Average Error: 40.0 → 16.2
Time: 7.3s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.6443088649730258 \cdot 10^{-8}:\\ \;\;\;\;\sqrt[3]{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3}} - \cos x\\ \mathbf{elif}\;\varepsilon \le 4.8831367818885993 \cdot 10^{-8}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3} - \left(\cos x \cdot \cos x\right) \cdot \cos x}{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right) + \cos x \cdot \cos x}\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -8.6443088649730258 \cdot 10^{-8}:\\
\;\;\;\;\sqrt[3]{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3}} - \cos x\\

\mathbf{elif}\;\varepsilon \le 4.8831367818885993 \cdot 10^{-8}:\\
\;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3} - \left(\cos x \cdot \cos x\right) \cdot \cos x}{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right) + \cos x \cdot \cos x}\\

\end{array}
double f(double x, double eps) {
        double r68751 = x;
        double r68752 = eps;
        double r68753 = r68751 + r68752;
        double r68754 = cos(r68753);
        double r68755 = cos(r68751);
        double r68756 = r68754 - r68755;
        return r68756;
}
↓
double f(double x, double eps) {
        double r68757 = eps;
        double r68758 = -8.644308864973026e-08;
        bool r68759 = r68757 <= r68758;
        double r68760 = x;
        double r68761 = cos(r68760);
        double r68762 = cos(r68757);
        double r68763 = r68761 * r68762;
        double r68764 = sin(r68760);
        double r68765 = sin(r68757);
        double r68766 = r68764 * r68765;
        double r68767 = r68763 - r68766;
        double r68768 = 3.0;
        double r68769 = pow(r68767, r68768);
        double r68770 = cbrt(r68769);
        double r68771 = r68770 - r68761;
        double r68772 = 4.883136781888599e-08;
        bool r68773 = r68757 <= r68772;
        double r68774 = 0.16666666666666666;
        double r68775 = pow(r68760, r68768);
        double r68776 = r68774 * r68775;
        double r68777 = r68776 - r68760;
        double r68778 = 0.5;
        double r68779 = r68757 * r68778;
        double r68780 = r68777 - r68779;
        double r68781 = r68757 * r68780;
        double r68782 = r68761 * r68761;
        double r68783 = r68782 * r68761;
        double r68784 = r68769 - r68783;
        double r68785 = r68762 * r68761;
        double r68786 = r68785 - r68766;
        double r68787 = r68767 + r68761;
        double r68788 = r68786 * r68787;
        double r68789 = r68788 + r68782;
        double r68790 = r68784 / r68789;
        double r68791 = r68773 ? r68781 : r68790;
        double r68792 = r68759 ? r68771 : r68791;
        return r68792;
}

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 3 regimes

  2. if eps < -8.644308864973026e-08

    1. Initial program 30.6

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

    3. Applied cos-sum1.1

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

    5. Applied add-cbrt-cube1.4

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

    6. Simplified1.4

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

    if -8.644308864973026e-08 < eps < 4.883136781888599e-08

    1. Initial program 49.9

      \[\cos \left(x + \varepsilon\right) - \cos x\]

    2. Taylor expanded around 0 31.6

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({x}^{3} \cdot \varepsilon\right) - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)}\]

    3. Simplified31.6

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)}\]

    if 4.883136781888599e-08 < eps

    1. Initial program 30.4

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

    3. Applied cos-sum1.2

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

    5. Applied flip3--1.4

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

    6. Simplified1.4

      \[\leadsto \frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3} - {\left(\cos x\right)}^{3}}{\color{blue}{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right) + \cos x \cdot \cos x}}\]
    7. Using strategy rm

    8. Applied add-cbrt-cube1.6

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

    9. Applied rem-cube-cbrt1.4

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

  4. Final simplification16.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -8.6443088649730258 \cdot 10^{-8}:\\ \;\;\;\;\sqrt[3]{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3}} - \cos x\\ \mathbf{elif}\;\varepsilon \le 4.8831367818885993 \cdot 10^{-8}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3} - \left(\cos x \cdot \cos x\right) \cdot \cos x}{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right) + \cos x \cdot \cos x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))