2cos (problem 3.3.5)

Average Error: 39.8 → 16.1

Time: 6.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -5.83896420831228018259913010307160519119 \cdot 10^{-7}:\\ \;\;\;\;\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos x \cdot \cos \varepsilon\right)\right) - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 2.120754527726823240979086562449373865036 \cdot 10^{-8}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{\mathsf{fma}\left(\sin x \cdot \sin \varepsilon, \mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \sin \varepsilon\right), \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)\right)} - \cos x\\ \end{array}\]

C

double f(double x, double eps) {
        double r48840 = x;
        double r48841 = eps;
        double r48842 = r48840 + r48841;
        double r48843 = cos(r48842);
        double r48844 = cos(r48840);
        double r48845 = r48843 - r48844;
        return r48845;
}

↓

double f(double x, double eps) {
        double r48846 = eps;
        double r48847 = -5.83896420831228e-07;
        bool r48848 = r48846 <= r48847;
        double r48849 = x;
        double r48850 = cos(r48849);
        double r48851 = cos(r48846);
        double r48852 = r48850 * r48851;
        double r48853 = expm1(r48852);
        double r48854 = log1p(r48853);
        double r48855 = sin(r48849);
        double r48856 = sin(r48846);
        double r48857 = r48855 * r48856;
        double r48858 = r48854 - r48857;
        double r48859 = r48858 - r48850;
        double r48860 = 2.1207545277268232e-08;
        bool r48861 = r48846 <= r48860;
        double r48862 = 0.16666666666666666;
        double r48863 = 3.0;
        double r48864 = pow(r48849, r48863);
        double r48865 = r48862 * r48864;
        double r48866 = r48865 - r48849;
        double r48867 = 0.5;
        double r48868 = r48846 * r48867;
        double r48869 = r48866 - r48868;
        double r48870 = r48846 * r48869;
        double r48871 = pow(r48852, r48863);
        double r48872 = pow(r48857, r48863);
        double r48873 = r48871 - r48872;
        double r48874 = fma(r48850, r48851, r48857);
        double r48875 = r48852 * r48852;
        double r48876 = fma(r48857, r48874, r48875);
        double r48877 = r48873 / r48876;
        double r48878 = r48877 - r48850;
        double r48879 = r48861 ? r48870 : r48878;
        double r48880 = r48848 ? r48859 : r48879;
        return r48880;
}

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 3 regimes

2. if eps < -5.83896420831228e-07

   1. Initial program 31.3

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

   3. Applied cos-sum 1.1

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

   5. Applied log1p-expm1-u 1.2

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

   if -5.83896420831228e-07 < eps < 2.1207545277268232e-08

   1. Initial program 49.4

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

   2. Taylor expanded around 0 32.3

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

   3. Simplified 32.3

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

   if 2.1207545277268232e-08 < eps

   1. Initial program 30.6

      \[\cos \left(x + \varepsilon\right) - \cos x\]
   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 flip3-- 1.5

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

   6. Simplified 1.4

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

4. Final simplification 16.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -5.83896420831228018259913010307160519119 \cdot 10^{-7}:\\ \;\;\;\;\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos x \cdot \cos \varepsilon\right)\right) - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 2.120754527726823240979086562449373865036 \cdot 10^{-8}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{\mathsf{fma}\left(\sin x \cdot \sin \varepsilon, \mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \sin \varepsilon\right), \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)\right)} - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2019344 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))