Average Error: 39.1 → 0.7
Time: 24.3s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.68449879533306243542534197388249594951 \cdot 10^{-4} \lor \neg \left(\varepsilon \le 3.028675491606891100851602083299241030545 \cdot 10^{-6}\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -2.68449879533306243542534197388249594951 \cdot 10^{-4} \lor \neg \left(\varepsilon \le 3.028675491606891100851602083299241030545 \cdot 10^{-6}\right):\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\mathbf{else}:\\
\;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\

\end{array}
double f(double x, double eps) {
        double r39101 = x;
        double r39102 = eps;
        double r39103 = r39101 + r39102;
        double r39104 = cos(r39103);
        double r39105 = cos(r39101);
        double r39106 = r39104 - r39105;
        return r39106;
}
↓
double f(double x, double eps) {
        double r39107 = eps;
        double r39108 = -0.00026844987953330624;
        bool r39109 = r39107 <= r39108;
        double r39110 = 3.028675491606891e-06;
        bool r39111 = r39107 <= r39110;
        double r39112 = !r39111;
        bool r39113 = r39109 || r39112;
        double r39114 = x;
        double r39115 = cos(r39114);
        double r39116 = cos(r39107);
        double r39117 = r39115 * r39116;
        double r39118 = sin(r39114);
        double r39119 = sin(r39107);
        double r39120 = r39118 * r39119;
        double r39121 = r39117 - r39120;
        double r39122 = r39121 - r39115;
        double r39123 = -2.0;
        double r39124 = 2.0;
        double r39125 = r39107 / r39124;
        double r39126 = sin(r39125);
        double r39127 = r39114 + r39107;
        double r39128 = r39127 + r39114;
        double r39129 = r39128 / r39124;
        double r39130 = sin(r39129);
        double r39131 = r39126 * r39130;
        double r39132 = r39123 * r39131;
        double r39133 = r39113 ? r39122 : r39132;
        return r39133;
}

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 < -0.00026844987953330624 or 3.028675491606891e-06 < eps

    1. Initial program 29.4

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

    3. Applied cos-sum1.0

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

    if -0.00026844987953330624 < eps < 3.028675491606891e-06

    1. Initial program 49.2

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

    3. Applied diff-cos38.3

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

    4. Simplified0.5

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.68449879533306243542534197388249594951 \cdot 10^{-4} \lor \neg \left(\varepsilon \le 3.028675491606891100851602083299241030545 \cdot 10^{-6}\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))