Average Error: 39.7 → 16.0
Time: 6.8s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -3.21001947811687037525689421214365347132 \cdot 10^{-11}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sqrt[3]{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 3.097623705192343513450383904461737927249 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{24} \cdot {\varepsilon}^{4} - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x}\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -3.21001947811687037525689421214365347132 \cdot 10^{-11}:\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sqrt[3]{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}\right) - \cos x\\

\mathbf{elif}\;\varepsilon \le 3.097623705192343513450383904461737927249 \cdot 10^{-6}:\\
\;\;\;\;\frac{1}{24} \cdot {\varepsilon}^{4} - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(e^{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x}\right)\\

\end{array}
double f(double x, double eps) {
        double r93018 = x;
        double r93019 = eps;
        double r93020 = r93018 + r93019;
        double r93021 = cos(r93020);
        double r93022 = cos(r93018);
        double r93023 = r93021 - r93022;
        return r93023;
}
↓
double f(double x, double eps) {
        double r93024 = eps;
        double r93025 = -3.2100194781168704e-11;
        bool r93026 = r93024 <= r93025;
        double r93027 = x;
        double r93028 = cos(r93027);
        double r93029 = cos(r93024);
        double r93030 = r93028 * r93029;
        double r93031 = sin(r93027);
        double r93032 = sin(r93024);
        double r93033 = r93031 * r93032;
        double r93034 = 3.0;
        double r93035 = pow(r93033, r93034);
        double r93036 = cbrt(r93035);
        double r93037 = r93030 - r93036;
        double r93038 = r93037 - r93028;
        double r93039 = 3.0976237051923435e-06;
        bool r93040 = r93024 <= r93039;
        double r93041 = 0.041666666666666664;
        double r93042 = 4.0;
        double r93043 = pow(r93024, r93042);
        double r93044 = r93041 * r93043;
        double r93045 = r93027 * r93024;
        double r93046 = 0.5;
        double r93047 = 2.0;
        double r93048 = pow(r93024, r93047);
        double r93049 = r93046 * r93048;
        double r93050 = r93045 + r93049;
        double r93051 = r93044 - r93050;
        double r93052 = r93030 - r93033;
        double r93053 = r93052 - r93028;
        double r93054 = exp(r93053);
        double r93055 = log(r93054);
        double r93056 = r93040 ? r93051 : r93055;
        double r93057 = r93026 ? r93038 : r93056;
        return r93057;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if eps < -3.2100194781168704e-11

    1. Initial program 30.5

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

    3. Applied cos-sum1.7

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

    5. Applied add-cbrt-cube1.7

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

    6. Applied add-cbrt-cube1.8

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

    7. Applied cbrt-unprod1.7

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

    8. Simplified1.7

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

    if -3.2100194781168704e-11 < eps < 3.0976237051923435e-06

    1. Initial program 49.4

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

    3. Applied cos-sum49.0

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

    4. Taylor expanded around inf 49.0

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

    5. Taylor expanded around 0 31.2

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

    if 3.0976237051923435e-06 < eps

    1. Initial program 30.4

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

    3. Applied cos-sum1.0

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

    5. Applied add-log-exp1.1

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

    6. Applied add-log-exp1.1

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

    7. Applied add-log-exp1.3

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

    8. Applied diff-log1.3

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

    9. Applied diff-log1.4

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

    10. Simplified1.1

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

  4. Final simplification16.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -3.21001947811687037525689421214365347132 \cdot 10^{-11}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sqrt[3]{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 3.097623705192343513450383904461737927249 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{24} \cdot {\varepsilon}^{4} - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))