Toniolo and Linder, Equation (7)

Average Accuracy: 32.1% → 85.8%

Time: 38.1s

Precision: binary64

Cost: 21200

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\left(t \cdot t\right) \cdot \left(1 + \frac{2}{x}\right) + \frac{\ell}{\frac{x}{\ell}}\right)}}\\ t_2 := \sqrt{\frac{x + -1}{x + 1}}\\ t_3 := -t_2\\ \mathbf{if}\;t \leq -4.6 \cdot 10^{+142}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -3.05 \cdot 10^{-164}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.3 \cdot 10^{-274}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 2.85 \cdot 10^{-269}:\\ \;\;\;\;\frac{t}{e^{0.5 \cdot \left(\left(\frac{1}{x} + \log \left(\ell \cdot \left(\left(\ell + \ell\right) \cdot 0.5\right)\right)\right) + \left(\frac{\frac{0.5}{x}}{x} - \log x\right)\right)}}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-207}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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 2.0)
          (/
           t
           (sqrt (* 2.0 (+ (* (* t t) (+ 1.0 (/ 2.0 x))) (/ l (/ x l))))))))
        (t_2 (sqrt (/ (+ x -1.0) (+ x 1.0))))
        (t_3 (- t_2)))
   (if (<= t -4.6e+142)
     t_3
     (if (<= t -3.05e-164)
       t_1
       (if (<= t -1.3e-274)
         t_3
         (if (<= t 2.85e-269)
           (/
            t
            (exp
             (*
              0.5
              (+
               (+ (/ 1.0 x) (log (* l (* (+ l l) 0.5))))
               (- (/ (/ 0.5 x) x) (log x))))))
           (if (<= t 2.1e-207) 1.0 (if (<= t 3.6e+100) t_1 t_2))))))))

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(2.0) * (t / sqrt((2.0 * (((t * t) * (1.0 + (2.0 / x))) + (l / (x / l))))));
	double t_2 = sqrt(((x + -1.0) / (x + 1.0)));
	double t_3 = -t_2;
	double tmp;
	if (t <= -4.6e+142) {
		tmp = t_3;
	} else if (t <= -3.05e-164) {
		tmp = t_1;
	} else if (t <= -1.3e-274) {
		tmp = t_3;
	} else if (t <= 2.85e-269) {
		tmp = t / exp((0.5 * (((1.0 / x) + log((l * ((l + l) * 0.5)))) + (((0.5 / x) / x) - log(x)))));
	} else if (t <= 2.1e-207) {
		tmp = 1.0;
	} else if (t <= 3.6e+100) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    code = (sqrt(2.0d0) * t) / sqrt(((((x + 1.0d0) / (x - 1.0d0)) * ((l * l) + (2.0d0 * (t * t)))) - (l * l)))
end function

↓

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt(2.0d0) * (t / sqrt((2.0d0 * (((t * t) * (1.0d0 + (2.0d0 / x))) + (l / (x / l))))))
    t_2 = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    t_3 = -t_2
    if (t <= (-4.6d+142)) then
        tmp = t_3
    else if (t <= (-3.05d-164)) then
        tmp = t_1
    else if (t <= (-1.3d-274)) then
        tmp = t_3
    else if (t <= 2.85d-269) then
        tmp = t / exp((0.5d0 * (((1.0d0 / x) + log((l * ((l + l) * 0.5d0)))) + (((0.5d0 / x) / x) - log(x)))))
    else if (t <= 2.1d-207) then
        tmp = 1.0d0
    else if (t <= 3.6d+100) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

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(2.0) * (t / Math.sqrt((2.0 * (((t * t) * (1.0 + (2.0 / x))) + (l / (x / l))))));
	double t_2 = Math.sqrt(((x + -1.0) / (x + 1.0)));
	double t_3 = -t_2;
	double tmp;
	if (t <= -4.6e+142) {
		tmp = t_3;
	} else if (t <= -3.05e-164) {
		tmp = t_1;
	} else if (t <= -1.3e-274) {
		tmp = t_3;
	} else if (t <= 2.85e-269) {
		tmp = t / Math.exp((0.5 * (((1.0 / x) + Math.log((l * ((l + l) * 0.5)))) + (((0.5 / x) / x) - Math.log(x)))));
	} else if (t <= 2.1e-207) {
		tmp = 1.0;
	} else if (t <= 3.6e+100) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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(2.0) * (t / math.sqrt((2.0 * (((t * t) * (1.0 + (2.0 / x))) + (l / (x / l))))))
	t_2 = math.sqrt(((x + -1.0) / (x + 1.0)))
	t_3 = -t_2
	tmp = 0
	if t <= -4.6e+142:
		tmp = t_3
	elif t <= -3.05e-164:
		tmp = t_1
	elif t <= -1.3e-274:
		tmp = t_3
	elif t <= 2.85e-269:
		tmp = t / math.exp((0.5 * (((1.0 / x) + math.log((l * ((l + l) * 0.5)))) + (((0.5 / x) / x) - math.log(x)))))
	elif t <= 2.1e-207:
		tmp = 1.0
	elif t <= 3.6e+100:
		tmp = t_1
	else:
		tmp = t_2
	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(2.0) * Float64(t / sqrt(Float64(2.0 * Float64(Float64(Float64(t * t) * Float64(1.0 + Float64(2.0 / x))) + Float64(l / Float64(x / l)))))))
	t_2 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	t_3 = Float64(-t_2)
	tmp = 0.0
	if (t <= -4.6e+142)
		tmp = t_3;
	elseif (t <= -3.05e-164)
		tmp = t_1;
	elseif (t <= -1.3e-274)
		tmp = t_3;
	elseif (t <= 2.85e-269)
		tmp = Float64(t / exp(Float64(0.5 * Float64(Float64(Float64(1.0 / x) + log(Float64(l * Float64(Float64(l + l) * 0.5)))) + Float64(Float64(Float64(0.5 / x) / x) - log(x))))));
	elseif (t <= 2.1e-207)
		tmp = 1.0;
	elseif (t <= 3.6e+100)
		tmp = t_1;
	else
		tmp = t_2;
	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(2.0) * (t / sqrt((2.0 * (((t * t) * (1.0 + (2.0 / x))) + (l / (x / l))))));
	t_2 = sqrt(((x + -1.0) / (x + 1.0)));
	t_3 = -t_2;
	tmp = 0.0;
	if (t <= -4.6e+142)
		tmp = t_3;
	elseif (t <= -3.05e-164)
		tmp = t_1;
	elseif (t <= -1.3e-274)
		tmp = t_3;
	elseif (t <= 2.85e-269)
		tmp = t / exp((0.5 * (((1.0 / x) + log((l * ((l + l) * 0.5)))) + (((0.5 / x) / x) - log(x)))));
	elseif (t <= 2.1e-207)
		tmp = 1.0;
	elseif (t <= 3.6e+100)
		tmp = t_1;
	else
		tmp = t_2;
	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[Sqrt[2.0], $MachinePrecision] * N[(t / N[Sqrt[N[(2.0 * N[(N[(N[(t * t), $MachinePrecision] * N[(1.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = (-t$95$2)}, If[LessEqual[t, -4.6e+142], t$95$3, If[LessEqual[t, -3.05e-164], t$95$1, If[LessEqual[t, -1.3e-274], t$95$3, If[LessEqual[t, 2.85e-269], N[(t / N[Exp[N[(0.5 * N[(N[(N[(1.0 / x), $MachinePrecision] + N[Log[N[(l * N[(N[(l + l), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.5 / x), $MachinePrecision] / x), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.1e-207], 1.0, If[LessEqual[t, 3.6e+100], t$95$1, t$95$2]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -4.60000000000000004e142 or -3.05000000000000007e-164 < t < -1.3e-274

   1. Initial program 6.0%

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

   2. Simplified 6.0%

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

      [Start] 6.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 6.0	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 6.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 6.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Applied egg-rr 6.0%

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

      No proof available- proof too large to flatten.

   4. Taylor expanded in t around -inf 82.1%

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

   5. Simplified 82.1%

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

      [Start] 82.1	\[ -1 \cdot \sqrt{\frac{x - 1}{1 + x}} \]
      mul-1-neg [=>] 82.1	\[ \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]
      sub-neg [=>] 82.1	\[ -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]
      metadata-eval [=>] 82.1	\[ -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]
      +-commutative [=>] 82.1	\[ -\sqrt{\frac{x + -1}{\color{blue}{x + 1}}} \]

   if -4.60000000000000004e142 < t < -3.05000000000000007e-164 or 2.10000000000000003e-207 < t < 3.6e100

   1. Initial program 55.1%

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

   2. Simplified 46.0%

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

      [Start] 55.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 55.0	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      associate-*l/ [=>] 42.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\left(x + 1\right) \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x - 1}} - \ell \cdot \ell}} \]
      associate-*r/ [<=] 45.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}} - \ell \cdot \ell}} \]
      *-lft-identity [<=] 45.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \left(x + 1\right)\right)} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} - \ell \cdot \ell}} \]
      associate-*r* [<=] 45.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{1 \cdot \left(\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 45.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{1 \cdot \color{blue}{\left(\frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} \cdot \left(x + 1\right)\right)} - \ell \cdot \ell}} \]
      associate-*r* [=>] 45.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right) \cdot \left(x + 1\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 45.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      fma-neg [=>] 46.0	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(x + 1, 1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}, -\ell \cdot \ell\right)}}} \]

   3. Taylor expanded in x around -inf 80.4%

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

   4. Simplified 80.4%

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

      [Start] 80.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + 2 \cdot {t}^{2}}} \]
      distribute-lft-out [=>] 80.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{2 \cdot \left(\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + {t}^{2}\right)}}} \]
      +-commutative [=>] 80.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 80.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x} + {t}^{2}\right)}} \]
      fma-udef [<=] 80.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 80.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 80.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} + \color{blue}{t \cdot t}\right)}} \]

   5. Taylor expanded in t around 0 80.4%

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

   6. Simplified 88.7%

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

      [Start] 80.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{{\ell}^{2}}{x} + {t}^{2} \cdot \left(1 + 2 \cdot \frac{1}{x}\right)\right)}} \]
      +-commutative [=>] 80.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \color{blue}{\left({t}^{2} \cdot \left(1 + 2 \cdot \frac{1}{x}\right) + \frac{{\ell}^{2}}{x}\right)}}} \]
      unpow2 [=>] 80.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \left(1 + 2 \cdot \frac{1}{x}\right) + \frac{{\ell}^{2}}{x}\right)}} \]
      associate-*r/ [=>] 80.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\left(t \cdot t\right) \cdot \left(1 + \color{blue}{\frac{2 \cdot 1}{x}}\right) + \frac{{\ell}^{2}}{x}\right)}} \]
      metadata-eval [=>] 80.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\left(t \cdot t\right) \cdot \left(1 + \frac{\color{blue}{2}}{x}\right) + \frac{{\ell}^{2}}{x}\right)}} \]
      unpow2 [=>] 80.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\left(t \cdot t\right) \cdot \left(1 + \frac{2}{x}\right) + \frac{\color{blue}{\ell \cdot \ell}}{x}\right)}} \]
      associate-/l* [=>] 88.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\left(t \cdot t\right) \cdot \left(1 + \frac{2}{x}\right) + \color{blue}{\frac{\ell}{\frac{x}{\ell}}}\right)}} \]

   if -1.3e-274 < t < 2.84999999999999985e-269

   1. Initial program 1.7%

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

   2. Simplified 1.7%

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

      [Start] 1.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 1.7	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 1.7	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 1.7	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Applied egg-rr 1.7%

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

      No proof available- proof too large to flatten.

   4. Taylor expanded in t around 0 1.7%

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

   5. Simplified 1.7%

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

      [Start] 1.7	\[ \frac{t}{\sqrt{\frac{\frac{x + 1}{x + -1} \cdot {\ell}^{2} - \ell \cdot \ell}{2}}} \]
      unpow2 [=>] 1.7	\[ \frac{t}{\sqrt{\frac{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell\right)} - \ell \cdot \ell}{2}}} \]

   6. Applied egg-rr 4.5%

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

      No proof available- proof too large to flatten.

   7. Taylor expanded in x around inf 63.2%

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

   8. Simplified 63.2%

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

      [Start] 63.2	\[ \frac{t}{e^{\left(\log \left(\frac{1}{x}\right) + \left(0.5 \cdot \frac{1}{{x}^{2}} + \left(\log \left(0.5 \cdot \left(\ell \cdot \left(\ell - -1 \cdot \ell\right)\right)\right) + \frac{1}{x}\right)\right)\right) \cdot 0.5}} \]
      associate-+r+ [=>] 63.2	\[ \frac{t}{e^{\color{blue}{\left(\left(\log \left(\frac{1}{x}\right) + 0.5 \cdot \frac{1}{{x}^{2}}\right) + \left(\log \left(0.5 \cdot \left(\ell \cdot \left(\ell - -1 \cdot \ell\right)\right)\right) + \frac{1}{x}\right)\right)} \cdot 0.5}} \]
      +-commutative [=>] 63.2	\[ \frac{t}{e^{\color{blue}{\left(\left(\log \left(0.5 \cdot \left(\ell \cdot \left(\ell - -1 \cdot \ell\right)\right)\right) + \frac{1}{x}\right) + \left(\log \left(\frac{1}{x}\right) + 0.5 \cdot \frac{1}{{x}^{2}}\right)\right)} \cdot 0.5}} \]
      +-commutative [=>] 63.2	\[ \frac{t}{e^{\left(\color{blue}{\left(\frac{1}{x} + \log \left(0.5 \cdot \left(\ell \cdot \left(\ell - -1 \cdot \ell\right)\right)\right)\right)} + \left(\log \left(\frac{1}{x}\right) + 0.5 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot 0.5}} \]
      *-commutative [=>] 63.2	\[ \frac{t}{e^{\left(\left(\frac{1}{x} + \log \color{blue}{\left(\left(\ell \cdot \left(\ell - -1 \cdot \ell\right)\right) \cdot 0.5\right)}\right) + \left(\log \left(\frac{1}{x}\right) + 0.5 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot 0.5}} \]
      associate-*l* [=>] 63.2	\[ \frac{t}{e^{\left(\left(\frac{1}{x} + \log \color{blue}{\left(\ell \cdot \left(\left(\ell - -1 \cdot \ell\right) \cdot 0.5\right)\right)}\right) + \left(\log \left(\frac{1}{x}\right) + 0.5 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot 0.5}} \]
      mul-1-neg [=>] 63.2	\[ \frac{t}{e^{\left(\left(\frac{1}{x} + \log \left(\ell \cdot \left(\left(\ell - \color{blue}{\left(-\ell\right)}\right) \cdot 0.5\right)\right)\right) + \left(\log \left(\frac{1}{x}\right) + 0.5 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot 0.5}} \]
      log-rec [=>] 63.2	\[ \frac{t}{e^{\left(\left(\frac{1}{x} + \log \left(\ell \cdot \left(\left(\ell - \left(-\ell\right)\right) \cdot 0.5\right)\right)\right) + \left(\color{blue}{\left(-\log x\right)} + 0.5 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot 0.5}} \]
      associate-*r/ [=>] 63.2	\[ \frac{t}{e^{\left(\left(\frac{1}{x} + \log \left(\ell \cdot \left(\left(\ell - \left(-\ell\right)\right) \cdot 0.5\right)\right)\right) + \left(\left(-\log x\right) + \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right)\right) \cdot 0.5}} \]
      metadata-eval [=>] 63.2	\[ \frac{t}{e^{\left(\left(\frac{1}{x} + \log \left(\ell \cdot \left(\left(\ell - \left(-\ell\right)\right) \cdot 0.5\right)\right)\right) + \left(\left(-\log x\right) + \frac{\color{blue}{0.5}}{{x}^{2}}\right)\right) \cdot 0.5}} \]
      unpow2 [=>] 63.2	\[ \frac{t}{e^{\left(\left(\frac{1}{x} + \log \left(\ell \cdot \left(\left(\ell - \left(-\ell\right)\right) \cdot 0.5\right)\right)\right) + \left(\left(-\log x\right) + \frac{0.5}{\color{blue}{x \cdot x}}\right)\right) \cdot 0.5}} \]
      associate-/r* [=>] 63.2	\[ \frac{t}{e^{\left(\left(\frac{1}{x} + \log \left(\ell \cdot \left(\left(\ell - \left(-\ell\right)\right) \cdot 0.5\right)\right)\right) + \left(\left(-\log x\right) + \color{blue}{\frac{\frac{0.5}{x}}{x}}\right)\right) \cdot 0.5}} \]

   if 2.84999999999999985e-269 < t < 2.10000000000000003e-207

   1. Initial program 2.4%

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

   2. Simplified 2.4%

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

      [Start] 2.4	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      *-lft-identity [<=] 2.4	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(1 \cdot \frac{x + 1}{x - 1}\right)} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      *-lft-identity [=>] 2.4	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{x + 1}{x - 1}} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      sub-neg [=>] 2.4	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      metadata-eval [=>] 2.4	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      fma-def [=>] 2.4	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}} \]

   3. Taylor expanded in x around inf 39.1%

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

   4. Applied egg-rr 39.6%

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

      No proof available- proof too large to flatten.

   if 3.6e100 < t

   1. Initial program 19.4%

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

   2. Simplified 19.4%

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

      [Start] 19.4	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 19.4	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 19.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 19.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Applied egg-rr 19.5%

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

      No proof available- proof too large to flatten.

   4. Taylor expanded in t around inf 96.5%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.6 \cdot 10^{+142}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -3.05 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\left(t \cdot t\right) \cdot \left(1 + \frac{2}{x}\right) + \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{elif}\;t \leq -1.3 \cdot 10^{-274}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 2.85 \cdot 10^{-269}:\\ \;\;\;\;\frac{t}{e^{0.5 \cdot \left(\left(\frac{1}{x} + \log \left(\ell \cdot \left(\left(\ell + \ell\right) \cdot 0.5\right)\right)\right) + \left(\frac{\frac{0.5}{x}}{x} - \log x\right)\right)}}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-207}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+100}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\left(t \cdot t\right) \cdot \left(1 + \frac{2}{x}\right) + \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	86.2%
Cost	28172

\[\begin{array}{l} t_1 := \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\left(t \cdot t\right) \cdot \left(1 + \frac{2}{x}\right) + \frac{\ell}{\frac{x}{\ell}}\right)}}\\ t_2 := \sqrt{\frac{x + -1}{x + 1}}\\ t_3 := 2 + \left(\frac{2}{x} + \frac{2}{x}\right)\\ \mathbf{if}\;t \leq -1.2 \cdot 10^{+144}:\\ \;\;\;\;-t_2\\ \mathbf{elif}\;t \leq -5.8 \cdot 10^{-163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-289}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{t_3}, \sqrt{\frac{1}{t_3}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{-203}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Accuracy	85.2%
Cost	14936

\[\begin{array}{l} t_1 := \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\left(t \cdot t\right) \cdot \left(1 + \frac{2}{x}\right) + \frac{\ell}{\frac{x}{\ell}}\right)}}\\ t_2 := \sqrt{\frac{x + -1}{x + 1}}\\ t_3 := -t_2\\ \mathbf{if}\;t \leq -2 \cdot 10^{+142}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -4.2 \cdot 10^{-164}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-274}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-289}:\\ \;\;\;\;t \cdot \sqrt{\frac{x}{\ell \cdot \ell}}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-204}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Accuracy	81.4%
Cost	14552

\[\begin{array}{l} t_1 := \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(t \cdot t + \frac{\ell \cdot \ell}{x}\right)}}\\ t_2 := \sqrt{\frac{x + -1}{x + 1}}\\ t_3 := -t_2\\ \mathbf{if}\;t \leq -2.8 \cdot 10^{-29}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -8.8 \cdot 10^{-163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -7.2 \cdot 10^{-275}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-288}:\\ \;\;\;\;t \cdot \sqrt{\frac{x}{\ell \cdot \ell}}\\ \mathbf{elif}\;t \leq 1.66 \cdot 10^{-209}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+55}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Accuracy	46.0%
Cost	7244

\[\begin{array}{l} t_1 := -\sqrt{x}\\ \mathbf{if}\;t \leq -5.5 \cdot 10^{-50}:\\ \;\;\;\;\frac{t_1}{\frac{\ell}{t}}\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{-141}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-287}:\\ \;\;\;\;t \cdot \frac{t_1}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 5
Accuracy	44.9%
Cost	7180

\[\begin{array}{l} t_1 := \sqrt{x} \cdot \frac{-t}{\ell}\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{-47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -5.2 \cdot 10^{-141}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{-292}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Accuracy	45.3%
Cost	7180

\[\begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-49}:\\ \;\;\;\;\sqrt{x} \cdot \frac{-t}{\ell}\\ \mathbf{elif}\;t \leq -3.8 \cdot 10^{-140}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-293}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Accuracy	45.4%
Cost	7180

\[\begin{array}{l} t_1 := -\sqrt{x}\\ \mathbf{if}\;t \leq -4 \cdot 10^{-48}:\\ \;\;\;\;\frac{t_1}{\frac{\ell}{t}}\\ \mathbf{elif}\;t \leq -1.7 \cdot 10^{-140}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-287}:\\ \;\;\;\;t \cdot \frac{t_1}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Accuracy	46.6%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq -2.15 \cdot 10^{-94}:\\ \;\;\;\;\frac{-\sqrt{x}}{\frac{\ell}{t}}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-287}:\\ \;\;\;\;t \cdot \sqrt{\frac{x}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 9
Accuracy	77.9%
Cost	7112

\[\begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -1.06 \cdot 10^{-274}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-287}:\\ \;\;\;\;t \cdot \sqrt{\frac{x}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Accuracy	48.0%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;t \leq 4.6 \cdot 10^{-290}:\\ \;\;\;\;t \cdot \sqrt{\frac{\frac{x}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 11
Accuracy	44.6%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;t \leq 6.7 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12
Accuracy	45.4%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;t \leq 2.15 \cdot 10^{-305}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13
Accuracy	45.4%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-306}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14
Accuracy	38.8%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023133 
(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)))))