Toniolo and Linder, Equation (7)

Average Error: 43.3 → 8.9

Time: 13.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := {\left(\frac{\ell}{x}\right)}^{2}\\ t_2 := \sqrt{2} \cdot x\\ t_3 := 2 + \left(\frac{4}{x} + \frac{4}{x \cdot x}\right)\\ t_4 := t \cdot \sqrt{2}\\ t_5 := t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}\\ t_6 := {\left(\frac{t}{x}\right)}^{2}\\ \mathbf{if}\;t \leq -0.008644579131846854:\\ \;\;\;\;\frac{t_4}{-t_5}\\ \mathbf{elif}\;t \leq -1.3699428530136365 \cdot 10^{-209}:\\ \;\;\;\;\frac{t_4}{{\left({\left(\mathsf{fma}\left(4, t_6, \mathsf{fma}\left(4, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t_1, 2 \cdot \mathsf{fma}\left(t, t, \frac{\ell}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{\ell}{\sqrt[3]{x}}\right)\right)\right)\right)\right)}^{0.25}\right)}^{2}}\\ \mathbf{elif}\;t \leq -5.760693106007019 \cdot 10^{-274}:\\ \;\;\;\;\frac{t_4}{-\left(\sqrt{\frac{1}{t_3}} \cdot \left(\frac{\ell \cdot \ell}{t \cdot x} + \frac{\ell \cdot \ell}{x \cdot \left(t \cdot x\right)}\right) + t \cdot \sqrt{t_3}\right)}\\ \mathbf{elif}\;t \leq 4.04367196613585 \cdot 10^{-310}:\\ \;\;\;\;\frac{t_4}{{\left({\left(\mathsf{fma}\left(4, t_6, \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{x} + \frac{2}{x \cdot x}\right)\right)\right)}^{0.25}\right)}^{2}}\\ \mathbf{elif}\;t \leq 1.6898595657411804 \cdot 10^{-186}:\\ \;\;\;\;\frac{t_4}{\mathsf{fma}\left(t, \sqrt{2}, \mathsf{fma}\left(2, \frac{t}{t_2}, \frac{\ell \cdot \ell}{t \cdot t_2}\right)\right)}\\ \mathbf{elif}\;t \leq 1.7217277470457817 \cdot 10^{+63}:\\ \;\;\;\;\frac{t_4}{{\left({\left(\mathsf{fma}\left(4, t_6, \mathsf{fma}\left(4, t \cdot \frac{t}{x}, \mathsf{fma}\left(2, t_1, 2 \cdot \mathsf{fma}\left(t, t, \ell \cdot \frac{\ell}{x}\right)\right)\right)\right)\right)}^{0.25}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_4}{t_5}\\ \end{array} \]

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 (pow (/ l x) 2.0))
        (t_2 (* (sqrt 2.0) x))
        (t_3 (+ 2.0 (+ (/ 4.0 x) (/ 4.0 (* x x)))))
        (t_4 (* t (sqrt 2.0)))
        (t_5 (* t (sqrt (+ (/ 2.0 (- x 1.0)) (* 2.0 (/ x (- x 1.0)))))))
        (t_6 (pow (/ t x) 2.0)))
   (if (<= t -0.008644579131846854)
     (/ t_4 (- t_5))
     (if (<= t -1.3699428530136365e-209)
       (/
        t_4
        (pow
         (pow
          (fma
           4.0
           t_6
           (fma
            4.0
            (/ (* t t) x)
            (fma
             2.0
             t_1
             (* 2.0 (fma t t (* (/ l (pow (cbrt x) 2.0)) (/ l (cbrt x))))))))
          0.25)
         2.0))
       (if (<= t -5.760693106007019e-274)
         (/
          t_4
          (-
           (+
            (*
             (sqrt (/ 1.0 t_3))
             (+ (/ (* l l) (* t x)) (/ (* l l) (* x (* t x)))))
            (* t (sqrt t_3)))))
         (if (<= t 4.04367196613585e-310)
           (/
            t_4
            (pow
             (pow (fma 4.0 t_6 (* (* l l) (+ (/ 2.0 x) (/ 2.0 (* x x))))) 0.25)
             2.0))
           (if (<= t 1.6898595657411804e-186)
             (/
              t_4
              (fma t (sqrt 2.0) (fma 2.0 (/ t t_2) (/ (* l l) (* t t_2)))))
             (if (<= t 1.7217277470457817e+63)
               (/
                t_4
                (pow
                 (pow
                  (fma
                   4.0
                   t_6
                   (fma
                    4.0
                    (* t (/ t x))
                    (fma 2.0 t_1 (* 2.0 (fma t t (* l (/ l x)))))))
                  0.25)
                 2.0))
               (/ t_4 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 = pow((l / x), 2.0);
	double t_2 = sqrt(2.0) * x;
	double t_3 = 2.0 + ((4.0 / x) + (4.0 / (x * x)));
	double t_4 = t * sqrt(2.0);
	double t_5 = t * sqrt(((2.0 / (x - 1.0)) + (2.0 * (x / (x - 1.0)))));
	double t_6 = pow((t / x), 2.0);
	double tmp;
	if (t <= -0.008644579131846854) {
		tmp = t_4 / -t_5;
	} else if (t <= -1.3699428530136365e-209) {
		tmp = t_4 / pow(pow(fma(4.0, t_6, fma(4.0, ((t * t) / x), fma(2.0, t_1, (2.0 * fma(t, t, ((l / pow(cbrt(x), 2.0)) * (l / cbrt(x)))))))), 0.25), 2.0);
	} else if (t <= -5.760693106007019e-274) {
		tmp = t_4 / -((sqrt((1.0 / t_3)) * (((l * l) / (t * x)) + ((l * l) / (x * (t * x))))) + (t * sqrt(t_3)));
	} else if (t <= 4.04367196613585e-310) {
		tmp = t_4 / pow(pow(fma(4.0, t_6, ((l * l) * ((2.0 / x) + (2.0 / (x * x))))), 0.25), 2.0);
	} else if (t <= 1.6898595657411804e-186) {
		tmp = t_4 / fma(t, sqrt(2.0), fma(2.0, (t / t_2), ((l * l) / (t * t_2))));
	} else if (t <= 1.7217277470457817e+63) {
		tmp = t_4 / pow(pow(fma(4.0, t_6, fma(4.0, (t * (t / x)), fma(2.0, t_1, (2.0 * fma(t, t, (l * (l / x))))))), 0.25), 2.0);
	} else {
		tmp = t_4 / 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(l / x) ^ 2.0
	t_2 = Float64(sqrt(2.0) * x)
	t_3 = Float64(2.0 + Float64(Float64(4.0 / x) + Float64(4.0 / Float64(x * x))))
	t_4 = Float64(t * sqrt(2.0))
	t_5 = Float64(t * sqrt(Float64(Float64(2.0 / Float64(x - 1.0)) + Float64(2.0 * Float64(x / Float64(x - 1.0))))))
	t_6 = Float64(t / x) ^ 2.0
	tmp = 0.0
	if (t <= -0.008644579131846854)
		tmp = Float64(t_4 / Float64(-t_5));
	elseif (t <= -1.3699428530136365e-209)
		tmp = Float64(t_4 / ((fma(4.0, t_6, fma(4.0, Float64(Float64(t * t) / x), fma(2.0, t_1, Float64(2.0 * fma(t, t, Float64(Float64(l / (cbrt(x) ^ 2.0)) * Float64(l / cbrt(x)))))))) ^ 0.25) ^ 2.0));
	elseif (t <= -5.760693106007019e-274)
		tmp = Float64(t_4 / Float64(-Float64(Float64(sqrt(Float64(1.0 / t_3)) * Float64(Float64(Float64(l * l) / Float64(t * x)) + Float64(Float64(l * l) / Float64(x * Float64(t * x))))) + Float64(t * sqrt(t_3)))));
	elseif (t <= 4.04367196613585e-310)
		tmp = Float64(t_4 / ((fma(4.0, t_6, Float64(Float64(l * l) * Float64(Float64(2.0 / x) + Float64(2.0 / Float64(x * x))))) ^ 0.25) ^ 2.0));
	elseif (t <= 1.6898595657411804e-186)
		tmp = Float64(t_4 / fma(t, sqrt(2.0), fma(2.0, Float64(t / t_2), Float64(Float64(l * l) / Float64(t * t_2)))));
	elseif (t <= 1.7217277470457817e+63)
		tmp = Float64(t_4 / ((fma(4.0, t_6, fma(4.0, Float64(t * Float64(t / x)), fma(2.0, t_1, Float64(2.0 * fma(t, t, Float64(l * Float64(l / x))))))) ^ 0.25) ^ 2.0));
	else
		tmp = Float64(t_4 / t_5);
	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[Power[N[(l / x), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + N[(N[(4.0 / x), $MachinePrecision] + N[(4.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = 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$6 = N[Power[N[(t / x), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, -0.008644579131846854], N[(t$95$4 / (-t$95$5)), $MachinePrecision], If[LessEqual[t, -1.3699428530136365e-209], N[(t$95$4 / N[Power[N[Power[N[(4.0 * t$95$6 + N[(4.0 * N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * t$95$1 + N[(2.0 * N[(t * t + N[(N[(l / N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -5.760693106007019e-274], N[(t$95$4 / (-N[(N[(N[Sqrt[N[(1.0 / t$95$3), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] / N[(t * x), $MachinePrecision]), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / N[(x * N[(t * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t * N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[t, 4.04367196613585e-310], N[(t$95$4 / N[Power[N[Power[N[(4.0 * t$95$6 + N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.6898595657411804e-186], N[(t$95$4 / N[(t * N[Sqrt[2.0], $MachinePrecision] + N[(2.0 * N[(t / t$95$2), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / N[(t * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.7217277470457817e+63], N[(t$95$4 / N[Power[N[Power[N[(4.0 * t$95$6 + N[(4.0 * N[(t * N[(t / x), $MachinePrecision]), $MachinePrecision] + N[(2.0 * t$95$1 + N[(2.0 * N[(t * t + N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(t$95$4 / t$95$5), $MachinePrecision]]]]]]]]]]]]]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

1. Split input into 7 regimes

2. if t < -0.00864457913184685407

   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

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

   3. Taylor expanded in t around -inf 5.2

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

   4. Simplified 5.2

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

   if -0.00864457913184685407 < t < -1.3699428530136365e-209

   1. Initial program 38.0

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

   2. Simplified 38.0

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

   3. Taylor expanded in x around inf 16.4

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

   4. Simplified 16.4

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

   5. Applied egg-rr 15.6

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

   6. Applied egg-rr 11.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\left({\left(\mathsf{fma}\left(4, {\left(\frac{t}{x}\right)}^{2}, \mathsf{fma}\left(4, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, {\left(\frac{\ell}{x}\right)}^{2}, 2 \cdot \mathsf{fma}\left(t, t, \color{blue}{\frac{\ell}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{\ell}{\sqrt[3]{x}}}\right)\right)\right)\right)\right)}^{0.25}\right)}^{2}} \]

   if -1.3699428530136365e-209 < t < -5.7606931060070187e-274

   1. Initial program 62.1

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

   2. Simplified 62.1

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

   3. Taylor expanded in x around inf 33.8

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

   4. Simplified 33.8

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

   5. Taylor expanded in t around -inf 29.5

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

   6. Simplified 26.5

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

   if -5.7606931060070187e-274 < t < 4.043671966135855e-310

   1. Initial program 63.1

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

   2. Simplified 63.1

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

   3. Taylor expanded in x around inf 38.1

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

   4. Simplified 38.1

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

   5. Applied egg-rr 29.4

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

   6. Taylor expanded in l around inf 29.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\left({\left(\mathsf{fma}\left(4, {\left(\frac{t}{x}\right)}^{2}, \color{blue}{{\ell}^{2} \cdot \left(2 \cdot \frac{1}{x} + 2 \cdot \frac{1}{{x}^{2}}\right)}\right)\right)}^{0.25}\right)}^{2}} \]

   7. Simplified 29.4

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

   if 4.043671966135855e-310 < t < 1.68985956574118043e-186

   1. Initial program 63.1

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

   2. Simplified 63.1

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

   3. Taylor expanded in x around inf 24.6

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

   4. Simplified 24.6

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

   if 1.68985956574118043e-186 < t < 1.7217277470457817e63

   1. Initial program 32.8

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

   2. Simplified 32.8

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

   3. Taylor expanded in x around inf 13.0

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

   4. Simplified 13.0

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

   5. Applied egg-rr 12.5

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

   6. Applied egg-rr 7.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\left({\left(\mathsf{fma}\left(4, {\left(\frac{t}{x}\right)}^{2}, \mathsf{fma}\left(4, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, {\left(\frac{\ell}{x}\right)}^{2}, 2 \cdot \mathsf{fma}\left(t, t, \color{blue}{{\left(\sqrt[3]{\ell \cdot \frac{\ell}{x}}\right)}^{3}}\right)\right)\right)\right)\right)}^{0.25}\right)}^{2}} \]

   7. Applied egg-rr 7.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\left({\left(\mathsf{fma}\left(4, {\left(\frac{t}{x}\right)}^{2}, \color{blue}{{\left(\mathsf{fma}\left(4, t \cdot \frac{t}{x}, \mathsf{fma}\left(2, {\left(\frac{\ell}{x}\right)}^{2}, 2 \cdot \mathsf{fma}\left(t, t, \ell \cdot \frac{\ell}{x}\right)\right)\right)\right)}^{1}}\right)\right)}^{0.25}\right)}^{2}} \]

   if 1.7217277470457817e63 < t

   1. Initial program 46.1

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

   2. Simplified 46.1

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

   3. Taylor expanded in t around inf 3.2

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

   4. Simplified 3.2

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

4. Final simplification 8.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.008644579131846854:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \mathbf{elif}\;t \leq -1.3699428530136365 \cdot 10^{-209}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{{\left({\left(\mathsf{fma}\left(4, {\left(\frac{t}{x}\right)}^{2}, \mathsf{fma}\left(4, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, {\left(\frac{\ell}{x}\right)}^{2}, 2 \cdot \mathsf{fma}\left(t, t, \frac{\ell}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{\ell}{\sqrt[3]{x}}\right)\right)\right)\right)\right)}^{0.25}\right)}^{2}}\\ \mathbf{elif}\;t \leq -5.760693106007019 \cdot 10^{-274}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-\left(\sqrt{\frac{1}{2 + \left(\frac{4}{x} + \frac{4}{x \cdot x}\right)}} \cdot \left(\frac{\ell \cdot \ell}{t \cdot x} + \frac{\ell \cdot \ell}{x \cdot \left(t \cdot x\right)}\right) + t \cdot \sqrt{2 + \left(\frac{4}{x} + \frac{4}{x \cdot x}\right)}\right)}\\ \mathbf{elif}\;t \leq 4.04367196613585 \cdot 10^{-310}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{{\left({\left(\mathsf{fma}\left(4, {\left(\frac{t}{x}\right)}^{2}, \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{x} + \frac{2}{x \cdot x}\right)\right)\right)}^{0.25}\right)}^{2}}\\ \mathbf{elif}\;t \leq 1.6898595657411804 \cdot 10^{-186}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, \frac{\ell \cdot \ell}{t \cdot \left(\sqrt{2} \cdot x\right)}\right)\right)}\\ \mathbf{elif}\;t \leq 1.7217277470457817 \cdot 10^{+63}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{{\left({\left(\mathsf{fma}\left(4, {\left(\frac{t}{x}\right)}^{2}, \mathsf{fma}\left(4, t \cdot \frac{t}{x}, \mathsf{fma}\left(2, {\left(\frac{\ell}{x}\right)}^{2}, 2 \cdot \mathsf{fma}\left(t, t, \ell \cdot \frac{\ell}{x}\right)\right)\right)\right)\right)}^{0.25}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022146 
(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)))))