Average Error: 39.6 → 16.4
Time: 9.3s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -6.5696410515067922 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 9.2579730885600051 \cdot 10^{-8}\right):\\ \;\;\;\;\log \left(e^{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -6.5696410515067922 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 9.2579730885600051 \cdot 10^{-8}\right):\\
\;\;\;\;\log \left(e^{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x}\right)\\

\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 r67495 = x;
        double r67496 = eps;
        double r67497 = r67495 + r67496;
        double r67498 = cos(r67497);
        double r67499 = cos(r67495);
        double r67500 = r67498 - r67499;
        return r67500;
}
↓
double f(double x, double eps) {
        double r67501 = eps;
        double r67502 = -6.569641051506792e-09;
        bool r67503 = r67501 <= r67502;
        double r67504 = 9.257973088560005e-08;
        bool r67505 = r67501 <= r67504;
        double r67506 = !r67505;
        bool r67507 = r67503 || r67506;
        double r67508 = x;
        double r67509 = cos(r67508);
        double r67510 = cos(r67501);
        double r67511 = r67509 * r67510;
        double r67512 = sin(r67508);
        double r67513 = sin(r67501);
        double r67514 = r67512 * r67513;
        double r67515 = r67511 - r67514;
        double r67516 = r67515 - r67509;
        double r67517 = exp(r67516);
        double r67518 = log(r67517);
        double r67519 = 0.16666666666666666;
        double r67520 = 3.0;
        double r67521 = pow(r67508, r67520);
        double r67522 = r67519 * r67521;
        double r67523 = r67522 - r67508;
        double r67524 = 0.5;
        double r67525 = r67501 * r67524;
        double r67526 = r67523 - r67525;
        double r67527 = r67501 * r67526;
        double r67528 = r67507 ? r67518 : r67527;
        return r67528;
}

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 < -6.569641051506792e-09 or 9.257973088560005e-08 < eps

    1. Initial program 30.4

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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-log-exp1.2

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

    6. Applied add-log-exp1.3

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

    7. Applied add-log-exp1.5

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

    8. Applied diff-log1.5

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

    9. Applied diff-log1.6

      \[\leadsto \color{blue}{\log \left(\frac{\frac{e^{\cos x \cdot \cos \varepsilon}}{e^{\sin x \cdot \sin \varepsilon}}}{e^{\cos x}}\right)}\]

    10. Simplified1.3

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

    if -6.569641051506792e-09 < eps < 9.257973088560005e-08

    1. Initial program 49.3

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

    2. Taylor expanded around 0 32.3

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

    3. Simplified32.3

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

  4. Final simplification16.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -6.5696410515067922 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 9.2579730885600051 \cdot 10^{-8}\right):\\ \;\;\;\;\log \left(e^{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))