2cos (problem 3.3.5)

Average Error: 40.0 → 16.4

Time: 7.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.153154815626698153605163400370998364602 \cdot 10^{-10} \lor \neg \left(\varepsilon \le 1.280988864612255259754335784405654408147 \cdot 10^{-6}\right):\\ \;\;\;\;\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos x \cdot \cos \varepsilon\right)\right) - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{24}, {\varepsilon}^{4}, -\mathsf{fma}\left(x, \varepsilon, \frac{1}{2} \cdot {\varepsilon}^{2}\right)\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right)}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x}\\ \end{array}\]

C

double f(double x, double eps) {
        double r62821 = x;
        double r62822 = eps;
        double r62823 = r62821 + r62822;
        double r62824 = cos(r62823);
        double r62825 = cos(r62821);
        double r62826 = r62824 - r62825;
        return r62826;
}

↓

double f(double x, double eps) {
        double r62827 = eps;
        double r62828 = -1.1531548156266982e-10;
        bool r62829 = r62827 <= r62828;
        double r62830 = 1.2809888646122553e-06;
        bool r62831 = r62827 <= r62830;
        double r62832 = !r62831;
        bool r62833 = r62829 || r62832;
        double r62834 = x;
        double r62835 = cos(r62834);
        double r62836 = cos(r62827);
        double r62837 = r62835 * r62836;
        double r62838 = expm1(r62837);
        double r62839 = log1p(r62838);
        double r62840 = sin(r62834);
        double r62841 = sin(r62827);
        double r62842 = r62840 * r62841;
        double r62843 = r62839 - r62842;
        double r62844 = r62843 - r62835;
        double r62845 = 0.041666666666666664;
        double r62846 = 4.0;
        double r62847 = pow(r62827, r62846);
        double r62848 = 0.5;
        double r62849 = 2.0;
        double r62850 = pow(r62827, r62849);
        double r62851 = r62848 * r62850;
        double r62852 = fma(r62834, r62827, r62851);
        double r62853 = -r62852;
        double r62854 = fma(r62845, r62847, r62853);
        double r62855 = r62837 - r62842;
        double r62856 = r62855 + r62835;
        double r62857 = r62854 * r62856;
        double r62858 = r62857 / r62856;
        double r62859 = r62833 ? r62844 : r62858;
        return r62859;
}

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 2 regimes

2. if eps < -1.1531548156266982e-10 or 1.2809888646122553e-06 < eps

   1. Initial program 30.5

      \[\cos \left(x + \varepsilon\right) - \cos x\]
   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 log1p-expm1-u 1.3

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

   if -1.1531548156266982e-10 < eps < 1.2809888646122553e-06

   1. Initial program 49.7

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

   3. Applied cos-sum 49.3

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

   5. Applied flip-- 49.3

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

   6. Simplified 49.3

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

   7. Taylor expanded around 0 32.0

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

   8. Simplified 32.0

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {\varepsilon}^{4}, -\mathsf{fma}\left(x, \varepsilon, \frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)} \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right)}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x}\]
3. Recombined 2 regimes into one program.

4. Final simplification 16.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.153154815626698153605163400370998364602 \cdot 10^{-10} \lor \neg \left(\varepsilon \le 1.280988864612255259754335784405654408147 \cdot 10^{-6}\right):\\ \;\;\;\;\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos x \cdot \cos \varepsilon\right)\right) - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{24}, {\varepsilon}^{4}, -\mathsf{fma}\left(x, \varepsilon, \frac{1}{2} \cdot {\varepsilon}^{2}\right)\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right)}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))