Toniolo and Linder, Equation (7)

Average Error: 43.4 → 8.7

Time: 3.2min

Precision: binary64

Cost: 47684

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -5.2919924128564495 \cdot 10^{+60}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \mathbf{elif}\;t \leq 4.192030017014753 \cdot 10^{-266}:\\ \;\;\;\;\frac{\sqrt{\sqrt{2}} \cdot \left(t \cdot \sqrt{\sqrt{2}}\right)}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right) + \left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right)}}\\ \mathbf{elif}\;t \leq 7.081498969282516 \cdot 10^{-185}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2 + \frac{4}{x}} + \sqrt{\frac{1}{2 + \frac{4}{x}}} \cdot \frac{\ell \cdot \ell}{t \cdot x}}\\ \mathbf{elif}\;t \leq 3.478259665751439 \cdot 10^{+54}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(t \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right) + \left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \end{array}\]

FPCore

(FPCore (x l t)
 :precision binary64
 (/
  (* (sqrt 2.0) t)
  (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))

↓

(FPCore (x l t)
 :precision binary64
 (if (<= t -5.2919924128564495e+60)
   (/
    (* t (sqrt 2.0))
    (- (* t (sqrt (+ (/ 2.0 (- x 1.0)) (* 2.0 (/ x (- x 1.0))))))))
   (if (<= t 4.192030017014753e-266)
     (/
      (* (sqrt (sqrt 2.0)) (* t (sqrt (sqrt 2.0))))
      (sqrt (+ (* 2.0 (* l (/ l x))) (* (* t t) (+ 2.0 (/ 4.0 x))))))
     (if (<= t 7.081498969282516e-185)
       (/
        (* t (sqrt 2.0))
        (+
         (* t (sqrt (+ 2.0 (/ 4.0 x))))
         (* (sqrt (/ 1.0 (+ 2.0 (/ 4.0 x)))) (/ (* l l) (* t x)))))
       (if (<= t 3.478259665751439e+54)
         (/
          (* (* (cbrt (sqrt 2.0)) (cbrt (sqrt 2.0))) (* t (cbrt (sqrt 2.0))))
          (sqrt (+ (* 2.0 (* l (/ l x))) (* (* t t) (+ 2.0 (/ 4.0 x))))))
         (/
          (* t (sqrt 2.0))
          (* t (sqrt (+ (/ 2.0 (- x 1.0)) (* 2.0 (/ x (- x 1.0))))))))))))

C

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 tmp;
	if (t <= -5.2919924128564495e+60) {
		tmp = (t * sqrt(2.0)) / -(t * sqrt((2.0 / (x - 1.0)) + (2.0 * (x / (x - 1.0)))));
	} else if (t <= 4.192030017014753e-266) {
		tmp = (sqrt(sqrt(2.0)) * (t * sqrt(sqrt(2.0)))) / sqrt((2.0 * (l * (l / x))) + ((t * t) * (2.0 + (4.0 / x))));
	} else if (t <= 7.081498969282516e-185) {
		tmp = (t * sqrt(2.0)) / ((t * sqrt(2.0 + (4.0 / x))) + (sqrt(1.0 / (2.0 + (4.0 / x))) * ((l * l) / (t * x))));
	} else if (t <= 3.478259665751439e+54) {
		tmp = ((cbrt(sqrt(2.0)) * cbrt(sqrt(2.0))) * (t * cbrt(sqrt(2.0)))) / sqrt((2.0 * (l * (l / x))) + ((t * t) * (2.0 + (4.0 / x))));
	} else {
		tmp = (t * sqrt(2.0)) / (t * sqrt((2.0 / (x - 1.0)) + (2.0 * (x / (x - 1.0)))));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Alternatives

Alternative 1
Error	8.7
Cost	34114

\[\begin{array}{l} \mathbf{if}\;t \leq -5.2919924128564495 \cdot 10^{+60}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \mathbf{elif}\;t \leq 4.192030017014753 \cdot 10^{-266}:\\ \;\;\;\;\frac{\sqrt{\sqrt{2}} \cdot \left(t \cdot \sqrt{\sqrt{2}}\right)}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right) + \left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right)}}\\ \mathbf{elif}\;t \leq 7.081498969282516 \cdot 10^{-185}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2 + \frac{4}{x}} + \sqrt{\frac{1}{2 + \frac{4}{x}}} \cdot \frac{\ell \cdot \ell}{t \cdot x}}\\ \mathbf{elif}\;t \leq 3.478259665751439 \cdot 10^{+54}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t\right) + 4 \cdot \frac{t \cdot t}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \end{array}\]

Alternative 2
Error	8.7
Cost	21891

\[\begin{array}{l} \mathbf{if}\;t \leq -5.2919924128564495 \cdot 10^{+60}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \mathbf{elif}\;t \leq 4.192030017014753 \cdot 10^{-266}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right) + \left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right)}}\\ \mathbf{elif}\;t \leq 7.081498969282516 \cdot 10^{-185}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2 + \frac{4}{x}} + \sqrt{\frac{1}{2 + \frac{4}{x}}} \cdot \frac{\ell \cdot \ell}{t \cdot x}}\\ \mathbf{elif}\;t \leq 3.478259665751439 \cdot 10^{+54}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t\right) + 4 \cdot \frac{t \cdot t}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \end{array}\]

Alternative 3
Error	9.2
Cost	15684

\[\begin{array}{l} \mathbf{if}\;t \leq -5.2919924128564495 \cdot 10^{+60}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \mathbf{elif}\;t \leq 5.224886030954151 \cdot 10^{-231}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right) + \left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right)}}\\ \mathbf{elif}\;t \leq 7.081498969282516 \cdot 10^{-185}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.2544254585346644 \cdot 10^{+54}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t\right) + 4 \cdot \frac{t \cdot t}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \end{array}\]

Alternative 4
Error	9.3
Cost	15300

\[\begin{array}{l} \mathbf{if}\;t \leq -5.2919924128564495 \cdot 10^{+60}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \mathbf{elif}\;t \leq 5.224886030954151 \cdot 10^{-231}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right) + \left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right)}}\\ \mathbf{elif}\;t \leq 7.081498969282516 \cdot 10^{-185}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.478259665751439 \cdot 10^{+54}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right) + 2 \cdot \left(t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \end{array}\]

Alternative 5
Error	9.3
Cost	15300

\[\begin{array}{l} \mathbf{if}\;t \leq -5.2919924128564495 \cdot 10^{+60}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \mathbf{elif}\;t \leq 5.224886030954151 \cdot 10^{-231}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right) + 2 \cdot \left(t \cdot t\right)}}\\ \mathbf{elif}\;t \leq 7.081498969282516 \cdot 10^{-185}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.478259665751439 \cdot 10^{+54}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right) + 2 \cdot \left(t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \end{array}\]

Alternative 6
Error	9.5
Cost	15300

\[\begin{array}{l} \mathbf{if}\;t \leq -1.108092975063467 \cdot 10^{+95}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{\frac{0.5}{\frac{x}{x - 1} + \frac{1}{x - 1}}}\\ \mathbf{elif}\;t \leq 5.224886030954151 \cdot 10^{-231}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right) + 2 \cdot \left(t \cdot t\right)}}\\ \mathbf{elif}\;t \leq 7.081498969282516 \cdot 10^{-185}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.478259665751439 \cdot 10^{+54}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right) + 2 \cdot \left(t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \end{array}\]

Alternative 7
Error	9.6
Cost	15172

\[\begin{array}{l} \mathbf{if}\;t \leq -1.108092975063467 \cdot 10^{+95}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{\frac{0.5}{\frac{x}{x - 1} + \frac{1}{x - 1}}}\\ \mathbf{elif}\;t \leq 5.224886030954151 \cdot 10^{-231}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right) + 2 \cdot \left(t \cdot t\right)}}\\ \mathbf{elif}\;t \leq 7.081498969282516 \cdot 10^{-185}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.3511096763505706 \cdot 10^{+54}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right) + 2 \cdot \left(t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2 + \frac{4}{x}}}\\ \end{array}\]

Alternative 8
Error	11.9
Cost	15172

\[\begin{array}{l} \mathbf{if}\;t \leq -7.88324907210378 \cdot 10^{+43}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{\frac{0.5}{\frac{x}{x - 1} + \frac{1}{x - 1}}}\\ \mathbf{elif}\;t \leq 5.224886030954151 \cdot 10^{-231}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot t\right) + 2 \cdot \frac{\ell \cdot \ell}{x}}}\\ \mathbf{elif}\;t \leq 7.081498969282516 \cdot 10^{-185}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.3403225210506321 \cdot 10^{+28}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot t\right) + 2 \cdot \frac{\ell \cdot \ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2 + \frac{4}{x}}}\\ \end{array}\]

Alternative 9
Error	15.3
Cost	14916

\[\begin{array}{l} \mathbf{if}\;t \leq -4.826401648043764 \cdot 10^{-173}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{\frac{0.5}{\frac{x}{x - 1} + \frac{1}{x - 1}}}\\ \mathbf{elif}\;t \leq 5.224886030954151 \cdot 10^{-231}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq 7.081498969282516 \cdot 10^{-185}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.86238128998016 \cdot 10^{-94}:\\ \;\;\;\;\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{2 \cdot \frac{\ell \cdot \ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2 + \frac{4}{x}}}\\ \end{array}\]

Alternative 10
Error	15.1
Cost	14916

\[\begin{array}{l} \mathbf{if}\;t \leq -4.826401648043764 \cdot 10^{-173}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{2 + \frac{4}{x}}}\\ \mathbf{elif}\;t \leq 5.224886030954151 \cdot 10^{-231}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq 7.081498969282516 \cdot 10^{-185}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.86238128998016 \cdot 10^{-94}:\\ \;\;\;\;\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{2 \cdot \frac{\ell \cdot \ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2 + \frac{4}{x}}}\\ \end{array}\]

Alternative 11
Error	14.8
Cost	14788

\[\begin{array}{l} \mathbf{if}\;t \leq -4.826401648043764 \cdot 10^{-173}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{2 + \frac{4}{x}}}\\ \mathbf{elif}\;t \leq 5.224886030954151 \cdot 10^{-231}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq 7.081498969282516 \cdot 10^{-185}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.8307532102252626 \cdot 10^{-96}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2 + \frac{4}{x}}}\\ \end{array}\]

Alternative 12
Error	15.0
Cost	14788

\[\begin{array}{l} \mathbf{if}\;t \leq -4.826401648043764 \cdot 10^{-173}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 5.224886030954151 \cdot 10^{-231}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq 7.081498969282516 \cdot 10^{-185}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.86238128998016 \cdot 10^{-94}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2 + \frac{4}{x}}}\\ \end{array}\]

Alternative 13
Error	16.4
Cost	14146

\[\begin{array}{l} \mathbf{if}\;t \leq -4.826401648043764 \cdot 10^{-173}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 4.57531098855737 \cdot 10^{-304}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2 + \frac{4}{x}}}\\ \end{array}\]

Alternative 14
Error	16.6
Cost	706

\[\begin{array}{l} \mathbf{if}\;t \leq -4.826401648043764 \cdot 10^{-173}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 4.57531098855737 \cdot 10^{-304}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Alternative 15
Error	37.4
Cost	385

\[\begin{array}{l} \mathbf{if}\;t \leq 4.57531098855737 \cdot 10^{-304}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Alternative 16
Error	39.4
Cost	64

\[1\]

Derivation

1. Split input into 5 regimes

2. if t < -5.2919924128564495e60

   1. Initial program 45.3

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

   2. Taylor expanded around -inf 3.6

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

   3. Simplified 3.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}}\]

   4. Simplified 3.6

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}}\]

   if -5.2919924128564495e60 < t < 4.19203001701475306e-266

   1. Initial program 42.1

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

   2. Taylor expanded around inf 19.1

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

   3. Simplified 19.1

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

   5. Applied *-un-lft-identity_binary64_78 19.1

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

   6. Applied times-frac_binary64_84 14.9

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

   7. Simplified 14.9

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

   8. Taylor expanded around 0 14.9

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

   9. Simplified 14.9

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

   11. Applied add-sqr-sqrt_binary64_100 15.0

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

   12. Applied associate-*l*_binary64_19 14.9

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

   13. Simplified 14.9

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

   14. Simplified 14.9

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

   if 4.19203001701475306e-266 < t < 7.0814989692825159e-185

   1. Initial program 63.2

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

   2. Taylor expanded around inf 33.3

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

   3. Simplified 33.3

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

   4. Taylor expanded around inf 23.7

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

   5. Simplified 23.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \sqrt{2 + \frac{4}{x}} + \sqrt{\frac{1}{2 + \frac{4}{x}}} \cdot \frac{\ell \cdot \ell}{x \cdot t}}}\]

   6. Simplified 23.7

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{t \cdot \sqrt{2 + \frac{4}{x}} + \sqrt{\frac{1}{2 + \frac{4}{x}}} \cdot \frac{\ell \cdot \ell}{x \cdot t}}}\]

   if 7.0814989692825159e-185 < t < 3.4782596657514389e54

   1. Initial program 33.5

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

   2. Taylor expanded around inf 13.7

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

   3. Simplified 13.7

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

   5. Applied *-un-lft-identity_binary64_78 13.7

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

   6. Applied times-frac_binary64_84 8.3

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

   7. Simplified 8.3

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

   8. Taylor expanded around 0 8.3

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

   9. Simplified 8.3

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

   11. Applied add-cube-cbrt_binary64_113 8.3

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

   12. Applied associate-*l*_binary64_19 8.2

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

   13. Simplified 8.2

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

   14. Simplified 8.2

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

   if 3.4782596657514389e54 < t

   1. Initial program 46.2

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

   2. Taylor expanded around inf 3.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \sqrt{2 \cdot \frac{1}{x - 1} + 2 \cdot \frac{x}{x - 1}}}}\]

   3. Simplified 3.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}}\]

   4. Simplified 3.2

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 8.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.2919924128564495 \cdot 10^{+60}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \mathbf{elif}\;t \leq 4.192030017014753 \cdot 10^{-266}:\\ \;\;\;\;\frac{\sqrt{\sqrt{2}} \cdot \left(t \cdot \sqrt{\sqrt{2}}\right)}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right) + \left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right)}}\\ \mathbf{elif}\;t \leq 7.081498969282516 \cdot 10^{-185}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2 + \frac{4}{x}} + \sqrt{\frac{1}{2 + \frac{4}{x}}} \cdot \frac{\ell \cdot \ell}{t \cdot x}}\\ \mathbf{elif}\;t \leq 3.478259665751439 \cdot 10^{+54}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(t \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right) + \left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021065 
(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)))))