Average Error: 31.5 → 0.3
Time: 14.0s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03155884382850993041813580930465832352638:\\ \;\;\;\;\frac{\frac{1 + \cos x}{\frac{x \cdot \frac{\left(1 - \cos x\right) \cdot \left(1 + \cos x\right)}{1 - \cos x}}{1 - \cos x}}}{x}\\ \mathbf{elif}\;x \le 0.03328058389703145159010944098554318770766:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{x} - \frac{\frac{\cos x}{x}}{x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03155884382850993041813580930465832352638:\\
\;\;\;\;\frac{\frac{1 + \cos x}{\frac{x \cdot \frac{\left(1 - \cos x\right) \cdot \left(1 + \cos x\right)}{1 - \cos x}}{1 - \cos x}}}{x}\\

\mathbf{elif}\;x \le 0.03328058389703145159010944098554318770766:\\
\;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\

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

\end{array}
double f(double x) {
        double r17986 = 1.0;
        double r17987 = x;
        double r17988 = cos(r17987);
        double r17989 = r17986 - r17988;
        double r17990 = r17987 * r17987;
        double r17991 = r17989 / r17990;
        return r17991;
}
↓
double f(double x) {
        double r17992 = x;
        double r17993 = -0.03155884382850993;
        bool r17994 = r17992 <= r17993;
        double r17995 = 1.0;
        double r17996 = cos(r17992);
        double r17997 = r17995 + r17996;
        double r17998 = r17995 - r17996;
        double r17999 = r17998 * r17997;
        double r18000 = r17999 / r17998;
        double r18001 = r17992 * r18000;
        double r18002 = r18001 / r17998;
        double r18003 = r17997 / r18002;
        double r18004 = r18003 / r17992;
        double r18005 = 0.03328058389703145;
        bool r18006 = r17992 <= r18005;
        double r18007 = 0.001388888888888889;
        double r18008 = 4.0;
        double r18009 = pow(r17992, r18008);
        double r18010 = r18007 * r18009;
        double r18011 = 0.5;
        double r18012 = r18010 + r18011;
        double r18013 = 0.041666666666666664;
        double r18014 = 2.0;
        double r18015 = pow(r17992, r18014);
        double r18016 = r18013 * r18015;
        double r18017 = r18012 - r18016;
        double r18018 = r17995 / r17992;
        double r18019 = r18018 / r17992;
        double r18020 = r17996 / r17992;
        double r18021 = r18020 / r17992;
        double r18022 = r18019 - r18021;
        double r18023 = r18006 ? r18017 : r18022;
        double r18024 = r17994 ? r18004 : r18023;
        return r18024;
}

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 < -0.03155884382850993

    1. Initial program 1.0

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

    3. Applied associate-/r*0.5

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

    5. Applied flip--0.6

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

    6. Applied associate-/l/0.7

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

    8. Applied difference-of-squares0.5

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

    9. Applied associate-/l*0.5

      \[\leadsto \frac{\color{blue}{\frac{1 + \cos x}{\frac{x \cdot \left(1 + \cos x\right)}{1 - \cos x}}}}{x}\]
    10. Using strategy rm

    11. Applied flip-+0.7

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

    12. Simplified0.5

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

    if -0.03155884382850993 < x < 0.03328058389703145

    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 0.03328058389703145 < x

    1. Initial program 0.9

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

    3. Applied associate-/r*0.5

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

    5. Applied div-sub0.6

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

    6. Applied div-sub0.6

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03155884382850993041813580930465832352638:\\ \;\;\;\;\frac{\frac{1 + \cos x}{\frac{x \cdot \frac{\left(1 - \cos x\right) \cdot \left(1 + \cos x\right)}{1 - \cos x}}{1 - \cos x}}}{x}\\ \mathbf{elif}\;x \le 0.03328058389703145159010944098554318770766:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{x} - \frac{\frac{\cos x}{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))