Average Error: 31.3 → 0.4
Time: 3.7s
Precision: binary64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0349912626132538249:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(\cos x \cdot \cos x\right) \cdot \left(\cos x \cdot \cos x\right)}{\left({\left(\cos x\right)}^{2} + 1 \cdot 1\right) \cdot \left(1 + \cos x\right)}}{x}\\ \mathbf{elif}\;x \le 0.0273076678347106937:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 \cdot x - x \cdot \cos x}{x}}{x \cdot x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0349912626132538249:\\
\;\;\;\;\frac{1}{x} \cdot \frac{\frac{\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(\cos x \cdot \cos x\right) \cdot \left(\cos x \cdot \cos x\right)}{\left({\left(\cos x\right)}^{2} + 1 \cdot 1\right) \cdot \left(1 + \cos x\right)}}{x}\\

\mathbf{elif}\;x \le 0.0273076678347106937:\\
\;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\

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

\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 <= -0.034991262613253825)) {
		VAR = ((double) (((double) (1.0 / x)) * ((double) (((double) (((double) (((double) (((double) (1.0 * 1.0)) * ((double) (1.0 * 1.0)))) - ((double) (((double) (((double) cos(x)) * ((double) cos(x)))) * ((double) (((double) cos(x)) * ((double) cos(x)))))))) / ((double) (((double) (((double) pow(((double) cos(x)), 2.0)) + ((double) (1.0 * 1.0)))) * ((double) (1.0 + ((double) cos(x)))))))) / x))));
	} else {
		double VAR_1;
		if ((x <= 0.027307667834710694)) {
			VAR_1 = ((double) (((double) (((double) (0.001388888888888889 * ((double) pow(x, 4.0)))) + 0.5)) - ((double) (0.041666666666666664 * ((double) pow(x, 2.0))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) (1.0 * x)) - ((double) (x * ((double) cos(x)))))) / x)) / ((double) (x * x))));
		}
		VAR = VAR_1;
	}
	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 3 regimes

  2. if x < -0.0349912626132538249

    1. Initial program 1.1

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

    3. Applied *-un-lft-identity1.1

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

    4. Applied times-frac0.5

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

    6. Applied flip--0.7

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

    8. Applied flip--0.8

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

    9. Applied associate-/l/0.8

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

    10. Simplified0.8

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

    if -0.0349912626132538249 < x < 0.0273076678347106937

    1. Initial program 62.3

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

    2. Taylor expanded around 0 0.0

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

    if 0.0273076678347106937 < 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-frac0.5

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

    6. Applied div-sub0.6

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

    8. Applied frac-sub1.0

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

    9. Applied associate-*r/1.0

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

    10. Simplified0.9

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0349912626132538249:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(\cos x \cdot \cos x\right) \cdot \left(\cos x \cdot \cos x\right)}{\left({\left(\cos x\right)}^{2} + 1 \cdot 1\right) \cdot \left(1 + \cos x\right)}}{x}\\ \mathbf{elif}\;x \le 0.0273076678347106937:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 \cdot x - x \cdot \cos x}{x}}{x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020153 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))