Average Error: 31.5 → 0.3
Time: 16.0s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03119874976006306588338645724434172734618:\\ \;\;\;\;\left(\frac{1}{x} \cdot \frac{1}{x}\right) \cdot \left(1 - \cos x\right)\\ \mathbf{elif}\;x \le 0.03381719709829058423400383048829098697752:\\ \;\;\;\;\left(\frac{1}{720} \cdot \left(x \cdot x\right) - \frac{1}{24}\right) \cdot \left(x \cdot x\right) + \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} \cdot \frac{1}{x}\right) \cdot \left(1 - \cos x\right)\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03119874976006306588338645724434172734618:\\
\;\;\;\;\left(\frac{1}{x} \cdot \frac{1}{x}\right) \cdot \left(1 - \cos x\right)\\

\mathbf{elif}\;x \le 0.03381719709829058423400383048829098697752:\\
\;\;\;\;\left(\frac{1}{720} \cdot \left(x \cdot x\right) - \frac{1}{24}\right) \cdot \left(x \cdot x\right) + \frac{1}{2}\\

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

\end{array}
double f(double x) {
        double r798445 = 1.0;
        double r798446 = x;
        double r798447 = cos(r798446);
        double r798448 = r798445 - r798447;
        double r798449 = r798446 * r798446;
        double r798450 = r798448 / r798449;
        return r798450;
}
↓
double f(double x) {
        double r798451 = x;
        double r798452 = -0.031198749760063066;
        bool r798453 = r798451 <= r798452;
        double r798454 = 1.0;
        double r798455 = r798454 / r798451;
        double r798456 = r798455 * r798455;
        double r798457 = 1.0;
        double r798458 = cos(r798451);
        double r798459 = r798457 - r798458;
        double r798460 = r798456 * r798459;
        double r798461 = 0.033817197098290584;
        bool r798462 = r798451 <= r798461;
        double r798463 = 0.001388888888888889;
        double r798464 = r798451 * r798451;
        double r798465 = r798463 * r798464;
        double r798466 = 0.041666666666666664;
        double r798467 = r798465 - r798466;
        double r798468 = r798467 * r798464;
        double r798469 = 0.5;
        double r798470 = r798468 + r798469;
        double r798471 = r798462 ? r798470 : r798460;
        double r798472 = r798453 ? r798460 : r798471;
        return r798472;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -0.031198749760063066 or 0.033817197098290584 < x

    1. Initial program 1.1

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

    3. Applied add-sqr-sqrt1.2

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

    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 div-inv0.6

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

    8. Applied swap-sqr0.6

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

    9. Simplified0.6

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

    if -0.031198749760063066 < x < 0.033817197098290584

    1. Initial program 62.4

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

    3. Applied add-sqr-sqrt62.4

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

    4. Applied times-frac61.4

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

    5. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}}\]

    6. Simplified0.0

      \[\leadsto \color{blue}{\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{1}{720} \cdot \left(x \cdot x\right) - \frac{1}{24}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03119874976006306588338645724434172734618:\\ \;\;\;\;\left(\frac{1}{x} \cdot \frac{1}{x}\right) \cdot \left(1 - \cos x\right)\\ \mathbf{elif}\;x \le 0.03381719709829058423400383048829098697752:\\ \;\;\;\;\left(\frac{1}{720} \cdot \left(x \cdot x\right) - \frac{1}{24}\right) \cdot \left(x \cdot x\right) + \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} \cdot \frac{1}{x}\right) \cdot \left(1 - \cos x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1.0 (cos x)) (* x x)))