Average Error: 43.0 → 9.7
Time: 8.5s
Precision: binary64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -9.89774605087060996 \cdot 10^{124}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \sqrt{2} \cdot t\right) - \frac{t}{\sqrt{2}} \cdot \left(\frac{2}{{x}^{2}} + \frac{2}{x}\right)}\\ \mathbf{elif}\;t \le -2.1552676557946121 \cdot 10^{-209}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(\sqrt{{t}^{2} + \ell \cdot \frac{\ell}{x}} \cdot \sqrt{{t}^{2} + \ell \cdot \frac{\ell}{x}}\right)}}\\ \mathbf{elif}\;t \le -1.27366272653752753 \cdot 10^{-279}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \sqrt{2} \cdot t\right) - \frac{t}{\sqrt{2}} \cdot \left(\frac{2}{{x}^{2}} + \frac{2}{x}\right)}\\ \mathbf{elif}\;t \le 3.2032712372167452 \cdot 10^{26}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(\sqrt{{t}^{2} + \ell \cdot \frac{\ell}{x}} \cdot \sqrt{{t}^{2} + \ell \cdot \frac{\ell}{x}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2} \cdot t + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + \frac{t}{{x}^{2}} \cdot \left(\frac{2}{\sqrt{2}} - \frac{2}{{\left(\sqrt{2}\right)}^{3}}\right)\right)}\\ \end{array}\]
\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -9.89774605087060996 \cdot 10^{124}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \sqrt{2} \cdot t\right) - \frac{t}{\sqrt{2}} \cdot \left(\frac{2}{{x}^{2}} + \frac{2}{x}\right)}\\

\mathbf{elif}\;t \le -2.1552676557946121 \cdot 10^{-209}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(\sqrt{{t}^{2} + \ell \cdot \frac{\ell}{x}} \cdot \sqrt{{t}^{2} + \ell \cdot \frac{\ell}{x}}\right)}}\\

\mathbf{elif}\;t \le -1.27366272653752753 \cdot 10^{-279}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \sqrt{2} \cdot t\right) - \frac{t}{\sqrt{2}} \cdot \left(\frac{2}{{x}^{2}} + \frac{2}{x}\right)}\\

\mathbf{elif}\;t \le 3.2032712372167452 \cdot 10^{26}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(\sqrt{{t}^{2} + \ell \cdot \frac{\ell}{x}} \cdot \sqrt{{t}^{2} + \ell \cdot \frac{\ell}{x}}\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2} \cdot t + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + \frac{t}{{x}^{2}} \cdot \left(\frac{2}{\sqrt{2}} - \frac{2}{{\left(\sqrt{2}\right)}^{3}}\right)\right)}\\

\end{array}
double code(double x, double l, double t) {
	return ((double) (((double) (((double) sqrt(2.0)) * t)) / ((double) sqrt(((double) (((double) (((double) (((double) (x + 1.0)) / ((double) (x - 1.0)))) * ((double) (((double) (l * l)) + ((double) (2.0 * ((double) (t * t)))))))) - ((double) (l * l))))))));
}
↓
double code(double x, double l, double t) {
	double VAR;
	if ((t <= -9.89774605087061e+124)) {
		VAR = ((double) (((double) (((double) sqrt(2.0)) * t)) / ((double) (((double) (((double) (2.0 * ((double) (t / ((double) (((double) pow(((double) sqrt(2.0)), 3.0)) * ((double) pow(x, 2.0)))))))) - ((double) (((double) sqrt(2.0)) * t)))) - ((double) (((double) (t / ((double) sqrt(2.0)))) * ((double) (((double) (2.0 / ((double) pow(x, 2.0)))) + ((double) (2.0 / x))))))))));
	} else {
		double VAR_1;
		if ((t <= -2.155267655794612e-209)) {
			VAR_1 = ((double) (((double) (((double) sqrt(2.0)) * t)) / ((double) sqrt(((double) (((double) (4.0 * ((double) (((double) pow(t, 2.0)) / x)))) + ((double) (2.0 * ((double) (((double) sqrt(((double) (((double) pow(t, 2.0)) + ((double) (l * ((double) (l / x)))))))) * ((double) sqrt(((double) (((double) pow(t, 2.0)) + ((double) (l * ((double) (l / x))))))))))))))))));
		} else {
			double VAR_2;
			if ((t <= -1.2736627265375275e-279)) {
				VAR_2 = ((double) (((double) (((double) sqrt(2.0)) * t)) / ((double) (((double) (((double) (2.0 * ((double) (t / ((double) (((double) pow(((double) sqrt(2.0)), 3.0)) * ((double) pow(x, 2.0)))))))) - ((double) (((double) sqrt(2.0)) * t)))) - ((double) (((double) (t / ((double) sqrt(2.0)))) * ((double) (((double) (2.0 / ((double) pow(x, 2.0)))) + ((double) (2.0 / x))))))))));
			} else {
				double VAR_3;
				if ((t <= 3.203271237216745e+26)) {
					VAR_3 = ((double) (((double) (((double) sqrt(2.0)) * t)) / ((double) sqrt(((double) (((double) (4.0 * ((double) (((double) pow(t, 2.0)) / x)))) + ((double) (2.0 * ((double) (((double) sqrt(((double) (((double) pow(t, 2.0)) + ((double) (l * ((double) (l / x)))))))) * ((double) sqrt(((double) (((double) pow(t, 2.0)) + ((double) (l * ((double) (l / x))))))))))))))))));
				} else {
					VAR_3 = ((double) (((double) (((double) sqrt(2.0)) * t)) / ((double) (((double) (((double) sqrt(2.0)) * t)) + ((double) (((double) (2.0 * ((double) (t / ((double) (((double) sqrt(2.0)) * x)))))) + ((double) (((double) (t / ((double) pow(x, 2.0)))) * ((double) (((double) (2.0 / ((double) sqrt(2.0)))) - ((double) (2.0 / ((double) pow(((double) sqrt(2.0)), 3.0))))))))))))));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if t < -9.89774605087060996e124 or -2.1552676557946121e-209 < t < -1.27366272653752753e-279

    1. Initial program 56.8

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around -inf 9.7

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

    3. Simplified9.7

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

    if -9.89774605087060996e124 < t < -2.1552676557946121e-209 or -1.27366272653752753e-279 < t < 3.2032712372167452e26

    1. Initial program 37.0

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around inf 17.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\]

    3. Simplified17.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{{\ell}^{2}}{x}\right)}}}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity17.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{{\ell}^{2}}{\color{blue}{1 \cdot x}}\right)}}\]

    6. Applied add-sqr-sqrt40.9

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{{\color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)}}^{2}}{1 \cdot x}\right)}}\]

    7. Applied unpow-prod-down40.9

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\color{blue}{{\left(\sqrt{\ell}\right)}^{2} \cdot {\left(\sqrt{\ell}\right)}^{2}}}{1 \cdot x}\right)}}\]

    8. Applied times-frac38.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \color{blue}{\frac{{\left(\sqrt{\ell}\right)}^{2}}{1} \cdot \frac{{\left(\sqrt{\ell}\right)}^{2}}{x}}\right)}}\]

    9. Simplified38.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \color{blue}{\ell} \cdot \frac{{\left(\sqrt{\ell}\right)}^{2}}{x}\right)}}\]

    10. Simplified12.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \ell \cdot \color{blue}{\frac{\ell}{x}}\right)}}\]
    11. Using strategy rm

    12. Applied add-sqr-sqrt12.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \color{blue}{\left(\sqrt{{t}^{2} + \ell \cdot \frac{\ell}{x}} \cdot \sqrt{{t}^{2} + \ell \cdot \frac{\ell}{x}}\right)}}}\]

    if 3.2032712372167452e26 < t

    1. Initial program 43.2

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around inf 4.8

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

    3. Simplified4.8

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

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -9.89774605087060996 \cdot 10^{124}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \sqrt{2} \cdot t\right) - \frac{t}{\sqrt{2}} \cdot \left(\frac{2}{{x}^{2}} + \frac{2}{x}\right)}\\ \mathbf{elif}\;t \le -2.1552676557946121 \cdot 10^{-209}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(\sqrt{{t}^{2} + \ell \cdot \frac{\ell}{x}} \cdot \sqrt{{t}^{2} + \ell \cdot \frac{\ell}{x}}\right)}}\\ \mathbf{elif}\;t \le -1.27366272653752753 \cdot 10^{-279}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \sqrt{2} \cdot t\right) - \frac{t}{\sqrt{2}} \cdot \left(\frac{2}{{x}^{2}} + \frac{2}{x}\right)}\\ \mathbf{elif}\;t \le 3.2032712372167452 \cdot 10^{26}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(\sqrt{{t}^{2} + \ell \cdot \frac{\ell}{x}} \cdot \sqrt{{t}^{2} + \ell \cdot \frac{\ell}{x}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2} \cdot t + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + \frac{t}{{x}^{2}} \cdot \left(\frac{2}{\sqrt{2}} - \frac{2}{{\left(\sqrt{2}\right)}^{3}}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020149 
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  :precision binary64
  (/ (* (sqrt 2.0) t) (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))