Toniolo and Linder, Equation (7)

Average Error: 42.9 → 9.1

Time: 25.9s

Precision: binary64

Cost: 28805

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -1.2351728058409963 \cdot 10^{-41}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \mathbf{elif}\;t \leq -1.5844530436624705 \cdot 10^{-154}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{elif}\;t \leq -4.222777320075483 \cdot 10^{-236}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(-t \cdot \sqrt{2 + \frac{4}{x}}\right) - \sqrt{\frac{1}{2 + \frac{4}{x}}} \cdot \frac{\ell \cdot \ell}{t \cdot x}}\\ \mathbf{elif}\;t \leq 1.673826578300523 \cdot 10^{-305}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{1}{x}} \cdot \left(\sqrt{2} \cdot \ell\right)}\\ \mathbf{elif}\;t \leq 3.385594735519463 \cdot 10^{-144}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + \frac{\ell \cdot \ell}{\left(t \cdot \sqrt{2}\right) \cdot x}\right)}\\ \mathbf{elif}\;t \leq 2.2674113060888642 \cdot 10^{+126}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(t \cdot t\right)\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \end{array}\]

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 -1.2351728058409963e-41)
   (/
    (* t (sqrt 2.0))
    (- (* t (sqrt (+ (/ 2.0 (- x 1.0)) (* 2.0 (/ x (- x 1.0))))))))
   (if (<= t -1.5844530436624705e-154)
     (/
      (* t (sqrt 2.0))
      (sqrt (+ (* 4.0 (/ (* t t) x)) (* 2.0 (+ (* t t) (/ l (/ x l)))))))
     (if (<= t -4.222777320075483e-236)
       (/
        (* 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 1.673826578300523e-305)
         (/ (* t (sqrt 2.0)) (* (sqrt (/ 1.0 x)) (* (sqrt 2.0) l)))
         (if (<= t 3.385594735519463e-144)
           (/
            (* t (sqrt 2.0))
            (+
             (* t (sqrt 2.0))
             (+
              (* 2.0 (/ t (* (sqrt 2.0) x)))
              (/ (* l l) (* (* t (sqrt 2.0)) x)))))
           (if (<= t 2.2674113060888642e+126)
             (/
              (* t (sqrt 2.0))
              (sqrt
               (+
                (+ (* 4.0 (/ (* t t) x)) (* 2.0 (* t t)))
                (* 2.0 (* l (/ l 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 <= -1.2351728058409963e-41) {
		tmp = (t * sqrt(2.0)) / -(t * sqrt((2.0 / (x - 1.0)) + (2.0 * (x / (x - 1.0)))));
	} else if (t <= -1.5844530436624705e-154) {
		tmp = (t * sqrt(2.0)) / sqrt((4.0 * ((t * t) / x)) + (2.0 * ((t * t) + (l / (x / l)))));
	} else if (t <= -4.222777320075483e-236) {
		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 <= 1.673826578300523e-305) {
		tmp = (t * sqrt(2.0)) / (sqrt(1.0 / x) * (sqrt(2.0) * l));
	} else if (t <= 3.385594735519463e-144) {
		tmp = (t * sqrt(2.0)) / ((t * sqrt(2.0)) + ((2.0 * (t / (sqrt(2.0) * x))) + ((l * l) / ((t * sqrt(2.0)) * x))));
	} else if (t <= 2.2674113060888642e+126) {
		tmp = (t * sqrt(2.0)) / sqrt(((4.0 * ((t * t) / x)) + (2.0 * (t * t))) + (2.0 * (l * (l / 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	10.2
Cost	21955

\[\begin{array}{l} \mathbf{if}\;t \leq -1.2351728058409963 \cdot 10^{-41}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \mathbf{elif}\;t \leq -1.5844530436624705 \cdot 10^{-154}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{elif}\;t \leq -4.222777320075483 \cdot 10^{-236}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(-t \cdot \sqrt{2 + \frac{4}{x}}\right) - \sqrt{\frac{1}{2 + \frac{4}{x}}} \cdot \frac{\ell \cdot \ell}{t \cdot x}}\\ \mathbf{elif}\;t \leq 4.901623676390601 \cdot 10^{-185}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{1}{x}} \cdot \left(\sqrt{2} \cdot \ell\right)}\\ \mathbf{elif}\;t \leq 3.385594735519463 \cdot 10^{-144}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2 + \frac{4}{x}}}\\ \mathbf{elif}\;t \leq 9.671127833094812 \cdot 10^{+125}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(t \cdot t\right)\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \end{array}\]

Alternative 2
Error	10.6
Cost	21188

\[\begin{array}{l} \mathbf{if}\;t \leq -1.2351728058409963 \cdot 10^{-41}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \mathbf{elif}\;t \leq -1.908450893682963 \cdot 10^{-175}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{elif}\;t \leq -4.222777320075483 \cdot 10^{-236}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 4.901623676390601 \cdot 10^{-185}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{1}{x}} \cdot \left(\sqrt{2} \cdot \ell\right)}\\ \mathbf{elif}\;t \leq 3.385594735519463 \cdot 10^{-144}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2 + \frac{4}{x}}}\\ \mathbf{elif}\;t \leq 2.2674113060888642 \cdot 10^{+126}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(t \cdot t\right)\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \end{array}\]

Alternative 3
Error	9.9
Cost	15684

\[\begin{array}{l} \mathbf{if}\;t \leq -1.2351728058409963 \cdot 10^{-41}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \mathbf{elif}\;t \leq -1.908450893682963 \cdot 10^{-175}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{elif}\;t \leq -4.222777320075483 \cdot 10^{-236}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 3.2861889216797514 \cdot 10^{+126}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(t \cdot t\right)\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \end{array}\]

Alternative 4
Error	9.9
Cost	15556

\[\begin{array}{l} \mathbf{if}\;t \leq -1.2351728058409963 \cdot 10^{-41}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \mathbf{elif}\;t \leq -1.908450893682963 \cdot 10^{-175}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{elif}\;t \leq -4.222777320075483 \cdot 10^{-236}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 2.221815944911774 \cdot 10^{+126}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \end{array}\]

Alternative 5
Error	11.8
Cost	15428

\[\begin{array}{l} \mathbf{if}\;t \leq -1.2351728058409963 \cdot 10^{-41}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \mathbf{elif}\;t \leq -1.908450893682963 \cdot 10^{-175}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot t\right) + 2 \cdot \frac{\ell \cdot \ell}{x}}}\\ \mathbf{elif}\;t \leq -4.222777320075483 \cdot 10^{-236}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1148938624868404.5:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell \cdot \ell}{x} + \left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \end{array}\]

Alternative 6
Error	12.1
Cost	15300

\[\begin{array}{l} \mathbf{if}\;t \leq -1.2351728058409963 \cdot 10^{-41}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \mathbf{elif}\;t \leq -1.908450893682963 \cdot 10^{-175}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot t\right) + 2 \cdot \frac{\ell \cdot \ell}{x}}}\\ \mathbf{elif}\;t \leq -4.222777320075483 \cdot 10^{-236}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1.597425099800245 \cdot 10^{-73}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot t\right) + 2 \cdot \frac{\ell \cdot \ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \end{array}\]

Alternative 7
Error	12.2
Cost	15300

\[\begin{array}{l} \mathbf{if}\;t \leq -1.2440943124107288 \cdot 10^{+92}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{\frac{0.5}{\frac{x}{x - 1} + \frac{1}{x - 1}}}\\ \mathbf{elif}\;t \leq -1.908450893682963 \cdot 10^{-175}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot t\right) + 2 \cdot \frac{\ell \cdot \ell}{x}}}\\ \mathbf{elif}\;t \leq -4.222777320075483 \cdot 10^{-236}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1.597425099800245 \cdot 10^{-73}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot t\right) + 2 \cdot \frac{\ell \cdot \ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \end{array}\]

Alternative 8
Error	12.0
Cost	15172

\[\begin{array}{l} \mathbf{if}\;t \leq -3.8006551229722496 \cdot 10^{+88}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{\frac{0.5}{\frac{x}{x - 1} + \frac{1}{x - 1}}}\\ \mathbf{elif}\;t \leq -1.908450893682963 \cdot 10^{-175}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot t\right) + 2 \cdot \frac{\ell \cdot \ell}{x}}}\\ \mathbf{elif}\;t \leq -4.222777320075483 \cdot 10^{-236}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1523336689259179.5:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot t\right) + 2 \cdot \frac{\ell \cdot \ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2 + \frac{4}{x}}}\\ \end{array}\]

Alternative 9
Error	14.2
Cost	14146

\[\begin{array}{l} \mathbf{if}\;t \leq -4.222777320075483 \cdot 10^{-236}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{2 + \frac{4}{x}}}\\ \mathbf{elif}\;t \leq 4.901623676390601 \cdot 10^{-185}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2 + \frac{4}{x}}}\\ \end{array}\]

Alternative 10
Error	14.4
Cost	14146

\[\begin{array}{l} \mathbf{if}\;t \leq -4.222777320075483 \cdot 10^{-236}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 4.901623676390601 \cdot 10^{-185}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2 + \frac{4}{x}}}\\ \end{array}\]

Alternative 11
Error	15.4
Cost	13825

\[\begin{array}{l} \mathbf{if}\;t \leq -6.2486584787246 \cdot 10^{-311}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2 + \frac{4}{x}}}\\ \end{array}\]

Alternative 12
Error	15.6
Cost	385

\[\begin{array}{l} \mathbf{if}\;t \leq -6.2486584787246 \cdot 10^{-311}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Alternative 13
Error	37.7
Cost	385

\[\begin{array}{l} \mathbf{if}\;t \leq 1.7261542091313192 \cdot 10^{-302}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Alternative 14
Error	39.7
Cost	64

\[1\]

Derivation

1. Split input into 7 regimes

2. if t < -1.2351728058409963e-41

   1. Initial program 39.8

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

   2. Taylor expanded around -inf 6.3

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

   3. Simplified 6.3

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

   4. Simplified 6.3

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

   if -1.2351728058409963e-41 < t < -1.58445304366247051e-154

   1. Initial program 32.6

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

   2. Taylor expanded around inf 10.0

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

   3. Simplified 10.0

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

   5. Applied add-cube-cbrt_binary64_113 10.0

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

   6. Applied times-frac_binary64_84 4.4

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

   8. Applied associate-+r+_binary64_10 4.4

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

   9. Simplified 4.3

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

   10. Simplified 4.3

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

   if -1.58445304366247051e-154 < t < -4.2227773200754832e-236

   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 36.4

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

   3. Simplified 36.4

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

   4. Taylor expanded around -inf 24.2

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

   5. Simplified 24.2

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

   6. Simplified 24.2

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

   if -4.2227773200754832e-236 < t < 1.67382657830052298e-305

   1. Initial program 62.9

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

   2. Taylor expanded around inf 29.8

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

   3. Simplified 29.8

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

   4. Taylor expanded around inf 32.7

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

   5. Simplified 32.7

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

   if 1.67382657830052298e-305 < t < 3.38559473551946292e-144

   1. Initial program 60.4

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

   2. Taylor expanded around inf 24.1

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

   3. Simplified 24.1

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

   4. Simplified 24.1

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

   if 3.38559473551946292e-144 < t < 2.26741130608886417e126

   1. Initial program 24.5

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

   2. Taylor expanded around inf 10.7

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

   3. Simplified 10.7

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

   5. Applied *-un-lft-identity_binary64_78 10.7

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

   6. Applied times-frac_binary64_84 5.1

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

   7. Simplified 5.1

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

   8. Simplified 5.1

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

   if 2.26741130608886417e126 < t

   1. Initial program 55.1

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

   2. Taylor expanded around inf 2.2

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

   3. Simplified 2.2

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

   4. Simplified 2.2

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}}\]
3. Recombined 7 regimes into one program.

4. Final simplification 9.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.2351728058409963 \cdot 10^{-41}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \mathbf{elif}\;t \leq -1.5844530436624705 \cdot 10^{-154}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{elif}\;t \leq -4.222777320075483 \cdot 10^{-236}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(-t \cdot \sqrt{2 + \frac{4}{x}}\right) - \sqrt{\frac{1}{2 + \frac{4}{x}}} \cdot \frac{\ell \cdot \ell}{t \cdot x}}\\ \mathbf{elif}\;t \leq 1.673826578300523 \cdot 10^{-305}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{1}{x}} \cdot \left(\sqrt{2} \cdot \ell\right)}\\ \mathbf{elif}\;t \leq 3.385594735519463 \cdot 10^{-144}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + \frac{\ell \cdot \ell}{\left(t \cdot \sqrt{2}\right) \cdot x}\right)}\\ \mathbf{elif}\;t \leq 2.2674113060888642 \cdot 10^{+126}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(t \cdot t\right)\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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)))))