Average Error: 39.6 → 16.1
Time: 7.2s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -6.5696410515067922 \cdot 10^{-9}:\\ \;\;\;\;\sqrt[3]{{\left(\mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\right)}^{3}}\\ \mathbf{elif}\;\varepsilon \le 9.2579730885600051 \cdot 10^{-8}:\\ \;\;\;\;\varepsilon \cdot \left({\varepsilon}^{3} \cdot \frac{1}{24} - \mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)\right)\right)\right) - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -6.5696410515067922 \cdot 10^{-9}:\\
\;\;\;\;\sqrt[3]{{\left(\mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\right)}^{3}}\\

\mathbf{elif}\;\varepsilon \le 9.2579730885600051 \cdot 10^{-8}:\\
\;\;\;\;\varepsilon \cdot \left({\varepsilon}^{3} \cdot \frac{1}{24} - \mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)\right)\right)\right) - \cos x\\

\end{array}
double f(double x, double eps) {
        double r37236 = x;
        double r37237 = eps;
        double r37238 = r37236 + r37237;
        double r37239 = cos(r37238);
        double r37240 = cos(r37236);
        double r37241 = r37239 - r37240;
        return r37241;
}
↓
double f(double x, double eps) {
        double r37242 = eps;
        double r37243 = -6.569641051506792e-09;
        bool r37244 = r37242 <= r37243;
        double r37245 = cos(r37242);
        double r37246 = x;
        double r37247 = cos(r37246);
        double r37248 = sin(r37246);
        double r37249 = sin(r37242);
        double r37250 = fma(r37248, r37249, r37247);
        double r37251 = -r37250;
        double r37252 = fma(r37245, r37247, r37251);
        double r37253 = 3.0;
        double r37254 = pow(r37252, r37253);
        double r37255 = cbrt(r37254);
        double r37256 = 9.257973088560005e-08;
        bool r37257 = r37242 <= r37256;
        double r37258 = pow(r37242, r37253);
        double r37259 = 0.041666666666666664;
        double r37260 = r37258 * r37259;
        double r37261 = 0.5;
        double r37262 = fma(r37261, r37242, r37246);
        double r37263 = r37260 - r37262;
        double r37264 = r37242 * r37263;
        double r37265 = r37247 * r37245;
        double r37266 = r37248 * r37249;
        double r37267 = r37265 - r37266;
        double r37268 = expm1(r37267);
        double r37269 = log1p(r37268);
        double r37270 = log1p(r37269);
        double r37271 = expm1(r37270);
        double r37272 = r37271 - r37247;
        double r37273 = r37257 ? r37264 : r37272;
        double r37274 = r37244 ? r37255 : r37273;
        return r37274;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 3 regimes

  2. if eps < -6.569641051506792e-09

    1. Initial program 30.2

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

    3. Applied cos-sum1.1

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

    5. Applied add-cbrt-cube1.3

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

    6. Simplified1.3

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

    if -6.569641051506792e-09 < eps < 9.257973088560005e-08

    1. Initial program 49.3

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

    3. Applied cos-sum48.9

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

    5. Applied add-log-exp49.2

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

    6. Applied add-log-exp49.2

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

    7. Applied add-log-exp49.0

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

    8. Applied diff-log49.0

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

    9. Applied diff-log49.0

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

    10. Simplified48.9

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

    11. Taylor expanded around 0 31.7

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

    12. Simplified31.7

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

    if 9.257973088560005e-08 < eps

    1. Initial program 30.6

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

    3. Applied cos-sum1.2

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

    5. Applied log1p-expm1-u1.3

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

    7. Applied expm1-log1p-u1.4

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)\right)\right)\right)} - \cos x\]
  3. Recombined 3 regimes into one program.

  4. Final simplification16.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -6.5696410515067922 \cdot 10^{-9}:\\ \;\;\;\;\sqrt[3]{{\left(\mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\right)}^{3}}\\ \mathbf{elif}\;\varepsilon \le 9.2579730885600051 \cdot 10^{-8}:\\ \;\;\;\;\varepsilon \cdot \left({\varepsilon}^{3} \cdot \frac{1}{24} - \mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)\right)\right)\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))