Average Error: 39.6 → 0.9
Time: 46.8s
Precision: 64
Internal Precision: 2432
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\sqrt[3]{\sin \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)}\right)\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\sin \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)}} \cdot \sqrt[3]{\sqrt[3]{\sin \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\sin \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)}}\right) \le -0.00564960816297931:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - (\left(\sin \varepsilon\right) \cdot \left(\sin x\right) + \left(\cos x\right))_*\\ \mathbf{if}\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\sqrt[3]{\sin \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)}\right)\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\sin \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)}} \cdot \sqrt[3]{\sqrt[3]{\sin \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\sin \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)}}\right) \le 0.0006679966275687113:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot (e^{\log_* (1 + \sin \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right))} - 1)^*\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

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 (cbrt (sin (/ (fma 2 x eps) 2))))) (cbrt (cbrt (sin (/ (fma 2 x eps) 2)))))) < -0.00564960816297931

    1. Initial program 60.6

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

    3. Applied cos-sum0.7

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

    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 -0.00564960816297931 < (* (* (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 (cbrt (sin (/ (fma 2 x eps) 2))))) (cbrt (cbrt (sin (/ (fma 2 x eps) 2)))))) < 0.0006679966275687113

    1. Initial program 48.9

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

    3. Applied diff-cos36.7

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

    4. Applied simplify0.9

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

    6. Applied expm1-log1p-u1.0

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

    if 0.0006679966275687113 < (* (* (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 (cbrt (sin (/ (fma 2 x eps) 2))))) (cbrt (cbrt (sin (/ (fma 2 x eps) 2))))))

    1. Initial program 25.2

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

Runtime

Time bar (total: 46.8s)Debug logProfile

herbie shell --seed '#(1064397287 3527694221 3797617954 1138343853 2854031332 1153838279)' +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))