2cos (problem 3.3.5)

Average Error: 39.9 → 15.6

Time: 6.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.39699580404506896243528445638072145972 \cdot 10^{-12} \lor \neg \left(\varepsilon \le 3.588703897944783975981375382800948881101 \cdot 10^{-9}\right):\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({\varepsilon}^{3} \cdot \frac{1}{24} - \mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r46989 = x;
        double r46990 = eps;
        double r46991 = r46989 + r46990;
        double r46992 = cos(r46991);
        double r46993 = cos(r46989);
        double r46994 = r46992 - r46993;
        return r46994;
}

↓

double f(double x, double eps) {
        double r46995 = eps;
        double r46996 = -8.396995804045069e-12;
        bool r46997 = r46995 <= r46996;
        double r46998 = 3.588703897944784e-09;
        bool r46999 = r46995 <= r46998;
        double r47000 = !r46999;
        bool r47001 = r46997 || r47000;
        double r47002 = 1.0;
        double r47003 = cos(r46995);
        double r47004 = x;
        double r47005 = cos(r47004);
        double r47006 = sin(r47004);
        double r47007 = sin(r46995);
        double r47008 = fma(r47006, r47007, r47005);
        double r47009 = expm1(r47008);
        double r47010 = log1p(r47009);
        double r47011 = -r47010;
        double r47012 = fma(r47003, r47005, r47011);
        double r47013 = r47002 * r47012;
        double r47014 = 3.0;
        double r47015 = pow(r46995, r47014);
        double r47016 = 0.041666666666666664;
        double r47017 = r47015 * r47016;
        double r47018 = 0.5;
        double r47019 = fma(r47018, r46995, r47004);
        double r47020 = r47017 - r47019;
        double r47021 = r46995 * r47020;
        double r47022 = r47001 ? r47013 : r47021;
        return r47022;
}

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 2 regimes

2. if eps < -8.396995804045069e-12 or 3.588703897944784e-09 < eps

   1. Initial program 30.6

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

   3. Applied cos-sum 1.4

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

   5. Applied *-un-lft-identity 1.4

      \[\leadsto \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \color{blue}{1 \cdot \cos x}\]

   6. Applied *-un-lft-identity 1.4

      \[\leadsto \color{blue}{1 \cdot \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - 1 \cdot \cos x\]

   7. Applied distribute-lft-out-- 1.4

      \[\leadsto \color{blue}{1 \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\right)}\]

   8. Simplified 1.3

      \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)}\]
   9. Using strategy rm

   10. Applied log1p-expm1-u 1.4

       \[\leadsto 1 \cdot \mathsf{fma}\left(\cos \varepsilon, \cos x, -\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\right)}\right)\]

   if -8.396995804045069e-12 < eps < 3.588703897944784e-09

   1. Initial program 49.8

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

   3. Applied cos-sum 49.6

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

   5. Applied add-log-exp 49.8

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

   6. Applied add-log-exp 49.8

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

   7. Applied add-log-exp 49.6

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

   8. Applied diff-log 49.6

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

   9. Applied diff-log 49.6

      \[\leadsto \color{blue}{\log \left(\frac{\frac{e^{\cos x \cdot \cos \varepsilon}}{e^{\sin x \cdot \sin \varepsilon}}}{e^{\cos x}}\right)}\]

   10. Simplified 49.6

       \[\leadsto \log \color{blue}{\left(e^{\mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)}\right)}\]

   11. Taylor expanded around 0 30.8

       \[\leadsto \color{blue}{\frac{1}{24} \cdot {\varepsilon}^{4} - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)}\]

   12. Simplified 30.8

       \[\leadsto \color{blue}{\varepsilon \cdot \left({\varepsilon}^{3} \cdot \frac{1}{24} - \mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 15.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -8.39699580404506896243528445638072145972 \cdot 10^{-12} \lor \neg \left(\varepsilon \le 3.588703897944783975981375382800948881101 \cdot 10^{-9}\right):\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({\varepsilon}^{3} \cdot \frac{1}{24} - \mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))