Average Error: 32.3 → 0.4
Time: 14.8s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0347406315074210436666390933169168420136:\\ \;\;\;\;\frac{\frac{1}{x}}{x} - \frac{\frac{\cos x}{x}}{x}\\ \mathbf{elif}\;x \le 0.03078339684086396285667142080910707591102:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \frac{-1}{24}, \mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right), \frac{1}{720}, \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{1 - \cos x}{x}}}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0347406315074210436666390933169168420136:\\
\;\;\;\;\frac{\frac{1}{x}}{x} - \frac{\frac{\cos x}{x}}{x}\\

\mathbf{elif}\;x \le 0.03078339684086396285667142080910707591102:\\
\;\;\;\;\mathsf{fma}\left(x \cdot x, \frac{-1}{24}, \mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right), \frac{1}{720}, \frac{1}{2}\right)\right)\\

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

\end{array}
double f(double x) {
        double r720203 = 1.0;
        double r720204 = x;
        double r720205 = cos(r720204);
        double r720206 = r720203 - r720205;
        double r720207 = r720204 * r720204;
        double r720208 = r720206 / r720207;
        return r720208;
}
↓
double f(double x) {
        double r720209 = x;
        double r720210 = -0.034740631507421044;
        bool r720211 = r720209 <= r720210;
        double r720212 = 1.0;
        double r720213 = r720212 / r720209;
        double r720214 = r720213 / r720209;
        double r720215 = cos(r720209);
        double r720216 = r720215 / r720209;
        double r720217 = r720216 / r720209;
        double r720218 = r720214 - r720217;
        double r720219 = 0.030783396840863963;
        bool r720220 = r720209 <= r720219;
        double r720221 = r720209 * r720209;
        double r720222 = -0.041666666666666664;
        double r720223 = r720221 * r720221;
        double r720224 = 0.001388888888888889;
        double r720225 = 0.5;
        double r720226 = fma(r720223, r720224, r720225);
        double r720227 = fma(r720221, r720222, r720226);
        double r720228 = 1.0;
        double r720229 = r720212 - r720215;
        double r720230 = r720229 / r720209;
        double r720231 = r720209 / r720230;
        double r720232 = r720228 / r720231;
        double r720233 = r720220 ? r720227 : r720232;
        double r720234 = r720211 ? r720218 : r720233;
        return r720234;
}

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -0.034740631507421044

    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 div-sub0.6

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

    6. Applied div-sub0.7

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

    if -0.034740631507421044 < x < 0.030783396840863963

    1. Initial program 62.3

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

    3. Applied associate-/r*61.3

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

    4. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}}\]

    5. Simplified0.0

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

    if 0.030783396840863963 < 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-num1.1

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0347406315074210436666390933169168420136:\\ \;\;\;\;\frac{\frac{1}{x}}{x} - \frac{\frac{\cos x}{x}}{x}\\ \mathbf{elif}\;x \le 0.03078339684086396285667142080910707591102:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \frac{-1}{24}, \mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right), \frac{1}{720}, \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{1 - \cos x}{x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1.0 (cos x)) (* x x)))