Average Error: 31.8 → 0.2
Time: 4.9s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0262744398468406255 \lor \neg \left(x \le 0.034123238941917128\right):\\ \;\;\;\;\frac{1 - \cos x}{x} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0262744398468406255 \lor \neg \left(x \le 0.034123238941917128\right):\\
\;\;\;\;\frac{1 - \cos x}{x} \cdot \frac{1}{x}\\

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

\end{array}
double f(double x) {
        double r25692 = 1.0;
        double r25693 = x;
        double r25694 = cos(r25693);
        double r25695 = r25692 - r25694;
        double r25696 = r25693 * r25693;
        double r25697 = r25695 / r25696;
        return r25697;
}
↓
double f(double x) {
        double r25698 = x;
        double r25699 = -0.026274439846840626;
        bool r25700 = r25698 <= r25699;
        double r25701 = 0.03412323894191713;
        bool r25702 = r25698 <= r25701;
        double r25703 = !r25702;
        bool r25704 = r25700 || r25703;
        double r25705 = 1.0;
        double r25706 = cos(r25698);
        double r25707 = r25705 - r25706;
        double r25708 = r25707 / r25698;
        double r25709 = 1.0;
        double r25710 = r25709 / r25698;
        double r25711 = r25708 * r25710;
        double r25712 = 4.0;
        double r25713 = pow(r25698, r25712);
        double r25714 = 0.001388888888888889;
        double r25715 = 0.5;
        double r25716 = 0.041666666666666664;
        double r25717 = 2.0;
        double r25718 = pow(r25698, r25717);
        double r25719 = r25716 * r25718;
        double r25720 = r25715 - r25719;
        double r25721 = fma(r25713, r25714, r25720);
        double r25722 = r25704 ? r25711 : r25721;
        return r25722;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -0.026274439846840626 or 0.03412323894191713 < x

    1. Initial program 1.0

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

    3. Applied *-un-lft-identity1.0

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

    4. Applied times-frac0.5

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

    6. Applied div-inv0.5

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

    7. Applied associate-*r*0.5

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

    8. Simplified0.5

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

    if -0.026274439846840626 < x < 0.03412323894191713

    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

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0262744398468406255 \lor \neg \left(x \le 0.034123238941917128\right):\\ \;\;\;\;\frac{1 - \cos x}{x} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))