Toniolo and Linder, Equation (7)

Average Error: 43.3 → 6.6

Time: 28.6s

Precision: binary64

Cost: 39628

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \frac{t_1}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{if}\;t \leq -1.25 \cdot 10^{+82}:\\ \;\;\;\;-\sqrt{\frac{1 - x}{-1 - x}}\\ \mathbf{elif}\;t \leq -1.15 \cdot 10^{-149}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-170}:\\ \;\;\;\;\frac{t_1}{\sqrt{2} \cdot \mathsf{hypot}\left(\mathsf{hypot}\left(t, \frac{t}{\sqrt{x}}\right), \frac{\ell}{\sqrt{x}}\right)}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+70}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (/
  (* (sqrt 2.0) t)
  (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))

↓

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* t (sqrt 2.0)))
        (t_2
         (/ t_1 (sqrt (* 2.0 (+ (* l (/ l x)) (+ (* t t) (/ (* t t) x))))))))
   (if (<= t -1.25e+82)
     (- (sqrt (/ (- 1.0 x) (- -1.0 x))))
     (if (<= t -1.15e-149)
       t_2
       (if (<= t 1.25e-170)
         (/ t_1 (* (sqrt 2.0) (hypot (hypot t (/ t (sqrt x))) (/ l (sqrt x)))))
         (if (<= t 3.4e+70) t_2 (+ 1.0 (+ (/ 0.5 (* x x)) (/ -1.0 x)))))))))

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);
	double t_2 = t_1 / sqrt((2.0 * ((l * (l / x)) + ((t * t) + ((t * t) / x)))));
	double tmp;
	if (t <= -1.25e+82) {
		tmp = -sqrt(((1.0 - x) / (-1.0 - x)));
	} else if (t <= -1.15e-149) {
		tmp = t_2;
	} else if (t <= 1.25e-170) {
		tmp = t_1 / (sqrt(2.0) * hypot(hypot(t, (t / sqrt(x))), (l / sqrt(x))));
	} else if (t <= 3.4e+70) {
		tmp = t_2;
	} else {
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x));
	}
	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);
	double t_2 = t_1 / Math.sqrt((2.0 * ((l * (l / x)) + ((t * t) + ((t * t) / x)))));
	double tmp;
	if (t <= -1.25e+82) {
		tmp = -Math.sqrt(((1.0 - x) / (-1.0 - x)));
	} else if (t <= -1.15e-149) {
		tmp = t_2;
	} else if (t <= 1.25e-170) {
		tmp = t_1 / (Math.sqrt(2.0) * Math.hypot(Math.hypot(t, (t / Math.sqrt(x))), (l / Math.sqrt(x))));
	} else if (t <= 3.4e+70) {
		tmp = t_2;
	} else {
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x));
	}
	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)
	t_2 = t_1 / math.sqrt((2.0 * ((l * (l / x)) + ((t * t) + ((t * t) / x)))))
	tmp = 0
	if t <= -1.25e+82:
		tmp = -math.sqrt(((1.0 - x) / (-1.0 - x)))
	elif t <= -1.15e-149:
		tmp = t_2
	elif t <= 1.25e-170:
		tmp = t_1 / (math.sqrt(2.0) * math.hypot(math.hypot(t, (t / math.sqrt(x))), (l / math.sqrt(x))))
	elif t <= 3.4e+70:
		tmp = t_2
	else:
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x))
	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 * sqrt(2.0))
	t_2 = Float64(t_1 / sqrt(Float64(2.0 * Float64(Float64(l * Float64(l / x)) + Float64(Float64(t * t) + Float64(Float64(t * t) / x))))))
	tmp = 0.0
	if (t <= -1.25e+82)
		tmp = Float64(-sqrt(Float64(Float64(1.0 - x) / Float64(-1.0 - x))));
	elseif (t <= -1.15e-149)
		tmp = t_2;
	elseif (t <= 1.25e-170)
		tmp = Float64(t_1 / Float64(sqrt(2.0) * hypot(hypot(t, Float64(t / sqrt(x))), Float64(l / sqrt(x)))));
	elseif (t <= 3.4e+70)
		tmp = t_2;
	else
		tmp = Float64(1.0 + Float64(Float64(0.5 / Float64(x * x)) + Float64(-1.0 / x)));
	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);
	t_2 = t_1 / sqrt((2.0 * ((l * (l / x)) + ((t * t) + ((t * t) / x)))));
	tmp = 0.0;
	if (t <= -1.25e+82)
		tmp = -sqrt(((1.0 - x) / (-1.0 - x)));
	elseif (t <= -1.15e-149)
		tmp = t_2;
	elseif (t <= 1.25e-170)
		tmp = t_1 / (sqrt(2.0) * hypot(hypot(t, (t / sqrt(x))), (l / sqrt(x))));
	elseif (t <= 3.4e+70)
		tmp = t_2;
	else
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x));
	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[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[Sqrt[N[(2.0 * N[(N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.25e+82], (-N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, -1.15e-149], t$95$2, If[LessEqual[t, 1.25e-170], N[(t$95$1 / N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[Sqrt[t ^ 2 + N[(t / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2 + N[(l / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.4e+70], t$95$2, N[(1.0 + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.25000000000000004e82

   1. Initial program 48.7

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

   2. Simplified 48.7

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

      [Start] 48.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 48.7	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      fma-neg [=>] 48.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]
      +-commutative [=>] 48.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}, -\ell \cdot \ell\right)}} \]
      fma-def [=>] 48.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}, -\ell \cdot \ell\right)}} \]
      distribute-rgt-neg-in [=>] 48.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \color{blue}{\ell \cdot \left(-\ell\right)}\right)}} \]

   3. Taylor expanded in t around -inf 4.0

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

   4. Simplified 4.0

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

      [Start] 4.0	\[ \sqrt{2} \cdot \left(-1 \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)\right) \]
      mul-1-neg [=>] 4.0	\[ \sqrt{2} \cdot \color{blue}{\left(-\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)} \]
      *-commutative [=>] 4.0	\[ \sqrt{2} \cdot \left(-\color{blue}{\sqrt{\frac{x - 1}{1 + x}} \cdot \sqrt{0.5}}\right) \]
      sub-neg [=>] 4.0	\[ \sqrt{2} \cdot \left(-\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \cdot \sqrt{0.5}\right) \]
      metadata-eval [=>] 4.0	\[ \sqrt{2} \cdot \left(-\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \cdot \sqrt{0.5}\right) \]
      +-commutative [=>] 4.0	\[ \sqrt{2} \cdot \left(-\sqrt{\frac{\color{blue}{-1 + x}}{1 + x}} \cdot \sqrt{0.5}\right) \]
      +-commutative [=>] 4.0	\[ \sqrt{2} \cdot \left(-\sqrt{\frac{-1 + x}{\color{blue}{x + 1}}} \cdot \sqrt{0.5}\right) \]

   5. Applied egg-rr 3.1

      \[\leadsto \color{blue}{\left(0 - e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(\frac{1 - x}{-1 - x} \cdot 0.5\right)}\right)}\right) + 1} \]

   6. Simplified 3.1

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

      [Start] 3.1	\[ \left(0 - e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(\frac{1 - x}{-1 - x} \cdot 0.5\right)}\right)}\right) + 1 \]
      associate-+l- [=>] 3.1	\[ \color{blue}{0 - \left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(\frac{1 - x}{-1 - x} \cdot 0.5\right)}\right)} - 1\right)} \]
      expm1-def [=>] 3.1	\[ 0 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \left(\frac{1 - x}{-1 - x} \cdot 0.5\right)}\right)\right)} \]
      expm1-log1p [=>] 3.1	\[ 0 - \color{blue}{\sqrt{2 \cdot \left(\frac{1 - x}{-1 - x} \cdot 0.5\right)}} \]
      sub0-neg [=>] 3.1	\[ \color{blue}{-\sqrt{2 \cdot \left(\frac{1 - x}{-1 - x} \cdot 0.5\right)}} \]
      *-commutative [=>] 3.1	\[ -\sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{1 - x}{-1 - x}\right)}} \]
      associate-*r* [=>] 3.1	\[ -\sqrt{\color{blue}{\left(2 \cdot 0.5\right) \cdot \frac{1 - x}{-1 - x}}} \]
      metadata-eval [=>] 3.1	\[ -\sqrt{\color{blue}{1} \cdot \frac{1 - x}{-1 - x}} \]
      associate-*r/ [=>] 3.1	\[ -\sqrt{\color{blue}{\frac{1 \cdot \left(1 - x\right)}{-1 - x}}} \]

   if -1.25000000000000004e82 < t < -1.15e-149 or 1.25000000000000003e-170 < t < 3.4000000000000001e70

   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. Taylor expanded in x around inf 10.8

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

   3. Simplified 10.8

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

      [Start] 10.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      associate--l+ [=>] 10.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]
      unpow2 [=>] 10.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-/l* [=>] 10.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}}} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      distribute-lft-out [=>] 10.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      +-commutative [=>] 10.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \color{blue}{\left({t}^{2} + \frac{{t}^{2}}{x}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 10.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(\color{blue}{t \cdot t} + \frac{{t}^{2}}{x}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 10.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{\color{blue}{t \cdot t}}{x}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-/l* [=>] 10.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \color{blue}{\frac{t}{\frac{x}{t}}}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-*r/ [=>] 10.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \]
      mul-1-neg [=>] 10.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \]
      +-commutative [=>] 10.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + {\ell}^{2}\right)}}{x}\right)}} \]
      unpow2 [=>] 10.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\left(2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}\right)}{x}\right)}} \]
      fma-udef [<=] 10.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)}} \]
      unpow2 [=>] 10.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)}} \]

   4. Taylor expanded in t around 0 11.1

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

   5. Simplified 11.1

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

      [Start] 11.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - -1 \cdot \frac{{\ell}^{2}}{x}\right)}} \]
      mul-1-neg [=>] 11.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \color{blue}{\left(-\frac{{\ell}^{2}}{x}\right)}\right)}} \]
      distribute-frac-neg [<=] 11.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \color{blue}{\frac{-{\ell}^{2}}{x}}\right)}} \]
      unpow2 [=>] 11.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\color{blue}{\ell \cdot \ell}}{x}\right)}} \]

   6. Taylor expanded in l around 0 11.1

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

   7. Simplified 6.2

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

      [Start] 11.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \frac{{\ell}^{2}}{x} + 2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)}} \]
      distribute-lft-out [=>] 11.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \left(\frac{{\ell}^{2}}{x} + \left(\frac{{t}^{2}}{x} + {t}^{2}\right)\right)}}} \]
      unpow2 [=>] 11.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\frac{{t}^{2}}{x} + {t}^{2}\right)\right)}} \]
      associate-*r/ [<=] 6.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\color{blue}{\ell \cdot \frac{\ell}{x}} + \left(\frac{{t}^{2}}{x} + {t}^{2}\right)\right)}} \]
      +-commutative [=>] 6.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + \color{blue}{\left({t}^{2} + \frac{{t}^{2}}{x}\right)}\right)}} \]
      unpow2 [=>] 6.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + \left(\color{blue}{t \cdot t} + \frac{{t}^{2}}{x}\right)\right)}} \]
      unpow2 [=>] 6.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + \left(t \cdot t + \frac{\color{blue}{t \cdot t}}{x}\right)\right)}} \]

   if -1.15e-149 < t < 1.25000000000000003e-170

   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. Taylor expanded in x around inf 34.0

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

   3. Simplified 34.0

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

      [Start] 34.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      associate--l+ [=>] 34.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]
      unpow2 [=>] 34.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-/l* [=>] 34.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}}} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      distribute-lft-out [=>] 34.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      +-commutative [=>] 34.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \color{blue}{\left({t}^{2} + \frac{{t}^{2}}{x}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 34.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(\color{blue}{t \cdot t} + \frac{{t}^{2}}{x}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 34.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{\color{blue}{t \cdot t}}{x}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-/l* [=>] 34.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \color{blue}{\frac{t}{\frac{x}{t}}}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-*r/ [=>] 34.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \]
      mul-1-neg [=>] 34.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \]
      +-commutative [=>] 34.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + {\ell}^{2}\right)}}{x}\right)}} \]
      unpow2 [=>] 34.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\left(2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}\right)}{x}\right)}} \]
      fma-udef [<=] 34.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)}} \]
      unpow2 [=>] 34.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)}} \]

   4. Taylor expanded in t around 0 34.0

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

   5. Simplified 34.0

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

      [Start] 34.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - -1 \cdot \frac{{\ell}^{2}}{x}\right)}} \]
      mul-1-neg [=>] 34.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \color{blue}{\left(-\frac{{\ell}^{2}}{x}\right)}\right)}} \]
      distribute-frac-neg [<=] 34.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \color{blue}{\frac{-{\ell}^{2}}{x}}\right)}} \]
      unpow2 [=>] 34.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\color{blue}{\ell \cdot \ell}}{x}\right)}} \]

   6. Taylor expanded in l around 0 34.0

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

   7. Simplified 31.8

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

      [Start] 34.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \frac{{\ell}^{2}}{x} + 2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)}} \]
      distribute-lft-out [=>] 34.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \left(\frac{{\ell}^{2}}{x} + \left(\frac{{t}^{2}}{x} + {t}^{2}\right)\right)}}} \]
      unpow2 [=>] 34.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\frac{{t}^{2}}{x} + {t}^{2}\right)\right)}} \]
      associate-*r/ [<=] 31.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\color{blue}{\ell \cdot \frac{\ell}{x}} + \left(\frac{{t}^{2}}{x} + {t}^{2}\right)\right)}} \]
      +-commutative [=>] 31.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + \color{blue}{\left({t}^{2} + \frac{{t}^{2}}{x}\right)}\right)}} \]
      unpow2 [=>] 31.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + \left(\color{blue}{t \cdot t} + \frac{{t}^{2}}{x}\right)\right)}} \]
      unpow2 [=>] 31.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + \left(t \cdot t + \frac{\color{blue}{t \cdot t}}{x}\right)\right)}} \]

   8. Applied egg-rr 14.8

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

   if 3.4000000000000001e70 < t

   1. Initial program 47.0

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

   2. Simplified 47.0

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

      [Start] 47.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 47.0	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      fma-neg [=>] 47.0	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]
      +-commutative [=>] 47.0	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}, -\ell \cdot \ell\right)}} \]
      fma-def [=>] 47.0	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}, -\ell \cdot \ell\right)}} \]
      distribute-rgt-neg-in [=>] 47.0	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \color{blue}{\ell \cdot \left(-\ell\right)}\right)}} \]

   3. Taylor expanded in t around -inf 63.0

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

   4. Simplified 63.0

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

      [Start] 63.0	\[ \sqrt{2} \cdot \left(-1 \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)\right) \]
      mul-1-neg [=>] 63.0	\[ \sqrt{2} \cdot \color{blue}{\left(-\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)} \]
      *-commutative [=>] 63.0	\[ \sqrt{2} \cdot \left(-\color{blue}{\sqrt{\frac{x - 1}{1 + x}} \cdot \sqrt{0.5}}\right) \]
      sub-neg [=>] 63.0	\[ \sqrt{2} \cdot \left(-\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \cdot \sqrt{0.5}\right) \]
      metadata-eval [=>] 63.0	\[ \sqrt{2} \cdot \left(-\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \cdot \sqrt{0.5}\right) \]
      +-commutative [=>] 63.0	\[ \sqrt{2} \cdot \left(-\sqrt{\frac{\color{blue}{-1 + x}}{1 + x}} \cdot \sqrt{0.5}\right) \]
      +-commutative [=>] 63.0	\[ \sqrt{2} \cdot \left(-\sqrt{\frac{-1 + x}{\color{blue}{x + 1}}} \cdot \sqrt{0.5}\right) \]

   5. Applied egg-rr 3.3

      \[\leadsto \color{blue}{0 + \sqrt{2 \cdot \left(\frac{1 - x}{-1 - x} \cdot 0.5\right)}} \]

   6. Simplified 3.3

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

      [Start] 3.3	\[ 0 + \sqrt{2 \cdot \left(\frac{1 - x}{-1 - x} \cdot 0.5\right)} \]
      +-lft-identity [=>] 3.3	\[ \color{blue}{\sqrt{2 \cdot \left(\frac{1 - x}{-1 - x} \cdot 0.5\right)}} \]
      *-commutative [=>] 3.3	\[ \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{1 - x}{-1 - x}\right)}} \]
      associate-*r* [=>] 3.3	\[ \sqrt{\color{blue}{\left(2 \cdot 0.5\right) \cdot \frac{1 - x}{-1 - x}}} \]
      metadata-eval [=>] 3.3	\[ \sqrt{\color{blue}{1} \cdot \frac{1 - x}{-1 - x}} \]
      associate-*r/ [=>] 3.3	\[ \sqrt{\color{blue}{\frac{1 \cdot \left(1 - x\right)}{-1 - x}}} \]

   7. Taylor expanded in x around -inf 3.6

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

   8. Simplified 3.6

      \[\leadsto \color{blue}{1 + \left(\frac{0.5}{x \cdot x} - \frac{1}{x}\right)} \]
      Proof

      [Start] 3.6	\[ \left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{x} \]
      sub-neg [=>] 3.6	\[ \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) + \left(-\frac{1}{x}\right)} \]
      associate-+l+ [=>] 3.6	\[ \color{blue}{1 + \left(0.5 \cdot \frac{1}{{x}^{2}} + \left(-\frac{1}{x}\right)\right)} \]
      sub-neg [<=] 3.6	\[ 1 + \color{blue}{\left(0.5 \cdot \frac{1}{{x}^{2}} - \frac{1}{x}\right)} \]
      associate-*r/ [=>] 3.6	\[ 1 + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} - \frac{1}{x}\right) \]
      metadata-eval [=>] 3.6	\[ 1 + \left(\frac{\color{blue}{0.5}}{{x}^{2}} - \frac{1}{x}\right) \]
      unpow2 [=>] 3.6	\[ 1 + \left(\frac{0.5}{\color{blue}{x \cdot x}} - \frac{1}{x}\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 6.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+82}:\\ \;\;\;\;-\sqrt{\frac{1 - x}{-1 - x}}\\ \mathbf{elif}\;t \leq -1.15 \cdot 10^{-149}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-170}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2} \cdot \mathsf{hypot}\left(\mathsf{hypot}\left(t, \frac{t}{\sqrt{x}}\right), \frac{\ell}{\sqrt{x}}\right)}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+70}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	9.7
Cost	14672

\[\begin{array}{l} t_1 := \frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{if}\;t \leq -6.6 \cdot 10^{+81}:\\ \;\;\;\;-\sqrt{\frac{1 - x}{-1 - x}}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-208}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-195}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{+70}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 2
Error	9.7
Cost	14288

\[\begin{array}{l} t_1 := \frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot t\right)}}\\ \mathbf{if}\;t \leq -8 \cdot 10^{+81}:\\ \;\;\;\;-\sqrt{\frac{1 - x}{-1 - x}}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-207}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-188}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+70}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 3
Error	14.3
Cost	14032

\[\begin{array}{l} t_1 := \sqrt{2} \cdot \frac{t}{\sqrt{\ell \cdot \frac{\ell + \ell}{x}}}\\ t_2 := \sqrt{\frac{1 - x}{-1 - x}}\\ \mathbf{if}\;t \leq -3.3 \cdot 10^{-191}:\\ \;\;\;\;-t_2\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-208}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-145}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t \leq 6.3 \cdot 10^{-117}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	14.3
Cost	14032

\[\begin{array}{l} t_1 := \sqrt{\frac{1 - x}{-1 - x}}\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{-192}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-208}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{elif}\;t \leq 1.16 \cdot 10^{-145}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t \leq 2.65 \cdot 10^{-117}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\ell \cdot \frac{\ell + \ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	14.3
Cost	13640

\[\begin{array}{l} t_1 := \sqrt{\frac{1 - x}{-1 - x}}\\ \mathbf{if}\;t \leq -4.6 \cdot 10^{-222}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-208}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	15.0
Cost	7044

\[\begin{array}{l} t_1 := \sqrt{\frac{1 - x}{-1 - x}}\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{-300}:\\ \;\;\;\;-t_1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	15.2
Cost	6980

\[\begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{-300}:\\ \;\;\;\;\frac{1}{x} + \left(-1 + \frac{\frac{-0.5}{x}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1 - x}{-1 - x}}\\ \end{array} \]

Alternative 8
Error	15.4
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{-300}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 9
Error	15.3
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{-300}:\\ \;\;\;\;\frac{1}{x} + \left(-1 + \frac{\frac{-0.5}{x}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 10
Error	15.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{-300}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Error	15.5
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{-300}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 12
Error	15.9
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{-300}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13
Error	39.6
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023057 
(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)))))