Average Error: 39.9 → 16.4
Time: 10.1s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -9.09194571234337580020110294837711693374 \cdot 10^{-27}:\\ \;\;\;\;\left(\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \cos \varepsilon\right) - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 3.672352851463477013948306603677469717173 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(e^{\cos x \cdot \cos \varepsilon}\right) - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -9.09194571234337580020110294837711693374 \cdot 10^{-27}:\\
\;\;\;\;\left(\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \cos \varepsilon\right) - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\mathbf{elif}\;\varepsilon \le 3.672352851463477013948306603677469717173 \cdot 10^{-9}:\\
\;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\

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

\end{array}
double f(double x, double eps) {
        double r76147 = x;
        double r76148 = eps;
        double r76149 = r76147 + r76148;
        double r76150 = cos(r76149);
        double r76151 = cos(r76147);
        double r76152 = r76150 - r76151;
        return r76152;
}
↓
double f(double x, double eps) {
        double r76153 = eps;
        double r76154 = -9.091945712343376e-27;
        bool r76155 = r76153 <= r76154;
        double r76156 = x;
        double r76157 = cos(r76156);
        double r76158 = cbrt(r76157);
        double r76159 = r76158 * r76158;
        double r76160 = cos(r76153);
        double r76161 = r76158 * r76160;
        double r76162 = r76159 * r76161;
        double r76163 = sin(r76156);
        double r76164 = sin(r76153);
        double r76165 = r76163 * r76164;
        double r76166 = r76162 - r76165;
        double r76167 = r76166 - r76157;
        double r76168 = 3.672352851463477e-09;
        bool r76169 = r76153 <= r76168;
        double r76170 = 0.16666666666666666;
        double r76171 = 3.0;
        double r76172 = pow(r76156, r76171);
        double r76173 = r76170 * r76172;
        double r76174 = r76173 - r76156;
        double r76175 = 0.5;
        double r76176 = r76153 * r76175;
        double r76177 = r76174 - r76176;
        double r76178 = r76153 * r76177;
        double r76179 = r76157 * r76160;
        double r76180 = exp(r76179);
        double r76181 = log(r76180);
        double r76182 = r76181 - r76165;
        double r76183 = r76182 - r76157;
        double r76184 = r76169 ? r76178 : r76183;
        double r76185 = r76155 ? r76167 : r76184;
        return r76185;
}

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 < -9.091945712343376e-27

    1. Initial program 32.2

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

    3. Applied cos-sum4.0

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

    5. Applied add-cube-cbrt4.3

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

    6. Applied associate-*l*4.3

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

    if -9.091945712343376e-27 < eps < 3.672352851463477e-09

    1. Initial program 49.5

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

    2. Taylor expanded around 0 31.5

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

    3. Simplified31.5

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

    if 3.672352851463477e-09 < eps

    1. Initial program 30.4

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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 add-log-exp1.5

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

  4. Final simplification16.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -9.09194571234337580020110294837711693374 \cdot 10^{-27}:\\ \;\;\;\;\left(\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \cos \varepsilon\right) - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 3.672352851463477013948306603677469717173 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(e^{\cos x \cdot \cos \varepsilon}\right) - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

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