Average Error: 40.2 → 1.2
Time: 38.7s
Precision: 64
Internal Precision: 2432
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot \sqrt[3]{{\left(\sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)}^{3}} \le -0.0015137841305649844:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - (\left(\sin \varepsilon\right) \cdot \left(\sin x\right) + \left(\cos x\right))_*\\ \mathbf{if}\;-2 \cdot \sqrt[3]{{\left(\sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)}^{3}} \le 2.1703749806955596 \cdot 10^{-06}:\\ \;\;\;\;-2 \cdot \left(\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right)\right) \cdot \sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - (\left(\sin \varepsilon\right) \cdot \left(\sin x\right) + \left(\cos x\right))_*\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if (* -2 (cbrt (pow (* (sin (/ (+ x (+ x eps)) 2)) (sin (/ eps 2))) 3))) < -0.0015137841305649844 or 2.1703749806955596e-06 < (* -2 (cbrt (pow (* (sin (/ (+ x (+ x eps)) 2)) (sin (/ eps 2))) 3)))

    1. Initial program 30.7

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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-0.8

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

    5. Applied simplify0.8

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

    if -0.0015137841305649844 < (* -2 (cbrt (pow (* (sin (/ (+ x (+ x eps)) 2)) (sin (/ eps 2))) 3))) < 2.1703749806955596e-06

    1. Initial program 49.4

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

      \[\leadsto -2 \cdot \color{blue}{\left(\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right)\right) \cdot \sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right)}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 38.7s)Debug logProfile

herbie shell --seed '#(1070386091 2509006183 1430610344 1025408621 36622005 1425925650)' +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))