Toniolo and Linder, Equation (7)

Average Accuracy: 32.5% → 87.3%

Time: 38.6s

Precision: binary64

Cost: 52752

Math

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

↓

\[\begin{array}{l} t_1 := -\sqrt{\frac{-1 + x}{x + 1}}\\ t_2 := \frac{\ell}{\sqrt{x}}\\ \mathbf{if}\;t \leq -2.8 \cdot 10^{+142}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.5 \cdot 10^{-165}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq -4.1 \cdot 10^{-201}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{-60}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{hypot}\left(\mathsf{hypot}\left(\sqrt{2} \cdot \mathsf{hypot}\left(t, \frac{t}{\sqrt{x}}\right), t_2\right), t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

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 (- (sqrt (/ (+ -1.0 x) (+ x 1.0))))) (t_2 (/ l (sqrt x))))
   (if (<= t -2.8e+142)
     t_1
     (if (<= t -2.5e-165)
       (*
        t
        (/ (sqrt 2.0) (sqrt (* 2.0 (+ (* l (/ l x)) (* t (+ t (/ t x))))))))
       (if (<= t -4.1e-201)
         t_1
         (if (<= t 1.12e-60)
           (/
            (* t (sqrt 2.0))
            (hypot (hypot (* (sqrt 2.0) (hypot t (/ t (sqrt x)))) t_2) t_2))
           (+ 1.0 (/ -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 = -sqrt(((-1.0 + x) / (x + 1.0)));
	double t_2 = l / sqrt(x);
	double tmp;
	if (t <= -2.8e+142) {
		tmp = t_1;
	} else if (t <= -2.5e-165) {
		tmp = t * (sqrt(2.0) / sqrt((2.0 * ((l * (l / x)) + (t * (t + (t / x)))))));
	} else if (t <= -4.1e-201) {
		tmp = t_1;
	} else if (t <= 1.12e-60) {
		tmp = (t * sqrt(2.0)) / hypot(hypot((sqrt(2.0) * hypot(t, (t / sqrt(x)))), t_2), t_2);
	} else {
		tmp = 1.0 + (-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 = -Math.sqrt(((-1.0 + x) / (x + 1.0)));
	double t_2 = l / Math.sqrt(x);
	double tmp;
	if (t <= -2.8e+142) {
		tmp = t_1;
	} else if (t <= -2.5e-165) {
		tmp = t * (Math.sqrt(2.0) / Math.sqrt((2.0 * ((l * (l / x)) + (t * (t + (t / x)))))));
	} else if (t <= -4.1e-201) {
		tmp = t_1;
	} else if (t <= 1.12e-60) {
		tmp = (t * Math.sqrt(2.0)) / Math.hypot(Math.hypot((Math.sqrt(2.0) * Math.hypot(t, (t / Math.sqrt(x)))), t_2), t_2);
	} else {
		tmp = 1.0 + (-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 = -math.sqrt(((-1.0 + x) / (x + 1.0)))
	t_2 = l / math.sqrt(x)
	tmp = 0
	if t <= -2.8e+142:
		tmp = t_1
	elif t <= -2.5e-165:
		tmp = t * (math.sqrt(2.0) / math.sqrt((2.0 * ((l * (l / x)) + (t * (t + (t / x)))))))
	elif t <= -4.1e-201:
		tmp = t_1
	elif t <= 1.12e-60:
		tmp = (t * math.sqrt(2.0)) / math.hypot(math.hypot((math.sqrt(2.0) * math.hypot(t, (t / math.sqrt(x)))), t_2), t_2)
	else:
		tmp = 1.0 + (-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(-sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0))))
	t_2 = Float64(l / sqrt(x))
	tmp = 0.0
	if (t <= -2.8e+142)
		tmp = t_1;
	elseif (t <= -2.5e-165)
		tmp = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(2.0 * Float64(Float64(l * Float64(l / x)) + Float64(t * Float64(t + Float64(t / x))))))));
	elseif (t <= -4.1e-201)
		tmp = t_1;
	elseif (t <= 1.12e-60)
		tmp = Float64(Float64(t * sqrt(2.0)) / hypot(hypot(Float64(sqrt(2.0) * hypot(t, Float64(t / sqrt(x)))), t_2), t_2));
	else
		tmp = Float64(1.0 + 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 = -sqrt(((-1.0 + x) / (x + 1.0)));
	t_2 = l / sqrt(x);
	tmp = 0.0;
	if (t <= -2.8e+142)
		tmp = t_1;
	elseif (t <= -2.5e-165)
		tmp = t * (sqrt(2.0) / sqrt((2.0 * ((l * (l / x)) + (t * (t + (t / x)))))));
	elseif (t <= -4.1e-201)
		tmp = t_1;
	elseif (t <= 1.12e-60)
		tmp = (t * sqrt(2.0)) / hypot(hypot((sqrt(2.0) * hypot(t, (t / sqrt(x)))), t_2), t_2);
	else
		tmp = 1.0 + (-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[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])}, Block[{t$95$2 = N[(l / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.8e+142], t$95$1, If[LessEqual[t, -2.5e-165], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(2.0 * N[(N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision] + N[(t * N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -4.1e-201], t$95$1, If[LessEqual[t, 1.12e-60], N[(N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[Sqrt[N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[t ^ 2 + N[(t / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] ^ 2 + t$95$2 ^ 2], $MachinePrecision] ^ 2 + t$95$2 ^ 2], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -2.8e142 or -2.4999999999999999e-165 < t < -4.10000000000000001e-201

   1. Initial program 6.9%

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

   2. Simplified 6.9%

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

      [Start] 6.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 6.9	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Taylor expanded in t around -inf 91.8%

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

   4. Simplified 91.8%

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

      [Start] 91.8	\[ \frac{\sqrt{2}}{-1 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)} \cdot t \]
      associate-*r* [=>] 91.8	\[ \frac{\sqrt{2}}{\color{blue}{\left(-1 \cdot \left(\sqrt{2} \cdot t\right)\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \cdot t \]
      *-commutative [<=] 91.8	\[ \frac{\sqrt{2}}{\left(-1 \cdot \color{blue}{\left(t \cdot \sqrt{2}\right)}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}} \cdot t \]
      neg-mul-1 [<=] 91.8	\[ \frac{\sqrt{2}}{\color{blue}{\left(-t \cdot \sqrt{2}\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \cdot t \]
      distribute-rgt-neg-in [=>] 91.8	\[ \frac{\sqrt{2}}{\color{blue}{\left(t \cdot \left(-\sqrt{2}\right)\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \cdot t \]
      +-commutative [<=] 91.8	\[ \frac{\sqrt{2}}{\left(t \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \cdot t \]
      sub-neg [=>] 91.8	\[ \frac{\sqrt{2}}{\left(t \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \cdot t \]
      metadata-eval [=>] 91.8	\[ \frac{\sqrt{2}}{\left(t \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \cdot t \]
      +-commutative [=>] 91.8	\[ \frac{\sqrt{2}}{\left(t \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \cdot t \]

   5. Taylor expanded in t around 0 92.4%

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

   6. Simplified 92.4%

      \[\leadsto \color{blue}{-\sqrt{\frac{-1 + x}{x + 1}}} \]
      Proof

      [Start] 92.4	\[ -1 \cdot \sqrt{\frac{x - 1}{1 + x}} \]
      mul-1-neg [=>] 92.4	\[ \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]
      sub-neg [=>] 92.4	\[ -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]
      metadata-eval [=>] 92.4	\[ -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]
      +-commutative [=>] 92.4	\[ -\sqrt{\frac{\color{blue}{-1 + x}}{1 + x}} \]
      +-commutative [<=] 92.4	\[ -\sqrt{\frac{-1 + x}{\color{blue}{x + 1}}} \]

   if -2.8e142 < t < -2.4999999999999999e-165

   1. Initial program 61.0%

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

   2. Simplified 61.1%

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

      [Start] 61.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 61.1	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Taylor expanded in x around inf 83.6%

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

   4. Simplified 83.6%

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

      [Start] 83.6	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 83.6	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 83.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 83.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 83.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 83.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 83.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      unpow2 [=>] 83.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}}{x}\right)\right)}} \cdot t \]
      +-commutative [=>] 83.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + \ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 83.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 83.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]

   5. Taylor expanded in t around 0 82.8%

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

   6. Simplified 82.8%

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

      [Start] 82.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{{\ell}^{2}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 82.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]

   7. Applied egg-rr 90.5%

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

      [Start] 82.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)\right)}} \cdot t \]
      pow1/2 [=>] 82.9	\[ \frac{\sqrt{2}}{\color{blue}{{\left(\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)\right)\right)}^{0.5}}} \cdot t \]
      sqr-pow [=>] 82.6	\[ \frac{\sqrt{2}}{\color{blue}{{\left(\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}}} \cdot t \]

   8. Simplified 90.7%

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

      [Start] 90.5	\[ \frac{\sqrt{2}}{{\left(\mathsf{fma}\left(2, \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right), \mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right)\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(2, \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right), \mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right)\right)\right)}^{0.25}} \cdot t \]
      pow-sqr [=>] 90.7	\[ \frac{\sqrt{2}}{\color{blue}{{\left(\mathsf{fma}\left(2, \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right), \mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right)\right)\right)}^{\left(2 \cdot 0.25\right)}}} \cdot t \]
      metadata-eval [=>] 90.7	\[ \frac{\sqrt{2}}{{\left(\mathsf{fma}\left(2, \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right), \mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right)\right)\right)}^{\color{blue}{0.5}}} \cdot t \]
      unpow1/2 [=>] 90.7	\[ \frac{\sqrt{2}}{\color{blue}{\sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right), \mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right)\right)}}} \cdot t \]
      fma-udef [=>] 90.7	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right) + \mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right)}}} \cdot t \]
      +-commutative [=>] 90.7	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right) + 2 \cdot \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right)}}} \cdot t \]
      fma-udef [=>] 90.7	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\frac{\ell}{x} \cdot \ell + \frac{\ell}{x} \cdot \ell\right)} + 2 \cdot \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right)}} \cdot t \]
      count-2 [=>] 90.7	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \left(\frac{\ell}{x} \cdot \ell\right)} + 2 \cdot \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right)}} \cdot t \]
      distribute-lft-out [=>] 90.7	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right)\right)}}} \cdot t \]
      *-commutative [=>] 90.7	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\color{blue}{\ell \cdot \frac{\ell}{x}} + \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right)\right)}} \cdot t \]
      fma-udef [=>] 90.7	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + \color{blue}{\left(\frac{t}{x} \cdot t + t \cdot t\right)}\right)}} \cdot t \]
      +-commutative [=>] 90.7	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + \color{blue}{\left(t \cdot t + \frac{t}{x} \cdot t\right)}\right)}} \cdot t \]
      distribute-rgt-out [=>] 90.7	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + \color{blue}{t \cdot \left(t + \frac{t}{x}\right)}\right)}} \cdot t \]

   if -4.10000000000000001e-201 < t < 1.12e-60

   1. Initial program 15.8%

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

   2. Taylor expanded in x around inf 60.3%

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

   3. Simplified 60.3%

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

      [Start] 60.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      associate--l+ [=>] 60.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]
      unpow2 [=>] 60.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      distribute-lft-out [=>] 60.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 60.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 60.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-*r/ [=>] 60.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \]
      mul-1-neg [=>] 60.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \]
      +-commutative [=>] 60.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + {\ell}^{2}\right)}}{x}\right)}} \]
      unpow2 [=>] 60.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(2 \cdot \color{blue}{\left(t \cdot t\right)} + {\ell}^{2}\right)}{x}\right)}} \]
      unpow2 [=>] 60.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(2 \cdot \left(t \cdot t\right) + \color{blue}{\ell \cdot \ell}\right)}{x}\right)}} \]
      fma-udef [<=] 60.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)}} \]

   4. Taylor expanded in t around 0 60.2%

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

   5. Simplified 60.2%

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

      [Start] 60.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - -1 \cdot \frac{{\ell}^{2}}{x}\right)}} \]
      associate-*r/ [=>] 60.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot {\ell}^{2}}{x}}\right)}} \]
      mul-1-neg [=>] 60.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-{\ell}^{2}}}{x}\right)}} \]
      unpow2 [=>] 60.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\ell \cdot \ell}}{x}\right)}} \]

   6. Applied egg-rr 76.3%

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

      [Start] 60.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}} \]
      +-commutative [=>] 60.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right) + \frac{\ell \cdot \ell}{x}}}} \]
      add-sqr-sqrt [=>] 60.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\sqrt{2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}} \cdot \sqrt{2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}}} + \frac{\ell \cdot \ell}{x}}} \]
      add-sqr-sqrt [=>] 58.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\sqrt{2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}} \cdot \sqrt{2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}} + \color{blue}{\sqrt{\frac{\ell \cdot \ell}{x}} \cdot \sqrt{\frac{\ell \cdot \ell}{x}}}}} \]
      hypot-def [=>] 58.1	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{\mathsf{hypot}\left(\sqrt{2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}}, \sqrt{\frac{\ell \cdot \ell}{x}}\right)}} \]

   if 1.12e-60 < t

   1. Initial program 36.8%

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

   2. Simplified 36.9%

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

      [Start] 36.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 36.9	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Taylor expanded in t around -inf 1.6%

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

   4. Simplified 1.6%

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

      [Start] 1.6	\[ \frac{\sqrt{2}}{-1 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)} \cdot t \]
      associate-*r* [=>] 1.6	\[ \frac{\sqrt{2}}{\color{blue}{\left(-1 \cdot \left(\sqrt{2} \cdot t\right)\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \cdot t \]
      *-commutative [<=] 1.6	\[ \frac{\sqrt{2}}{\left(-1 \cdot \color{blue}{\left(t \cdot \sqrt{2}\right)}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}} \cdot t \]
      neg-mul-1 [<=] 1.6	\[ \frac{\sqrt{2}}{\color{blue}{\left(-t \cdot \sqrt{2}\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \cdot t \]
      distribute-rgt-neg-in [=>] 1.6	\[ \frac{\sqrt{2}}{\color{blue}{\left(t \cdot \left(-\sqrt{2}\right)\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \cdot t \]
      +-commutative [<=] 1.6	\[ \frac{\sqrt{2}}{\left(t \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \cdot t \]
      sub-neg [=>] 1.6	\[ \frac{\sqrt{2}}{\left(t \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \cdot t \]
      metadata-eval [=>] 1.6	\[ \frac{\sqrt{2}}{\left(t \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \cdot t \]
      +-commutative [=>] 1.6	\[ \frac{\sqrt{2}}{\left(t \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \cdot t \]

   5. Applied egg-rr 89.8%

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

      No proof available- proof too large to flatten.

   6. Simplified 89.9%

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

      [Start] 89.8	\[ \left(\sqrt{2} \cdot \frac{\frac{1}{t}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{x + -1}}}\right) \cdot t \]
      associate-*r/ [=>] 89.8	\[ \color{blue}{\frac{\sqrt{2} \cdot \frac{1}{t}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{x + -1}}}} \cdot t \]
      times-frac [=>] 89.9	\[ \color{blue}{\left(\frac{\sqrt{2}}{\sqrt{2}} \cdot \frac{\frac{1}{t}}{\sqrt{\frac{x + 1}{x + -1}}}\right)} \cdot t \]
      *-inverses [=>] 89.9	\[ \left(\color{blue}{1} \cdot \frac{\frac{1}{t}}{\sqrt{\frac{x + 1}{x + -1}}}\right) \cdot t \]
      *-lft-identity [=>] 89.9	\[ \color{blue}{\frac{\frac{1}{t}}{\sqrt{\frac{x + 1}{x + -1}}}} \cdot t \]
      +-commutative [=>] 89.9	\[ \frac{\frac{1}{t}}{\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \cdot t \]

   7. Taylor expanded in x around inf 89.3%

      \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{+142}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq -2.5 \cdot 10^{-165}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq -4.1 \cdot 10^{-201}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{-60}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{hypot}\left(\mathsf{hypot}\left(\sqrt{2} \cdot \mathsf{hypot}\left(t, \frac{t}{\sqrt{x}}\right), \frac{\ell}{\sqrt{x}}\right), \frac{\ell}{\sqrt{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	87.3%
Cost	52752

\[\begin{array}{l} t_1 := -\sqrt{\frac{-1 + x}{x + 1}}\\ t_2 := \frac{\ell}{\sqrt{x}}\\ \mathbf{if}\;t \leq -2.6 \cdot 10^{+142}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.32 \cdot 10^{-165}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq -6.4 \cdot 10^{-202}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-55}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{hypot}\left(\mathsf{hypot}\left(\sqrt{2} \cdot \mathsf{hypot}\left(t, \frac{t}{\sqrt{x}}\right), t_2\right), t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 2
Accuracy	86.2%
Cost	34132

\[\begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ t_2 := \frac{2}{x} + \left(2 + \frac{2}{x}\right)\\ \mathbf{if}\;t \leq -1 \cdot 10^{+147}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{-168}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3 \cdot 10^{-266}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-\mathsf{fma}\left(\sqrt{t_2}, t, \sqrt{\frac{1}{t_2}} \cdot \frac{\ell \cdot \ell}{t \cdot x}\right)}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-213}:\\ \;\;\;\;t \cdot \left(-\frac{\sqrt{x}}{\ell}\right)\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-158}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\mathsf{fma}\left(t, \sqrt{2}, \frac{0.5 \cdot \left(2 \cdot \mathsf{fma}\left(\ell, \ell, t \cdot \left(t \cdot 2\right)\right)\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}\right)}{t}}\\ \mathbf{elif}\;t \leq 2200:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 3
Accuracy	86.6%
Cost	33296

\[\begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ t_2 := \frac{2}{x} + \left(2 + \frac{2}{x}\right)\\ \mathbf{if}\;t \leq -2.3 \cdot 10^{+142}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq -1.55 \cdot 10^{-168}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.9 \cdot 10^{-228}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-\mathsf{fma}\left(\sqrt{t_2}, t, \sqrt{\frac{1}{t_2}} \cdot \frac{\ell \cdot \ell}{t \cdot x}\right)}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-186}:\\ \;\;\;\;t \cdot \left(\sqrt{2} \cdot {\left(e^{-0.5}\right)}^{\left(\log \left(\ell \cdot \left(2 \cdot \ell\right)\right) - \log x\right)}\right)\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{-163}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 175:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 4
Accuracy	86.2%
Cost	28172

\[\begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ t_2 := \frac{2}{x} + \left(2 + \frac{2}{x}\right)\\ \mathbf{if}\;t \leq -5 \cdot 10^{+142}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-171}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3 \cdot 10^{-271}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-\mathsf{fma}\left(\sqrt{t_2}, t, \sqrt{\frac{1}{t_2}} \cdot \frac{\ell \cdot \ell}{t \cdot x}\right)}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-186}:\\ \;\;\;\;t \cdot \sqrt{2 \cdot \frac{\frac{x}{\ell}}{\ell + \ell} - \frac{2}{\ell \cdot \left(\ell + \ell\right)}}\\ \mathbf{elif}\;t \leq 1.06 \cdot 10^{-160}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2200:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 5
Accuracy	84.7%
Cost	14808

\[\begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ t_2 := -\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{if}\;t \leq -5.2 \cdot 10^{+144}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.2 \cdot 10^{-164}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3.1 \cdot 10^{-214}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-189}:\\ \;\;\;\;t \cdot \left(-\frac{\sqrt{x}}{\ell}\right)\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-164}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2200:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 6
Accuracy	77.3%
Cost	14032

\[\begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-214}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-190}:\\ \;\;\;\;t \cdot \left(-\frac{\sqrt{x}}{\ell}\right)\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-86}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 7
Accuracy	77.2%
Cost	8144

\[\begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{-214}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-187}:\\ \;\;\;\;t \cdot \left(-\frac{\sqrt{x}}{\ell}\right)\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-156}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-82}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\ell \cdot \left(\frac{\ell + \ell}{x \cdot x} + \frac{\ell + \ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 8
Accuracy	77.7%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{-214}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-190}:\\ \;\;\;\;\sqrt{x} \cdot \frac{-t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 9
Accuracy	78.1%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;t \leq -1.45 \cdot 10^{-214}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-188}:\\ \;\;\;\;t \cdot \left(-\frac{\sqrt{x}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 10
Accuracy	78.4%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{-208}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-190}:\\ \;\;\;\;t \cdot \left(-\frac{\sqrt{x}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 11
Accuracy	78.4%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{-220}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-186}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 12
Accuracy	78.4%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{-224}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-186}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 13
Accuracy	76.8%
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-293}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 14
Accuracy	76.3%
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-293}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 15
Accuracy	76.7%
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-293}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 16
Accuracy	76.0%
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-293}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 17
Accuracy	39.0%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023146 
(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)))))