Toniolo and Linder, Equation (7)

Average Error: 42.7 → 11.1

Time: 18.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{t \cdot t}{x}\\ t_2 := t \cdot \sqrt{2}\\ t_3 := \sqrt{2} \cdot x\\ t_4 := t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}\\ t_5 := \frac{t_2}{-t_4}\\ t_6 := 2 \cdot \left(t \cdot t + \frac{\ell \cdot \ell}{x}\right)\\ t_7 := \frac{t_2}{\sqrt{\mathsf{fma}\left(4, t_1, t_6\right)}}\\ t_8 := \sqrt{\sqrt{\mathsf{fma}\left(4, \frac{t \cdot t}{x \cdot x}, \mathsf{fma}\left(4, t_1, \mathsf{fma}\left(4, \frac{t \cdot t}{{x}^{3}}, \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{x \cdot x}, \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{{x}^{3}}, t_6\right)\right)\right)\right)\right)}}\\ \mathbf{if}\;t \leq -5.617995743088974 \cdot 10^{+23}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;t \leq -1.9442302404542524 \cdot 10^{-154}:\\ \;\;\;\;\frac{t_2}{t_8 \cdot t_8}\\ \mathbf{elif}\;t \leq -1.4178637038882688 \cdot 10^{-186}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;t \leq -1.2062010810487966 \cdot 10^{-222}:\\ \;\;\;\;\frac{t_2}{\ell \cdot \sqrt{\frac{2}{x} + \left(\frac{2}{{x}^{3}} + \frac{2}{x \cdot x}\right)}}\\ \mathbf{elif}\;t \leq 2.7904332859958544 \cdot 10^{-275}:\\ \;\;\;\;t_7\\ \mathbf{elif}\;t \leq 8.418248280997763 \cdot 10^{-203}:\\ \;\;\;\;\frac{t_2}{\mathsf{fma}\left(t, \sqrt{2}, \mathsf{fma}\left(2, \frac{t}{t_3}, \frac{\ell \cdot \ell}{t \cdot t_3}\right)\right)}\\ \mathbf{elif}\;t \leq 5.140420865775031 \cdot 10^{-43}:\\ \;\;\;\;t_7\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{t_4}\\ \end{array} \]

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 (/ (* t t) x))
        (t_2 (* t (sqrt 2.0)))
        (t_3 (* (sqrt 2.0) x))
        (t_4 (* t (sqrt (+ (/ 2.0 (- x 1.0)) (* 2.0 (/ x (- x 1.0)))))))
        (t_5 (/ t_2 (- t_4)))
        (t_6 (* 2.0 (+ (* t t) (/ (* l l) x))))
        (t_7 (/ t_2 (sqrt (fma 4.0 t_1 t_6))))
        (t_8
         (sqrt
          (sqrt
           (fma
            4.0
            (/ (* t t) (* x x))
            (fma
             4.0
             t_1
             (fma
              4.0
              (/ (* t t) (pow x 3.0))
              (fma
               2.0
               (/ (* l l) (* x x))
               (fma 2.0 (/ (* l l) (pow x 3.0)) t_6)))))))))
   (if (<= t -5.617995743088974e+23)
     t_5
     (if (<= t -1.9442302404542524e-154)
       (/ t_2 (* t_8 t_8))
       (if (<= t -1.4178637038882688e-186)
         t_5
         (if (<= t -1.2062010810487966e-222)
           (/
            t_2
            (* l (sqrt (+ (/ 2.0 x) (+ (/ 2.0 (pow x 3.0)) (/ 2.0 (* x x)))))))
           (if (<= t 2.7904332859958544e-275)
             t_7
             (if (<= t 8.418248280997763e-203)
               (/
                t_2
                (fma t (sqrt 2.0) (fma 2.0 (/ t t_3) (/ (* l l) (* t t_3)))))
               (if (<= t 5.140420865775031e-43) t_7 (/ t_2 t_4))))))))))

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 = (t * t) / x;
	double t_2 = t * sqrt(2.0);
	double t_3 = sqrt(2.0) * x;
	double t_4 = t * sqrt(((2.0 / (x - 1.0)) + (2.0 * (x / (x - 1.0)))));
	double t_5 = t_2 / -t_4;
	double t_6 = 2.0 * ((t * t) + ((l * l) / x));
	double t_7 = t_2 / sqrt(fma(4.0, t_1, t_6));
	double t_8 = sqrt(sqrt(fma(4.0, ((t * t) / (x * x)), fma(4.0, t_1, fma(4.0, ((t * t) / pow(x, 3.0)), fma(2.0, ((l * l) / (x * x)), fma(2.0, ((l * l) / pow(x, 3.0)), t_6)))))));
	double tmp;
	if (t <= -5.617995743088974e+23) {
		tmp = t_5;
	} else if (t <= -1.9442302404542524e-154) {
		tmp = t_2 / (t_8 * t_8);
	} else if (t <= -1.4178637038882688e-186) {
		tmp = t_5;
	} else if (t <= -1.2062010810487966e-222) {
		tmp = t_2 / (l * sqrt(((2.0 / x) + ((2.0 / pow(x, 3.0)) + (2.0 / (x * x))))));
	} else if (t <= 2.7904332859958544e-275) {
		tmp = t_7;
	} else if (t <= 8.418248280997763e-203) {
		tmp = t_2 / fma(t, sqrt(2.0), fma(2.0, (t / t_3), ((l * l) / (t * t_3))));
	} else if (t <= 5.140420865775031e-43) {
		tmp = t_7;
	} else {
		tmp = t_2 / t_4;
	}
	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(t * t) / x)
	t_2 = Float64(t * sqrt(2.0))
	t_3 = Float64(sqrt(2.0) * x)
	t_4 = Float64(t * sqrt(Float64(Float64(2.0 / Float64(x - 1.0)) + Float64(2.0 * Float64(x / Float64(x - 1.0))))))
	t_5 = Float64(t_2 / Float64(-t_4))
	t_6 = Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(l * l) / x)))
	t_7 = Float64(t_2 / sqrt(fma(4.0, t_1, t_6)))
	t_8 = sqrt(sqrt(fma(4.0, Float64(Float64(t * t) / Float64(x * x)), fma(4.0, t_1, fma(4.0, Float64(Float64(t * t) / (x ^ 3.0)), fma(2.0, Float64(Float64(l * l) / Float64(x * x)), fma(2.0, Float64(Float64(l * l) / (x ^ 3.0)), t_6)))))))
	tmp = 0.0
	if (t <= -5.617995743088974e+23)
		tmp = t_5;
	elseif (t <= -1.9442302404542524e-154)
		tmp = Float64(t_2 / Float64(t_8 * t_8));
	elseif (t <= -1.4178637038882688e-186)
		tmp = t_5;
	elseif (t <= -1.2062010810487966e-222)
		tmp = Float64(t_2 / Float64(l * sqrt(Float64(Float64(2.0 / x) + Float64(Float64(2.0 / (x ^ 3.0)) + Float64(2.0 / Float64(x * x)))))));
	elseif (t <= 2.7904332859958544e-275)
		tmp = t_7;
	elseif (t <= 8.418248280997763e-203)
		tmp = Float64(t_2 / fma(t, sqrt(2.0), fma(2.0, Float64(t / t_3), Float64(Float64(l * l) / Float64(t * t_3)))));
	elseif (t <= 5.140420865775031e-43)
		tmp = t_7;
	else
		tmp = Float64(t_2 / t_4);
	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[(t * t), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$4 = 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]}, Block[{t$95$5 = N[(t$95$2 / (-t$95$4)), $MachinePrecision]}, Block[{t$95$6 = N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$2 / N[Sqrt[N[(4.0 * t$95$1 + t$95$6), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[Sqrt[N[Sqrt[N[(4.0 * N[(N[(t * t), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(4.0 * t$95$1 + N[(4.0 * N[(N[(t * t), $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(l * l), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(l * l), $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -5.617995743088974e+23], t$95$5, If[LessEqual[t, -1.9442302404542524e-154], N[(t$95$2 / N[(t$95$8 * t$95$8), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.4178637038882688e-186], t$95$5, If[LessEqual[t, -1.2062010810487966e-222], N[(t$95$2 / N[(l * N[Sqrt[N[(N[(2.0 / x), $MachinePrecision] + N[(N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.7904332859958544e-275], t$95$7, If[LessEqual[t, 8.418248280997763e-203], N[(t$95$2 / N[(t * N[Sqrt[2.0], $MachinePrecision] + N[(2.0 * N[(t / t$95$3), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / N[(t * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.140420865775031e-43], t$95$7, N[(t$95$2 / t$95$4), $MachinePrecision]]]]]]]]]]]]]]]]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

1. Split input into 6 regimes

2. if t < -5.61799574308897423e23 or -1.94423024045425239e-154 < t < -1.4178637038882688e-186

   1. Initial program 44.1

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

   2. Simplified 44.1

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

   3. Taylor expanded in t around -inf 6.3

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

   4. Simplified 6.3

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

   if -5.61799574308897423e23 < t < -1.94423024045425239e-154

   1. Initial program 29.4

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

   2. Simplified 29.4

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

   3. Taylor expanded in x around inf 9.3

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

   4. Simplified 9.3

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

   5. Applied add-sqr-sqrt_binary64 9.5

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

   if -1.4178637038882688e-186 < t < -1.2062010810487966e-222

   1. Initial program 63.9

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

   2. Simplified 63.9

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

   3. Taylor expanded in x around inf 43.4

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

   4. Simplified 43.4

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

   5. Taylor expanded in l around inf 40.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\ell \cdot \sqrt{2 \cdot \frac{1}{x} + \left(2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{2}}\right)}}} \]

   6. Simplified 40.0

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

   if -1.2062010810487966e-222 < t < 2.79043328599585439e-275 or 8.4182482809977629e-203 < t < 5.1404208657750308e-43

   1. Initial program 48.2

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

   2. Simplified 48.2

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

   3. Taylor expanded in x around inf 21.7

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

   4. Simplified 21.7

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

   if 2.79043328599585439e-275 < t < 8.4182482809977629e-203

   1. Initial program 63.4

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

   2. Simplified 63.4

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

   3. Taylor expanded in x around inf 22.5

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

   4. Simplified 22.5

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

   if 5.1404208657750308e-43 < t

   1. Initial program 39.6

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

   2. Simplified 39.6

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

   3. Taylor expanded in t around inf 6.0

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

   4. Simplified 6.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}} \cdot t}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 11.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.617995743088974 \cdot 10^{+23}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \mathbf{elif}\;t \leq -1.9442302404542524 \cdot 10^{-154}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\sqrt{\mathsf{fma}\left(4, \frac{t \cdot t}{x \cdot x}, \mathsf{fma}\left(4, \frac{t \cdot t}{x}, \mathsf{fma}\left(4, \frac{t \cdot t}{{x}^{3}}, \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{x \cdot x}, \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{{x}^{3}}, 2 \cdot \left(t \cdot t + \frac{\ell \cdot \ell}{x}\right)\right)\right)\right)\right)\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(4, \frac{t \cdot t}{x \cdot x}, \mathsf{fma}\left(4, \frac{t \cdot t}{x}, \mathsf{fma}\left(4, \frac{t \cdot t}{{x}^{3}}, \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{x \cdot x}, \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{{x}^{3}}, 2 \cdot \left(t \cdot t + \frac{\ell \cdot \ell}{x}\right)\right)\right)\right)\right)\right)}}}\\ \mathbf{elif}\;t \leq -1.4178637038882688 \cdot 10^{-186}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \mathbf{elif}\;t \leq -1.2062010810487966 \cdot 10^{-222}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \left(\frac{2}{{x}^{3}} + \frac{2}{x \cdot x}\right)}}\\ \mathbf{elif}\;t \leq 2.7904332859958544 \cdot 10^{-275}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\mathsf{fma}\left(4, \frac{t \cdot t}{x}, 2 \cdot \left(t \cdot t + \frac{\ell \cdot \ell}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq 8.418248280997763 \cdot 10^{-203}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, \frac{\ell \cdot \ell}{t \cdot \left(\sqrt{2} \cdot x\right)}\right)\right)}\\ \mathbf{elif}\;t \leq 5.140420865775031 \cdot 10^{-43}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\mathsf{fma}\left(4, \frac{t \cdot t}{x}, 2 \cdot \left(t \cdot t + \frac{\ell \cdot \ell}{x}\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 2022129 
(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)))))