Average Error: 39.6 → 15.9
Time: 8.0s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.13315499271887297 \cdot 10^{-10} \lor \neg \left(\varepsilon \le 3.28455979101293219 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x \cdot \cos \varepsilon\right) + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)} - \cos x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.13315499271887297 \cdot 10^{-10} \lor \neg \left(\varepsilon \le 3.28455979101293219 \cdot 10^{-7}\right):\\
\;\;\;\;\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x \cdot \cos \varepsilon\right) + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)} - \cos x\\

\mathbf{else}:\\
\;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\

\end{array}
double f(double x, double eps) {
        double r64597 = x;
        double r64598 = eps;
        double r64599 = r64597 + r64598;
        double r64600 = cos(r64599);
        double r64601 = cos(r64597);
        double r64602 = r64600 - r64601;
        return r64602;
}
↓
double f(double x, double eps) {
        double r64603 = eps;
        double r64604 = -1.133154992718873e-10;
        bool r64605 = r64603 <= r64604;
        double r64606 = 3.284559791012932e-07;
        bool r64607 = r64603 <= r64606;
        double r64608 = !r64607;
        bool r64609 = r64605 || r64608;
        double r64610 = x;
        double r64611 = cos(r64610);
        double r64612 = cos(r64603);
        double r64613 = r64611 * r64612;
        double r64614 = 3.0;
        double r64615 = pow(r64613, r64614);
        double r64616 = sin(r64610);
        double r64617 = sin(r64603);
        double r64618 = r64616 * r64617;
        double r64619 = pow(r64618, r64614);
        double r64620 = r64615 - r64619;
        double r64621 = r64618 + r64613;
        double r64622 = r64618 * r64621;
        double r64623 = r64613 * r64613;
        double r64624 = r64622 + r64623;
        double r64625 = r64620 / r64624;
        double r64626 = r64625 - r64611;
        double r64627 = 0.16666666666666666;
        double r64628 = pow(r64610, r64614);
        double r64629 = r64627 * r64628;
        double r64630 = r64629 - r64610;
        double r64631 = 0.5;
        double r64632 = r64603 * r64631;
        double r64633 = r64630 - r64632;
        double r64634 = r64603 * r64633;
        double r64635 = r64609 ? r64626 : r64634;
        return r64635;
}

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 < -1.133154992718873e-10 or 3.284559791012932e-07 < eps

    1. Initial program 30.5

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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 flip3--1.3

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

    6. Simplified1.3

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

    if -1.133154992718873e-10 < eps < 3.284559791012932e-07

    1. Initial program 49.3

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

    2. Taylor expanded around 0 31.7

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

    3. Simplified31.7

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification15.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.13315499271887297 \cdot 10^{-10} \lor \neg \left(\varepsilon \le 3.28455979101293219 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x \cdot \cos \varepsilon\right) + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)} - \cos x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))