Toniolo and Linder, Equation (7)

Average Error: 43.0 → 11.9

Time: 10.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := 2 \cdot {t}^{2}\\ t_2 := {\ell}^{2} + t_1\\ t_3 := t \cdot \sqrt{2}\\ t_4 := \mathsf{hypot}\left(\ell, t_3\right)\\ t_5 := \sqrt{\frac{x + 1}{x + -1}}\\ t_6 := {t}^{2} \cdot -2 - {\ell}^{2}\\ \mathbf{if}\;t \leq -2.9 \cdot 10^{-126}:\\ \;\;\;\;\frac{t_3}{t_3 \cdot \left(-t_5\right)}\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{-280}:\\ \;\;\;\;\frac{t_3}{\mathsf{hypot}\left(\mathsf{hypot}\left(\frac{\ell}{\sqrt{x}}, \mathsf{hypot}\left(\sqrt{2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)}, \frac{\mathsf{hypot}\left(t_4, t_4\right)}{x}\right)\right), \frac{t_4}{\sqrt{x}}\right)}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-194}:\\ \;\;\;\;\frac{t_3}{t_3 + \frac{t_6 + t_6}{\sqrt{2} \cdot \left(t \cdot x\right)} \cdot -0.5}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-79}:\\ \;\;\;\;\frac{t_3}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(\frac{t_2 + t_2}{{x}^{2}} + \left(t_1 + 2 \cdot \frac{{t}^{2}}{x}\right)\right)\right) + \frac{t_2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_3}{\sqrt{2} \cdot \left(t \cdot t_5\right)}\\ \end{array} \]

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 (* 2.0 (pow t 2.0)))
        (t_2 (+ (pow l 2.0) t_1))
        (t_3 (* t (sqrt 2.0)))
        (t_4 (hypot l t_3))
        (t_5 (sqrt (/ (+ x 1.0) (+ x -1.0))))
        (t_6 (- (* (pow t 2.0) -2.0) (pow l 2.0))))
   (if (<= t -2.9e-126)
     (/ t_3 (* t_3 (- t_5)))
     (if (<= t 6.4e-280)
       (/
        t_3
        (hypot
         (hypot
          (/ l (sqrt x))
          (hypot
           (sqrt (* 2.0 (+ (* t t) (/ (* t t) x))))
           (/ (hypot t_4 t_4) x)))
         (/ t_4 (sqrt x))))
       (if (<= t 1.3e-194)
         (/ t_3 (+ t_3 (* (/ (+ t_6 t_6) (* (sqrt 2.0) (* t x))) -0.5)))
         (if (<= t 1.35e-79)
           (/
            t_3
            (sqrt
             (+
              (+
               (/ (pow l 2.0) x)
               (+
                (/ (+ t_2 t_2) (pow x 2.0))
                (+ t_1 (* 2.0 (/ (pow t 2.0) x)))))
              (/ t_2 x))))
           (/ t_3 (* (sqrt 2.0) (* t t_5)))))))))

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 = 2.0 * pow(t, 2.0);
	double t_2 = pow(l, 2.0) + t_1;
	double t_3 = t * sqrt(2.0);
	double t_4 = hypot(l, t_3);
	double t_5 = sqrt(((x + 1.0) / (x + -1.0)));
	double t_6 = (pow(t, 2.0) * -2.0) - pow(l, 2.0);
	double tmp;
	if (t <= -2.9e-126) {
		tmp = t_3 / (t_3 * -t_5);
	} else if (t <= 6.4e-280) {
		tmp = t_3 / hypot(hypot((l / sqrt(x)), hypot(sqrt((2.0 * ((t * t) + ((t * t) / x)))), (hypot(t_4, t_4) / x))), (t_4 / sqrt(x)));
	} else if (t <= 1.3e-194) {
		tmp = t_3 / (t_3 + (((t_6 + t_6) / (sqrt(2.0) * (t * x))) * -0.5));
	} else if (t <= 1.35e-79) {
		tmp = t_3 / sqrt((((pow(l, 2.0) / x) + (((t_2 + t_2) / pow(x, 2.0)) + (t_1 + (2.0 * (pow(t, 2.0) / x))))) + (t_2 / x)));
	} else {
		tmp = t_3 / (sqrt(2.0) * (t * t_5));
	}
	return tmp;
}

Java

public static double code(double x, double l, double t) {
	return (Math.sqrt(2.0) * t) / Math.sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
}

↓

public static double code(double x, double l, double t) {
	double t_1 = 2.0 * Math.pow(t, 2.0);
	double t_2 = Math.pow(l, 2.0) + t_1;
	double t_3 = t * Math.sqrt(2.0);
	double t_4 = Math.hypot(l, t_3);
	double t_5 = Math.sqrt(((x + 1.0) / (x + -1.0)));
	double t_6 = (Math.pow(t, 2.0) * -2.0) - Math.pow(l, 2.0);
	double tmp;
	if (t <= -2.9e-126) {
		tmp = t_3 / (t_3 * -t_5);
	} else if (t <= 6.4e-280) {
		tmp = t_3 / Math.hypot(Math.hypot((l / Math.sqrt(x)), Math.hypot(Math.sqrt((2.0 * ((t * t) + ((t * t) / x)))), (Math.hypot(t_4, t_4) / x))), (t_4 / Math.sqrt(x)));
	} else if (t <= 1.3e-194) {
		tmp = t_3 / (t_3 + (((t_6 + t_6) / (Math.sqrt(2.0) * (t * x))) * -0.5));
	} else if (t <= 1.35e-79) {
		tmp = t_3 / Math.sqrt((((Math.pow(l, 2.0) / x) + (((t_2 + t_2) / Math.pow(x, 2.0)) + (t_1 + (2.0 * (Math.pow(t, 2.0) / x))))) + (t_2 / x)));
	} else {
		tmp = t_3 / (Math.sqrt(2.0) * (t * t_5));
	}
	return tmp;
}

Python

def code(x, l, t):
	return (math.sqrt(2.0) * t) / math.sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)))

↓

def code(x, l, t):
	t_1 = 2.0 * math.pow(t, 2.0)
	t_2 = math.pow(l, 2.0) + t_1
	t_3 = t * math.sqrt(2.0)
	t_4 = math.hypot(l, t_3)
	t_5 = math.sqrt(((x + 1.0) / (x + -1.0)))
	t_6 = (math.pow(t, 2.0) * -2.0) - math.pow(l, 2.0)
	tmp = 0
	if t <= -2.9e-126:
		tmp = t_3 / (t_3 * -t_5)
	elif t <= 6.4e-280:
		tmp = t_3 / math.hypot(math.hypot((l / math.sqrt(x)), math.hypot(math.sqrt((2.0 * ((t * t) + ((t * t) / x)))), (math.hypot(t_4, t_4) / x))), (t_4 / math.sqrt(x)))
	elif t <= 1.3e-194:
		tmp = t_3 / (t_3 + (((t_6 + t_6) / (math.sqrt(2.0) * (t * x))) * -0.5))
	elif t <= 1.35e-79:
		tmp = t_3 / math.sqrt((((math.pow(l, 2.0) / x) + (((t_2 + t_2) / math.pow(x, 2.0)) + (t_1 + (2.0 * (math.pow(t, 2.0) / x))))) + (t_2 / x)))
	else:
		tmp = t_3 / (math.sqrt(2.0) * (t * t_5))
	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(2.0 * (t ^ 2.0))
	t_2 = Float64((l ^ 2.0) + t_1)
	t_3 = Float64(t * sqrt(2.0))
	t_4 = hypot(l, t_3)
	t_5 = sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0)))
	t_6 = Float64(Float64((t ^ 2.0) * -2.0) - (l ^ 2.0))
	tmp = 0.0
	if (t <= -2.9e-126)
		tmp = Float64(t_3 / Float64(t_3 * Float64(-t_5)));
	elseif (t <= 6.4e-280)
		tmp = Float64(t_3 / hypot(hypot(Float64(l / sqrt(x)), hypot(sqrt(Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x)))), Float64(hypot(t_4, t_4) / x))), Float64(t_4 / sqrt(x))));
	elseif (t <= 1.3e-194)
		tmp = Float64(t_3 / Float64(t_3 + Float64(Float64(Float64(t_6 + t_6) / Float64(sqrt(2.0) * Float64(t * x))) * -0.5)));
	elseif (t <= 1.35e-79)
		tmp = Float64(t_3 / sqrt(Float64(Float64(Float64((l ^ 2.0) / x) + Float64(Float64(Float64(t_2 + t_2) / (x ^ 2.0)) + Float64(t_1 + Float64(2.0 * Float64((t ^ 2.0) / x))))) + Float64(t_2 / x))));
	else
		tmp = Float64(t_3 / Float64(sqrt(2.0) * Float64(t * t_5)));
	end
	return tmp
end

MATLAB

function tmp = code(x, l, t)
	tmp = (sqrt(2.0) * t) / sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
end

↓

function tmp_2 = code(x, l, t)
	t_1 = 2.0 * (t ^ 2.0);
	t_2 = (l ^ 2.0) + t_1;
	t_3 = t * sqrt(2.0);
	t_4 = hypot(l, t_3);
	t_5 = sqrt(((x + 1.0) / (x + -1.0)));
	t_6 = ((t ^ 2.0) * -2.0) - (l ^ 2.0);
	tmp = 0.0;
	if (t <= -2.9e-126)
		tmp = t_3 / (t_3 * -t_5);
	elseif (t <= 6.4e-280)
		tmp = t_3 / hypot(hypot((l / sqrt(x)), hypot(sqrt((2.0 * ((t * t) + ((t * t) / x)))), (hypot(t_4, t_4) / x))), (t_4 / sqrt(x)));
	elseif (t <= 1.3e-194)
		tmp = t_3 / (t_3 + (((t_6 + t_6) / (sqrt(2.0) * (t * x))) * -0.5));
	elseif (t <= 1.35e-79)
		tmp = t_3 / sqrt(((((l ^ 2.0) / x) + (((t_2 + t_2) / (x ^ 2.0)) + (t_1 + (2.0 * ((t ^ 2.0) / x))))) + (t_2 / x)));
	else
		tmp = t_3 / (sqrt(2.0) * (t * t_5));
	end
	tmp_2 = 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[(2.0 * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[l, 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[l ^ 2 + t$95$3 ^ 2], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[Power[t, 2.0], $MachinePrecision] * -2.0), $MachinePrecision] - N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.9e-126], N[(t$95$3 / N[(t$95$3 * (-t$95$5)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.4e-280], N[(t$95$3 / N[Sqrt[N[Sqrt[N[(l / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] ^ 2 + N[Sqrt[N[Sqrt[N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2 + N[(N[Sqrt[t$95$4 ^ 2 + t$95$4 ^ 2], $MachinePrecision] / x), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision] ^ 2 + N[(t$95$4 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.3e-194], N[(t$95$3 / N[(t$95$3 + N[(N[(N[(t$95$6 + t$95$6), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[(t * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.35e-79], N[(t$95$3 / N[Sqrt[N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision] + N[(N[(N[(t$95$2 + t$95$2), $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(2.0 * N[(N[Power[t, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$3 / N[(N[Sqrt[2.0], $MachinePrecision] * N[(t * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -2.89999999999999988e-126

   1. Initial program 38.3

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

   2. Taylor expanded in t around -inf 8.6

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

   3. Simplified 8.6

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

   if -2.89999999999999988e-126 < t < 6.4000000000000001e-280

   1. Initial program 59.0

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

   2. Taylor expanded in x around -inf 33.4

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

   3. Applied egg-rr 33.4

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

   4. Applied egg-rr 23.2

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

   if 6.4000000000000001e-280 < t < 1.30000000000000001e-194

   1. Initial program 63.0

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

   2. Taylor expanded in x around -inf 39.6

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

   3. Applied egg-rr 39.7

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

   4. Taylor expanded in x around inf 23.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{0.5 \cdot \frac{\left({\ell}^{2} + 2 \cdot {t}^{2}\right) - -1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{\sqrt{2} \cdot \left(t \cdot x\right)} + \sqrt{2} \cdot t}} \]

   if 1.30000000000000001e-194 < t < 1.3500000000000001e-79

   1. Initial program 43.1

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

   2. Taylor expanded in x around -inf 18.2

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

   if 1.3500000000000001e-79 < t

   1. Initial program 39.1

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

   2. Taylor expanded in l around 0 8.0

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

   3. Simplified 8.0

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

4. Final simplification 11.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-126}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{-280}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{hypot}\left(\mathsf{hypot}\left(\frac{\ell}{\sqrt{x}}, \mathsf{hypot}\left(\sqrt{2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)}, \frac{\mathsf{hypot}\left(\mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right)\right)}{x}\right)\right), \frac{\mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right)}{\sqrt{x}}\right)}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-194}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + \frac{\left({t}^{2} \cdot -2 - {\ell}^{2}\right) + \left({t}^{2} \cdot -2 - {\ell}^{2}\right)}{\sqrt{2} \cdot \left(t \cdot x\right)} \cdot -0.5}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-79}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(\frac{\left({\ell}^{2} + 2 \cdot {t}^{2}\right) + \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{{x}^{2}} + \left(2 \cdot {t}^{2} + 2 \cdot \frac{{t}^{2}}{x}\right)\right)\right) + \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{x + 1}{x + -1}}\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(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)))))