Toniolo and Linder, Equation (7)

Average Error: 43.6 → 9.5

Time: 27.3s

Precision: binary64

Cost: 15056

Math

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

↓

\[\begin{array}{l} t_1 := \frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right)\right)}}\\ \mathbf{if}\;t \leq -6.8 \cdot 10^{+44}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{-117}:\\ \;\;\;\;\frac{t}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\\ \mathbf{elif}\;t \leq 0.38:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+48}:\\ \;\;\;\;\frac{t}{\mathsf{hypot}\left(t, \ell \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

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 (sqrt 2.0))
          (sqrt
           (+
            (/ l (/ x l))
            (+ (/ (* l l) x) (* 2.0 (+ (* t t) (/ t (/ x t))))))))))
   (if (<= t -6.8e+44)
     -1.0
     (if (<= t -1.8e-159)
       t_1
       (if (<= t 8.2e-117)
         (/ t (hypot t (/ l (sqrt x))))
         (if (<= t 0.38)
           t_1
           (if (<= t 2.5e+48) (/ t (hypot t (* l (sqrt (/ 1.0 x))))) 1.0)))))))

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 * sqrt(2.0)) / sqrt(((l / (x / l)) + (((l * l) / x) + (2.0 * ((t * t) + (t / (x / t)))))));
	double tmp;
	if (t <= -6.8e+44) {
		tmp = -1.0;
	} else if (t <= -1.8e-159) {
		tmp = t_1;
	} else if (t <= 8.2e-117) {
		tmp = t / hypot(t, (l / sqrt(x)));
	} else if (t <= 0.38) {
		tmp = t_1;
	} else if (t <= 2.5e+48) {
		tmp = t / hypot(t, (l * sqrt((1.0 / x))));
	} else {
		tmp = 1.0;
	}
	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 = (t * Math.sqrt(2.0)) / Math.sqrt(((l / (x / l)) + (((l * l) / x) + (2.0 * ((t * t) + (t / (x / t)))))));
	double tmp;
	if (t <= -6.8e+44) {
		tmp = -1.0;
	} else if (t <= -1.8e-159) {
		tmp = t_1;
	} else if (t <= 8.2e-117) {
		tmp = t / Math.hypot(t, (l / Math.sqrt(x)));
	} else if (t <= 0.38) {
		tmp = t_1;
	} else if (t <= 2.5e+48) {
		tmp = t / Math.hypot(t, (l * Math.sqrt((1.0 / x))));
	} else {
		tmp = 1.0;
	}
	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 = (t * math.sqrt(2.0)) / math.sqrt(((l / (x / l)) + (((l * l) / x) + (2.0 * ((t * t) + (t / (x / t)))))))
	tmp = 0
	if t <= -6.8e+44:
		tmp = -1.0
	elif t <= -1.8e-159:
		tmp = t_1
	elif t <= 8.2e-117:
		tmp = t / math.hypot(t, (l / math.sqrt(x)))
	elif t <= 0.38:
		tmp = t_1
	elif t <= 2.5e+48:
		tmp = t / math.hypot(t, (l * math.sqrt((1.0 / x))))
	else:
		tmp = 1.0
	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 * sqrt(2.0)) / sqrt(Float64(Float64(l / Float64(x / l)) + Float64(Float64(Float64(l * l) / x) + Float64(2.0 * Float64(Float64(t * t) + Float64(t / Float64(x / t))))))))
	tmp = 0.0
	if (t <= -6.8e+44)
		tmp = -1.0;
	elseif (t <= -1.8e-159)
		tmp = t_1;
	elseif (t <= 8.2e-117)
		tmp = Float64(t / hypot(t, Float64(l / sqrt(x))));
	elseif (t <= 0.38)
		tmp = t_1;
	elseif (t <= 2.5e+48)
		tmp = Float64(t / hypot(t, Float64(l * sqrt(Float64(1.0 / x)))));
	else
		tmp = 1.0;
	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 = (t * sqrt(2.0)) / sqrt(((l / (x / l)) + (((l * l) / x) + (2.0 * ((t * t) + (t / (x / t)))))));
	tmp = 0.0;
	if (t <= -6.8e+44)
		tmp = -1.0;
	elseif (t <= -1.8e-159)
		tmp = t_1;
	elseif (t <= 8.2e-117)
		tmp = t / hypot(t, (l / sqrt(x)));
	elseif (t <= 0.38)
		tmp = t_1;
	elseif (t <= 2.5e+48)
		tmp = t / hypot(t, (l * sqrt((1.0 / x))));
	else
		tmp = 1.0;
	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[(N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(t / N[(x / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -6.8e+44], -1.0, If[LessEqual[t, -1.8e-159], t$95$1, If[LessEqual[t, 8.2e-117], N[(t / N[Sqrt[t ^ 2 + N[(l / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.38], t$95$1, If[LessEqual[t, 2.5e+48], N[(t / N[Sqrt[t ^ 2 + N[(l * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], 1.0]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -6.8e44

   1. Initial program 44.0

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

   2. Simplified 44.0

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

      [Start] 44.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 44.0	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 44.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 44.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Taylor expanded in t around -inf 3.8

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

   4. Simplified 3.8

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

      [Start] 3.8	\[ \frac{\sqrt{2}}{-1 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)} \cdot t \]
      associate-*r* [=>] 3.8	\[ \frac{\sqrt{2}}{\color{blue}{\left(-1 \cdot \left(\sqrt{2} \cdot t\right)\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \cdot t \]
      neg-mul-1 [<=] 3.8	\[ \frac{\sqrt{2}}{\color{blue}{\left(-\sqrt{2} \cdot t\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \cdot t \]
      *-commutative [<=] 3.8	\[ \frac{\sqrt{2}}{\left(-\color{blue}{t \cdot \sqrt{2}}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}} \cdot t \]
      distribute-rgt-neg-in [=>] 3.8	\[ \frac{\sqrt{2}}{\color{blue}{\left(t \cdot \left(-\sqrt{2}\right)\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \cdot t \]
      sub-neg [=>] 3.8	\[ \frac{\sqrt{2}}{\left(t \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{1 + x}{\color{blue}{x + \left(-1\right)}}}} \cdot t \]
      metadata-eval [=>] 3.8	\[ \frac{\sqrt{2}}{\left(t \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{1 + x}{x + \color{blue}{-1}}}} \cdot t \]
      +-commutative [=>] 3.8	\[ \frac{\sqrt{2}}{\left(t \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x + -1}}} \cdot t \]
      +-commutative [=>] 3.8	\[ \frac{\sqrt{2}}{\left(t \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \cdot t \]

   5. Taylor expanded in x around inf 4.6

      \[\leadsto \color{blue}{-1} \]

   if -6.8e44 < t < -1.80000000000000011e-159 or 8.20000000000000063e-117 < t < 0.38

   1. Initial program 30.5

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

   2. Taylor expanded in x around inf 11.0

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

   3. Simplified 11.0

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

      [Start] 11.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      associate--l+ [=>] 11.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]
      unpow2 [=>] 11.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-/l* [=>] 11.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}}} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      distribute-lft-out [=>] 11.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      +-commutative [=>] 11.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \color{blue}{\left({t}^{2} + \frac{{t}^{2}}{x}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 11.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(\color{blue}{t \cdot t} + \frac{{t}^{2}}{x}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 11.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{\color{blue}{t \cdot t}}{x}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-/l* [=>] 11.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \color{blue}{\frac{t}{\frac{x}{t}}}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-*r/ [=>] 11.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \]
      mul-1-neg [=>] 11.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \]
      +-commutative [=>] 11.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + {\ell}^{2}\right)}}{x}\right)}} \]
      unpow2 [=>] 11.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\left(2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}\right)}{x}\right)}} \]
      fma-udef [<=] 11.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)}} \]
      unpow2 [=>] 11.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)}} \]

   4. Taylor expanded in t around 0 11.3

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

   5. Simplified 11.3

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

      [Start] 11.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - -1 \cdot \frac{{\ell}^{2}}{x}\right)}} \]
      mul-1-neg [=>] 11.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \color{blue}{\left(-\frac{{\ell}^{2}}{x}\right)}\right)}} \]
      distribute-frac-neg [<=] 11.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \color{blue}{\frac{-{\ell}^{2}}{x}}\right)}} \]
      unpow2 [=>] 11.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\color{blue}{\ell \cdot \ell}}{x}\right)}} \]

   if -1.80000000000000011e-159 < t < 8.20000000000000063e-117

   1. Initial program 59.7

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

   2. Simplified 61.7

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

      [Start] 59.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 59.7	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      associate-*l/ [=>] 58.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\left(x + 1\right) \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x - 1}} - \ell \cdot \ell}} \]
      associate-*r/ [<=] 62.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}} - \ell \cdot \ell}} \]
      *-lft-identity [<=] 62.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \left(x + 1\right)\right)} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} - \ell \cdot \ell}} \]
      associate-*r* [<=] 62.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{1 \cdot \left(\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 62.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{1 \cdot \color{blue}{\left(\frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} \cdot \left(x + 1\right)\right)} - \ell \cdot \ell}} \]
      associate-*r* [=>] 62.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right) \cdot \left(x + 1\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 62.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      fma-neg [=>] 61.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(x + 1, 1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}, -\ell \cdot \ell\right)}}} \]

   3. Taylor expanded in x around -inf 31.2

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

   4. Simplified 31.2

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

      [Start] 31.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + 2 \cdot {t}^{2}}} \]
      distribute-lft-out [=>] 31.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{2 \cdot \left(\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + {t}^{2}\right)}}} \]
      +-commutative [=>] 31.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 31.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x} + {t}^{2}\right)}} \]
      fma-udef [<=] 31.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 31.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 31.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} + \color{blue}{t \cdot t}\right)}} \]

   5. Taylor expanded in t around 0 31.2

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

   6. Simplified 31.2

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

      [Start] 31.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{{\ell}^{2}}{x} + t \cdot t\right)}} \]
      unpow2 [=>] 31.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{x} + t \cdot t\right)}} \]

   7. Applied egg-rr 38.6

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{t \cdot \sqrt{2}}{\sqrt{2}}}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\right)} - 1} \]

   8. Simplified 15.5

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

      [Start] 38.6	\[ e^{\mathsf{log1p}\left(\frac{\frac{t \cdot \sqrt{2}}{\sqrt{2}}}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\right)} - 1 \]
      expm1-def [=>] 15.8	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{t \cdot \sqrt{2}}{\sqrt{2}}}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\right)\right)} \]
      expm1-log1p [=>] 15.6	\[ \color{blue}{\frac{\frac{t \cdot \sqrt{2}}{\sqrt{2}}}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}} \]
      associate-/l* [=>] 15.5	\[ \frac{\color{blue}{\frac{t}{\frac{\sqrt{2}}{\sqrt{2}}}}}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)} \]
      associate-/l/ [=>] 15.5	\[ \color{blue}{\frac{t}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right) \cdot \frac{\sqrt{2}}{\sqrt{2}}}} \]
      *-inverses [=>] 15.5	\[ \frac{t}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right) \cdot \color{blue}{1}} \]
      *-rgt-identity [=>] 15.5	\[ \frac{t}{\color{blue}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}} \]

   if 0.38 < t < 2.49999999999999987e48

   1. Initial program 24.6

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

   2. Simplified 24.1

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

      [Start] 24.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 24.6	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      associate-*l/ [=>] 31.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\left(x + 1\right) \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x - 1}} - \ell \cdot \ell}} \]
      associate-*r/ [<=] 24.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}} - \ell \cdot \ell}} \]
      *-lft-identity [<=] 24.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \left(x + 1\right)\right)} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} - \ell \cdot \ell}} \]
      associate-*r* [<=] 24.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{1 \cdot \left(\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 24.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{1 \cdot \color{blue}{\left(\frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} \cdot \left(x + 1\right)\right)} - \ell \cdot \ell}} \]
      associate-*r* [=>] 24.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right) \cdot \left(x + 1\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 24.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      fma-neg [=>] 24.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(x + 1, 1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}, -\ell \cdot \ell\right)}}} \]

   3. Taylor expanded in x around -inf 11.5

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

   4. Simplified 11.5

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

      [Start] 11.5	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + 2 \cdot {t}^{2}}} \]
      distribute-lft-out [=>] 11.5	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{2 \cdot \left(\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + {t}^{2}\right)}}} \]
      +-commutative [=>] 11.5	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 11.5	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x} + {t}^{2}\right)}} \]
      fma-udef [<=] 11.5	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 11.5	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 11.5	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} + \color{blue}{t \cdot t}\right)}} \]

   5. Applied egg-rr 34.7

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

   6. Simplified 34.7

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

      [Start] 34.7	\[ \frac{\frac{t}{\frac{\sqrt{2}}{\sqrt{2}}}}{\mathsf{hypot}\left(t, \sqrt{\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}\right)} \]
      associate-/l/ [=>] 34.7	\[ \color{blue}{\frac{t}{\mathsf{hypot}\left(t, \sqrt{\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}\right) \cdot \frac{\sqrt{2}}{\sqrt{2}}}} \]
      *-inverses [=>] 34.7	\[ \frac{t}{\mathsf{hypot}\left(t, \sqrt{\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}\right) \cdot \color{blue}{1}} \]

   7. Taylor expanded in t around 0 24.7

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

   if 2.49999999999999987e48 < t

   1. Initial program 44.8

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

   2. Simplified 44.8

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

      [Start] 44.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 44.8	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      associate-*l/ [=>] 56.9	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\left(x + 1\right) \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x - 1}} - \ell \cdot \ell}} \]
      associate-*r/ [<=] 44.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}} - \ell \cdot \ell}} \]
      *-lft-identity [<=] 44.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \left(x + 1\right)\right)} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} - \ell \cdot \ell}} \]
      associate-*r* [<=] 44.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{1 \cdot \left(\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 44.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{1 \cdot \color{blue}{\left(\frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} \cdot \left(x + 1\right)\right)} - \ell \cdot \ell}} \]
      associate-*r* [=>] 44.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right) \cdot \left(x + 1\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 44.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      fma-neg [=>] 44.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(x + 1, 1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}, -\ell \cdot \ell\right)}}} \]

   3. Taylor expanded in x around inf 6.2

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]

   4. Applied egg-rr 5.3

      \[\leadsto \color{blue}{1} \]
3. Recombined 5 regimes into one program.

4. Final simplification 9.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.8 \cdot 10^{+44}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-159}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right)\right)}}\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{-117}:\\ \;\;\;\;\frac{t}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\\ \mathbf{elif}\;t \leq 0.38:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right)\right)}}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+48}:\\ \;\;\;\;\frac{t}{\mathsf{hypot}\left(t, \ell \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternatives

Alternative 1
Error	8.6
Cost	21448

\[\begin{array}{l} t_1 := \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\\ \mathbf{if}\;t \leq -6 \cdot 10^{+65}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq -1.75 \cdot 10^{-108}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + t_1\right)}}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-69}:\\ \;\;\;\;\frac{t}{\mathsf{hypot}\left(t, \sqrt{t_1}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2
Error	8.7
Cost	20680

\[\begin{array}{l} t_1 := \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\\ \mathbf{if}\;t \leq -1.05 \cdot 10^{+45}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq -1.26 \cdot 10^{-108}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(t \cdot t + t_1\right)}}\\ \mathbf{elif}\;t \leq 8.4 \cdot 10^{-69}:\\ \;\;\;\;\frac{t}{\mathsf{hypot}\left(t, \sqrt{t_1}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3
Error	8.7
Cost	20364

\[\begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{+43}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-108}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right)\right)}}\\ \mathbf{elif}\;t \leq 10^{-68}:\\ \;\;\;\;\frac{t}{\mathsf{hypot}\left(t, \sqrt{\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Error	9.5
Cost	14288

\[\begin{array}{l} t_1 := \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\ell \cdot \ell}{x} + t \cdot t\right)}}\\ \mathbf{if}\;t \leq -1.35 \cdot 10^{+45}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-117}:\\ \;\;\;\;\frac{t}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\\ \mathbf{elif}\;t \leq 7.2:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+48}:\\ \;\;\;\;\frac{t}{\mathsf{hypot}\left(t, \ell \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	13.6
Cost	13712

\[\begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+296}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -4 \cdot 10^{+176}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -3 \cdot 10^{+130}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -2.1191957900803896 \cdot 10^{+20}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\\ \end{array} \]

Alternative 6
Error	14.9
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-231}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-244}:\\ \;\;\;\;\frac{t}{\frac{1}{\frac{\sqrt{x}}{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	15.1
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{-229}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-243}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Error	14.9
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -1.95 \cdot 10^{-233}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-242}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Error	15.5
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	15.8
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Error	39.4
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023045 
(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)))))