Average Error: 55.3 → 0.5
Time: 17.3s
Precision: 64
\[\cos \left(x + 1\right) - \cos 1\]
\[\begin{array}{l} \mathbf{if}\;x \le -17243517846109498:\\ \;\;\;\;\cos x \cdot \cos 1 - \left(\cos 1 + \sin x \cdot \sin 1\right)\\ \mathbf{elif}\;x \le 401435.715085908654145896434783935546875:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{x}{2}\right) \cdot \sin \left(\frac{1 + \left(x + 1\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos 1 - \sin x \cdot \sin 1\right) - \cos 1\\ \end{array}\]
\cos \left(x + 1\right) - \cos 1
\begin{array}{l}
\mathbf{if}\;x \le -17243517846109498:\\
\;\;\;\;\cos x \cdot \cos 1 - \left(\cos 1 + \sin x \cdot \sin 1\right)\\

\mathbf{elif}\;x \le 401435.715085908654145896434783935546875:\\
\;\;\;\;-2 \cdot \left(\sin \left(\frac{x}{2}\right) \cdot \sin \left(\frac{1 + \left(x + 1\right)}{2}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\cos x \cdot \cos 1 - \sin x \cdot \sin 1\right) - \cos 1\\

\end{array}
double f(double x) {
        double r42592769 = x;
        double r42592770 = 1.0;
        double r42592771 = r42592769 + r42592770;
        double r42592772 = cos(r42592771);
        double r42592773 = cos(r42592770);
        double r42592774 = r42592772 - r42592773;
        return r42592774;
}

double f(double x) {
        double r42592775 = x;
        double r42592776 = -17243517846109498.0;
        bool r42592777 = r42592775 <= r42592776;
        double r42592778 = cos(r42592775);
        double r42592779 = 1.0;
        double r42592780 = cos(r42592779);
        double r42592781 = r42592778 * r42592780;
        double r42592782 = sin(r42592775);
        double r42592783 = sin(r42592779);
        double r42592784 = r42592782 * r42592783;
        double r42592785 = r42592780 + r42592784;
        double r42592786 = r42592781 - r42592785;
        double r42592787 = 401435.71508590865;
        bool r42592788 = r42592775 <= r42592787;
        double r42592789 = -2.0;
        double r42592790 = 2.0;
        double r42592791 = r42592775 / r42592790;
        double r42592792 = sin(r42592791);
        double r42592793 = r42592775 + r42592779;
        double r42592794 = r42592779 + r42592793;
        double r42592795 = r42592794 / r42592790;
        double r42592796 = sin(r42592795);
        double r42592797 = r42592792 * r42592796;
        double r42592798 = r42592789 * r42592797;
        double r42592799 = r42592781 - r42592784;
        double r42592800 = r42592799 - r42592780;
        double r42592801 = r42592788 ? r42592798 : r42592800;
        double r42592802 = r42592777 ? r42592786 : r42592801;
        return r42592802;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes
  2. if x < -17243517846109498.0

    1. Initial program 55.2

      \[\cos \left(x + 1\right) - \cos 1\]
    2. Using strategy rm
    3. Applied cos-sum0.9

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos 1 - \sin x \cdot \sin 1\right)} - \cos 1\]
    4. Applied associate--l-0.9

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

    if -17243517846109498.0 < x < 401435.71508590865

    1. Initial program 56.3

      \[\cos \left(x + 1\right) - \cos 1\]
    2. Using strategy rm
    3. Applied diff-cos56.3

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + 1\right) - 1}{2}\right) \cdot \sin \left(\frac{\left(x + 1\right) + 1}{2}\right)\right)}\]
    4. Simplified0.1

      \[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\frac{x}{2}\right) \cdot \sin \left(\frac{1 + \left(x + 1\right)}{2}\right)\right)}\]

    if 401435.71508590865 < x

    1. Initial program 53.1

      \[\cos \left(x + 1\right) - \cos 1\]
    2. Using strategy rm
    3. Applied cos-sum0.9

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos 1 - \sin x \cdot \sin 1\right)} - \cos 1\]
  3. Recombined 3 regimes into one program.
  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -17243517846109498:\\ \;\;\;\;\cos x \cdot \cos 1 - \left(\cos 1 + \sin x \cdot \sin 1\right)\\ \mathbf{elif}\;x \le 401435.715085908654145896434783935546875:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{x}{2}\right) \cdot \sin \left(\frac{1 + \left(x + 1\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos 1 - \sin x \cdot \sin 1\right) - \cos 1\\ \end{array}\]

Reproduce

herbie shell --seed 1 
(FPCore (x)
  :name "cos(x+1)-cos(1)"
  (- (cos (+ x 1.0)) (cos 1.0)))