Average Error: 31.5 → 0.3
Time: 16.9s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03119874976006306588338645724434172734618:\\ \;\;\;\;\left(\frac{1}{x} \cdot \frac{1}{x}\right) \cdot \left(1 - \cos x\right)\\ \mathbf{elif}\;x \le 0.03381719709829058423400383048829098697752:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \frac{-1}{24}, \mathsf{fma}\left(\frac{1}{720}, \left(x \cdot x\right) \cdot \left(x \cdot x\right), \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} \cdot \frac{1}{x}\right) \cdot \left(1 - \cos x\right)\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03119874976006306588338645724434172734618:\\
\;\;\;\;\left(\frac{1}{x} \cdot \frac{1}{x}\right) \cdot \left(1 - \cos x\right)\\

\mathbf{elif}\;x \le 0.03381719709829058423400383048829098697752:\\
\;\;\;\;\mathsf{fma}\left(x \cdot x, \frac{-1}{24}, \mathsf{fma}\left(\frac{1}{720}, \left(x \cdot x\right) \cdot \left(x \cdot x\right), \frac{1}{2}\right)\right)\\

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

\end{array}
double f(double x) {
        double r720921 = 1.0;
        double r720922 = x;
        double r720923 = cos(r720922);
        double r720924 = r720921 - r720923;
        double r720925 = r720922 * r720922;
        double r720926 = r720924 / r720925;
        return r720926;
}
↓
double f(double x) {
        double r720927 = x;
        double r720928 = -0.031198749760063066;
        bool r720929 = r720927 <= r720928;
        double r720930 = 1.0;
        double r720931 = r720930 / r720927;
        double r720932 = r720931 * r720931;
        double r720933 = 1.0;
        double r720934 = cos(r720927);
        double r720935 = r720933 - r720934;
        double r720936 = r720932 * r720935;
        double r720937 = 0.033817197098290584;
        bool r720938 = r720927 <= r720937;
        double r720939 = r720927 * r720927;
        double r720940 = -0.041666666666666664;
        double r720941 = 0.001388888888888889;
        double r720942 = r720939 * r720939;
        double r720943 = 0.5;
        double r720944 = fma(r720941, r720942, r720943);
        double r720945 = fma(r720939, r720940, r720944);
        double r720946 = r720938 ? r720945 : r720936;
        double r720947 = r720929 ? r720936 : r720946;
        return r720947;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -0.031198749760063066 or 0.033817197098290584 < x

    1. Initial program 1.1

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

    3. Applied add-sqr-sqrt1.2

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

    4. Applied times-frac0.6

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

    6. Applied div-inv0.6

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

    7. Applied div-inv0.6

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

    8. Applied swap-sqr0.6

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

    9. Simplified0.6

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

    if -0.031198749760063066 < x < 0.033817197098290584

    1. Initial program 62.4

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

    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

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{-1}{24}, \mathsf{fma}\left(\frac{1}{720}, \left(x \cdot x\right) \cdot \left(x \cdot x\right), \frac{1}{2}\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03119874976006306588338645724434172734618:\\ \;\;\;\;\left(\frac{1}{x} \cdot \frac{1}{x}\right) \cdot \left(1 - \cos x\right)\\ \mathbf{elif}\;x \le 0.03381719709829058423400383048829098697752:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \frac{-1}{24}, \mathsf{fma}\left(\frac{1}{720}, \left(x \cdot x\right) \cdot \left(x \cdot x\right), \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} \cdot \frac{1}{x}\right) \cdot \left(1 - \cos x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1.0 (cos x)) (* x x)))