Average Error: 39.1 → 15.5
Time: 7.8s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.14002637841950426 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt[3]{{\left({\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3}\right)}^{3}} - {\left(\cos x\right)}^{3}}{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right) + \cos x \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 2.673419722608438 \cdot 10^{-8}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(e^{\cos x \cdot \cos \varepsilon}\right) - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -2.14002637841950426 \cdot 10^{-7}:\\
\;\;\;\;\frac{\sqrt[3]{{\left({\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3}\right)}^{3}} - {\left(\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}\\

\mathbf{elif}\;\varepsilon \le 2.673419722608438 \cdot 10^{-8}:\\
\;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\log \left(e^{\cos x \cdot \cos \varepsilon}\right) - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\end{array}
double f(double x, double eps) {
        double r60485 = x;
        double r60486 = eps;
        double r60487 = r60485 + r60486;
        double r60488 = cos(r60487);
        double r60489 = cos(r60485);
        double r60490 = r60488 - r60489;
        return r60490;
}
↓
double f(double x, double eps) {
        double r60491 = eps;
        double r60492 = -2.1400263784195043e-07;
        bool r60493 = r60491 <= r60492;
        double r60494 = x;
        double r60495 = cos(r60494);
        double r60496 = cos(r60491);
        double r60497 = r60495 * r60496;
        double r60498 = sin(r60494);
        double r60499 = sin(r60491);
        double r60500 = r60498 * r60499;
        double r60501 = r60497 - r60500;
        double r60502 = 3.0;
        double r60503 = pow(r60501, r60502);
        double r60504 = pow(r60503, r60502);
        double r60505 = cbrt(r60504);
        double r60506 = pow(r60495, r60502);
        double r60507 = r60505 - r60506;
        double r60508 = r60496 * r60495;
        double r60509 = r60508 - r60500;
        double r60510 = r60501 + r60495;
        double r60511 = r60509 * r60510;
        double r60512 = r60495 * r60495;
        double r60513 = r60511 + r60512;
        double r60514 = r60507 / r60513;
        double r60515 = 2.6734197226084376e-08;
        bool r60516 = r60491 <= r60515;
        double r60517 = 0.16666666666666666;
        double r60518 = pow(r60494, r60502);
        double r60519 = r60517 * r60518;
        double r60520 = r60519 - r60494;
        double r60521 = 0.5;
        double r60522 = r60491 * r60521;
        double r60523 = r60520 - r60522;
        double r60524 = r60491 * r60523;
        double r60525 = exp(r60497);
        double r60526 = log(r60525);
        double r60527 = r60526 - r60500;
        double r60528 = r60527 - r60495;
        double r60529 = r60516 ? r60524 : r60528;
        double r60530 = r60493 ? r60514 : r60529;
        return r60530;
}

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 < -2.1400263784195043e-07

    1. Initial program 31.0

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

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

    6. Simplified1.3

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

    8. Applied add-cbrt-cube1.3

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

    9. Simplified1.3

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

    if -2.1400263784195043e-07 < eps < 2.6734197226084376e-08

    1. Initial program 48.7

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

    2. Taylor expanded around 0 30.9

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

    3. Simplified30.9

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

    if 2.6734197226084376e-08 < eps

    1. Initial program 29.7

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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 add-log-exp1.4

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

  4. Final simplification15.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.14002637841950426 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt[3]{{\left({\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3}\right)}^{3}} - {\left(\cos x\right)}^{3}}{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right) + \cos x \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 2.673419722608438 \cdot 10^{-8}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(e^{\cos x \cdot \cos \varepsilon}\right) - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))