Average Error: 39.6 → 0.7
Time: 24.8s
Precision: 64
Internal Precision: 2368
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\cos \varepsilon \cdot \cos x - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \le -4.865588194673254 \cdot 10^{-08}:\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\ \mathbf{elif}\;\cos \varepsilon \cdot \cos x - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \le 2.8298628035149804 \cdot 10^{-11}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot (e^{\log_* (1 + \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right))} - 1)^*\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

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))) < -4.865588194673254e-08

    1. Initial program 21.2

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

    3. Applied cos-sum0.8

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

    4. Applied associate--l-0.9

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

    if -4.865588194673254e-08 < (- (* (cos x) (cos eps)) (+ (* (sin x) (sin eps)) (cos x))) < 2.8298628035149804e-11

    1. Initial program 49.6

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

    3. Applied diff-cos37.8

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

    4. Simplified0.3

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

    6. Applied expm1-log1p-u0.3

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

    if 2.8298628035149804e-11 < (- (* (cos x) (cos eps)) (+ (* (sin x) (sin eps)) (cos x)))

    1. Initial program 58.5

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

    3. Applied cos-sum1.3

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \varepsilon \cdot \cos x - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \le -4.865588194673254 \cdot 10^{-08}:\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\ \mathbf{elif}\;\cos \varepsilon \cdot \cos x - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \le 2.8298628035149804 \cdot 10^{-11}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot (e^{\log_* (1 + \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right))} - 1)^*\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Runtime

Time bar (total: 24.8s)Debug logProfile

herbie shell --seed 2018217 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))