Average Error: 39.9 → 0.8
Time: 23.7s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -6443.306290177518349082674831151962280273:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 1.459775390859078191777703503717589228472 \cdot 10^{-4}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]
\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{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{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 r1775031 = x;
        double r1775032 = eps;
        double r1775033 = r1775031 + r1775032;
        double r1775034 = cos(r1775033);
        double r1775035 = cos(r1775031);
        double r1775036 = r1775034 - r1775035;
        return r1775036;
}
↓
double f(double x, double eps) {
        double r1775037 = eps;
        double r1775038 = -6443.306290177518;
        bool r1775039 = r1775037 <= r1775038;
        double r1775040 = x;
        double r1775041 = cos(r1775040);
        double r1775042 = cos(r1775037);
        double r1775043 = r1775041 * r1775042;
        double r1775044 = sin(r1775040);
        double r1775045 = sin(r1775037);
        double r1775046 = r1775044 * r1775045;
        double r1775047 = r1775043 - r1775046;
        double r1775048 = r1775047 - r1775041;
        double r1775049 = 0.00014597753908590782;
        bool r1775050 = r1775037 <= r1775049;
        double r1775051 = -2.0;
        double r1775052 = 2.0;
        double r1775053 = r1775037 / r1775052;
        double r1775054 = sin(r1775053);
        double r1775055 = r1775040 + r1775037;
        double r1775056 = r1775055 + r1775040;
        double r1775057 = r1775056 / r1775052;
        double r1775058 = sin(r1775057);
        double r1775059 = r1775054 * r1775058;
        double r1775060 = r1775051 * r1775059;
        double r1775061 = r1775050 ? r1775060 : r1775048;
        double r1775062 = r1775039 ? r1775048 : r1775061;
        return r1775062;
}

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

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

  4. Final simplification0.8

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

Reproduce

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