Average Error: 39.9 → 0.8
Time: 18.3s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -6443.306290177518349082674831151962280273:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 1.459775390859078191777703503717589228472 \cdot 10^{-4}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -6443.306290177518349082674831151962280273:\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\mathbf{elif}\;\varepsilon \le 1.459775390859078191777703503717589228472 \cdot 10^{-4}:\\
\;\;\;\;-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\

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

\end{array}
double f(double x, double eps) {
        double r2041190 = x;
        double r2041191 = eps;
        double r2041192 = r2041190 + r2041191;
        double r2041193 = cos(r2041192);
        double r2041194 = cos(r2041190);
        double r2041195 = r2041193 - r2041194;
        return r2041195;
}
↓
double f(double x, double eps) {
        double r2041196 = eps;
        double r2041197 = -6443.306290177518;
        bool r2041198 = r2041196 <= r2041197;
        double r2041199 = x;
        double r2041200 = cos(r2041199);
        double r2041201 = cos(r2041196);
        double r2041202 = r2041200 * r2041201;
        double r2041203 = sin(r2041199);
        double r2041204 = sin(r2041196);
        double r2041205 = r2041203 * r2041204;
        double r2041206 = r2041202 - r2041205;
        double r2041207 = r2041206 - r2041200;
        double r2041208 = 0.00014597753908590782;
        bool r2041209 = r2041196 <= r2041208;
        double r2041210 = -2.0;
        double r2041211 = r2041199 + r2041196;
        double r2041212 = r2041211 + r2041199;
        double r2041213 = 2.0;
        double r2041214 = r2041212 / r2041213;
        double r2041215 = sin(r2041214);
        double r2041216 = r2041196 / r2041213;
        double r2041217 = sin(r2041216);
        double r2041218 = r2041215 * r2041217;
        double r2041219 = r2041210 * r2041218;
        double r2041220 = r2041209 ? r2041219 : r2041207;
        double r2041221 = r2041198 ? r2041207 : r2041220;
        return r2041221;
}

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 < -6443.306290177518 or 0.00014597753908590782 < eps

    1. Initial program 30.3

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

    3. Applied cos-sum0.9

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

    if -6443.306290177518 < eps < 0.00014597753908590782

    1. Initial program 49.7

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

    3. Applied diff-cos37.7

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]

    4. Simplified0.8

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -6443.306290177518349082674831151962280273:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 1.459775390859078191777703503717589228472 \cdot 10^{-4}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))