Average Error: 31.3 → 0.3
Time: 4.3s
Precision: binary64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0301990296518355426 \lor \neg \left(x \le 0.0362985163383400602\right):\\ \;\;\;\;\frac{\sqrt{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)}}}{x} \cdot \frac{\sqrt{\log \left(e^{1 - \cos x}\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \frac{1}{720} + \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{-1}{24}\right)\right)\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0301990296518355426 \lor \neg \left(x \le 0.0362985163383400602\right):\\
\;\;\;\;\frac{\sqrt{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)}}}{x} \cdot \frac{\sqrt{\log \left(e^{1 - \cos x}\right)}}{x}\\

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

\end{array}
double code(double x) {
	return ((double) (((double) (1.0 - ((double) cos(x)))) / ((double) (x * x))));
}
↓
double code(double x) {
	double VAR;
	if (((x <= -0.030199029651835543) || !(x <= 0.03629851633834006))) {
		VAR = ((double) (((double) (((double) sqrt(((double) (((double) (((double) pow(1.0, 3.0)) - ((double) pow(((double) cos(x)), 3.0)))) / ((double) (((double) (1.0 * 1.0)) + ((double) (((double) cos(x)) * ((double) (1.0 + ((double) cos(x)))))))))))) / x)) * ((double) (((double) sqrt(((double) log(((double) exp(((double) (1.0 - ((double) cos(x)))))))))) / x))));
	} else {
		VAR = ((double) (((double) (((double) pow(x, 4.0)) * 0.001388888888888889)) + ((double) (0.5 + ((double) (x * ((double) (x * -0.041666666666666664))))))));
	}
	return VAR;
}

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 < -0.0301990296518355426 or 0.0362985163383400602 < x

    1. Initial program 1.0

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

    3. Applied add-sqr-sqrt1.1

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

    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 add-log-exp0.6

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

    7. Applied add-log-exp0.6

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

    8. Applied diff-log0.6

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

    9. Simplified0.6

      \[\leadsto \frac{\sqrt{1 - \cos x}}{x} \cdot \frac{\sqrt{\log \color{blue}{\left(e^{1 - \cos x}\right)}}}{x}\]
    10. Using strategy rm

    11. Applied flip3--0.6

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

    12. Simplified0.6

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

    if -0.0301990296518355426 < x < 0.0362985163383400602

    1. Initial program 62.3

      \[\frac{1 - \cos x}{x \cdot x}\]

    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

      \[\leadsto \color{blue}{{x}^{4} \cdot \frac{1}{720} + \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{-1}{24}\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0301990296518355426 \lor \neg \left(x \le 0.0362985163383400602\right):\\ \;\;\;\;\frac{\sqrt{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)}}}{x} \cdot \frac{\sqrt{\log \left(e^{1 - \cos x}\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \frac{1}{720} + \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{-1}{24}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020185 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))