Average Error: 32.0 → 0.4
Time: 4.6s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.031598206850451019:\\ \;\;\;\;\frac{e^{\log \left(1 - \cos x\right)}}{x \cdot x}\\ \mathbf{elif}\;x \le 0.0302961560572222073:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{e^{\log \left(1 \cdot 1 - \cos x \cdot \cos x\right)}}}{\sqrt{1 + \cos x} \cdot x} \cdot \left(\sqrt{1 - \cos x} \cdot \frac{1}{x}\right)\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.031598206850451019:\\
\;\;\;\;\frac{e^{\log \left(1 - \cos x\right)}}{x \cdot x}\\

\mathbf{elif}\;x \le 0.0302961560572222073:\\
\;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\

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

\end{array}
double code(double x) {
	return ((1.0 - cos(x)) / (x * x));
}
↓
double code(double x) {
	double temp;
	if ((x <= -0.03159820685045102)) {
		temp = (exp(log((1.0 - cos(x)))) / (x * x));
	} else {
		double temp_1;
		if ((x <= 0.030296156057222207)) {
			temp_1 = (((0.001388888888888889 * pow(x, 4.0)) + 0.5) - (0.041666666666666664 * pow(x, 2.0)));
		} else {
			temp_1 = ((sqrt(exp(log(((1.0 * 1.0) - (cos(x) * cos(x)))))) / (sqrt((1.0 + cos(x))) * x)) * (sqrt((1.0 - cos(x))) * (1.0 / x)));
		}
		temp = temp_1;
	}
	return temp;
}

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 < -0.03159820685045102

    1. Initial program 1.0

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

    3. Applied add-exp-log1.0

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

    if -0.03159820685045102 < x < 0.030296156057222207

    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 0.030296156057222207 < 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. Using strategy rm

    8. Applied flip--0.7

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

    9. Applied sqrt-div0.7

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

    10. Applied associate-/l/0.7

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

    11. Simplified0.7

      \[\leadsto \frac{\sqrt{1 \cdot 1 - \cos x \cdot \cos x}}{\color{blue}{\sqrt{1 + \cos x} \cdot x}} \cdot \left(\sqrt{1 - \cos x} \cdot \frac{1}{x}\right)\]
    12. Using strategy rm

    13. Applied add-exp-log0.7

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.031598206850451019:\\ \;\;\;\;\frac{e^{\log \left(1 - \cos x\right)}}{x \cdot x}\\ \mathbf{elif}\;x \le 0.0302961560572222073:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{e^{\log \left(1 \cdot 1 - \cos x \cdot \cos x\right)}}}{\sqrt{1 + \cos x} \cdot x} \cdot \left(\sqrt{1 - \cos x} \cdot \frac{1}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))