Average Error: 39.6 → 0.9
Time: 37.9s
Precision: 64
Internal Precision: 2368
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\left(\sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right)\right) \le -1.374836816675505 \cdot 10^{-07}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{if}\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\left(\sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right)\right) \le 0.04112801369640989:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - (\left(\sin \varepsilon\right) \cdot \left(\sin x\right) + \left(\cos x\right))_*\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 3 regimes

  2. if (* -2 (* (sin (/ eps 2)) (* (* (cbrt (sin (/ (+ x (+ eps x)) 2))) (cbrt (sin (/ (+ x (+ eps x)) 2)))) (cbrt (sin (/ (+ x (+ eps x)) 2)))))) < -1.374836816675505e-07

    1. Initial program 24.4

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

    3. Applied cos-sum0.8

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

    if -1.374836816675505e-07 < (* -2 (* (sin (/ eps 2)) (* (* (cbrt (sin (/ (+ x (+ eps x)) 2))) (cbrt (sin (/ (+ x (+ eps x)) 2)))) (cbrt (sin (/ (+ x (+ eps x)) 2)))))) < 0.04112801369640989

    1. Initial program 49.7

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

    3. Applied diff-cos38.5

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

    4. Applied simplify1.0

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

    if 0.04112801369640989 < (* -2 (* (sin (/ eps 2)) (* (* (cbrt (sin (/ (+ x (+ eps x)) 2))) (cbrt (sin (/ (+ x (+ eps x)) 2)))) (cbrt (sin (/ (+ x (+ eps x)) 2))))))

    1. Initial program 60.8

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

    3. Applied cos-sum1.0

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

    4. Applied associate--l-1.0

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

    5. Applied simplify1.0

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

Runtime

Time bar (total: 37.9s)Debug logProfile

herbie shell --seed '#(1071948828 1180510430 2986424009 997076509 406109801 420189285)' +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))