Average Error: 31.6 → 0.4
Time: 4.3s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02993031743019760812507534808446507668123:\\ \;\;\;\;\frac{\frac{\log \left(e^{1 - \cos x}\right)}{x}}{x}\\ \mathbf{elif}\;x \le 0.03087561509222142355768347954381169984117:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \frac{1}{2}, \frac{1}{720} \cdot {x}^{5} - \frac{1}{24} \cdot {x}^{3}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{1 - \cos x}{x}}}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02993031743019760812507534808446507668123:\\
\;\;\;\;\frac{\frac{\log \left(e^{1 - \cos x}\right)}{x}}{x}\\

\mathbf{elif}\;x \le 0.03087561509222142355768347954381169984117:\\
\;\;\;\;\frac{\mathsf{fma}\left(x, \frac{1}{2}, \frac{1}{720} \cdot {x}^{5} - \frac{1}{24} \cdot {x}^{3}\right)}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{x}{\frac{1 - \cos x}{x}}}\\

\end{array}
double f(double x) {
        double r33379 = 1.0;
        double r33380 = x;
        double r33381 = cos(r33380);
        double r33382 = r33379 - r33381;
        double r33383 = r33380 * r33380;
        double r33384 = r33382 / r33383;
        return r33384;
}
↓
double f(double x) {
        double r33385 = x;
        double r33386 = -0.029930317430197608;
        bool r33387 = r33385 <= r33386;
        double r33388 = 1.0;
        double r33389 = cos(r33385);
        double r33390 = r33388 - r33389;
        double r33391 = exp(r33390);
        double r33392 = log(r33391);
        double r33393 = r33392 / r33385;
        double r33394 = r33393 / r33385;
        double r33395 = 0.030875615092221424;
        bool r33396 = r33385 <= r33395;
        double r33397 = 0.5;
        double r33398 = 0.001388888888888889;
        double r33399 = 5.0;
        double r33400 = pow(r33385, r33399);
        double r33401 = r33398 * r33400;
        double r33402 = 0.041666666666666664;
        double r33403 = 3.0;
        double r33404 = pow(r33385, r33403);
        double r33405 = r33402 * r33404;
        double r33406 = r33401 - r33405;
        double r33407 = fma(r33385, r33397, r33406);
        double r33408 = r33407 / r33385;
        double r33409 = 1.0;
        double r33410 = r33390 / r33385;
        double r33411 = r33385 / r33410;
        double r33412 = r33409 / r33411;
        double r33413 = r33396 ? r33408 : r33412;
        double r33414 = r33387 ? r33394 : r33413;
        return r33414;
}

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -0.029930317430197608

    1. Initial program 1.1

      \[\frac{1 - \cos x}{x \cdot x}\]
    2. Using strategy rm

    3. Applied associate-/r*0.5

      \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}}\]
    4. Using strategy rm

    5. Applied add-log-exp0.6

      \[\leadsto \frac{\frac{1 - \color{blue}{\log \left(e^{\cos x}\right)}}{x}}{x}\]

    6. Applied add-log-exp0.6

      \[\leadsto \frac{\frac{\color{blue}{\log \left(e^{1}\right)} - \log \left(e^{\cos x}\right)}{x}}{x}\]

    7. Applied diff-log0.7

      \[\leadsto \frac{\frac{\color{blue}{\log \left(\frac{e^{1}}{e^{\cos x}}\right)}}{x}}{x}\]

    8. Simplified0.6

      \[\leadsto \frac{\frac{\log \color{blue}{\left(e^{1 - \cos x}\right)}}{x}}{x}\]

    if -0.029930317430197608 < x < 0.030875615092221424

    1. Initial program 62.2

      \[\frac{1 - \cos x}{x \cdot x}\]
    2. Using strategy rm

    3. Applied associate-/r*61.2

      \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}}\]

    4. Taylor expanded around 0 0.0

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot x + \frac{1}{720} \cdot {x}^{5}\right) - \frac{1}{24} \cdot {x}^{3}}}{x}\]

    5. Simplified0.0

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, \frac{1}{720} \cdot {x}^{5} - \frac{1}{24} \cdot {x}^{3}\right)}}{x}\]

    if 0.030875615092221424 < x

    1. Initial program 1.0

      \[\frac{1 - \cos x}{x \cdot x}\]
    2. Using strategy rm

    3. Applied associate-/r*0.5

      \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}}\]
    4. Using strategy rm

    5. Applied clear-num1.0

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{1 - \cos x}{x}}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02993031743019760812507534808446507668123:\\ \;\;\;\;\frac{\frac{\log \left(e^{1 - \cos x}\right)}{x}}{x}\\ \mathbf{elif}\;x \le 0.03087561509222142355768347954381169984117:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \frac{1}{2}, \frac{1}{720} \cdot {x}^{5} - \frac{1}{24} \cdot {x}^{3}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{1 - \cos x}{x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))