Average Error: 32.2 → 0.3
Time: 9.9s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02623489072028305918449220257571141701192:\\ \;\;\;\;\frac{1}{x} \cdot \left(\sqrt{1 - \cos x} \cdot \frac{\sqrt{1 - \cos x}}{x}\right)\\ \mathbf{elif}\;x \le 0.03474417225563055516834864988595654722303:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\log \left(e^{1 - \cos x}\right)}{x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02623489072028305918449220257571141701192:\\
\;\;\;\;\frac{1}{x} \cdot \left(\sqrt{1 - \cos x} \cdot \frac{\sqrt{1 - \cos x}}{x}\right)\\

\mathbf{elif}\;x \le 0.03474417225563055516834864988595654722303:\\
\;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\

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

\end{array}
double f(double x) {
        double r24255 = 1.0;
        double r24256 = x;
        double r24257 = cos(r24256);
        double r24258 = r24255 - r24257;
        double r24259 = r24256 * r24256;
        double r24260 = r24258 / r24259;
        return r24260;
}
↓
double f(double x) {
        double r24261 = x;
        double r24262 = -0.02623489072028306;
        bool r24263 = r24261 <= r24262;
        double r24264 = 1.0;
        double r24265 = r24264 / r24261;
        double r24266 = 1.0;
        double r24267 = cos(r24261);
        double r24268 = r24266 - r24267;
        double r24269 = sqrt(r24268);
        double r24270 = r24269 / r24261;
        double r24271 = r24269 * r24270;
        double r24272 = r24265 * r24271;
        double r24273 = 0.034744172255630555;
        bool r24274 = r24261 <= r24273;
        double r24275 = 0.001388888888888889;
        double r24276 = 4.0;
        double r24277 = pow(r24261, r24276);
        double r24278 = r24275 * r24277;
        double r24279 = 0.5;
        double r24280 = r24278 + r24279;
        double r24281 = 0.041666666666666664;
        double r24282 = 2.0;
        double r24283 = pow(r24261, r24282);
        double r24284 = r24281 * r24283;
        double r24285 = r24280 - r24284;
        double r24286 = exp(r24268);
        double r24287 = log(r24286);
        double r24288 = r24287 / r24261;
        double r24289 = r24265 * r24288;
        double r24290 = r24274 ? r24285 : r24289;
        double r24291 = r24263 ? r24272 : r24290;
        return r24291;
}

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 < -0.02623489072028306

    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-frac0.6

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

    6. Applied *-un-lft-identity0.6

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

    7. Applied add-sqr-sqrt0.7

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

    8. Applied times-frac0.7

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

    9. Simplified0.7

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

    if -0.02623489072028306 < x < 0.034744172255630555

    1. Initial program 62.4

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

    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 0.034744172255630555 < 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 add-log-exp0.6

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

    7. Applied add-log-exp0.6

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

    8. Applied diff-log0.7

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

    9. Simplified0.6

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02623489072028305918449220257571141701192:\\ \;\;\;\;\frac{1}{x} \cdot \left(\sqrt{1 - \cos x} \cdot \frac{\sqrt{1 - \cos x}}{x}\right)\\ \mathbf{elif}\;x \le 0.03474417225563055516834864988595654722303:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\log \left(e^{1 - \cos x}\right)}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))