Average Error: 40.0 → 1.1
Time: 39.7s
Precision: 64
Internal Precision: 2368
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot \left(\left(\sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\left(\sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right)}\right) \le -0.00802840710461251:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{if}\;-2 \cdot \left(\left(\sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\left(\sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right)}\right) \le 0.017558590530484378:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if (* -2 (* (* (cbrt (* (sin (/ eps 2)) (sin (/ (+ x (+ eps x)) 2)))) (cbrt (* (sin (/ eps 2)) (sin (/ (+ x (+ eps x)) 2))))) (cbrt (* (sin (/ eps 2)) (* (* (cbrt (sin (/ (+ x (+ eps x)) 2))) (cbrt (sin (/ (+ x (+ eps x)) 2)))) (cbrt (sin (/ (+ x (+ eps x)) 2)))))))) < -0.00802840710461251 or 0.017558590530484378 < (* -2 (* (* (cbrt (* (sin (/ eps 2)) (sin (/ (+ x (+ eps x)) 2)))) (cbrt (* (sin (/ eps 2)) (sin (/ (+ x (+ eps x)) 2))))) (cbrt (* (sin (/ eps 2)) (* (* (cbrt (sin (/ (+ x (+ eps x)) 2))) (cbrt (sin (/ (+ x (+ eps x)) 2)))) (cbrt (sin (/ (+ x (+ eps x)) 2))))))))

    1. Initial program 30.8

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

    3. Applied cos-sum0.6

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

    if -0.00802840710461251 < (* -2 (* (* (cbrt (* (sin (/ eps 2)) (sin (/ (+ x (+ eps x)) 2)))) (cbrt (* (sin (/ eps 2)) (sin (/ (+ x (+ eps x)) 2))))) (cbrt (* (sin (/ eps 2)) (* (* (cbrt (sin (/ (+ x (+ eps x)) 2))) (cbrt (sin (/ (+ x (+ eps x)) 2)))) (cbrt (sin (/ (+ x (+ eps x)) 2)))))))) < 0.017558590530484378

    1. Initial program 48.6

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

    3. Applied diff-cos37.3

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

    4. Applied simplify1.5

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

Runtime

Time bar (total: 39.7s)Debug logProfile

herbie shell --seed '#(1071852389 864846987 1238109217 3425890003 4124793586 650694553)' +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))