Average Error: 31.7 → 0.5
Time: 4.2s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.036038474483329114:\\ \;\;\;\;\frac{\sqrt{\log \left(e^{1 - \cos x}\right)}}{x} \cdot \frac{\sqrt{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\cos x \cdot \left(\left(\sqrt[3]{\cos x + 1} \cdot \sqrt[3]{\cos x + 1}\right) \cdot \sqrt[3]{\cos x + 1}\right) + 1 \cdot 1}}}{x}\\ \mathbf{elif}\;x \le 0.033100735860822239:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \cos x\right) \cdot \frac{1}{x \cdot x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.036038474483329114:\\
\;\;\;\;\frac{\sqrt{\log \left(e^{1 - \cos x}\right)}}{x} \cdot \frac{\sqrt{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\cos x \cdot \left(\left(\sqrt[3]{\cos x + 1} \cdot \sqrt[3]{\cos x + 1}\right) \cdot \sqrt[3]{\cos x + 1}\right) + 1 \cdot 1}}}{x}\\

\mathbf{elif}\;x \le 0.033100735860822239:\\
\;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\

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

\end{array}
double f(double x) {
        double r18529 = 1.0;
        double r18530 = x;
        double r18531 = cos(r18530);
        double r18532 = r18529 - r18531;
        double r18533 = r18530 * r18530;
        double r18534 = r18532 / r18533;
        return r18534;
}
↓
double f(double x) {
        double r18535 = x;
        double r18536 = -0.036038474483329114;
        bool r18537 = r18535 <= r18536;
        double r18538 = 1.0;
        double r18539 = cos(r18535);
        double r18540 = r18538 - r18539;
        double r18541 = exp(r18540);
        double r18542 = log(r18541);
        double r18543 = sqrt(r18542);
        double r18544 = r18543 / r18535;
        double r18545 = 3.0;
        double r18546 = pow(r18538, r18545);
        double r18547 = pow(r18539, r18545);
        double r18548 = r18546 - r18547;
        double r18549 = r18539 + r18538;
        double r18550 = cbrt(r18549);
        double r18551 = r18550 * r18550;
        double r18552 = r18551 * r18550;
        double r18553 = r18539 * r18552;
        double r18554 = r18538 * r18538;
        double r18555 = r18553 + r18554;
        double r18556 = r18548 / r18555;
        double r18557 = sqrt(r18556);
        double r18558 = r18557 / r18535;
        double r18559 = r18544 * r18558;
        double r18560 = 0.03310073586082224;
        bool r18561 = r18535 <= r18560;
        double r18562 = 0.001388888888888889;
        double r18563 = 4.0;
        double r18564 = pow(r18535, r18563);
        double r18565 = r18562 * r18564;
        double r18566 = 0.5;
        double r18567 = r18565 + r18566;
        double r18568 = 0.041666666666666664;
        double r18569 = 2.0;
        double r18570 = pow(r18535, r18569);
        double r18571 = r18568 * r18570;
        double r18572 = r18567 - r18571;
        double r18573 = 1.0;
        double r18574 = r18535 * r18535;
        double r18575 = r18573 / r18574;
        double r18576 = r18540 * r18575;
        double r18577 = r18561 ? r18572 : r18576;
        double r18578 = r18537 ? r18559 : r18577;
        return r18578;
}

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 < -0.036038474483329114

    1. Initial program 1.0

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

    3. Applied add-sqr-sqrt1.1

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

    4. Applied times-frac0.6

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

    6. Applied add-log-exp0.6

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

    7. Applied add-log-exp0.6

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

    8. Applied diff-log0.6

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

    9. Simplified0.6

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

    11. Applied flip3--0.6

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

    12. Simplified0.6

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

    14. Applied add-cube-cbrt0.6

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

    if -0.036038474483329114 < x < 0.03310073586082224

    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 0.03310073586082224 < x

    1. Initial program 1.2

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

    3. Applied div-inv1.3

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.036038474483329114:\\ \;\;\;\;\frac{\sqrt{\log \left(e^{1 - \cos x}\right)}}{x} \cdot \frac{\sqrt{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\cos x \cdot \left(\left(\sqrt[3]{\cos x + 1} \cdot \sqrt[3]{\cos x + 1}\right) \cdot \sqrt[3]{\cos x + 1}\right) + 1 \cdot 1}}}{x}\\ \mathbf{elif}\;x \le 0.033100735860822239:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \cos x\right) \cdot \frac{1}{x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))