Average Error: 39.6 → 0.9
Time: 44.6s
Precision: 64
Internal Precision: 2432
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)} \le -0.004943407142606899:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{if}\;\left(\sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)} \le 0.0012931088770952348:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if (* (* (cbrt (* (sin (/ eps 2)) (sin (/ (+ eps (+ x x)) 2)))) (cbrt (* (sin (/ eps 2)) (sin (/ (+ eps (+ x x)) 2))))) (cbrt (* (sin (/ eps 2)) (sin (/ (+ eps (+ x x)) 2))))) < -0.004943407142606899 or 0.0012931088770952348 < (* (* (cbrt (* (sin (/ eps 2)) (sin (/ (+ eps (+ x x)) 2)))) (cbrt (* (sin (/ eps 2)) (sin (/ (+ eps (+ x x)) 2))))) (cbrt (* (sin (/ eps 2)) (sin (/ (+ eps (+ x x)) 2)))))

    1. Initial program 30.3

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

    3. Applied cos-sum0.7

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

    if -0.004943407142606899 < (* (* (cbrt (* (sin (/ eps 2)) (sin (/ (+ eps (+ x x)) 2)))) (cbrt (* (sin (/ eps 2)) (sin (/ (+ eps (+ x x)) 2))))) (cbrt (* (sin (/ eps 2)) (sin (/ (+ eps (+ x x)) 2))))) < 0.0012931088770952348

    1. Initial program 48.8

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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 simplify1.0

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

Runtime

Time bar (total: 44.6s)Debug logProfile

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