Average Error: 31.5 → 0.3
Time: 14.9s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03059660473543995840017828413692768663168:\\ \;\;\;\;\frac{\sqrt{1 - \cos x}}{x} \cdot \frac{\sqrt{1 - \cos x}}{x}\\ \mathbf{elif}\;x \le 0.03249397456549534685121827237708203028888:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{e^{\log \left(\frac{\left(1 \cdot 1 + \cos x \cdot \left(\cos x + 1\right)\right) \cdot \left(1 - \cos x\right)}{\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1}\right)}}{x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03059660473543995840017828413692768663168:\\
\;\;\;\;\frac{\sqrt{1 - \cos x}}{x} \cdot \frac{\sqrt{1 - \cos x}}{x}\\

\mathbf{elif}\;x \le 0.03249397456549534685121827237708203028888:\\
\;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\

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

\end{array}
double f(double x) {
        double r24505 = 1.0;
        double r24506 = x;
        double r24507 = cos(r24506);
        double r24508 = r24505 - r24507;
        double r24509 = r24506 * r24506;
        double r24510 = r24508 / r24509;
        return r24510;
}
↓
double f(double x) {
        double r24511 = x;
        double r24512 = -0.03059660473543996;
        bool r24513 = r24511 <= r24512;
        double r24514 = 1.0;
        double r24515 = cos(r24511);
        double r24516 = r24514 - r24515;
        double r24517 = sqrt(r24516);
        double r24518 = r24517 / r24511;
        double r24519 = r24518 * r24518;
        double r24520 = 0.03249397456549535;
        bool r24521 = r24511 <= r24520;
        double r24522 = 0.001388888888888889;
        double r24523 = 4.0;
        double r24524 = pow(r24511, r24523);
        double r24525 = r24522 * r24524;
        double r24526 = 0.5;
        double r24527 = r24525 + r24526;
        double r24528 = 0.041666666666666664;
        double r24529 = 2.0;
        double r24530 = pow(r24511, r24529);
        double r24531 = r24528 * r24530;
        double r24532 = r24527 - r24531;
        double r24533 = 1.0;
        double r24534 = r24533 / r24511;
        double r24535 = r24514 * r24514;
        double r24536 = r24515 + r24514;
        double r24537 = r24515 * r24536;
        double r24538 = r24535 + r24537;
        double r24539 = r24538 * r24516;
        double r24540 = r24514 + r24515;
        double r24541 = r24515 * r24540;
        double r24542 = r24541 + r24535;
        double r24543 = r24539 / r24542;
        double r24544 = log(r24543);
        double r24545 = exp(r24544);
        double r24546 = r24545 / r24511;
        double r24547 = r24534 * r24546;
        double r24548 = r24521 ? r24532 : r24547;
        double r24549 = r24513 ? r24519 : r24548;
        return r24549;
}

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 < -0.03059660473543996

    1. Initial program 1.1

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

    3. Applied add-sqr-sqrt1.2

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

    4. Applied times-frac0.6

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

    if -0.03059660473543996 < x < 0.03249397456549535

    1. Initial program 62.2

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

    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 0.03249397456549535 < 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 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

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

    9. Applied add-exp-log0.5

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

    10. Applied add-exp-log0.5

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

    11. Applied div-exp0.5

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

    12. Simplified0.5

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

    14. Applied difference-cubes0.5

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

    15. Simplified0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03059660473543995840017828413692768663168:\\ \;\;\;\;\frac{\sqrt{1 - \cos x}}{x} \cdot \frac{\sqrt{1 - \cos x}}{x}\\ \mathbf{elif}\;x \le 0.03249397456549534685121827237708203028888:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{e^{\log \left(\frac{\left(1 \cdot 1 + \cos x \cdot \left(\cos x + 1\right)\right) \cdot \left(1 - \cos x\right)}{\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1}\right)}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))