Average Error: 43.0 → 9.7
Time: 7.4s
Precision: binary64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.09637139580299768 \cdot 10^{108}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{t}{x \cdot x} \cdot \left(\frac{2}{2 \cdot \sqrt{2}} - \frac{2}{\sqrt{2}}\right) - \left(t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\ \mathbf{elif}\;t \le 1.53206292407734662 \cdot 10^{-283}:\\ \;\;\;\;\frac{\sqrt{\sqrt{2}} \cdot \left(t \cdot \sqrt{\sqrt{2}}\right)}{\sqrt{4 \cdot \left(t \cdot \frac{t}{x}\right) + 2 \cdot \left(t \cdot t + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \le 7.28019137526069 \cdot 10^{-175} \lor \neg \left(t \le 1.1060627253666669 \cdot 10^{45}\right):\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + \frac{t}{x \cdot x} \cdot \left(\frac{2}{\sqrt{2}} - \frac{2}{2 \cdot \sqrt{2}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(t \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{4 \cdot \left(t \cdot \frac{t}{x}\right) + 2 \cdot \left(t \cdot t + \ell \cdot \frac{\ell}{x}\right)}}\\ \end{array}\]
\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 -1.09637139580299768 \cdot 10^{108}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{t}{x \cdot x} \cdot \left(\frac{2}{2 \cdot \sqrt{2}} - \frac{2}{\sqrt{2}}\right) - \left(t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\

\mathbf{elif}\;t \le 1.53206292407734662 \cdot 10^{-283}:\\
\;\;\;\;\frac{\sqrt{\sqrt{2}} \cdot \left(t \cdot \sqrt{\sqrt{2}}\right)}{\sqrt{4 \cdot \left(t \cdot \frac{t}{x}\right) + 2 \cdot \left(t \cdot t + \ell \cdot \frac{\ell}{x}\right)}}\\

\mathbf{elif}\;t \le 7.28019137526069 \cdot 10^{-175} \lor \neg \left(t \le 1.1060627253666669 \cdot 10^{45}\right):\\
\;\;\;\;\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)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(t \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{4 \cdot \left(t \cdot \frac{t}{x}\right) + 2 \cdot \left(t \cdot t + \ell \cdot \frac{\ell}{x}\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 <= -1.0963713958029977e+108)) {
		VAR = ((double) (((double) (t * ((double) sqrt(2.0)))) / ((double) (((double) (((double) (t / ((double) (x * x)))) * ((double) (((double) (2.0 / ((double) (2.0 * ((double) sqrt(2.0)))))) - ((double) (2.0 / ((double) sqrt(2.0)))))))) - ((double) (((double) (t * ((double) sqrt(2.0)))) + ((double) (2.0 * ((double) (t / ((double) (((double) sqrt(2.0)) * x))))))))))));
	} else {
		double VAR_1;
		if ((t <= 1.5320629240773466e-283)) {
			VAR_1 = ((double) (((double) (((double) sqrt(((double) sqrt(2.0)))) * ((double) (t * ((double) sqrt(((double) sqrt(2.0)))))))) / ((double) sqrt(((double) (((double) (4.0 * ((double) (t * ((double) (t / x)))))) + ((double) (2.0 * ((double) (((double) (t * t)) + ((double) (l * ((double) (l / x))))))))))))));
		} else {
			double VAR_2;
			if (((t <= 7.28019137526069e-175) || !(t <= 1.106062725366667e+45))) {
				VAR_2 = ((double) (((double) (t * ((double) sqrt(2.0)))) / ((double) (((double) (t * ((double) sqrt(2.0)))) + ((double) (((double) (2.0 * ((double) (t / ((double) (((double) sqrt(2.0)) * x)))))) + ((double) (((double) (t / ((double) (x * x)))) * ((double) (((double) (2.0 / ((double) sqrt(2.0)))) - ((double) (2.0 / ((double) (2.0 * ((double) sqrt(2.0))))))))))))))));
			} else {
				VAR_2 = ((double) (((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 * ((double) (t * ((double) (t / x)))))) + ((double) (2.0 * ((double) (((double) (t * t)) + ((double) (l * ((double) (l / x))))))))))))));
			}
			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 4 regimes

  2. if t < -1.09637139580299768e108

    1. Initial program 52.3

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

    2. Taylor expanded around -inf 2.9

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

    3. Simplified2.9

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

    if -1.09637139580299768e108 < t < 1.53206292407734662e-283

    1. Initial program 37.8

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

    2. Taylor expanded around inf 17.4

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

    3. Simplified13.6

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

    5. Applied add-sqr-sqrt13.6

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

    6. Applied sqrt-prod13.7

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

    7. Applied associate-*l*13.7

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

    8. Simplified13.7

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

    if 1.53206292407734662e-283 < t < 7.28019137526069e-175 or 1.1060627253666669e45 < t

    1. Initial program 48.9

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

    2. Taylor expanded around inf 11.4

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

    3. Simplified11.4

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

    if 7.28019137526069e-175 < t < 1.1060627253666669e45

    1. Initial program 30.2

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

    2. Taylor expanded around inf 11.7

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

    3. Simplified7.2

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

    5. Applied add-cube-cbrt7.2

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

    6. Applied associate-*l*7.2

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

    7. Simplified7.2

      \[\leadsto \frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \color{blue}{\left(t \cdot \sqrt[3]{\sqrt{2}}\right)}}{\sqrt{4 \cdot \left(t \cdot \frac{t}{x}\right) + 2 \cdot \left(t \cdot t + \ell \cdot \frac{\ell}{x}\right)}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.09637139580299768 \cdot 10^{108}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{t}{x \cdot x} \cdot \left(\frac{2}{2 \cdot \sqrt{2}} - \frac{2}{\sqrt{2}}\right) - \left(t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\ \mathbf{elif}\;t \le 1.53206292407734662 \cdot 10^{-283}:\\ \;\;\;\;\frac{\sqrt{\sqrt{2}} \cdot \left(t \cdot \sqrt{\sqrt{2}}\right)}{\sqrt{4 \cdot \left(t \cdot \frac{t}{x}\right) + 2 \cdot \left(t \cdot t + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \le 7.28019137526069 \cdot 10^{-175} \lor \neg \left(t \le 1.1060627253666669 \cdot 10^{45}\right):\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + \frac{t}{x \cdot x} \cdot \left(\frac{2}{\sqrt{2}} - \frac{2}{2 \cdot \sqrt{2}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(t \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{4 \cdot \left(t \cdot \frac{t}{x}\right) + 2 \cdot \left(t \cdot t + \ell \cdot \frac{\ell}{x}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(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)))))