Average Error: 31.6 → 0.3
Time: 7.3s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0266096722673715806:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos x \cdot \frac{{1}^{3} + {\left(\cos x\right)}^{3}}{1 \cdot 1 + \cos x \cdot \left(\cos x - 1\right)} + 1 \cdot 1\right) \cdot x}\\ \mathbf{elif}\;x \le 0.0284410596258715502:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\log \left(e^{{1}^{3} - {\left(\cos x\right)}^{3}}\right)}{\left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right) \cdot x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0266096722673715806:\\
\;\;\;\;\frac{1}{x} \cdot \frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos x \cdot \frac{{1}^{3} + {\left(\cos x\right)}^{3}}{1 \cdot 1 + \cos x \cdot \left(\cos x - 1\right)} + 1 \cdot 1\right) \cdot x}\\

\mathbf{elif}\;x \le 0.0284410596258715502:\\
\;\;\;\;\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}^{3} - {\left(\cos x\right)}^{3}}\right)}{\left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right) \cdot x}\\

\end{array}
double f(double x) {
        double r26548 = 1.0;
        double r26549 = x;
        double r26550 = cos(r26549);
        double r26551 = r26548 - r26550;
        double r26552 = r26549 * r26549;
        double r26553 = r26551 / r26552;
        return r26553;
}
↓
double f(double x) {
        double r26554 = x;
        double r26555 = -0.02660967226737158;
        bool r26556 = r26554 <= r26555;
        double r26557 = 1.0;
        double r26558 = r26557 / r26554;
        double r26559 = 1.0;
        double r26560 = 3.0;
        double r26561 = pow(r26559, r26560);
        double r26562 = cos(r26554);
        double r26563 = pow(r26562, r26560);
        double r26564 = r26561 - r26563;
        double r26565 = r26561 + r26563;
        double r26566 = r26559 * r26559;
        double r26567 = r26562 - r26559;
        double r26568 = r26562 * r26567;
        double r26569 = r26566 + r26568;
        double r26570 = r26565 / r26569;
        double r26571 = r26562 * r26570;
        double r26572 = r26571 + r26566;
        double r26573 = r26572 * r26554;
        double r26574 = r26564 / r26573;
        double r26575 = r26558 * r26574;
        double r26576 = 0.02844105962587155;
        bool r26577 = r26554 <= r26576;
        double r26578 = 0.001388888888888889;
        double r26579 = 4.0;
        double r26580 = pow(r26554, r26579);
        double r26581 = r26578 * r26580;
        double r26582 = 0.5;
        double r26583 = r26581 + r26582;
        double r26584 = 0.041666666666666664;
        double r26585 = 2.0;
        double r26586 = pow(r26554, r26585);
        double r26587 = r26584 * r26586;
        double r26588 = r26583 - r26587;
        double r26589 = exp(r26564);
        double r26590 = log(r26589);
        double r26591 = r26559 + r26562;
        double r26592 = r26562 * r26591;
        double r26593 = r26592 + r26566;
        double r26594 = r26593 * r26554;
        double r26595 = r26590 / r26594;
        double r26596 = r26558 * r26595;
        double r26597 = r26577 ? r26588 : r26596;
        double r26598 = r26556 ? r26575 : r26597;
        return r26598;
}

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 < -0.02660967226737158

    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. Applied associate-/l/0.5

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

    8. Simplified0.5

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

    10. Applied flip3-+0.5

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

    11. Simplified0.5

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

    if -0.02660967226737158 < x < 0.02844105962587155

    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 0.02844105962587155 < 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. Applied associate-/l/0.5

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

    8. Simplified0.5

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

    10. Applied add-log-exp0.5

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

    11. Applied add-log-exp0.5

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

    12. Applied diff-log0.6

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

    13. Simplified0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0266096722673715806:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos x \cdot \frac{{1}^{3} + {\left(\cos x\right)}^{3}}{1 \cdot 1 + \cos x \cdot \left(\cos x - 1\right)} + 1 \cdot 1\right) \cdot x}\\ \mathbf{elif}\;x \le 0.0284410596258715502:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\log \left(e^{{1}^{3} - {\left(\cos x\right)}^{3}}\right)}{\left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right) \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))