Average Error: 39.5 → 17.1
Time: 6.9s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -3.2377418339921979 \cdot 10^{-63} \lor \neg \left(\varepsilon \le 9.62382077211877741 \cdot 10^{-17}\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sqrt[3]{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{4} - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -3.2377418339921979 \cdot 10^{-63} \lor \neg \left(\varepsilon \le 9.62382077211877741 \cdot 10^{-17}\right):\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sqrt[3]{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}\right) - \cos x\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{4} - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)\\

\end{array}
double f(double x, double eps) {
        double r30236 = x;
        double r30237 = eps;
        double r30238 = r30236 + r30237;
        double r30239 = cos(r30238);
        double r30240 = cos(r30236);
        double r30241 = r30239 - r30240;
        return r30241;
}
↓
double f(double x, double eps) {
        double r30242 = eps;
        double r30243 = -3.237741833992198e-63;
        bool r30244 = r30242 <= r30243;
        double r30245 = 9.623820772118777e-17;
        bool r30246 = r30242 <= r30245;
        double r30247 = !r30246;
        bool r30248 = r30244 || r30247;
        double r30249 = x;
        double r30250 = cos(r30249);
        double r30251 = cos(r30242);
        double r30252 = r30250 * r30251;
        double r30253 = sin(r30249);
        double r30254 = sin(r30242);
        double r30255 = r30253 * r30254;
        double r30256 = 3.0;
        double r30257 = pow(r30255, r30256);
        double r30258 = cbrt(r30257);
        double r30259 = r30252 - r30258;
        double r30260 = r30259 - r30250;
        double r30261 = 1.0;
        double r30262 = 0.041666666666666664;
        double r30263 = 4.0;
        double r30264 = pow(r30242, r30263);
        double r30265 = r30262 * r30264;
        double r30266 = r30249 * r30242;
        double r30267 = 0.5;
        double r30268 = 2.0;
        double r30269 = pow(r30242, r30268);
        double r30270 = r30267 * r30269;
        double r30271 = r30266 + r30270;
        double r30272 = r30265 - r30271;
        double r30273 = r30261 * r30272;
        double r30274 = r30248 ? r30260 : r30273;
        return r30274;
}

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 2 regimes

  2. if eps < -3.237741833992198e-63 or 9.623820772118777e-17 < eps

    1. Initial program 33.2

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

    3. Applied cos-sum6.5

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

    5. Applied add-cbrt-cube6.5

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

    6. Applied add-cbrt-cube6.6

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

    7. Applied cbrt-unprod6.6

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

    8. Simplified6.6

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

    if -3.237741833992198e-63 < eps < 9.623820772118777e-17

    1. Initial program 47.7

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

    3. Applied cos-sum47.7

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

    5. Applied *-un-lft-identity47.7

      \[\leadsto \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \color{blue}{1 \cdot \cos x}\]

    6. Applied *-un-lft-identity47.7

      \[\leadsto \color{blue}{1 \cdot \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - 1 \cdot \cos x\]

    7. Applied distribute-lft-out--47.7

      \[\leadsto \color{blue}{1 \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\right)}\]

    8. Simplified47.7

      \[\leadsto 1 \cdot \color{blue}{\left(\cos \varepsilon \cdot \cos x - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}\]

    9. Taylor expanded around 0 30.7

      \[\leadsto 1 \cdot \color{blue}{\left(\frac{1}{24} \cdot {\varepsilon}^{4} - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification17.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -3.2377418339921979 \cdot 10^{-63} \lor \neg \left(\varepsilon \le 9.62382077211877741 \cdot 10^{-17}\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sqrt[3]{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{4} - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))