Average Error: 30.9 → 0.3
Time: 4.4s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02739833596085251843366137336488463915884:\\ \;\;\;\;\frac{\frac{\frac{{1}^{3} - \sqrt[3]{{\left({\left(\cos x\right)}^{3}\right)}^{3}}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{x}}{x}\\ \mathbf{elif}\;x \le 0.03037211695605043088042052090713696088642:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\log \left(e^{{1}^{3} - {\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{x}}{x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02739833596085251843366137336488463915884:\\
\;\;\;\;\frac{\frac{\frac{{1}^{3} - \sqrt[3]{{\left({\left(\cos x\right)}^{3}\right)}^{3}}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{x}}{x}\\

\mathbf{elif}\;x \le 0.03037211695605043088042052090713696088642:\\
\;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\

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

\end{array}
double f(double x) {
        double r20507 = 1.0;
        double r20508 = x;
        double r20509 = cos(r20508);
        double r20510 = r20507 - r20509;
        double r20511 = r20508 * r20508;
        double r20512 = r20510 / r20511;
        return r20512;
}
↓
double f(double x) {
        double r20513 = x;
        double r20514 = -0.02739833596085252;
        bool r20515 = r20513 <= r20514;
        double r20516 = 1.0;
        double r20517 = 3.0;
        double r20518 = pow(r20516, r20517);
        double r20519 = cos(r20513);
        double r20520 = pow(r20519, r20517);
        double r20521 = pow(r20520, r20517);
        double r20522 = cbrt(r20521);
        double r20523 = r20518 - r20522;
        double r20524 = r20519 + r20516;
        double r20525 = r20519 * r20524;
        double r20526 = r20516 * r20516;
        double r20527 = r20525 + r20526;
        double r20528 = r20523 / r20527;
        double r20529 = r20528 / r20513;
        double r20530 = r20529 / r20513;
        double r20531 = 0.03037211695605043;
        bool r20532 = r20513 <= r20531;
        double r20533 = 0.001388888888888889;
        double r20534 = 4.0;
        double r20535 = pow(r20513, r20534);
        double r20536 = r20533 * r20535;
        double r20537 = 0.5;
        double r20538 = r20536 + r20537;
        double r20539 = 0.041666666666666664;
        double r20540 = 2.0;
        double r20541 = pow(r20513, r20540);
        double r20542 = r20539 * r20541;
        double r20543 = r20538 - r20542;
        double r20544 = r20518 - r20520;
        double r20545 = exp(r20544);
        double r20546 = log(r20545);
        double r20547 = r20546 / r20527;
        double r20548 = r20547 / r20513;
        double r20549 = r20548 / r20513;
        double r20550 = r20532 ? r20543 : r20549;
        double r20551 = r20515 ? r20530 : r20550;
        return r20551;
}

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 < -0.02739833596085252

    1. Initial program 1.0

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

    3. Applied associate-/r*0.4

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

    5. Applied flip3--0.5

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

    6. Simplified0.5

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

    8. Applied add-cbrt-cube0.5

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

    9. Simplified0.5

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

    if -0.02739833596085252 < x < 0.03037211695605043

    1. Initial program 62.1

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

    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 0.03037211695605043 < x

    1. Initial program 0.9

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

    3. Applied associate-/r*0.5

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

    5. Applied flip3--0.5

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

    6. Simplified0.5

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

    8. Applied add-log-exp0.6

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

    9. Applied add-log-exp0.6

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

    10. Applied diff-log0.6

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

    11. Simplified0.6

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02739833596085251843366137336488463915884:\\ \;\;\;\;\frac{\frac{\frac{{1}^{3} - \sqrt[3]{{\left({\left(\cos x\right)}^{3}\right)}^{3}}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{x}}{x}\\ \mathbf{elif}\;x \le 0.03037211695605043088042052090713696088642:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\log \left(e^{{1}^{3} - {\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019344 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))