Toniolo and Linder, Equation (7)

Average Error: 42.5 → 8.9

Time: 10.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{t}{\frac{\ell}{t}}\\ t_2 := \frac{2}{x} + \left(\frac{2}{{x}^{3}} + \frac{2}{x \cdot x}\right)\\ t_3 := \sqrt{\frac{2}{x + -1} + 2 \cdot \frac{x}{x + -1}}\\ t_4 := t \cdot \sqrt{2}\\ t_5 := \frac{t_4}{t_3 \cdot \left(-t\right)}\\ t_6 := \frac{t_4}{\sqrt{\mathsf{fma}\left(4, {\left(\frac{t}{x}\right)}^{2}, \mathsf{fma}\left(4, t \cdot \frac{t}{x}, \mathsf{fma}\left(4, t \cdot \frac{t}{{x}^{3}}, \mathsf{fma}\left(2, {\left(\frac{\ell}{x}\right)}^{2}, \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{{x}^{3}}, 2 \cdot \mathsf{fma}\left(t, t, \ell \cdot \frac{\ell}{x}\right)\right)\right)\right)\right)\right)}}\\ t_7 := \sqrt{\frac{1}{t_2}}\\ \mathbf{if}\;t \leq -2.2 \cdot 10^{+133}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;t \leq -2.6 \cdot 10^{-172}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;t \leq -1.02 \cdot 10^{-221}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-225}:\\ \;\;\;\;\frac{t_4}{t_7 \cdot \left(t_1 + 2 \cdot \frac{t_1}{x}\right) + \left(t_7 \cdot \left(2 \cdot \frac{\frac{t}{x} \cdot \frac{t}{x}}{\ell} + 2 \cdot \frac{t_1}{{x}^{3}}\right) + \ell \cdot \sqrt{t_2}\right)}\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{-194}:\\ \;\;\;\;\frac{t_4}{\mathsf{fma}\left(t, \sqrt{2}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, \frac{\ell}{\frac{\sqrt{2} \cdot \left(t \cdot x\right)}{\ell}}\right)\right)}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+104}:\\ \;\;\;\;t_6\\ \mathbf{else}:\\ \;\;\;\;\frac{t_4}{t \cdot t_3}\\ \end{array} \]

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 (/ l t)))
        (t_2 (+ (/ 2.0 x) (+ (/ 2.0 (pow x 3.0)) (/ 2.0 (* x x)))))
        (t_3 (sqrt (+ (/ 2.0 (+ x -1.0)) (* 2.0 (/ x (+ x -1.0))))))
        (t_4 (* t (sqrt 2.0)))
        (t_5 (/ t_4 (* t_3 (- t))))
        (t_6
         (/
          t_4
          (sqrt
           (fma
            4.0
            (pow (/ t x) 2.0)
            (fma
             4.0
             (* t (/ t x))
             (fma
              4.0
              (* t (/ t (pow x 3.0)))
              (fma
               2.0
               (pow (/ l x) 2.0)
               (fma
                2.0
                (* l (/ l (pow x 3.0)))
                (* 2.0 (fma t t (* l (/ l x))))))))))))
        (t_7 (sqrt (/ 1.0 t_2))))
   (if (<= t -2.2e+133)
     t_5
     (if (<= t -2.6e-172)
       t_6
       (if (<= t -1.02e-221)
         t_5
         (if (<= t 2.8e-225)
           (/
            t_4
            (+
             (* t_7 (+ t_1 (* 2.0 (/ t_1 x))))
             (+
              (*
               t_7
               (+
                (* 2.0 (/ (* (/ t x) (/ t x)) l))
                (* 2.0 (/ t_1 (pow x 3.0)))))
              (* l (sqrt t_2)))))
           (if (<= t 1.12e-194)
             (/
              t_4
              (fma
               t
               (sqrt 2.0)
               (fma
                2.0
                (/ t (* (sqrt 2.0) x))
                (/ l (/ (* (sqrt 2.0) (* t x)) l)))))
             (if (<= t 4e+104) t_6 (/ t_4 (* t t_3))))))))))

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 / (l / t);
	double t_2 = (2.0 / x) + ((2.0 / pow(x, 3.0)) + (2.0 / (x * x)));
	double t_3 = sqrt(((2.0 / (x + -1.0)) + (2.0 * (x / (x + -1.0)))));
	double t_4 = t * sqrt(2.0);
	double t_5 = t_4 / (t_3 * -t);
	double t_6 = t_4 / sqrt(fma(4.0, pow((t / x), 2.0), fma(4.0, (t * (t / x)), fma(4.0, (t * (t / pow(x, 3.0))), fma(2.0, pow((l / x), 2.0), fma(2.0, (l * (l / pow(x, 3.0))), (2.0 * fma(t, t, (l * (l / x))))))))));
	double t_7 = sqrt((1.0 / t_2));
	double tmp;
	if (t <= -2.2e+133) {
		tmp = t_5;
	} else if (t <= -2.6e-172) {
		tmp = t_6;
	} else if (t <= -1.02e-221) {
		tmp = t_5;
	} else if (t <= 2.8e-225) {
		tmp = t_4 / ((t_7 * (t_1 + (2.0 * (t_1 / x)))) + ((t_7 * ((2.0 * (((t / x) * (t / x)) / l)) + (2.0 * (t_1 / pow(x, 3.0))))) + (l * sqrt(t_2))));
	} else if (t <= 1.12e-194) {
		tmp = t_4 / fma(t, sqrt(2.0), fma(2.0, (t / (sqrt(2.0) * x)), (l / ((sqrt(2.0) * (t * x)) / l))));
	} else if (t <= 4e+104) {
		tmp = t_6;
	} else {
		tmp = t_4 / (t * t_3);
	}
	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(t / Float64(l / t))
	t_2 = Float64(Float64(2.0 / x) + Float64(Float64(2.0 / (x ^ 3.0)) + Float64(2.0 / Float64(x * x))))
	t_3 = sqrt(Float64(Float64(2.0 / Float64(x + -1.0)) + Float64(2.0 * Float64(x / Float64(x + -1.0)))))
	t_4 = Float64(t * sqrt(2.0))
	t_5 = Float64(t_4 / Float64(t_3 * Float64(-t)))
	t_6 = Float64(t_4 / sqrt(fma(4.0, (Float64(t / x) ^ 2.0), fma(4.0, Float64(t * Float64(t / x)), fma(4.0, Float64(t * Float64(t / (x ^ 3.0))), fma(2.0, (Float64(l / x) ^ 2.0), fma(2.0, Float64(l * Float64(l / (x ^ 3.0))), Float64(2.0 * fma(t, t, Float64(l * Float64(l / x)))))))))))
	t_7 = sqrt(Float64(1.0 / t_2))
	tmp = 0.0
	if (t <= -2.2e+133)
		tmp = t_5;
	elseif (t <= -2.6e-172)
		tmp = t_6;
	elseif (t <= -1.02e-221)
		tmp = t_5;
	elseif (t <= 2.8e-225)
		tmp = Float64(t_4 / Float64(Float64(t_7 * Float64(t_1 + Float64(2.0 * Float64(t_1 / x)))) + Float64(Float64(t_7 * Float64(Float64(2.0 * Float64(Float64(Float64(t / x) * Float64(t / x)) / l)) + Float64(2.0 * Float64(t_1 / (x ^ 3.0))))) + Float64(l * sqrt(t_2)))));
	elseif (t <= 1.12e-194)
		tmp = Float64(t_4 / 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 <= 4e+104)
		tmp = t_6;
	else
		tmp = Float64(t_4 / Float64(t * t_3));
	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[(t / N[(l / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, Block[{t$95$3 = 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]}, Block[{t$95$4 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 / N[(t$95$3 * (-t)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$4 / 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[(4.0 * N[(t * N[(t / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[Power[N[(l / x), $MachinePrecision], 2.0], $MachinePrecision] + N[(2.0 * N[(l * N[(l / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(t * t + N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[N[(1.0 / t$95$2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -2.2e+133], t$95$5, If[LessEqual[t, -2.6e-172], t$95$6, If[LessEqual[t, -1.02e-221], t$95$5, If[LessEqual[t, 2.8e-225], N[(t$95$4 / N[(N[(t$95$7 * N[(t$95$1 + N[(2.0 * N[(t$95$1 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$7 * N[(N[(2.0 * N[(N[(N[(t / x), $MachinePrecision] * N[(t / x), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(t$95$1 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(l * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.12e-194], N[(t$95$4 / 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, 4e+104], t$95$6, N[(t$95$4 / N[(t * t$95$3), $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 < -2.2e133 or -2.5999999999999998e-172 < t < -1.02e-221

   1. Initial program 58.3

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

   2. Simplified 58.3

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

   3. Taylor expanded in t around -inf 6.9

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

   4. Simplified 6.9

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

   if -2.2e133 < t < -2.5999999999999998e-172 or 1.12000000000000001e-194 < t < 4e104

   1. Initial program 28.1

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

   2. Simplified 28.1

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

   3. Taylor expanded in x around inf 12.3

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

   4. Simplified 12.3

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

   5. Applied egg-rr 6.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{{\left(\mathsf{fma}\left(4, \frac{t}{x} \cdot \frac{t}{x}, \mathsf{fma}\left(4, \frac{t}{\frac{x}{t}}, \mathsf{fma}\left(4, \frac{t}{\frac{{x}^{3}}{t}}, \mathsf{fma}\left(2, \frac{\ell}{x} \cdot \frac{\ell}{x}, \mathsf{fma}\left(2, \frac{\ell}{\frac{{x}^{3}}{\ell}}, 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right)\right)\right)\right)\right)\right)\right)}^{0.5}}} \]

   6. Applied egg-rr 6.8

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

   if -1.02e-221 < t < 2.8e-225

   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. Simplified 62.9

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

   3. Taylor expanded in x around inf 40.1

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

   4. Simplified 40.1

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

   5. Taylor expanded in l around inf 32.8

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

   6. Simplified 32.6

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

   if 2.8e-225 < t < 1.12000000000000001e-194

   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. Simplified 62.9

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

   3. Taylor expanded in x around inf 42.1

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

   4. Simplified 42.1

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

   5. Taylor expanded in x around inf 22.0

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

   6. Simplified 21.9

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

   if 4e104 < t

   1. Initial program 49.4

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

   2. Simplified 49.4

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

   3. Taylor expanded in t around inf 2.4

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

   4. Simplified 2.4

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

4. Final simplification 8.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{2}{x + -1} + 2 \cdot \frac{x}{x + -1}} \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq -2.6 \cdot 10^{-172}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\mathsf{fma}\left(4, {\left(\frac{t}{x}\right)}^{2}, \mathsf{fma}\left(4, t \cdot \frac{t}{x}, \mathsf{fma}\left(4, t \cdot \frac{t}{{x}^{3}}, \mathsf{fma}\left(2, {\left(\frac{\ell}{x}\right)}^{2}, \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{{x}^{3}}, 2 \cdot \mathsf{fma}\left(t, t, \ell \cdot \frac{\ell}{x}\right)\right)\right)\right)\right)\right)}}\\ \mathbf{elif}\;t \leq -1.02 \cdot 10^{-221}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{2}{x + -1} + 2 \cdot \frac{x}{x + -1}} \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-225}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{1}{\frac{2}{x} + \left(\frac{2}{{x}^{3}} + \frac{2}{x \cdot x}\right)}} \cdot \left(\frac{t}{\frac{\ell}{t}} + 2 \cdot \frac{\frac{t}{\frac{\ell}{t}}}{x}\right) + \left(\sqrt{\frac{1}{\frac{2}{x} + \left(\frac{2}{{x}^{3}} + \frac{2}{x \cdot x}\right)}} \cdot \left(2 \cdot \frac{\frac{t}{x} \cdot \frac{t}{x}}{\ell} + 2 \cdot \frac{\frac{t}{\frac{\ell}{t}}}{{x}^{3}}\right) + \ell \cdot \sqrt{\frac{2}{x} + \left(\frac{2}{{x}^{3}} + \frac{2}{x \cdot x}\right)}\right)}\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{-194}:\\ \;\;\;\;\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 4 \cdot 10^{+104}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\mathsf{fma}\left(4, {\left(\frac{t}{x}\right)}^{2}, \mathsf{fma}\left(4, t \cdot \frac{t}{x}, \mathsf{fma}\left(4, t \cdot \frac{t}{{x}^{3}}, \mathsf{fma}\left(2, {\left(\frac{\ell}{x}\right)}^{2}, \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{{x}^{3}}, 2 \cdot \mathsf{fma}\left(t, t, \ell \cdot \frac{\ell}{x}\right)\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 2022166 
(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)))))