Average Error: 31.5 → 0.6
Time: 5.2s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.029466469328828359:\\ \;\;\;\;\frac{\frac{{1}^{3} - \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos x\right)}^{3}\right)\right)}{\mathsf{fma}\left(\cos x, 1 + \cos x, 1 \cdot 1\right)}}{x \cdot x}\\ \mathbf{elif}\;x \le 0.03308393611677984:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot 1 - \cos x \cdot \cos x}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.029466469328828359:\\
\;\;\;\;\frac{\frac{{1}^{3} - \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos x\right)}^{3}\right)\right)}{\mathsf{fma}\left(\cos x, 1 + \cos x, 1 \cdot 1\right)}}{x \cdot x}\\

\mathbf{elif}\;x \le 0.03308393611677984:\\
\;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\

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

\end{array}
double f(double x) {
        double r33113 = 1.0;
        double r33114 = x;
        double r33115 = cos(r33114);
        double r33116 = r33113 - r33115;
        double r33117 = r33114 * r33114;
        double r33118 = r33116 / r33117;
        return r33118;
}
↓
double f(double x) {
        double r33119 = x;
        double r33120 = -0.02946646932882836;
        bool r33121 = r33119 <= r33120;
        double r33122 = 1.0;
        double r33123 = 3.0;
        double r33124 = pow(r33122, r33123);
        double r33125 = cos(r33119);
        double r33126 = pow(r33125, r33123);
        double r33127 = log1p(r33126);
        double r33128 = expm1(r33127);
        double r33129 = r33124 - r33128;
        double r33130 = r33122 + r33125;
        double r33131 = r33122 * r33122;
        double r33132 = fma(r33125, r33130, r33131);
        double r33133 = r33129 / r33132;
        double r33134 = r33119 * r33119;
        double r33135 = r33133 / r33134;
        double r33136 = 0.033083936116779844;
        bool r33137 = r33119 <= r33136;
        double r33138 = 4.0;
        double r33139 = pow(r33119, r33138);
        double r33140 = 0.001388888888888889;
        double r33141 = 0.5;
        double r33142 = 0.041666666666666664;
        double r33143 = 2.0;
        double r33144 = pow(r33119, r33143);
        double r33145 = r33142 * r33144;
        double r33146 = r33141 - r33145;
        double r33147 = fma(r33139, r33140, r33146);
        double r33148 = r33125 * r33125;
        double r33149 = r33131 - r33148;
        double r33150 = r33134 * r33130;
        double r33151 = r33149 / r33150;
        double r33152 = r33137 ? r33147 : r33151;
        double r33153 = r33121 ? r33135 : r33152;
        return r33153;
}

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -0.02946646932882836

    1. Initial program 1.0

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

    3. Applied flip3--1.0

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

    4. Simplified1.0

      \[\leadsto \frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\color{blue}{\mathsf{fma}\left(\cos x, 1 + \cos x, 1 \cdot 1\right)}}}{x \cdot x}\]
    5. Using strategy rm

    6. Applied expm1-log1p-u1.0

      \[\leadsto \frac{\frac{{1}^{3} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos x\right)}^{3}\right)\right)}}{\mathsf{fma}\left(\cos x, 1 + \cos x, 1 \cdot 1\right)}}{x \cdot x}\]

    if -0.02946646932882836 < x < 0.033083936116779844

    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 0.033083936116779844 < x

    1. Initial program 1.2

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

    3. Applied flip--1.4

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

    4. Applied associate-/l/1.4

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.029466469328828359:\\ \;\;\;\;\frac{\frac{{1}^{3} - \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos x\right)}^{3}\right)\right)}{\mathsf{fma}\left(\cos x, 1 + \cos x, 1 \cdot 1\right)}}{x \cdot x}\\ \mathbf{elif}\;x \le 0.03308393611677984:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot 1 - \cos x \cdot \cos x}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))