Average Error: 31.6 → 0.4
Time: 5.1s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0338582795407828333:\\ \;\;\;\;\frac{\log \left(e^{1 - \cos x}\right)}{x \cdot x}\\ \mathbf{elif}\;x \le 0.025388615584991975:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{1}}{\frac{x}{\sqrt{1 - \cos x}}} \cdot \frac{\sqrt{1 - \cos x}}{x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0338582795407828333:\\
\;\;\;\;\frac{\log \left(e^{1 - \cos x}\right)}{x \cdot x}\\

\mathbf{elif}\;x \le 0.025388615584991975:\\
\;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{1}}{\frac{x}{\sqrt{1 - \cos x}}} \cdot \frac{\sqrt{1 - \cos x}}{x}\\

\end{array}
double f(double x) {
        double r29713 = 1.0;
        double r29714 = x;
        double r29715 = cos(r29714);
        double r29716 = r29713 - r29715;
        double r29717 = r29714 * r29714;
        double r29718 = r29716 / r29717;
        return r29718;
}
↓
double f(double x) {
        double r29719 = x;
        double r29720 = -0.03385827954078283;
        bool r29721 = r29719 <= r29720;
        double r29722 = 1.0;
        double r29723 = cos(r29719);
        double r29724 = r29722 - r29723;
        double r29725 = exp(r29724);
        double r29726 = log(r29725);
        double r29727 = r29719 * r29719;
        double r29728 = r29726 / r29727;
        double r29729 = 0.025388615584991975;
        bool r29730 = r29719 <= r29729;
        double r29731 = 0.001388888888888889;
        double r29732 = 4.0;
        double r29733 = pow(r29719, r29732);
        double r29734 = r29731 * r29733;
        double r29735 = 0.5;
        double r29736 = r29734 + r29735;
        double r29737 = 0.041666666666666664;
        double r29738 = 2.0;
        double r29739 = pow(r29719, r29738);
        double r29740 = r29737 * r29739;
        double r29741 = r29736 - r29740;
        double r29742 = 1.0;
        double r29743 = sqrt(r29742);
        double r29744 = sqrt(r29724);
        double r29745 = r29719 / r29744;
        double r29746 = r29743 / r29745;
        double r29747 = r29744 / r29719;
        double r29748 = r29746 * r29747;
        double r29749 = r29730 ? r29741 : r29748;
        double r29750 = r29721 ? r29728 : r29749;
        return r29750;
}

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 < -0.03385827954078283

    1. Initial program 0.9

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

    3. Applied add-log-exp1.1

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

    4. Applied add-log-exp1.1

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

    5. Applied diff-log1.2

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

    6. Simplified1.1

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

    if -0.03385827954078283 < x < 0.025388615584991975

    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 0.025388615584991975 < 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

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

    7. Applied sqrt-prod0.6

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

    8. Applied associate-/l*0.6

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0338582795407828333:\\ \;\;\;\;\frac{\log \left(e^{1 - \cos x}\right)}{x \cdot x}\\ \mathbf{elif}\;x \le 0.025388615584991975:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{1}}{\frac{x}{\sqrt{1 - \cos x}}} \cdot \frac{\sqrt{1 - \cos x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))