2cos (problem 3.3.5)

Average Error: 39.7 → 16.3

Time: 7.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.9118824918033931 \cdot 10^{-16}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\\ \mathbf{elif}\;\varepsilon \le 2.2379838828549683 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon + \left(-\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r50871 = x;
        double r50872 = eps;
        double r50873 = r50871 + r50872;
        double r50874 = cos(r50873);
        double r50875 = cos(r50871);
        double r50876 = r50874 - r50875;
        return r50876;
}

↓

double f(double x, double eps) {
        double r50877 = eps;
        double r50878 = -8.911882491803393e-16;
        bool r50879 = r50877 <= r50878;
        double r50880 = 1.0;
        double r50881 = cos(r50877);
        double r50882 = x;
        double r50883 = cos(r50882);
        double r50884 = sin(r50882);
        double r50885 = sin(r50877);
        double r50886 = fma(r50884, r50885, r50883);
        double r50887 = -r50886;
        double r50888 = fma(r50881, r50883, r50887);
        double r50889 = r50880 * r50888;
        double r50890 = 2.2379838828549683e-07;
        bool r50891 = r50877 <= r50890;
        double r50892 = 0.16666666666666666;
        double r50893 = 3.0;
        double r50894 = pow(r50882, r50893);
        double r50895 = r50892 * r50894;
        double r50896 = r50895 - r50882;
        double r50897 = 0.5;
        double r50898 = r50877 * r50897;
        double r50899 = r50896 - r50898;
        double r50900 = r50877 * r50899;
        double r50901 = r50883 * r50881;
        double r50902 = r50901 + r50887;
        double r50903 = r50891 ? r50900 : r50902;
        double r50904 = r50879 ? r50889 : r50903;
        return r50904;
}

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 3 regimes

2. if eps < -8.911882491803393e-16

   1. Initial program 31.0

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

   3. Applied cos-sum 2.2

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

   5. Applied fma-neg 2.2

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

   6. Simplified 2.2

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

   8. Applied log1p-expm1-u 2.3

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\cos x, \cos \varepsilon, -\sin \varepsilon \cdot \sin x\right)\right)\right)} - \cos x\]
   9. Using strategy rm

   10. Applied *-un-lft-identity 2.3

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

   11. Applied *-un-lft-identity 2.3

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

   12. Applied distribute-lft-out-- 2.3

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

   13. Simplified 2.2

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

   if -8.911882491803393e-16 < eps < 2.2379838828549683e-07

   1. Initial program 48.9

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

   2. Taylor expanded around 0 31.8

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

   3. Simplified 31.8

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

   if 2.2379838828549683e-07 < eps

   1. Initial program 31.2

      \[\cos \left(x + \varepsilon\right) - \cos x\]
   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 sub-neg 1.2

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

   6. Applied associate--l+ 1.2

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

   7. Simplified 1.1

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

4. Final simplification 16.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -8.9118824918033931 \cdot 10^{-16}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\\ \mathbf{elif}\;\varepsilon \le 2.2379838828549683 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon + \left(-\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))