Average Error: 39.6 → 0.8
Time: 53.7s
Precision: 64
Internal Precision: 2432
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\left(\sqrt[3]{\sin \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sin \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)}} \cdot \sqrt[3]{\sqrt[3]{\sin \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)}}\right)\right) \le -0.0027770781152483773:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - (\left(\sin \varepsilon\right) \cdot \left(\sin x\right) + \left(\cos x\right))_*\\ \mathbf{if}\;\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\left(\sqrt[3]{\sin \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sin \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)}} \cdot \sqrt[3]{\sqrt[3]{\sin \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)}}\right)\right) \le 9.895187161013057 \cdot 10^{-07}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 3 regimes

  2. if (* (sin (/ eps 2)) (* (* (cbrt (sin (/ (fma 2 x eps) 2))) (cbrt (sin (/ (fma 2 x eps) 2)))) (* (cbrt (* (cbrt (sin (/ (fma 2 x eps) 2))) (cbrt (sin (/ (fma 2 x eps) 2))))) (cbrt (cbrt (sin (/ (fma 2 x eps) 2))))))) < -0.0027770781152483773

    1. Initial program 60.4

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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 -0.0027770781152483773 < (* (sin (/ eps 2)) (* (* (cbrt (sin (/ (fma 2 x eps) 2))) (cbrt (sin (/ (fma 2 x eps) 2)))) (* (cbrt (* (cbrt (sin (/ (fma 2 x eps) 2))) (cbrt (sin (/ (fma 2 x eps) 2))))) (cbrt (cbrt (sin (/ (fma 2 x eps) 2))))))) < 9.895187161013057e-07

    1. Initial program 49.3

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

    3. Applied diff-cos37.9

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

    4. Applied simplify0.7

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

    if 9.895187161013057e-07 < (* (sin (/ eps 2)) (* (* (cbrt (sin (/ (fma 2 x eps) 2))) (cbrt (sin (/ (fma 2 x eps) 2)))) (* (cbrt (* (cbrt (sin (/ (fma 2 x eps) 2))) (cbrt (sin (/ (fma 2 x eps) 2))))) (cbrt (cbrt (sin (/ (fma 2 x eps) 2)))))))

    1. Initial program 25.1

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

    3. Applied cos-sum0.9

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

Runtime

Time bar (total: 53.7s)Debug logProfile

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))