Average Error: 31.8 → 0.4
Time: 4.8s
Precision: binary64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0301085241973035427:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot {x}^{2}}\\ \mathbf{elif}\;x \le 0.0278442900979120854:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{\frac{{\left({1}^{3}\right)}^{3} - {\left({\left(\cos x\right)}^{3}\right)}^{3}}{\left({1}^{6} + {\left(\cos x\right)}^{6}\right) + {1}^{3} \cdot {\left(\cos x\right)}^{3}}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}}{x} \cdot \frac{\sqrt{\log \left(e^{1 - \cos x}\right)}}{x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0301085241973035427:\\
\;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot {x}^{2}}\\

\mathbf{elif}\;x \le 0.0278442900979120854:\\
\;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{\frac{\frac{{\left({1}^{3}\right)}^{3} - {\left({\left(\cos x\right)}^{3}\right)}^{3}}{\left({1}^{6} + {\left(\cos x\right)}^{6}\right) + {1}^{3} \cdot {\left(\cos x\right)}^{3}}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}}{x} \cdot \frac{\sqrt{\log \left(e^{1 - \cos x}\right)}}{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 <= -0.030108524197303543)) {
		VAR = ((double) (((double) (((double) pow(1.0, 3.0)) - ((double) pow(((double) cos(x)), 3.0)))) / ((double) (((double) (((double) (((double) cos(x)) * ((double) (((double) cos(x)) + 1.0)))) + ((double) (1.0 * 1.0)))) * ((double) pow(x, 2.0))))));
	} else {
		double VAR_1;
		if ((x <= 0.027844290097912085)) {
			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) sqrt(((double) (((double) (((double) (((double) pow(((double) pow(1.0, 3.0)), 3.0)) - ((double) pow(((double) pow(((double) cos(x)), 3.0)), 3.0)))) / ((double) (((double) (((double) pow(1.0, 6.0)) + ((double) pow(((double) cos(x)), 6.0)))) + ((double) (((double) pow(1.0, 3.0)) * ((double) pow(((double) cos(x)), 3.0)))))))) / ((double) (((double) (((double) cos(x)) * ((double) (((double) cos(x)) + 1.0)))) + ((double) (1.0 * 1.0)))))))) / x)) * ((double) (((double) sqrt(((double) log(((double) exp(((double) (1.0 - ((double) cos(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 < -0.0301085241973035427

    1. Initial program 1.0

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

    3. Applied flip3--1.1

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

    4. Applied associate-/l/1.1

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

    5. Simplified1.1

      \[\leadsto \frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\color{blue}{\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot {x}^{2}}}\]

    if -0.0301085241973035427 < x < 0.0278442900979120854

    1. Initial program 62.4

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

    2. Taylor expanded around 0 0.0

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

    if 0.0278442900979120854 < x

    1. Initial program 1.0

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

    3. Applied add-sqr-sqrt1.0

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

    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

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

    14. Applied flip3--0.6

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

    15. Simplified0.6

      \[\leadsto \frac{\sqrt{\frac{\frac{{\left({1}^{3}\right)}^{3} - {\left({\left(\cos x\right)}^{3}\right)}^{3}}{\color{blue}{\left({1}^{6} + {\left(\cos x\right)}^{6}\right) + {1}^{3} \cdot {\left(\cos x\right)}^{3}}}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}}{x} \cdot \frac{\sqrt{\log \left(e^{1 - \cos x}\right)}}{x}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0301085241973035427:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot {x}^{2}}\\ \mathbf{elif}\;x \le 0.0278442900979120854:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{\frac{{\left({1}^{3}\right)}^{3} - {\left({\left(\cos x\right)}^{3}\right)}^{3}}{\left({1}^{6} + {\left(\cos x\right)}^{6}\right) + {1}^{3} \cdot {\left(\cos x\right)}^{3}}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}}{x} \cdot \frac{\sqrt{\log \left(e^{1 - \cos x}\right)}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020152 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))