Average Error: 39.7 → 1.2
Time: 30.7s
Precision: 64
Internal Precision: 2368
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\cos x \cdot \cos \varepsilon - \sin \varepsilon \cdot \sin x\right) - \cos x \le -0.06772419700563675:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{elif}\;\left(\cos x \cdot \cos \varepsilon - \sin \varepsilon \cdot \sin x\right) - \cos x \le 1.3011456344553308 \cdot 10^{-05}:\\ \;\;\;\;\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot -2\right) \cdot \sin \left((x \cdot 2 + \varepsilon)_* \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin \varepsilon \cdot \sin x\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 3 regimes

  2. if (- (- (* (cos eps) (cos x)) (* (sin eps) (sin x))) (cos x)) < -0.06772419700563675

    1. Initial program 21.0

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

    2. Initial simplification21.0

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

    4. Applied cos-sum0.4

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

    if -0.06772419700563675 < (- (- (* (cos eps) (cos x)) (* (sin eps) (sin x))) (cos x)) < 1.3011456344553308e-05

    1. Initial program 46.9

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

    2. Initial simplification46.9

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

    4. Applied diff-cos36.3

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

    5. Simplified1.8

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

    6. Taylor expanded around -inf 1.8

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

    7. Simplified1.8

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

    if 1.3011456344553308e-05 < (- (- (* (cos eps) (cos x)) (* (sin eps) (sin x))) (cos x))

    1. Initial program 58.2

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

    2. Initial simplification58.2

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

    4. Applied cos-sum0.9

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

    5. Applied associate--l-0.9

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

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos x \cdot \cos \varepsilon - \sin \varepsilon \cdot \sin x\right) - \cos x \le -0.06772419700563675:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{elif}\;\left(\cos x \cdot \cos \varepsilon - \sin \varepsilon \cdot \sin x\right) - \cos x \le 1.3011456344553308 \cdot 10^{-05}:\\ \;\;\;\;\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot -2\right) \cdot \sin \left((x \cdot 2 + \varepsilon)_* \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin \varepsilon \cdot \sin x\right)\\ \end{array}\]

Runtime

Time bar (total: 30.7s)Debug logProfile

herbie shell --seed 2018225 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))