Average Error: 43.5 → 10.2
Time: 6.7s
Precision: binary64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq -2.844712975366527 \cdot 10^{+153}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{2 \cdot \left(\frac{t}{x \cdot \left(x \cdot \left(2 \cdot \sqrt{2}\right)\right)} - \frac{t}{\sqrt{2} \cdot x}\right) - t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq -1.4984775607051275 \cdot 10^{-146}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(t \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{4 \cdot \frac{t}{\frac{x}{t}} + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{elif}\;t \leq -1.7459689189787066 \cdot 10^{-212}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{2 \cdot \left(\frac{t}{x \cdot \left(x \cdot \left(2 \cdot \sqrt{2}\right)\right)} - \frac{t}{\sqrt{2} \cdot x}\right) - t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 9.84043843066752 \cdot 10^{-258} \lor \neg \left(t \leq 1.4464152816202683 \cdot 10^{-161}\right) \land t \leq 1.0144513413400185 \cdot 10^{+38}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(t \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{4 \cdot \frac{t}{\frac{x}{t}} + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + \frac{t}{x \cdot x} \cdot \left(\frac{2}{\sqrt{2}} - \frac{2}{2 \cdot \sqrt{2}}\right)\right)}\\ \end{array}\]
\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 \leq -2.844712975366527 \cdot 10^{+153}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{2 \cdot \left(\frac{t}{x \cdot \left(x \cdot \left(2 \cdot \sqrt{2}\right)\right)} - \frac{t}{\sqrt{2} \cdot x}\right) - t \cdot \sqrt{2}}\\

\mathbf{elif}\;t \leq -1.4984775607051275 \cdot 10^{-146}:\\
\;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(t \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{4 \cdot \frac{t}{\frac{x}{t}} + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right)}}\\

\mathbf{elif}\;t \leq -1.7459689189787066 \cdot 10^{-212}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{2 \cdot \left(\frac{t}{x \cdot \left(x \cdot \left(2 \cdot \sqrt{2}\right)\right)} - \frac{t}{\sqrt{2} \cdot x}\right) - t \cdot \sqrt{2}}\\

\mathbf{elif}\;t \leq 9.84043843066752 \cdot 10^{-258} \lor \neg \left(t \leq 1.4464152816202683 \cdot 10^{-161}\right) \land t \leq 1.0144513413400185 \cdot 10^{+38}:\\
\;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(t \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{4 \cdot \frac{t}{\frac{x}{t}} + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right)}}\\

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

\end{array}
double code(double x, double l, double t) {
	return (((double) (((double) sqrt(2.0)) * t)) / ((double) sqrt(((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 <= -2.844712975366527e+153)) {
		VAR = (((double) (t * ((double) sqrt(2.0)))) / ((double) (((double) (2.0 * ((double) ((t / ((double) (x * ((double) (x * ((double) (2.0 * ((double) sqrt(2.0))))))))) - (t / ((double) (((double) sqrt(2.0)) * x))))))) - ((double) (t * ((double) sqrt(2.0)))))));
	} else {
		double VAR_1;
		if ((t <= -1.4984775607051275e-146)) {
			VAR_1 = (((double) (((double) (((double) cbrt(((double) sqrt(2.0)))) * ((double) cbrt(((double) sqrt(2.0)))))) * ((double) (t * ((double) cbrt(((double) sqrt(2.0)))))))) / ((double) sqrt(((double) (((double) (4.0 * (t / (x / t)))) + ((double) (2.0 * ((double) (((double) (t * t)) + (l / (x / l)))))))))));
		} else {
			double VAR_2;
			if ((t <= -1.7459689189787066e-212)) {
				VAR_2 = (((double) (t * ((double) sqrt(2.0)))) / ((double) (((double) (2.0 * ((double) ((t / ((double) (x * ((double) (x * ((double) (2.0 * ((double) sqrt(2.0))))))))) - (t / ((double) (((double) sqrt(2.0)) * x))))))) - ((double) (t * ((double) sqrt(2.0)))))));
			} else {
				double VAR_3;
				if (((t <= 9.84043843066752e-258) || (!(t <= 1.4464152816202683e-161) && (t <= 1.0144513413400185e+38)))) {
					VAR_3 = (((double) (((double) (((double) cbrt(((double) sqrt(2.0)))) * ((double) cbrt(((double) sqrt(2.0)))))) * ((double) (t * ((double) cbrt(((double) sqrt(2.0)))))))) / ((double) sqrt(((double) (((double) (4.0 * (t / (x / t)))) + ((double) (2.0 * ((double) (((double) (t * t)) + (l / (x / l)))))))))));
				} else {
					VAR_3 = (((double) (t * ((double) sqrt(2.0)))) / ((double) (((double) (t * ((double) sqrt(2.0)))) + ((double) (((double) (2.0 * (t / ((double) (((double) sqrt(2.0)) * x))))) + ((double) ((t / ((double) (x * x))) * ((double) ((2.0 / ((double) sqrt(2.0))) - (2.0 / ((double) (2.0 * ((double) sqrt(2.0))))))))))))));
				}
				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 < -2.8447129753665268e153 or -1.4984775607051275e-146 < t < -1.74596891897870663e-212

    1. Initial program 61.6

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

    2. Taylor expanded around inf 56.1

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

    3. Simplified55.0

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

    4. Taylor expanded around -inf 8.4

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

    5. Simplified8.4

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

    if -2.8447129753665268e153 < t < -1.4984775607051275e-146 or -1.74596891897870663e-212 < t < 9.8404384306675198e-258 or 1.4464152816202683e-161 < t < 1.01445134134001845e38

    1. Initial program 33.0

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

    2. Taylor expanded around inf 15.1

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

    3. Simplified10.4

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

    5. Applied add-cube-cbrt10.4

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

    6. Applied associate-*l*10.4

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

    7. Simplified10.4

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

    if 9.8404384306675198e-258 < t < 1.4464152816202683e-161 or 1.01445134134001845e38 < t

    1. Initial program 48.5

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

    2. Taylor expanded around inf 11.0

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

    3. Simplified11.0

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.844712975366527 \cdot 10^{+153}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{2 \cdot \left(\frac{t}{x \cdot \left(x \cdot \left(2 \cdot \sqrt{2}\right)\right)} - \frac{t}{\sqrt{2} \cdot x}\right) - t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq -1.4984775607051275 \cdot 10^{-146}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(t \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{4 \cdot \frac{t}{\frac{x}{t}} + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{elif}\;t \leq -1.7459689189787066 \cdot 10^{-212}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{2 \cdot \left(\frac{t}{x \cdot \left(x \cdot \left(2 \cdot \sqrt{2}\right)\right)} - \frac{t}{\sqrt{2} \cdot x}\right) - t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 9.84043843066752 \cdot 10^{-258} \lor \neg \left(t \leq 1.4464152816202683 \cdot 10^{-161}\right) \land t \leq 1.0144513413400185 \cdot 10^{+38}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(t \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{4 \cdot \frac{t}{\frac{x}{t}} + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + \frac{t}{x \cdot x} \cdot \left(\frac{2}{\sqrt{2}} - \frac{2}{2 \cdot \sqrt{2}}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(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)))))