Average Error: 31.5 → 0.3
Time: 4.2s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.033994064032605062:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\log \left(e^{1 - \cos x}\right)}{x}\\ \mathbf{elif}\;x \le 0.032573444318334172:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1}{\frac{x}{1 - \cos x}}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.033994064032605062:\\
\;\;\;\;\frac{1}{x} \cdot \frac{\log \left(e^{1 - \cos x}\right)}{x}\\

\mathbf{elif}\;x \le 0.032573444318334172:\\
\;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\

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

\end{array}
double f(double x) {
        double r26425 = 1.0;
        double r26426 = x;
        double r26427 = cos(r26426);
        double r26428 = r26425 - r26427;
        double r26429 = r26426 * r26426;
        double r26430 = r26428 / r26429;
        return r26430;
}
↓
double f(double x) {
        double r26431 = x;
        double r26432 = -0.03399406403260506;
        bool r26433 = r26431 <= r26432;
        double r26434 = 1.0;
        double r26435 = r26434 / r26431;
        double r26436 = 1.0;
        double r26437 = cos(r26431);
        double r26438 = r26436 - r26437;
        double r26439 = exp(r26438);
        double r26440 = log(r26439);
        double r26441 = r26440 / r26431;
        double r26442 = r26435 * r26441;
        double r26443 = 0.03257344431833417;
        bool r26444 = r26431 <= r26443;
        double r26445 = 0.001388888888888889;
        double r26446 = 4.0;
        double r26447 = pow(r26431, r26446);
        double r26448 = r26445 * r26447;
        double r26449 = 0.5;
        double r26450 = r26448 + r26449;
        double r26451 = 0.041666666666666664;
        double r26452 = 2.0;
        double r26453 = pow(r26431, r26452);
        double r26454 = r26451 * r26453;
        double r26455 = r26450 - r26454;
        double r26456 = r26431 / r26438;
        double r26457 = r26434 / r26456;
        double r26458 = r26435 * r26457;
        double r26459 = r26444 ? r26455 : r26458;
        double r26460 = r26433 ? r26442 : r26459;
        return r26460;
}

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 < -0.03399406403260506

    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-frac0.4

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

    6. Applied add-log-exp0.5

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

    7. Applied add-log-exp0.5

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

    8. Applied diff-log0.6

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

    9. Simplified0.5

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

    if -0.03399406403260506 < x < 0.03257344431833417

    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 0.03257344431833417 < x

    1. Initial program 0.9

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

    3. Applied *-un-lft-identity0.9

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

    4. Applied times-frac0.4

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

    6. Applied clear-num0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.033994064032605062:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\log \left(e^{1 - \cos x}\right)}{x}\\ \mathbf{elif}\;x \le 0.032573444318334172:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1}{\frac{x}{1 - \cos x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))