Average Error: 31.5 → 0.2
Time: 12.2s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0328836280217894294519709319501998834312:\\ \;\;\;\;1 \cdot \left(\left(1 - \cos x\right) \cdot \frac{\frac{1}{x}}{x}\right)\\ \mathbf{elif}\;x \le 0.03074851402437170019843470925025030737743:\\ \;\;\;\;1 \cdot \left(\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos x \cdot \frac{{\left(\cos x\right)}^{2} - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1\right) \cdot x}}{x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0328836280217894294519709319501998834312:\\
\;\;\;\;1 \cdot \left(\left(1 - \cos x\right) \cdot \frac{\frac{1}{x}}{x}\right)\\

\mathbf{elif}\;x \le 0.03074851402437170019843470925025030737743:\\
\;\;\;\;1 \cdot \left(\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\right)\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos x \cdot \frac{{\left(\cos x\right)}^{2} - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1\right) \cdot x}}{x}\\

\end{array}
double f(double x) {
        double r20234 = 1.0;
        double r20235 = x;
        double r20236 = cos(r20235);
        double r20237 = r20234 - r20236;
        double r20238 = r20235 * r20235;
        double r20239 = r20237 / r20238;
        return r20239;
}
↓
double f(double x) {
        double r20240 = x;
        double r20241 = -0.03288362802178943;
        bool r20242 = r20240 <= r20241;
        double r20243 = 1.0;
        double r20244 = 1.0;
        double r20245 = cos(r20240);
        double r20246 = r20244 - r20245;
        double r20247 = r20243 / r20240;
        double r20248 = r20247 / r20240;
        double r20249 = r20246 * r20248;
        double r20250 = r20243 * r20249;
        double r20251 = 0.0307485140243717;
        bool r20252 = r20240 <= r20251;
        double r20253 = 0.001388888888888889;
        double r20254 = 4.0;
        double r20255 = pow(r20240, r20254);
        double r20256 = r20253 * r20255;
        double r20257 = 0.5;
        double r20258 = r20256 + r20257;
        double r20259 = 0.041666666666666664;
        double r20260 = 2.0;
        double r20261 = pow(r20240, r20260);
        double r20262 = r20259 * r20261;
        double r20263 = r20258 - r20262;
        double r20264 = r20243 * r20263;
        double r20265 = 3.0;
        double r20266 = pow(r20244, r20265);
        double r20267 = pow(r20245, r20265);
        double r20268 = r20266 - r20267;
        double r20269 = pow(r20245, r20260);
        double r20270 = r20244 * r20244;
        double r20271 = r20269 - r20270;
        double r20272 = r20245 - r20244;
        double r20273 = r20271 / r20272;
        double r20274 = r20245 * r20273;
        double r20275 = r20274 + r20270;
        double r20276 = r20275 * r20240;
        double r20277 = r20268 / r20276;
        double r20278 = r20277 / r20240;
        double r20279 = r20243 * r20278;
        double r20280 = r20252 ? r20264 : r20279;
        double r20281 = r20242 ? r20250 : r20280;
        return r20281;
}

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 < -0.03288362802178943

    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 *-un-lft-identity0.5

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

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

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

    8. Applied times-frac0.5

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

    9. Applied associate-*l*0.5

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

    10. Simplified0.4

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

    12. Applied *-un-lft-identity0.4

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

    13. Applied div-inv0.5

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

    14. Applied times-frac0.5

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

    15. Simplified0.5

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

    if -0.03288362802178943 < x < 0.0307485140243717

    1. Initial program 62.3

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

    3. Applied *-un-lft-identity62.3

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

    4. Applied times-frac61.3

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

    6. Applied *-un-lft-identity61.3

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

    7. Applied *-un-lft-identity61.3

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

    8. Applied times-frac61.3

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

    9. Applied associate-*l*61.3

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

    10. Simplified61.3

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

    11. Taylor expanded around 0 0.0

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

    if 0.0307485140243717 < x

    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 *-un-lft-identity0.5

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

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

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

    8. Applied times-frac0.5

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

    9. Applied associate-*l*0.5

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

    10. Simplified0.5

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

    12. Applied flip3--0.5

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

    13. Applied associate-/l/0.5

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

    14. Simplified0.5

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

    16. Applied flip-+0.5

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

    17. Simplified0.5

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0328836280217894294519709319501998834312:\\ \;\;\;\;1 \cdot \left(\left(1 - \cos x\right) \cdot \frac{\frac{1}{x}}{x}\right)\\ \mathbf{elif}\;x \le 0.03074851402437170019843470925025030737743:\\ \;\;\;\;1 \cdot \left(\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos x \cdot \frac{{\left(\cos x\right)}^{2} - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1\right) \cdot x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019291 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))