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

↓

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ \mathbf{if}\;t \leq -7.548505597060446 \cdot 10^{+27}:\\ \;\;\;\;\frac{t_1}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(\sqrt{2} \cdot \left(-t\right)\right)}\\ \mathbf{elif}\;t \leq -1.25 \cdot 10^{-272}:\\ \;\;\;\;\frac{t_1}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \mathsf{fma}\left(\ell, \ell, t \cdot \left(t \cdot 2\right)\right) \cdot \frac{1}{x}\right)}}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-145}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\mathsf{fma}\left(t, \sqrt{2}, \frac{0.5}{\sqrt{2}} \cdot \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{t \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

(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 (sqrt 2.0))))
   (if (<= t -7.548505597060446e+27)
     (/ t_1 (* (sqrt (/ (+ x 1.0) (+ x -1.0))) (* (sqrt 2.0) (- t))))
     (if (<= t -1.25e-272)
       (/
        t_1
        (sqrt
         (+
          (/ l (/ x l))
          (fma
           2.0
           (+ (/ t (/ x t)) (* t t))
           (* (fma l l (* t (* t 2.0))) (/ 1.0 x))))))
       (if (<= t 9.2e-145)
         (*
          (sqrt 2.0)
          (/
           t
           (fma
            t
            (sqrt 2.0)
            (*
             (/ 0.5 (sqrt 2.0))
             (/ (* 2.0 (fma 2.0 (* t t) (* l l))) (* t x))))))
         (sqrt (/ (+ x -1.0) (+ x 1.0))))))))

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 * sqrt(2.0);
	double tmp;
	if (t <= -7.548505597060446e+27) {
		tmp = t_1 / (sqrt(((x + 1.0) / (x + -1.0))) * (sqrt(2.0) * -t));
	} else if (t <= -1.25e-272) {
		tmp = t_1 / sqrt(((l / (x / l)) + fma(2.0, ((t / (x / t)) + (t * t)), (fma(l, l, (t * (t * 2.0))) * (1.0 / x)))));
	} else if (t <= 9.2e-145) {
		tmp = sqrt(2.0) * (t / fma(t, sqrt(2.0), ((0.5 / sqrt(2.0)) * ((2.0 * fma(2.0, (t * t), (l * l))) / (t * x)))));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

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 * sqrt(2.0))
	tmp = 0.0
	if (t <= -7.548505597060446e+27)
		tmp = Float64(t_1 / Float64(sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))) * Float64(sqrt(2.0) * Float64(-t))));
	elseif (t <= -1.25e-272)
		tmp = Float64(t_1 / sqrt(Float64(Float64(l / Float64(x / l)) + fma(2.0, Float64(Float64(t / Float64(x / t)) + Float64(t * t)), Float64(fma(l, l, Float64(t * Float64(t * 2.0))) * Float64(1.0 / x))))));
	elseif (t <= 9.2e-145)
		tmp = Float64(sqrt(2.0) * Float64(t / fma(t, sqrt(2.0), Float64(Float64(0.5 / sqrt(2.0)) * Float64(Float64(2.0 * fma(2.0, Float64(t * t), Float64(l * l))) / Float64(t * x))))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return tmp
end

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[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -7.548505597060446e+27], N[(t$95$1 / N[(N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.25e-272], N[(t$95$1 / N[Sqrt[N[(N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(t / N[(x / t), $MachinePrecision]), $MachinePrecision] + N[(t * t), $MachinePrecision]), $MachinePrecision] + N[(N[(l * l + N[(t * N[(t * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.2e-145], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[(t * N[Sqrt[2.0], $MachinePrecision] + N[(N[(0.5 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

\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 := t \cdot \sqrt{2}\\
\mathbf{if}\;t \leq -7.548505597060446 \cdot 10^{+27}:\\
\;\;\;\;\frac{t_1}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(\sqrt{2} \cdot \left(-t\right)\right)}\\

\mathbf{elif}\;t \leq -1.25 \cdot 10^{-272}:\\
\;\;\;\;\frac{t_1}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \mathsf{fma}\left(\ell, \ell, t \cdot \left(t \cdot 2\right)\right) \cdot \frac{1}{x}\right)}}\\

\mathbf{elif}\;t \leq 9.2 \cdot 10^{-145}:\\
\;\;\;\;\sqrt{2} \cdot \frac{t}{\mathsf{fma}\left(t, \sqrt{2}, \frac{0.5}{\sqrt{2}} \cdot \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{t \cdot x}\right)}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\


\end{array}

Derivation

Split input into 4 regimes
if t < -7.5485055970604461e27
1. Initial program 43.9
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Taylor expanded in t around -inf 4.4
  \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{-1 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
3. Simplified4.4
  \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(-t \cdot \sqrt{2}\right)}} \]
  Proof
  (*.f64 (sqrt.f64 (/.f64 (+.f64 x 1) (+.f64 x -1))) (neg.f64 (*.f64 t (sqrt.f64 2)))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 x)) (+.f64 x -1))) (neg.f64 (*.f64 t (sqrt.f64 2)))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 (/.f64 (+.f64 1 x) (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))))) (neg.f64 (*.f64 t (sqrt.f64 2)))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 (/.f64 (+.f64 1 x) (Rewrite<= sub-neg_binary64 (-.f64 x 1)))) (neg.f64 (*.f64 t (sqrt.f64 2)))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 (/.f64 (+.f64 1 x) (-.f64 x 1))) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 2) t)))): 0 points increase in error, 0 points decrease in error
  (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (sqrt.f64 (/.f64 (+.f64 1 x) (-.f64 x 1))) (*.f64 (sqrt.f64 2) t)))): 0 points increase in error, 0 points decrease in error
  (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (/.f64 (+.f64 1 x) (-.f64 x 1)))))): 0 points increase in error, 0 points decrease in error
  (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (/.f64 (+.f64 1 x) (-.f64 x 1)))))): 0 points increase in error, 0 points decrease in error
if -7.5485055970604461e27 < t < -1.24999999999999995e-272
1. Initial program 41.3
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Taylor expanded in x around inf 19.1
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}} \]
3. Simplified19.1
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \]
  Proof
  (+.f64 (/.f64 l (/.f64 x l)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
  (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 3 points increase in error, 1 points decrease in error
  (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
  (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 t t) x)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
  (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) x) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
  (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (Rewrite<= unpow2_binary64 (pow.f64 t 2))) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
  (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (fma.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
  (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (fma.f64 2 (pow.f64 t 2) (Rewrite<= unpow2_binary64 (pow.f64 l 2))) x))): 0 points increase in error, 0 points decrease in error
  (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (pow.f64 t 2)) (pow.f64 l 2))) x))): 0 points increase in error, 0 points decrease in error
  (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) x))): 0 points increase in error, 0 points decrease in error
  (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))))): 0 points increase in error, 0 points decrease in error
  (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))))): 0 points increase in error, 0 points decrease in error
  (+.f64 (/.f64 (pow.f64 l 2) x) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x))))): 0 points increase in error, 0 points decrease in error
  (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))): 0 points increase in error, 0 points decrease in error
  (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))): 0 points increase in error, 0 points decrease in error
4. Applied egg-rr19.1
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \color{blue}{\mathsf{fma}\left(\ell, \ell, \left(2 \cdot t\right) \cdot t\right) \cdot \frac{1}{x}}\right)}} \]
if -1.24999999999999995e-272 < t < 9.20000000000000028e-145
1. Initial program 61.3
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Simplified62.7
  \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\frac{x + -1}{\mathsf{fma}\left(t, 2 \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}}} \]
  Proof
  (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (+.f64 x -1) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1)) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 t (*.f64 2 t)) (*.f64 l l))))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 2 t) t)) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 t t))) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t))))) (*.f64 l l))))): 3 points increase in error, 20 points decrease in error
  (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l))))): 1 points increase in error, 10 points decrease in error
3. Taylor expanded in x around inf 31.2
  \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{0.5 \cdot \frac{\left({\ell}^{2} + 2 \cdot {t}^{2}\right) - -1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{\sqrt{2} \cdot \left(t \cdot x\right)} + \sqrt{2} \cdot t}} \]
4. Simplified31.2
  \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\mathsf{fma}\left(t, \sqrt{2}, \frac{0.5}{\sqrt{2}} \cdot \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{t \cdot x}\right)}} \]
  Proof
  (fma.f64 t (sqrt.f64 2) (*.f64 (/.f64 1/2 (sqrt.f64 2)) (/.f64 (*.f64 2 (fma.f64 2 (*.f64 t t) (*.f64 l l))) (*.f64 t x)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 t (sqrt.f64 2) (*.f64 (/.f64 1/2 (sqrt.f64 2)) (/.f64 (*.f64 (Rewrite<= metadata-eval (+.f64 1 1)) (fma.f64 2 (*.f64 t t) (*.f64 l l))) (*.f64 t x)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 t (sqrt.f64 2) (*.f64 (/.f64 1/2 (sqrt.f64 2)) (/.f64 (*.f64 (+.f64 1 1) (fma.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (*.f64 l l))) (*.f64 t x)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 t (sqrt.f64 2) (*.f64 (/.f64 1/2 (sqrt.f64 2)) (/.f64 (*.f64 (+.f64 1 1) (fma.f64 2 (pow.f64 t 2) (Rewrite<= unpow2_binary64 (pow.f64 l 2)))) (*.f64 t x)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 t (sqrt.f64 2) (*.f64 (/.f64 1/2 (sqrt.f64 2)) (/.f64 (*.f64 (+.f64 1 1) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (pow.f64 t 2)) (pow.f64 l 2)))) (*.f64 t x)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 t (sqrt.f64 2) (*.f64 (/.f64 1/2 (sqrt.f64 2)) (/.f64 (*.f64 (+.f64 1 1) (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 t x)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 t (sqrt.f64 2) (*.f64 (/.f64 1/2 (sqrt.f64 2)) (/.f64 (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))))) (*.f64 t x)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 t (sqrt.f64 2) (*.f64 (/.f64 1/2 (sqrt.f64 2)) (/.f64 (+.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 (Rewrite<= metadata-eval (neg.f64 -1)) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 t x)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 t (sqrt.f64 2) (*.f64 (/.f64 1/2 (sqrt.f64 2)) (/.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))))) (*.f64 t x)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 t (sqrt.f64 2) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 1/2 (-.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))))) (*.f64 (sqrt.f64 2) (*.f64 t x))))): 7 points increase in error, 1 points decrease in error
  (fma.f64 t (sqrt.f64 2) (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 (-.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 (sqrt.f64 2) (*.f64 t x)))))): 0 points increase in error, 0 points decrease in error
  (Rewrite<= fma-def_binary64 (+.f64 (*.f64 t (sqrt.f64 2)) (*.f64 1/2 (/.f64 (-.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 (sqrt.f64 2) (*.f64 t x)))))): 0 points increase in error, 0 points decrease in error
  (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 2) t)) (*.f64 1/2 (/.f64 (-.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 (sqrt.f64 2) (*.f64 t x))))): 0 points increase in error, 0 points decrease in error
  (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 1/2 (/.f64 (-.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 (sqrt.f64 2) (*.f64 t x)))) (*.f64 (sqrt.f64 2) t))): 0 points increase in error, 0 points decrease in error
if 9.20000000000000028e-145 < t
1. Initial program 38.5
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Simplified42.3
  \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\frac{x + -1}{\mathsf{fma}\left(t, 2 \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}}} \]
  Proof
  (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (+.f64 x -1) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1)) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 t (*.f64 2 t)) (*.f64 l l))))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 2 t) t)) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 t t))) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t))))) (*.f64 l l))))): 3 points increase in error, 20 points decrease in error
  (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l))))): 1 points increase in error, 10 points decrease in error
3. Taylor expanded in t around inf 9.3
  \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
4. Simplified9.3
  \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{x + 1}{-1 + x}}\right)}} \]
  Proof
  (*.f64 (sqrt.f64 2) (*.f64 t (sqrt.f64 (/.f64 (+.f64 x 1) (+.f64 -1 x))))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 2) (*.f64 t (sqrt.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 x)) (+.f64 -1 x))))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 2) (*.f64 t (sqrt.f64 (/.f64 (+.f64 1 x) (Rewrite<= +-commutative_binary64 (+.f64 x -1)))))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 2) (*.f64 t (sqrt.f64 (/.f64 (+.f64 1 x) (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))))))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 2) (*.f64 t (sqrt.f64 (/.f64 (+.f64 1 x) (Rewrite<= sub-neg_binary64 (-.f64 x 1)))))): 0 points increase in error, 0 points decrease in error
  (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (/.f64 (+.f64 1 x) (-.f64 x 1))))): 1 points increase in error, 2 points decrease in error
5. Taylor expanded in t around 0 9.2
  \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
Recombined 4 regimes into one program.
Final simplification12.8
\[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.548505597060446 \cdot 10^{+27}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(\sqrt{2} \cdot \left(-t\right)\right)}\\ \mathbf{elif}\;t \leq -1.25 \cdot 10^{-272}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \mathsf{fma}\left(\ell, \ell, t \cdot \left(t \cdot 2\right)\right) \cdot \frac{1}{x}\right)}}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-145}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\mathsf{fma}\left(t, \sqrt{2}, \frac{0.5}{\sqrt{2}} \cdot \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{t \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 1
Error	14.3
Cost	7048

Alternative 2
Error	15.2
Cost	452

Alternative 3
Error	15.6
Cost	196

Error

Derivation

`if t < -7.5485055970604461e27`

`if -7.5485055970604461e27 < t < -1.24999999999999995e-272`

`if -1.24999999999999995e-272 < t < 9.20000000000000028e-145`

`if 9.20000000000000028e-145 < t`

Alternatives

Error

Reproduce