e^{\sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)} - 1)^*)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 3 regimes

  2. if eps < -0.00010087453472037896

    1. Initial program 30.6

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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-1.0

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

    if -0.00010087453472037896 < eps < 3.441463654237003e-05

    1. Initial program 49.2

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

    3. Applied diff-cos37.4

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

    4. Simplified0.5

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

    6. Applied associate-*r*0.5

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

    8. Applied log1p-expm1-u0.5

      \[\leadsto \left(-2 \cdot \color{blue}{\log_* (1 + (e^{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} - 1)^*)}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\]

    if 3.441463654237003e-05 < eps

    1. Initial program 30.1

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

    3. Applied cos-sum0.9

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.00010087453472037896:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \le 3.441463654237003 \cdot 10^{-05}:\\ \;\;\;\;\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(-2 \cdot \log_* (1 + (e^{\sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)} - 1)^*)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2019100 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))