Average Error: 31.7 → 0.3
Time: 13.0s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03020423764333628538492071413656958611682:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{{1}^{3} - \log \left(e^{{\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1}}{x}\\ \mathbf{elif}\;x \le 0.03141600742422194503244980978706735186279:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{{\left({1}^{3}\right)}^{3} - {\left({\left(\cos x\right)}^{3}\right)}^{3}}{\left({1}^{6} + {\left(\cos x\right)}^{3} \cdot \left({1}^{3} + {\left(\cos x\right)}^{3}\right)\right) \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)}}{x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03020423764333628538492071413656958611682:\\
\;\;\;\;\frac{1}{x} \cdot \frac{\frac{{1}^{3} - \log \left(e^{{\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1}}{x}\\

\mathbf{elif}\;x \le 0.03141600742422194503244980978706735186279:\\
\;\;\;\;\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{{\left({1}^{3}\right)}^{3} - {\left({\left(\cos x\right)}^{3}\right)}^{3}}{\left({1}^{6} + {\left(\cos x\right)}^{3} \cdot \left({1}^{3} + {\left(\cos x\right)}^{3}\right)\right) \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)}}{x}\\

\end{array}
double f(double x) {
        double r20661 = 1.0;
        double r20662 = x;
        double r20663 = cos(r20662);
        double r20664 = r20661 - r20663;
        double r20665 = r20662 * r20662;
        double r20666 = r20664 / r20665;
        return r20666;
}
↓
double f(double x) {
        double r20667 = x;
        double r20668 = -0.030204237643336285;
        bool r20669 = r20667 <= r20668;
        double r20670 = 1.0;
        double r20671 = r20670 / r20667;
        double r20672 = 1.0;
        double r20673 = 3.0;
        double r20674 = pow(r20672, r20673);
        double r20675 = cos(r20667);
        double r20676 = pow(r20675, r20673);
        double r20677 = exp(r20676);
        double r20678 = log(r20677);
        double r20679 = r20674 - r20678;
        double r20680 = r20672 + r20675;
        double r20681 = r20675 * r20680;
        double r20682 = r20672 * r20672;
        double r20683 = r20681 + r20682;
        double r20684 = r20679 / r20683;
        double r20685 = r20684 / r20667;
        double r20686 = r20671 * r20685;
        double r20687 = 0.031416007424221945;
        bool r20688 = r20667 <= r20687;
        double r20689 = 0.001388888888888889;
        double r20690 = 4.0;
        double r20691 = pow(r20667, r20690);
        double r20692 = r20689 * r20691;
        double r20693 = 0.5;
        double r20694 = r20692 + r20693;
        double r20695 = 0.041666666666666664;
        double r20696 = 2.0;
        double r20697 = pow(r20667, r20696);
        double r20698 = r20695 * r20697;
        double r20699 = r20694 - r20698;
        double r20700 = pow(r20674, r20673);
        double r20701 = pow(r20676, r20673);
        double r20702 = r20700 - r20701;
        double r20703 = 6.0;
        double r20704 = pow(r20672, r20703);
        double r20705 = r20674 + r20676;
        double r20706 = r20676 * r20705;
        double r20707 = r20704 + r20706;
        double r20708 = r20707 * r20683;
        double r20709 = r20702 / r20708;
        double r20710 = r20709 / r20667;
        double r20711 = r20671 * r20710;
        double r20712 = r20688 ? r20699 : r20711;
        double r20713 = r20669 ? r20686 : r20712;
        return r20713;
}

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 < -0.030204237643336285

    1. Initial program 1.1

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

    3. Applied *-un-lft-identity1.1

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

    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-log-exp0.5

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

    if -0.030204237643336285 < x < 0.031416007424221945

    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 0.031416007424221945 < 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 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 flip3--0.5

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

    10. Applied associate-/l/0.5

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

    11. Simplified0.5

      \[\leadsto \frac{1}{x} \cdot \frac{\frac{{\left({1}^{3}\right)}^{3} - {\left({\left(\cos x\right)}^{3}\right)}^{3}}{\color{blue}{\left({1}^{6} + {\left(\cos x\right)}^{3} \cdot \left({1}^{3} + {\left(\cos x\right)}^{3}\right)\right) \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)}}}{x}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03020423764333628538492071413656958611682:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{{1}^{3} - \log \left(e^{{\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1}}{x}\\ \mathbf{elif}\;x \le 0.03141600742422194503244980978706735186279:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{{\left({1}^{3}\right)}^{3} - {\left({\left(\cos x\right)}^{3}\right)}^{3}}{\left({1}^{6} + {\left(\cos x\right)}^{3} \cdot \left({1}^{3} + {\left(\cos x\right)}^{3}\right)\right) \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))