Average Error: 31.6 → 0.3
Time: 3.6s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03391785124353622477011427349680161569268:\\ \;\;\;\;\left(1 - \cos x\right) \cdot \frac{\frac{1}{x}}{x}\\ \mathbf{elif}\;x \le 0.03384107969428518797316840505118307191879:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \frac{\cos x - 1}{x}}{x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03391785124353622477011427349680161569268:\\
\;\;\;\;\left(1 - \cos x\right) \cdot \frac{\frac{1}{x}}{x}\\

\mathbf{elif}\;x \le 0.03384107969428518797316840505118307191879:\\
\;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\

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

\end{array}
double f(double x) {
        double r19919 = 1.0;
        double r19920 = x;
        double r19921 = cos(r19920);
        double r19922 = r19919 - r19921;
        double r19923 = r19920 * r19920;
        double r19924 = r19922 / r19923;
        return r19924;
}
↓
double f(double x) {
        double r19925 = x;
        double r19926 = -0.033917851243536225;
        bool r19927 = r19925 <= r19926;
        double r19928 = 1.0;
        double r19929 = cos(r19925);
        double r19930 = r19928 - r19929;
        double r19931 = 1.0;
        double r19932 = r19931 / r19925;
        double r19933 = r19932 / r19925;
        double r19934 = r19930 * r19933;
        double r19935 = 0.03384107969428519;
        bool r19936 = r19925 <= r19935;
        double r19937 = 4.0;
        double r19938 = pow(r19925, r19937);
        double r19939 = 0.001388888888888889;
        double r19940 = 0.5;
        double r19941 = 0.041666666666666664;
        double r19942 = 2.0;
        double r19943 = pow(r19925, r19942);
        double r19944 = r19941 * r19943;
        double r19945 = r19940 - r19944;
        double r19946 = fma(r19938, r19939, r19945);
        double r19947 = -1.0;
        double r19948 = r19929 - r19928;
        double r19949 = r19948 / r19925;
        double r19950 = r19947 * r19949;
        double r19951 = r19950 / r19925;
        double r19952 = r19936 ? r19946 : r19951;
        double r19953 = r19927 ? r19934 : r19952;
        return r19953;
}

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -0.033917851243536225

    1. Initial program 1.3

      \[\frac{1 - \cos x}{x \cdot x}\]
    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 *-un-lft-identity0.5

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

    6. Applied div-inv0.5

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

    7. Applied times-frac0.6

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

    8. Simplified0.6

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

    if -0.033917851243536225 < x < 0.03384107969428519

    1. Initial program 62.3

      \[\frac{1 - \cos x}{x \cdot x}\]

    2. Taylor expanded around 0 0.0

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

    3. Simplified0.0

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

    if 0.03384107969428519 < x

    1. Initial program 1.0

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

    3. Applied associate-/r*0.4

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

    4. Taylor expanded around -inf 0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03391785124353622477011427349680161569268:\\ \;\;\;\;\left(1 - \cos x\right) \cdot \frac{\frac{1}{x}}{x}\\ \mathbf{elif}\;x \le 0.03384107969428518797316840505118307191879:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \frac{\cos x - 1}{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))