Average Error: 31.5 → 0.3
Time: 13.1s
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 r830101 = 1.0;
        double r830102 = x;
        double r830103 = cos(r830102);
        double r830104 = r830101 - r830103;
        double r830105 = r830102 * r830102;
        double r830106 = r830104 / r830105;
        return r830106;
}
↓
double f(double x) {
        double r830107 = x;
        double r830108 = -0.031198749760063066;
        bool r830109 = r830107 <= r830108;
        double r830110 = 1.0;
        double r830111 = r830110 / r830107;
        double r830112 = r830111 * r830111;
        double r830113 = 1.0;
        double r830114 = cos(r830107);
        double r830115 = r830113 - r830114;
        double r830116 = r830112 * r830115;
        double r830117 = 0.033817197098290584;
        bool r830118 = r830107 <= r830117;
        double r830119 = 0.001388888888888889;
        double r830120 = r830107 * r830107;
        double r830121 = r830119 * r830120;
        double r830122 = -0.041666666666666664;
        double r830123 = r830121 + r830122;
        double r830124 = r830123 * r830120;
        double r830125 = 0.5;
        double r830126 = r830124 + r830125;
        double r830127 = r830118 ? r830126 : r830116;
        double r830128 = r830109 ? r830116 : r830127;
        return r830128;
}

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. Taylor expanded around 0 0.0

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

    3. 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)))