Average Error: 31.6 → 0.3
Time: 4.5s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0203295408193157160769981572912001865916:\\ \;\;\;\;\frac{1 - \cos x}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \le 0.02765964358927324243109246992844418855384:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{1 - \cos x}}{x} \cdot \frac{\sqrt{1}}{\frac{x}{\sqrt{\log \left(e^{1 - \cos x}\right)}}}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0203295408193157160769981572912001865916:\\
\;\;\;\;\frac{1 - \cos x}{x} \cdot \frac{1}{x}\\

\mathbf{elif}\;x \le 0.02765964358927324243109246992844418855384:\\
\;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\

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

\end{array}
double f(double x) {
        double r28701 = 1.0;
        double r28702 = x;
        double r28703 = cos(r28702);
        double r28704 = r28701 - r28703;
        double r28705 = r28702 * r28702;
        double r28706 = r28704 / r28705;
        return r28706;
}
↓
double f(double x) {
        double r28707 = x;
        double r28708 = -0.020329540819315716;
        bool r28709 = r28707 <= r28708;
        double r28710 = 1.0;
        double r28711 = cos(r28707);
        double r28712 = r28710 - r28711;
        double r28713 = r28712 / r28707;
        double r28714 = 1.0;
        double r28715 = r28714 / r28707;
        double r28716 = r28713 * r28715;
        double r28717 = 0.027659643589273242;
        bool r28718 = r28707 <= r28717;
        double r28719 = 0.001388888888888889;
        double r28720 = 4.0;
        double r28721 = pow(r28707, r28720);
        double r28722 = r28719 * r28721;
        double r28723 = 0.5;
        double r28724 = r28722 + r28723;
        double r28725 = 0.041666666666666664;
        double r28726 = 2.0;
        double r28727 = pow(r28707, r28726);
        double r28728 = r28725 * r28727;
        double r28729 = r28724 - r28728;
        double r28730 = sqrt(r28712);
        double r28731 = r28730 / r28707;
        double r28732 = sqrt(r28714);
        double r28733 = exp(r28712);
        double r28734 = log(r28733);
        double r28735 = sqrt(r28734);
        double r28736 = r28707 / r28735;
        double r28737 = r28732 / r28736;
        double r28738 = r28731 * r28737;
        double r28739 = r28718 ? r28729 : r28738;
        double r28740 = r28709 ? r28716 : r28739;
        return r28740;
}

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 < -0.020329540819315716

    1. Initial program 1.2

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

    3. Applied add-sqr-sqrt1.3

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

    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 div-inv0.6

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

    7. Applied associate-*r*0.6

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

    8. Simplified0.5

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

    if -0.020329540819315716 < x < 0.027659643589273242

    1. Initial program 62.3

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

    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 0.027659643589273242 < x

    1. Initial program 1.2

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

    3. Applied add-sqr-sqrt1.3

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

    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 *-un-lft-identity0.6

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

    7. Applied sqrt-prod0.6

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

    8. Applied associate-/l*0.6

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

    10. Applied add-log-exp0.6

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

    11. Applied add-log-exp0.6

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

    12. Applied diff-log0.6

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

    13. Simplified0.6

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0203295408193157160769981572912001865916:\\ \;\;\;\;\frac{1 - \cos x}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \le 0.02765964358927324243109246992844418855384:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{1 - \cos x}}{x} \cdot \frac{\sqrt{1}}{\frac{x}{\sqrt{\log \left(e^{1 - \cos x}\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))