Average Error: 42.7 → 9.8
Time: 10.3s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -8.08541959572897258 \cdot 10^{99}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}, -\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot {x}^{2}}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right)\right)\right)}\\ \mathbf{elif}\;t \le 5.31686358457741622 \cdot 10^{-279}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \left(\sqrt[3]{{\left(\sqrt[3]{\ell}\right)}^{4}} \cdot \sqrt[3]{{\left(\sqrt[3]{\ell}\right)}^{4}}\right) \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\ell}\right)}^{4}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{x}\right), 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\ \mathbf{elif}\;t \le 8.3630275522259312 \cdot 10^{-198}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot {x}^{2}}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \mathbf{elif}\;t \le 1.32955370740893664 \cdot 10^{133}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \left(\sqrt[3]{{\left(\sqrt[3]{\ell}\right)}^{4}} \cdot \sqrt[3]{{\left(\sqrt[3]{\ell}\right)}^{4}}\right) \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\ell}\right)}^{4}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{x}\right), 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot {x}^{2}}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \end{array}\]
\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 -8.08541959572897258 \cdot 10^{99}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}, -\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot {x}^{2}}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right)\right)\right)}\\

\mathbf{elif}\;t \le 5.31686358457741622 \cdot 10^{-279}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \left(\sqrt[3]{{\left(\sqrt[3]{\ell}\right)}^{4}} \cdot \sqrt[3]{{\left(\sqrt[3]{\ell}\right)}^{4}}\right) \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\ell}\right)}^{4}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{x}\right), 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\

\mathbf{elif}\;t \le 8.3630275522259312 \cdot 10^{-198}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot {x}^{2}}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\

\mathbf{elif}\;t \le 1.32955370740893664 \cdot 10^{133}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \left(\sqrt[3]{{\left(\sqrt[3]{\ell}\right)}^{4}} \cdot \sqrt[3]{{\left(\sqrt[3]{\ell}\right)}^{4}}\right) \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\ell}\right)}^{4}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{x}\right), 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\

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

\end{array}
double code(double x, double l, double t) {
	return ((sqrt(2.0) * t) / sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l))));
}
↓
double code(double x, double l, double t) {
	double temp;
	if ((t <= -8.085419595728973e+99)) {
		temp = ((sqrt(2.0) * t) / fma(2.0, (t / (pow(sqrt(2.0), 3.0) * pow(x, 2.0))), -fma(2.0, (t / (sqrt(2.0) * pow(x, 2.0))), fma(2.0, (t / (sqrt(2.0) * x)), (t * sqrt(2.0))))));
	} else {
		double temp_1;
		if ((t <= 5.316863584577416e-279)) {
			temp_1 = ((sqrt(2.0) * t) / sqrt(fma(2.0, pow(t, 2.0), fma(2.0, ((cbrt(pow(cbrt(l), 4.0)) * cbrt(pow(cbrt(l), 4.0))) * (cbrt(pow(cbrt(l), 4.0)) * (pow(cbrt(l), 2.0) / x))), (4.0 * (pow(t, 2.0) / x))))));
		} else {
			double temp_2;
			if ((t <= 8.363027552225931e-198)) {
				temp_2 = ((sqrt(2.0) * t) / fma(2.0, (t / (sqrt(2.0) * pow(x, 2.0))), (fma(2.0, (t / (sqrt(2.0) * x)), (t * sqrt(2.0))) - (2.0 * (t / (pow(sqrt(2.0), 3.0) * pow(x, 2.0)))))));
			} else {
				double temp_3;
				if ((t <= 1.3295537074089366e+133)) {
					temp_3 = ((sqrt(2.0) * t) / sqrt(fma(2.0, pow(t, 2.0), fma(2.0, ((cbrt(pow(cbrt(l), 4.0)) * cbrt(pow(cbrt(l), 4.0))) * (cbrt(pow(cbrt(l), 4.0)) * (pow(cbrt(l), 2.0) / x))), (4.0 * (pow(t, 2.0) / x))))));
				} else {
					temp_3 = ((sqrt(2.0) * t) / fma(2.0, (t / (sqrt(2.0) * pow(x, 2.0))), (fma(2.0, (t / (sqrt(2.0) * x)), (t * sqrt(2.0))) - (2.0 * (t / (pow(sqrt(2.0), 3.0) * pow(x, 2.0)))))));
				}
				temp_2 = temp_3;
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

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 < -8.085419595728973e+99

    1. Initial program 50.0

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

    2. Taylor expanded around -inf 3.3

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

    3. Simplified3.3

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

    if -8.085419595728973e+99 < t < 5.316863584577416e-279 or 8.363027552225931e-198 < t < 1.3295537074089366e+133

    1. Initial program 34.1

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

    2. Taylor expanded around inf 15.2

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

    3. Simplified15.2

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

    5. Applied *-un-lft-identity15.2

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

    6. Applied add-cube-cbrt15.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \frac{{\color{blue}{\left(\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}\right)}}^{2}}{1 \cdot x}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\]

    7. Applied unpow-prod-down15.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \frac{\color{blue}{{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)}^{2} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}}{1 \cdot x}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\]

    8. Applied times-frac12.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \color{blue}{\frac{{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)}^{2}}{1} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{x}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\]

    9. Simplified12.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \color{blue}{{\left(\sqrt[3]{\ell}\right)}^{4}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{x}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\]
    10. Using strategy rm

    11. Applied add-cube-cbrt12.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \color{blue}{\left(\left(\sqrt[3]{{\left(\sqrt[3]{\ell}\right)}^{4}} \cdot \sqrt[3]{{\left(\sqrt[3]{\ell}\right)}^{4}}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{\ell}\right)}^{4}}\right)} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{x}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\]

    12. Applied associate-*l*12.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \color{blue}{\left(\sqrt[3]{{\left(\sqrt[3]{\ell}\right)}^{4}} \cdot \sqrt[3]{{\left(\sqrt[3]{\ell}\right)}^{4}}\right) \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\ell}\right)}^{4}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{x}\right)}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\]

    if 5.316863584577416e-279 < t < 8.363027552225931e-198 or 1.3295537074089366e+133 < t

    1. Initial program 58.2

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

    2. Taylor expanded around inf 9.7

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

    3. Simplified9.7

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -8.08541959572897258 \cdot 10^{99}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}, -\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot {x}^{2}}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right)\right)\right)}\\ \mathbf{elif}\;t \le 5.31686358457741622 \cdot 10^{-279}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \left(\sqrt[3]{{\left(\sqrt[3]{\ell}\right)}^{4}} \cdot \sqrt[3]{{\left(\sqrt[3]{\ell}\right)}^{4}}\right) \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\ell}\right)}^{4}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{x}\right), 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\ \mathbf{elif}\;t \le 8.3630275522259312 \cdot 10^{-198}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot {x}^{2}}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \mathbf{elif}\;t \le 1.32955370740893664 \cdot 10^{133}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \left(\sqrt[3]{{\left(\sqrt[3]{\ell}\right)}^{4}} \cdot \sqrt[3]{{\left(\sqrt[3]{\ell}\right)}^{4}}\right) \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\ell}\right)}^{4}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{x}\right), 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot {x}^{2}}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 +o rules:numerics
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  :precision binary64
  (/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l)))))