Average Error: 31.5 → 0.3
Time: 17.9s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03155884382850993041813580930465832352638:\\ \;\;\;\;\frac{\frac{\sqrt[3]{{\left(1 - \cos x\right)}^{3}}}{x}}{x}\\ \mathbf{elif}\;x \le 0.02733685303105121144895406359864864498377:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{24}, x \cdot x, \mathsf{fma}\left(\frac{1}{720}, {x}^{4}, \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \cos x\right) \cdot \frac{\frac{1}{x}}{x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03155884382850993041813580930465832352638:\\
\;\;\;\;\frac{\frac{\sqrt[3]{{\left(1 - \cos x\right)}^{3}}}{x}}{x}\\

\mathbf{elif}\;x \le 0.02733685303105121144895406359864864498377:\\
\;\;\;\;\mathsf{fma}\left(\frac{-1}{24}, x \cdot x, \mathsf{fma}\left(\frac{1}{720}, {x}^{4}, \frac{1}{2}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(1 - \cos x\right) \cdot \frac{\frac{1}{x}}{x}\\

\end{array}
double f(double x) {
        double r21931 = 1.0;
        double r21932 = x;
        double r21933 = cos(r21932);
        double r21934 = r21931 - r21933;
        double r21935 = r21932 * r21932;
        double r21936 = r21934 / r21935;
        return r21936;
}
↓
double f(double x) {
        double r21937 = x;
        double r21938 = -0.03155884382850993;
        bool r21939 = r21937 <= r21938;
        double r21940 = 1.0;
        double r21941 = cos(r21937);
        double r21942 = r21940 - r21941;
        double r21943 = 3.0;
        double r21944 = pow(r21942, r21943);
        double r21945 = cbrt(r21944);
        double r21946 = r21945 / r21937;
        double r21947 = r21946 / r21937;
        double r21948 = 0.02733685303105121;
        bool r21949 = r21937 <= r21948;
        double r21950 = -0.041666666666666664;
        double r21951 = r21937 * r21937;
        double r21952 = 0.001388888888888889;
        double r21953 = 4.0;
        double r21954 = pow(r21937, r21953);
        double r21955 = 0.5;
        double r21956 = fma(r21952, r21954, r21955);
        double r21957 = fma(r21950, r21951, r21956);
        double r21958 = 1.0;
        double r21959 = r21958 / r21937;
        double r21960 = r21959 / r21937;
        double r21961 = r21942 * r21960;
        double r21962 = r21949 ? r21957 : r21961;
        double r21963 = r21939 ? r21947 : r21962;
        return r21963;
}

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -0.03155884382850993

    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 add-cbrt-cube0.6

      \[\leadsto \frac{\frac{\color{blue}{\sqrt[3]{\left(\left(1 - \cos x\right) \cdot \left(1 - \cos x\right)\right) \cdot \left(1 - \cos x\right)}}}{x}}{x}\]

    6. Simplified0.6

      \[\leadsto \frac{\frac{\sqrt[3]{\color{blue}{{\left(1 - \cos x\right)}^{3}}}}{x}}{x}\]

    if -0.03155884382850993 < x < 0.02733685303105121

    1. Initial program 62.3

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

    3. Applied associate-/r*61.4

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

    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

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

    if 0.02733685303105121 < x

    1. Initial program 0.9

      \[\frac{1 - \cos x}{x \cdot x}\]
    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-frac0.5

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

    8. Simplified0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03155884382850993041813580930465832352638:\\ \;\;\;\;\frac{\frac{\sqrt[3]{{\left(1 - \cos x\right)}^{3}}}{x}}{x}\\ \mathbf{elif}\;x \le 0.02733685303105121144895406359864864498377:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{24}, x \cdot x, \mathsf{fma}\left(\frac{1}{720}, {x}^{4}, \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \cos x\right) \cdot \frac{\frac{1}{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))