Average Error: 39.8 → 0.9
Time: 32.0s
Precision: 64
Internal Precision: 2368
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\cos \varepsilon \cdot \cos x - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \le -0.01804128847162817:\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\ \mathbf{elif}\;\cos \varepsilon \cdot \cos x - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \le 0.0008125685956268052:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot (e^{\log_* (1 + \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right))} - 1)^*\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

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))) < -0.01804128847162817

    1. Initial program 21.4

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

    3. Applied cos-sum0.6

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

    4. Applied associate--l-0.6

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

    if -0.01804128847162817 < (- (* (cos x) (cos eps)) (+ (* (sin x) (sin eps)) (cos x))) < 0.0008125685956268052

    1. Initial program 47.9

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

    3. Applied diff-cos36.6

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

    4. Simplified1.1

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

    6. Applied expm1-log1p-u1.1

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

    if 0.0008125685956268052 < (- (* (cos x) (cos eps)) (+ (* (sin x) (sin eps)) (cos x)))

    1. Initial program 58.4

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

    3. Applied cos-sum0.7

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \varepsilon \cdot \cos x - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \le -0.01804128847162817:\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\ \mathbf{elif}\;\cos \varepsilon \cdot \cos x - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \le 0.0008125685956268052:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot (e^{\log_* (1 + \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right))} - 1)^*\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Runtime

Time bar (total: 32.0s)Debug logProfile

herbie shell --seed 2018216 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))