Average Error: 39.7 → 1.5
Time: 40.8s
Precision: 64
Internal Precision: 2368
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right) \cdot \left(\sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)}\right)\right) \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)} \le -2.5065994025001616 \cdot 10^{-07}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - (\left(\sin \varepsilon\right) \cdot \left(\sin x\right) + \left(\cos x\right))_*\\ \mathbf{if}\;\left(\sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right) \cdot \left(\sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)}\right)\right) \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)} \le 0.0278628380812515:\\ \;\;\;\;\left(-2 \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 3 regimes

  2. if (* (* (sin (/ (+ eps (+ x x)) 2)) (* (cbrt (sin (/ eps 2))) (cbrt (sin (/ eps 2))))) (cbrt (sin (/ eps 2)))) < -2.5065994025001616e-07

    1. Initial program 60.6

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

    3. Applied cos-sum1.1

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

    4. Applied associate--l-1.1

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

    5. Applied simplify1.1

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

    if -2.5065994025001616e-07 < (* (* (sin (/ (+ eps (+ x x)) 2)) (* (cbrt (sin (/ eps 2))) (cbrt (sin (/ eps 2))))) (cbrt (sin (/ eps 2)))) < 0.0278628380812515

    1. Initial program 47.3

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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 simplify2.3

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

    6. Applied associate-*r*2.3

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

    if 0.0278628380812515 < (* (* (sin (/ (+ eps (+ x x)) 2)) (* (cbrt (sin (/ eps 2))) (cbrt (sin (/ eps 2))))) (cbrt (sin (/ eps 2))))

    1. Initial program 25.1

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

    3. Applied cos-sum0.5

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

Runtime

Time bar (total: 40.8s)Debug logProfile

herbie shell --seed 2020178 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))