Average Error: 39.3 → 1.2
Time: 2.0m
Precision: 64
Internal Precision: 8192
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x \le -0.007263882223791929:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{if}\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x \le 6.726306235604884 \cdot 10^{-08}:\\ \;\;\;\;-2 \cdot \left(\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right)\right) \cdot \sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - (\left(\sin \varepsilon\right) \cdot \left(\sin x\right) + \left(\cos x\right))_*\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 3 regimes

  2. if (- (- (* (cos x) (cos eps)) (* (sin x) (sin eps))) (cos x)) < -0.007263882223791929

    1. Initial program 20.3

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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 -0.007263882223791929 < (- (- (* (cos x) (cos eps)) (* (sin x) (sin eps))) (cos x)) < 6.726306235604884e-08

    1. Initial program 48.1

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

    3. Applied diff-cos36.9

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

    4. Applied simplify0.7

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

    6. Applied add-cube-cbrt1.7

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right)}\right)\]

    7. Applied associate-*r*1.7

      \[\leadsto -2 \cdot \color{blue}{\left(\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right)\right) \cdot \sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right)}\]

    if 6.726306235604884e-08 < (- (- (* (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-sum0.9

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

    4. Applied associate--l-1.0

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

    5. Applied simplify0.9

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

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))