2cos (problem 3.3.5)

Average Error: 39.7 → 16.5

Time: 26.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -3.934754991620558305595464982165609946207 \cdot 10^{-8}:\\ \;\;\;\;\frac{\left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right) - \left(\sqrt[3]{\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)} \cdot \sqrt[3]{\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)}\right) \cdot \sqrt[3]{\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)}}{\sin \varepsilon \cdot \sin x + \cos x \cdot \cos \varepsilon} - \cos x\\ \mathbf{elif}\;\varepsilon \le 9.775851168574081187156640883099244730994 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{6} \cdot \left(\left(x \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) - \mathsf{fma}\left(\frac{1}{2}, \varepsilon \cdot \varepsilon, x \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;0 \cdot \cos x + \left(\frac{\left(\left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right) - \left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)\right) \cdot \mathsf{fma}\left(\sin \varepsilon \cdot \sin x, \sin \varepsilon \cdot \sin x - \cos x \cdot \cos \varepsilon, \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)\right)}{\mathsf{fma}\left(\cos x \cdot \cos \varepsilon, \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right), \left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)\right)\right)} - \cos x\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r2437729 = x;
        double r2437730 = eps;
        double r2437731 = r2437729 + r2437730;
        double r2437732 = cos(r2437731);
        double r2437733 = cos(r2437729);
        double r2437734 = r2437732 - r2437733;
        return r2437734;
}

↓

double f(double x, double eps) {
        double r2437735 = eps;
        double r2437736 = -3.934754991620558e-08;
        bool r2437737 = r2437735 <= r2437736;
        double r2437738 = x;
        double r2437739 = cos(r2437738);
        double r2437740 = cos(r2437735);
        double r2437741 = r2437739 * r2437740;
        double r2437742 = r2437741 * r2437741;
        double r2437743 = sin(r2437735);
        double r2437744 = sin(r2437738);
        double r2437745 = r2437743 * r2437744;
        double r2437746 = r2437745 * r2437745;
        double r2437747 = cbrt(r2437746);
        double r2437748 = r2437747 * r2437747;
        double r2437749 = r2437748 * r2437747;
        double r2437750 = r2437742 - r2437749;
        double r2437751 = r2437745 + r2437741;
        double r2437752 = r2437750 / r2437751;
        double r2437753 = r2437752 - r2437739;
        double r2437754 = 9.775851168574081e-26;
        bool r2437755 = r2437735 <= r2437754;
        double r2437756 = 0.16666666666666666;
        double r2437757 = r2437738 * r2437735;
        double r2437758 = r2437738 * r2437738;
        double r2437759 = r2437757 * r2437758;
        double r2437760 = r2437756 * r2437759;
        double r2437761 = 0.5;
        double r2437762 = r2437735 * r2437735;
        double r2437763 = fma(r2437761, r2437762, r2437757);
        double r2437764 = r2437760 - r2437763;
        double r2437765 = 0.0;
        double r2437766 = r2437765 * r2437739;
        double r2437767 = r2437742 - r2437746;
        double r2437768 = r2437745 - r2437741;
        double r2437769 = fma(r2437745, r2437768, r2437742);
        double r2437770 = r2437767 * r2437769;
        double r2437771 = r2437745 * r2437746;
        double r2437772 = fma(r2437741, r2437742, r2437771);
        double r2437773 = r2437770 / r2437772;
        double r2437774 = r2437773 - r2437739;
        double r2437775 = r2437766 + r2437774;
        double r2437776 = r2437755 ? r2437764 : r2437775;
        double r2437777 = r2437737 ? r2437753 : r2437776;
        return r2437777;
}

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 3 regimes

2. if eps < -3.934754991620558e-08

   1. Initial program 30.9

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

   3. Applied cos-sum 1.2

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   4. Using strategy rm

   5. Applied flip-- 1.3

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

   7. Applied add-cube-cbrt 1.5

      \[\leadsto \frac{\left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right) - \color{blue}{\left(\sqrt[3]{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)} \cdot \sqrt[3]{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)}\right) \cdot \sqrt[3]{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)}}}{\cos x \cdot \cos \varepsilon + \sin x \cdot \sin \varepsilon} - \cos x\]

   if -3.934754991620558e-08 < eps < 9.775851168574081e-26

   1. Initial program 48.8

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

   3. Applied cos-sum 48.7

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   4. Using strategy rm

   5. Applied flip-- 48.7

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

   6. Taylor expanded around 0 31.6

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({x}^{3} \cdot \varepsilon\right) - \left(\frac{1}{2} \cdot {\varepsilon}^{2} + x \cdot \varepsilon\right)}\]

   7. Simplified 31.5

      \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{6} - \mathsf{fma}\left(\frac{1}{2}, \varepsilon \cdot \varepsilon, \varepsilon \cdot x\right)}\]

   if 9.775851168574081e-26 < eps

   1. Initial program 31.7

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

   3. Applied cos-sum 3.6

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   4. Using strategy rm

   5. Applied flip-- 3.7

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

   7. Applied add-cube-cbrt 4.0

      \[\leadsto \frac{\left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right) - \left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)}{\cos x \cdot \cos \varepsilon + \sin x \cdot \sin \varepsilon} - \color{blue}{\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \sqrt[3]{\cos x}}\]

   8. Applied flip3-+ 4.2

      \[\leadsto \frac{\left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right) - \left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)}{\color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} + {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{\left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right) + \left(\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right) - \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)}}} - \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \sqrt[3]{\cos x}\]

   9. Applied associate-/r/ 4.2

      \[\leadsto \color{blue}{\frac{\left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right) - \left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} + {\left(\sin x \cdot \sin \varepsilon\right)}^{3}} \cdot \left(\left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right) + \left(\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right) - \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)\right)} - \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \sqrt[3]{\cos x}\]

   10. Applied prod-diff 4.2

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right) - \left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} + {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}, \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right) + \left(\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right) - \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right), -\sqrt[3]{\cos x} \cdot \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\cos x}, \sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}, \sqrt[3]{\cos x} \cdot \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right)\right)}\]

   11. Simplified 3.8

       \[\leadsto \color{blue}{\left(\frac{\left(\left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right) - \left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)\right) \cdot \mathsf{fma}\left(\sin \varepsilon \cdot \sin x, \sin \varepsilon \cdot \sin x - \cos \varepsilon \cdot \cos x, \left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)\right)}{\mathsf{fma}\left(\cos \varepsilon \cdot \cos x, \left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right), \left(\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)\right)} - \cos x\right)} + \mathsf{fma}\left(-\sqrt[3]{\cos x}, \sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}, \sqrt[3]{\cos x} \cdot \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right)\right)\]

   12. Simplified 3.8

       \[\leadsto \left(\frac{\left(\left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right) - \left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)\right) \cdot \mathsf{fma}\left(\sin \varepsilon \cdot \sin x, \sin \varepsilon \cdot \sin x - \cos \varepsilon \cdot \cos x, \left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)\right)}{\mathsf{fma}\left(\cos \varepsilon \cdot \cos x, \left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right), \left(\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)\right)} - \cos x\right) + \color{blue}{\cos x \cdot 0}\]
3. Recombined 3 regimes into one program.

4. Final simplification 16.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -3.934754991620558305595464982165609946207 \cdot 10^{-8}:\\ \;\;\;\;\frac{\left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right) - \left(\sqrt[3]{\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)} \cdot \sqrt[3]{\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)}\right) \cdot \sqrt[3]{\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)}}{\sin \varepsilon \cdot \sin x + \cos x \cdot \cos \varepsilon} - \cos x\\ \mathbf{elif}\;\varepsilon \le 9.775851168574081187156640883099244730994 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{6} \cdot \left(\left(x \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) - \mathsf{fma}\left(\frac{1}{2}, \varepsilon \cdot \varepsilon, x \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;0 \cdot \cos x + \left(\frac{\left(\left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right) - \left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)\right) \cdot \mathsf{fma}\left(\sin \varepsilon \cdot \sin x, \sin \varepsilon \cdot \sin x - \cos x \cdot \cos \varepsilon, \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)\right)}{\mathsf{fma}\left(\cos x \cdot \cos \varepsilon, \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right), \left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)\right)\right)} - \cos x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))