Average Error: 31.5 → 0.3
Time: 5.1s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0354600816886072659:\\ \;\;\;\;\frac{1 \cdot \left({1}^{3} - {\left(\cos x\right)}^{3}\right)}{x} \cdot \frac{\frac{1}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{x}\\ \mathbf{elif}\;x \le 0.030404078865847541:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{{1}^{3} - \log \left(e^{{\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \frac{{\left(\cos x\right)}^{2} - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1}}{x}\\ \end{array}\]
\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 0.030404078865847541:\\
\;\;\;\;\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 \frac{{\left(\cos x\right)}^{2} - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1}}{x}\\

\end{array}
double f(double x) {
        double r34357 = 1.0;
        double r34358 = x;
        double r34359 = cos(r34358);
        double r34360 = r34357 - r34359;
        double r34361 = r34358 * r34358;
        double r34362 = r34360 / r34361;
        return r34362;
}
↓
double f(double x) {
        double r34363 = x;
        double r34364 = -0.035460081688607266;
        bool r34365 = r34363 <= r34364;
        double r34366 = 1.0;
        double r34367 = 1.0;
        double r34368 = 3.0;
        double r34369 = pow(r34367, r34368);
        double r34370 = cos(r34363);
        double r34371 = pow(r34370, r34368);
        double r34372 = r34369 - r34371;
        double r34373 = r34366 * r34372;
        double r34374 = r34373 / r34363;
        double r34375 = r34370 + r34367;
        double r34376 = r34370 * r34375;
        double r34377 = r34367 * r34367;
        double r34378 = r34376 + r34377;
        double r34379 = r34366 / r34378;
        double r34380 = r34379 / r34363;
        double r34381 = r34374 * r34380;
        double r34382 = 0.03040407886584754;
        bool r34383 = r34363 <= r34382;
        double r34384 = 0.001388888888888889;
        double r34385 = 4.0;
        double r34386 = pow(r34363, r34385);
        double r34387 = r34384 * r34386;
        double r34388 = 0.5;
        double r34389 = r34387 + r34388;
        double r34390 = 0.041666666666666664;
        double r34391 = 2.0;
        double r34392 = pow(r34363, r34391);
        double r34393 = r34390 * r34392;
        double r34394 = r34389 - r34393;
        double r34395 = r34366 / r34363;
        double r34396 = exp(r34371);
        double r34397 = log(r34396);
        double r34398 = r34369 - r34397;
        double r34399 = pow(r34370, r34391);
        double r34400 = r34399 - r34377;
        double r34401 = r34370 - r34367;
        double r34402 = r34400 / r34401;
        double r34403 = r34370 * r34402;
        double r34404 = r34403 + r34377;
        double r34405 = r34398 / r34404;
        double r34406 = r34405 / r34363;
        double r34407 = r34395 * r34406;
        double r34408 = r34383 ? r34394 : r34407;
        double r34409 = r34365 ? r34381 : r34408;
        return r34409;
}

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 < 0.03040407886584754

    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 0.03040407886584754 < 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

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

    11. Applied flip-+0.5

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

    12. Simplified0.5

      \[\leadsto \frac{1}{x} \cdot \frac{\frac{{1}^{3} - \log \left(e^{{\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \frac{\color{blue}{{\left(\cos x\right)}^{2} - 1 \cdot 1}}{\cos x - 1} + 1 \cdot 1}}{x}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0354600816886072659:\\ \;\;\;\;\frac{1 \cdot \left({1}^{3} - {\left(\cos x\right)}^{3}\right)}{x} \cdot \frac{\frac{1}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{x}\\ \mathbf{elif}\;x \le 0.030404078865847541:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{{1}^{3} - \log \left(e^{{\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \frac{{\left(\cos x\right)}^{2} - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1}}{x}\\ \end{array}\]

Reproduce

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