Average Error: 31.8 → 0.4
Time: 5.9s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0359263355856600727:\\ \;\;\;\;\frac{1}{x \cdot x} - \frac{\cos x}{x \cdot x}\\ \mathbf{elif}\;x \le 0.034123238941917128:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{1}{x}}{\frac{1}{1 - \cos x}}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0359263355856600727:\\
\;\;\;\;\frac{1}{x \cdot x} - \frac{\cos x}{x \cdot x}\\

\mathbf{elif}\;x \le 0.034123238941917128:\\
\;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\

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

\end{array}
double f(double x) {
        double r25758 = 1.0;
        double r25759 = x;
        double r25760 = cos(r25759);
        double r25761 = r25758 - r25760;
        double r25762 = r25759 * r25759;
        double r25763 = r25761 / r25762;
        return r25763;
}
↓
double f(double x) {
        double r25764 = x;
        double r25765 = -0.03592633558566007;
        bool r25766 = r25764 <= r25765;
        double r25767 = 1.0;
        double r25768 = r25764 * r25764;
        double r25769 = r25767 / r25768;
        double r25770 = cos(r25764);
        double r25771 = r25770 / r25768;
        double r25772 = r25769 - r25771;
        double r25773 = 0.03412323894191713;
        bool r25774 = r25764 <= r25773;
        double r25775 = 0.001388888888888889;
        double r25776 = 4.0;
        double r25777 = pow(r25764, r25776);
        double r25778 = r25775 * r25777;
        double r25779 = 0.5;
        double r25780 = r25778 + r25779;
        double r25781 = 0.041666666666666664;
        double r25782 = 2.0;
        double r25783 = pow(r25764, r25782);
        double r25784 = r25781 * r25783;
        double r25785 = r25780 - r25784;
        double r25786 = 1.0;
        double r25787 = r25786 / r25764;
        double r25788 = r25767 - r25770;
        double r25789 = r25786 / r25788;
        double r25790 = r25787 / r25789;
        double r25791 = r25787 * r25790;
        double r25792 = r25774 ? r25785 : r25791;
        double r25793 = r25766 ? r25772 : r25792;
        return r25793;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -0.03592633558566007

    1. Initial program 0.9

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

    3. Applied div-sub1.0

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

    if -0.03592633558566007 < x < 0.03412323894191713

    1. Initial program 62.3

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

    2. Taylor expanded around 0 0.0

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

    if 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 clear-num0.5

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

    8. Applied div-inv0.5

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

    9. Applied associate-/r*0.6

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0359263355856600727:\\ \;\;\;\;\frac{1}{x \cdot x} - \frac{\cos x}{x \cdot x}\\ \mathbf{elif}\;x \le 0.034123238941917128:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{1}{x}}{\frac{1}{1 - \cos x}}\\ \end{array}\]

Reproduce

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