Average Error: 39.2 → 0.8
Time: 37.0s
Precision: 64
Internal Precision: 2432
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot \left(\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\sqrt[3]{\sin \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)}\right)\right) \cdot \sqrt[3]{\sin \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)}\right) \le -4.960559738300167 \cdot 10^{-06}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{if}\;-2 \cdot \left(\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\sqrt[3]{\sin \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)}\right)\right) \cdot \sqrt[3]{\sin \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)}\right) \le 0.014495001146888918:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - (\left(\sin \varepsilon\right) \cdot \left(\sin x\right) + \left(\cos x\right))_*\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 3 regimes

  2. if (* -2 (* (* (sin (/ eps 2)) (* (cbrt (sin (/ (fma 2 x eps) 2))) (cbrt (sin (/ (fma 2 x eps) 2))))) (cbrt (sin (/ (fma 2 x eps) 2))))) < -4.960559738300167e-06

    1. Initial program 25.0

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

    3. Applied cos-sum0.8

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

    if -4.960559738300167e-06 < (* -2 (* (* (sin (/ eps 2)) (* (cbrt (sin (/ (fma 2 x eps) 2))) (cbrt (sin (/ (fma 2 x eps) 2))))) (cbrt (sin (/ (fma 2 x eps) 2))))) < 0.014495001146888918

    1. Initial program 49.1

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

    3. Applied diff-cos37.2

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

    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 0.014495001146888918 < (* -2 (* (* (sin (/ eps 2)) (* (cbrt (sin (/ (fma 2 x eps) 2))) (cbrt (sin (/ (fma 2 x eps) 2))))) (cbrt (sin (/ (fma 2 x eps) 2)))))

    1. Initial program 60.4

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

    3. Applied cos-sum1.2

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

    4. Applied associate--l-1.2

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

    5. Applied simplify1.2

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

Runtime

Time bar (total: 37.0s)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))