Average Error: 31.5 → 0.3
Time: 5.1s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0354600816886072659:\\ \;\;\;\;\frac{1 \cdot \left({1}^{3} - {\left(\cos x\right)}^{3}\right)}{x} \cdot \frac{\frac{1}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{x}\\ \mathbf{elif}\;x \le 0.032956962191434971:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{{1}^{3} - \log \left(e^{{\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0354600816886072659:\\
\;\;\;\;\frac{1 \cdot \left({1}^{3} - {\left(\cos x\right)}^{3}\right)}{x} \cdot \frac{\frac{1}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{x}\\

\mathbf{elif}\;x \le 0.032956962191434971:\\
\;\;\;\;\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}^{3} - \log \left(e^{{\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{x}\\

\end{array}
double f(double x) {
        double r28392 = 1.0;
        double r28393 = x;
        double r28394 = cos(r28393);
        double r28395 = r28392 - r28394;
        double r28396 = r28393 * r28393;
        double r28397 = r28395 / r28396;
        return r28397;
}
↓
double f(double x) {
        double r28398 = x;
        double r28399 = -0.035460081688607266;
        bool r28400 = r28398 <= r28399;
        double r28401 = 1.0;
        double r28402 = 1.0;
        double r28403 = 3.0;
        double r28404 = pow(r28402, r28403);
        double r28405 = cos(r28398);
        double r28406 = pow(r28405, r28403);
        double r28407 = r28404 - r28406;
        double r28408 = r28401 * r28407;
        double r28409 = r28408 / r28398;
        double r28410 = r28405 + r28402;
        double r28411 = r28405 * r28410;
        double r28412 = r28402 * r28402;
        double r28413 = r28411 + r28412;
        double r28414 = r28401 / r28413;
        double r28415 = r28414 / r28398;
        double r28416 = r28409 * r28415;
        double r28417 = 0.03295696219143497;
        bool r28418 = r28398 <= r28417;
        double r28419 = 0.001388888888888889;
        double r28420 = 4.0;
        double r28421 = pow(r28398, r28420);
        double r28422 = r28419 * r28421;
        double r28423 = 0.5;
        double r28424 = r28422 + r28423;
        double r28425 = 0.041666666666666664;
        double r28426 = 2.0;
        double r28427 = pow(r28398, r28426);
        double r28428 = r28425 * r28427;
        double r28429 = r28424 - r28428;
        double r28430 = r28401 / r28398;
        double r28431 = exp(r28406);
        double r28432 = log(r28431);
        double r28433 = r28404 - r28432;
        double r28434 = r28433 / r28413;
        double r28435 = r28434 / r28398;
        double r28436 = r28430 * r28435;
        double r28437 = r28418 ? r28429 : r28436;
        double r28438 = r28400 ? r28416 : r28437;
        return r28438;
}

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 < -0.035460081688607266

    1. Initial program 1.2

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

    3. Applied *-un-lft-identity1.2

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

    4. Applied times-frac0.5

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

    6. Applied flip3--0.5

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

    7. Simplified0.5

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

    9. Applied *-un-lft-identity0.5

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

    10. Applied div-inv0.5

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

    11. Applied times-frac0.6

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

    12. Applied associate-*r*0.6

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

    13. Simplified0.5

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

    if -0.035460081688607266 < x < 0.03295696219143497

    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 0.03295696219143497 < x

    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 flip3--0.5

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

    7. Simplified0.5

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

    9. Applied add-log-exp0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0354600816886072659:\\ \;\;\;\;\frac{1 \cdot \left({1}^{3} - {\left(\cos x\right)}^{3}\right)}{x} \cdot \frac{\frac{1}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{x}\\ \mathbf{elif}\;x \le 0.032956962191434971:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{{1}^{3} - \log \left(e^{{\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))