Toniolo and Linder, Equation (7)

Average Error: 42.9 → 8.6

Time: 13.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{4}{x \cdot x} + \left(2 + \frac{4}{x}\right)\\ t_2 := t \cdot \left(-\sqrt{t_1}\right)\\ t_3 := t \cdot \sqrt{2}\\ t_4 := \frac{t_3}{\left|\sqrt{\mathsf{fma}\left(4, {\left(\frac{t}{x}\right)}^{2}, \mathsf{fma}\left(4, t \cdot \frac{t}{x}, \mathsf{fma}\left(2, {\left(\frac{\ell}{x}\right)}^{2}, 2 \cdot \mathsf{fma}\left(t, t, \ell \cdot \frac{\ell}{x}\right)\right)\right)\right)}\right|}\\ \mathbf{if}\;t \leq -7.313146234677325:\\ \;\;\;\;\frac{t_3}{t_2}\\ \mathbf{elif}\;t \leq -4.598917439395751 \cdot 10^{-161}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq -1.1070675823535772 \cdot 10^{-218}:\\ \;\;\;\;\frac{t_3}{t_2 - \sqrt{\frac{1}{t_1}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t} + \frac{\frac{\ell}{x} \cdot \frac{\ell}{x}}{t}\right)}\\ \mathbf{elif}\;t \leq -1.1119314911646242 \cdot 10^{-301}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 2.898770924274869 \cdot 10^{-166}:\\ \;\;\;\;\frac{t_3}{\mathsf{fma}\left(t, \sqrt{2}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, \frac{\ell}{\frac{\sqrt{2} \cdot \left(t \cdot x\right)}{\ell}}\right)\right)}\\ \mathbf{elif}\;t \leq 1.7505159449100466 \cdot 10^{+61}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{t_3}{t \cdot \sqrt{\frac{2}{x + -1} + 2 \cdot \frac{x}{x + -1}}}\\ \end{array} \]

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
 (let* ((t_1 (+ (/ 4.0 (* x x)) (+ 2.0 (/ 4.0 x))))
        (t_2 (* t (- (sqrt t_1))))
        (t_3 (* t (sqrt 2.0)))
        (t_4
         (/
          t_3
          (fabs
           (sqrt
            (fma
             4.0
             (pow (/ t x) 2.0)
             (fma
              4.0
              (* t (/ t x))
              (fma
               2.0
               (pow (/ l x) 2.0)
               (* 2.0 (fma t t (* l (/ l x))))))))))))
   (if (<= t -7.313146234677325)
     (/ t_3 t_2)
     (if (<= t -4.598917439395751e-161)
       t_4
       (if (<= t -1.1070675823535772e-218)
         (/
          t_3
          (-
           t_2
           (*
            (sqrt (/ 1.0 t_1))
            (+ (* (/ l x) (/ l t)) (/ (* (/ l x) (/ l x)) t)))))
         (if (<= t -1.1119314911646242e-301)
           t_4
           (if (<= t 2.898770924274869e-166)
             (/
              t_3
              (fma
               t
               (sqrt 2.0)
               (fma
                2.0
                (/ t (* (sqrt 2.0) x))
                (/ l (/ (* (sqrt 2.0) (* t x)) l)))))
             (if (<= t 1.7505159449100466e+61)
               t_4
               (/
                t_3
                (*
                 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 t_1 = (4.0 / (x * x)) + (2.0 + (4.0 / x));
	double t_2 = t * -sqrt(t_1);
	double t_3 = t * sqrt(2.0);
	double t_4 = t_3 / fabs(sqrt(fma(4.0, pow((t / x), 2.0), fma(4.0, (t * (t / x)), fma(2.0, pow((l / x), 2.0), (2.0 * fma(t, t, (l * (l / x)))))))));
	double tmp;
	if (t <= -7.313146234677325) {
		tmp = t_3 / t_2;
	} else if (t <= -4.598917439395751e-161) {
		tmp = t_4;
	} else if (t <= -1.1070675823535772e-218) {
		tmp = t_3 / (t_2 - (sqrt((1.0 / t_1)) * (((l / x) * (l / t)) + (((l / x) * (l / x)) / t))));
	} else if (t <= -1.1119314911646242e-301) {
		tmp = t_4;
	} else if (t <= 2.898770924274869e-166) {
		tmp = t_3 / fma(t, sqrt(2.0), fma(2.0, (t / (sqrt(2.0) * x)), (l / ((sqrt(2.0) * (t * x)) / l))));
	} else if (t <= 1.7505159449100466e+61) {
		tmp = t_4;
	} else {
		tmp = t_3 / (t * sqrt(((2.0 / (x + -1.0)) + (2.0 * (x / (x + -1.0))))));
	}
	return tmp;
}

Julia

function code(x, l, t)
	return Float64(Float64(sqrt(2.0) * t) / sqrt(Float64(Float64(Float64(Float64(x + 1.0) / Float64(x - 1.0)) * Float64(Float64(l * l) + Float64(2.0 * Float64(t * t)))) - Float64(l * l))))
end

↓

function code(x, l, t)
	t_1 = Float64(Float64(4.0 / Float64(x * x)) + Float64(2.0 + Float64(4.0 / x)))
	t_2 = Float64(t * Float64(-sqrt(t_1)))
	t_3 = Float64(t * sqrt(2.0))
	t_4 = Float64(t_3 / abs(sqrt(fma(4.0, (Float64(t / x) ^ 2.0), fma(4.0, Float64(t * Float64(t / x)), fma(2.0, (Float64(l / x) ^ 2.0), Float64(2.0 * fma(t, t, Float64(l * Float64(l / x))))))))))
	tmp = 0.0
	if (t <= -7.313146234677325)
		tmp = Float64(t_3 / t_2);
	elseif (t <= -4.598917439395751e-161)
		tmp = t_4;
	elseif (t <= -1.1070675823535772e-218)
		tmp = Float64(t_3 / Float64(t_2 - Float64(sqrt(Float64(1.0 / t_1)) * Float64(Float64(Float64(l / x) * Float64(l / t)) + Float64(Float64(Float64(l / x) * Float64(l / x)) / t)))));
	elseif (t <= -1.1119314911646242e-301)
		tmp = t_4;
	elseif (t <= 2.898770924274869e-166)
		tmp = Float64(t_3 / fma(t, sqrt(2.0), fma(2.0, Float64(t / Float64(sqrt(2.0) * x)), Float64(l / Float64(Float64(sqrt(2.0) * Float64(t * x)) / l)))));
	elseif (t <= 1.7505159449100466e+61)
		tmp = t_4;
	else
		tmp = Float64(t_3 / Float64(t * sqrt(Float64(Float64(2.0 / Float64(x + -1.0)) + Float64(2.0 * Float64(x / Float64(x + -1.0)))))));
	end
	return tmp
end

Wolfram

code[x_, l_, t_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x_, l_, t_] := Block[{t$95$1 = N[(N[(4.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * (-N[Sqrt[t$95$1], $MachinePrecision])), $MachinePrecision]}, Block[{t$95$3 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / N[Abs[N[Sqrt[N[(4.0 * N[Power[N[(t / x), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(t * N[(t / x), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[Power[N[(l / x), $MachinePrecision], 2.0], $MachinePrecision] + N[(2.0 * N[(t * t + N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -7.313146234677325], N[(t$95$3 / t$95$2), $MachinePrecision], If[LessEqual[t, -4.598917439395751e-161], t$95$4, If[LessEqual[t, -1.1070675823535772e-218], N[(t$95$3 / N[(t$95$2 - N[(N[Sqrt[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(l / x), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(l / x), $MachinePrecision] * N[(l / x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.1119314911646242e-301], t$95$4, If[LessEqual[t, 2.898770924274869e-166], N[(t$95$3 / N[(t * N[Sqrt[2.0], $MachinePrecision] + N[(2.0 * N[(t / N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + N[(l / N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t * x), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.7505159449100466e+61], t$95$4, N[(t$95$3 / N[(t * N[Sqrt[N[(N[(2.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(x / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

1. Split input into 5 regimes

2. if t < -7.31314623467732527

   1. Initial program 41.5

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

   2. Simplified 41.5

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

   3. Taylor expanded in x around inf 38.8

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

   4. Simplified 38.8

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

   5. Taylor expanded in t around -inf 5.7

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

   6. Simplified 5.7

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

   if -7.31314623467732527 < t < -4.5989174393957506e-161 or -1.1070675823535772e-218 < t < -1.11193149116462419e-301 or 2.898770924274869e-166 < t < 1.7505159449100466e61

   1. Initial program 35.0

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

   2. Simplified 35.0

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

   3. Taylor expanded in x around inf 14.1

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

   4. Simplified 14.1

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

   5. Applied egg-rr 8.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{{\left({\left(\mathsf{fma}\left(4, {\left(\frac{t}{x}\right)}^{2}, \mathsf{fma}\left(4, \frac{t}{\frac{x}{t}}, \mathsf{fma}\left(2, {\left(\frac{\ell}{x}\right)}^{2}, 2 \cdot \mathsf{fma}\left(t, t, \frac{\ell}{\frac{x}{\ell}}\right)\right)\right)\right)\right)}^{0.25}\right)}^{2}}} \]

   6. Applied egg-rr 8.4

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

   if -4.5989174393957506e-161 < t < -1.1070675823535772e-218

   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. Simplified 63.2

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

   3. Taylor expanded in x around inf 41.8

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

   4. Simplified 41.8

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

   5. Taylor expanded in t around -inf 25.9

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

   6. Simplified 22.0

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

   if -1.11193149116462419e-301 < t < 2.898770924274869e-166

   1. Initial program 62.7

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

   2. Simplified 62.7

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

   3. Taylor expanded in x around inf 37.5

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

   4. Simplified 37.5

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

   5. Taylor expanded in x around inf 26.4

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

   6. Simplified 26.3

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

   if 1.7505159449100466e61 < t

   1. Initial program 44.8

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

   2. Simplified 44.8

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

   3. Taylor expanded in t around inf 3.0

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

   4. Simplified 3.0

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

4. Final simplification 8.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.313146234677325:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \left(-\sqrt{\frac{4}{x \cdot x} + \left(2 + \frac{4}{x}\right)}\right)}\\ \mathbf{elif}\;t \leq -4.598917439395751 \cdot 10^{-161}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left|\sqrt{\mathsf{fma}\left(4, {\left(\frac{t}{x}\right)}^{2}, \mathsf{fma}\left(4, t \cdot \frac{t}{x}, \mathsf{fma}\left(2, {\left(\frac{\ell}{x}\right)}^{2}, 2 \cdot \mathsf{fma}\left(t, t, \ell \cdot \frac{\ell}{x}\right)\right)\right)\right)}\right|}\\ \mathbf{elif}\;t \leq -1.1070675823535772 \cdot 10^{-218}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \left(-\sqrt{\frac{4}{x \cdot x} + \left(2 + \frac{4}{x}\right)}\right) - \sqrt{\frac{1}{\frac{4}{x \cdot x} + \left(2 + \frac{4}{x}\right)}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t} + \frac{\frac{\ell}{x} \cdot \frac{\ell}{x}}{t}\right)}\\ \mathbf{elif}\;t \leq -1.1119314911646242 \cdot 10^{-301}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left|\sqrt{\mathsf{fma}\left(4, {\left(\frac{t}{x}\right)}^{2}, \mathsf{fma}\left(4, t \cdot \frac{t}{x}, \mathsf{fma}\left(2, {\left(\frac{\ell}{x}\right)}^{2}, 2 \cdot \mathsf{fma}\left(t, t, \ell \cdot \frac{\ell}{x}\right)\right)\right)\right)}\right|}\\ \mathbf{elif}\;t \leq 2.898770924274869 \cdot 10^{-166}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, \frac{\ell}{\frac{\sqrt{2} \cdot \left(t \cdot x\right)}{\ell}}\right)\right)}\\ \mathbf{elif}\;t \leq 1.7505159449100466 \cdot 10^{+61}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left|\sqrt{\mathsf{fma}\left(4, {\left(\frac{t}{x}\right)}^{2}, \mathsf{fma}\left(4, t \cdot \frac{t}{x}, \mathsf{fma}\left(2, {\left(\frac{\ell}{x}\right)}^{2}, 2 \cdot \mathsf{fma}\left(t, t, \ell \cdot \frac{\ell}{x}\right)\right)\right)\right)}\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{2}{x + -1} + 2 \cdot \frac{x}{x + -1}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(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)))))