Average Error: 31.6 → 0.2
Time: 3.8s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0266096722673715806:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(1 - \cos x\right)\right)}{x}\\ \mathbf{elif}\;x \le 0.0240834512536182467:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0266096722673715806:\\
\;\;\;\;\frac{1}{x} \cdot \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(1 - \cos x\right)\right)}{x}\\

\mathbf{elif}\;x \le 0.0240834512536182467:\\
\;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\

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

\end{array}
double f(double x) {
        double r27704 = 1.0;
        double r27705 = x;
        double r27706 = cos(r27705);
        double r27707 = r27704 - r27706;
        double r27708 = r27705 * r27705;
        double r27709 = r27707 / r27708;
        return r27709;
}
↓
double f(double x) {
        double r27710 = x;
        double r27711 = -0.02660967226737158;
        bool r27712 = r27710 <= r27711;
        double r27713 = 1.0;
        double r27714 = r27713 / r27710;
        double r27715 = 1.0;
        double r27716 = cos(r27710);
        double r27717 = r27715 - r27716;
        double r27718 = expm1(r27717);
        double r27719 = log1p(r27718);
        double r27720 = r27719 / r27710;
        double r27721 = r27714 * r27720;
        double r27722 = 0.024083451253618247;
        bool r27723 = r27710 <= r27722;
        double r27724 = 4.0;
        double r27725 = pow(r27710, r27724);
        double r27726 = 0.001388888888888889;
        double r27727 = 0.5;
        double r27728 = 0.041666666666666664;
        double r27729 = 2.0;
        double r27730 = pow(r27710, r27729);
        double r27731 = r27728 * r27730;
        double r27732 = r27727 - r27731;
        double r27733 = fma(r27725, r27726, r27732);
        double r27734 = r27717 / r27710;
        double r27735 = r27734 / r27710;
        double r27736 = r27723 ? r27733 : r27735;
        double r27737 = r27712 ? r27721 : r27736;
        return r27737;
}

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -0.02660967226737158

    1. Initial program 0.9

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

    3. Applied *-un-lft-identity0.9

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

    4. Applied times-frac0.5

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

    6. Applied log1p-expm1-u0.5

      \[\leadsto \frac{1}{x} \cdot \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(1 - \cos x\right)\right)}}{x}\]

    if -0.02660967226737158 < x < 0.024083451253618247

    1. Initial program 62.2

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

    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 0.024083451253618247 < 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 log1p-expm1-u0.5

      \[\leadsto \frac{1}{x} \cdot \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(1 - \cos x\right)\right)}}{x}\]
    7. Using strategy rm

    8. Applied associate-*r/0.4

      \[\leadsto \color{blue}{\frac{\frac{1}{x} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(1 - \cos x\right)\right)}{x}}\]

    9. Simplified0.4

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0266096722673715806:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(1 - \cos x\right)\right)}{x}\\ \mathbf{elif}\;x \le 0.0240834512536182467:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))