Average Error: 39.9 → 15.6
Time: 6.5s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.39699580404506896243528445638072145972 \cdot 10^{-12} \lor \neg \left(\varepsilon \le 3.588703897944783975981375382800948881101 \cdot 10^{-9}\right):\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\right)\right) + \mathsf{fma}\left(-\cos x, 1, \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({\varepsilon}^{3} \cdot \frac{1}{24} - \mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)\\ \end{array}\]
\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):\\
\;\;\;\;\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) + \mathsf{fma}\left(-\cos x, 1, \cos x\right)\\

\mathbf{else}:\\
\;\;\;\;\varepsilon \cdot \left({\varepsilon}^{3} \cdot \frac{1}{24} - \mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)\\

\end{array}
double f(double x, double eps) {
        double r53899 = x;
        double r53900 = eps;
        double r53901 = r53899 + r53900;
        double r53902 = cos(r53901);
        double r53903 = cos(r53899);
        double r53904 = r53902 - r53903;
        return r53904;
}
↓
double f(double x, double eps) {
        double r53905 = eps;
        double r53906 = -8.396995804045069e-12;
        bool r53907 = r53905 <= r53906;
        double r53908 = 3.588703897944784e-09;
        bool r53909 = r53905 <= r53908;
        double r53910 = !r53909;
        bool r53911 = r53907 || r53910;
        double r53912 = cos(r53905);
        double r53913 = x;
        double r53914 = cos(r53913);
        double r53915 = sin(r53913);
        double r53916 = sin(r53905);
        double r53917 = fma(r53915, r53916, r53914);
        double r53918 = expm1(r53917);
        double r53919 = log1p(r53918);
        double r53920 = -r53919;
        double r53921 = fma(r53912, r53914, r53920);
        double r53922 = -r53914;
        double r53923 = 1.0;
        double r53924 = fma(r53922, r53923, r53914);
        double r53925 = r53921 + r53924;
        double r53926 = 3.0;
        double r53927 = pow(r53905, r53926);
        double r53928 = 0.041666666666666664;
        double r53929 = r53927 * r53928;
        double r53930 = 0.5;
        double r53931 = fma(r53930, r53905, r53913);
        double r53932 = r53929 - r53931;
        double r53933 = r53905 * r53932;
        double r53934 = r53911 ? r53925 : r53933;
        return r53934;
}

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-sum1.4

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

    5. Applied add-cube-cbrt1.8

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

    6. Applied add-sqr-sqrt32.6

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

    7. Applied prod-diff32.6

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

    8. Simplified1.4

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

    9. Simplified1.3

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

    11. Applied log1p-expm1-u1.4

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

    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-sum49.6

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

    5. Applied add-log-exp49.8

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

    6. Applied add-log-exp49.8

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

    7. Applied add-log-exp49.6

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

    8. Applied diff-log49.6

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

    9. Applied diff-log49.6

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

    10. Simplified49.6

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

    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. Simplified30.8

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

  4. Final simplification15.6

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

Reproduce

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