Toniolo and Linder, Equation (7)

Average Accuracy: 32.7% → 90.3%

Time: 33.1s

Precision: binary64

Cost: 52620

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ t_2 := \sqrt{\frac{x + 1}{x + -1}}\\ t_3 := \frac{\ell}{\sqrt{x}}\\ \mathbf{if}\;t \leq -3.55 \cdot 10^{+121}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2} \cdot \left(-t_2\right)}\\ \mathbf{elif}\;t \leq -3.3 \cdot 10^{-139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{-144}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{hypot}\left(\mathsf{hypot}\left(\sqrt{2} \cdot \mathsf{hypot}\left(t, \frac{t}{\sqrt{x}}\right), t_3\right), t_3\right)}\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2} \cdot t_2}\\ \end{array} \]

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
         (*
          t
          (/ (sqrt 2.0) (sqrt (* 2.0 (+ (* l (/ l x)) (* t (+ t (/ t x)))))))))
        (t_2 (sqrt (/ (+ x 1.0) (+ x -1.0))))
        (t_3 (/ l (sqrt x))))
   (if (<= t -3.55e+121)
     (/ (sqrt 2.0) (* (sqrt 2.0) (- t_2)))
     (if (<= t -3.3e-139)
       t_1
       (if (<= t 2.15e-144)
         (*
          t
          (/
           (sqrt 2.0)
           (hypot (hypot (* (sqrt 2.0) (hypot t (/ t (sqrt x)))) t_3) t_3)))
         (if (<= t 8.8e+16) t_1 (/ (sqrt 2.0) (* (sqrt 2.0) 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 = t * (sqrt(2.0) / sqrt((2.0 * ((l * (l / x)) + (t * (t + (t / x)))))));
	double t_2 = sqrt(((x + 1.0) / (x + -1.0)));
	double t_3 = l / sqrt(x);
	double tmp;
	if (t <= -3.55e+121) {
		tmp = sqrt(2.0) / (sqrt(2.0) * -t_2);
	} else if (t <= -3.3e-139) {
		tmp = t_1;
	} else if (t <= 2.15e-144) {
		tmp = t * (sqrt(2.0) / hypot(hypot((sqrt(2.0) * hypot(t, (t / sqrt(x)))), t_3), t_3));
	} else if (t <= 8.8e+16) {
		tmp = t_1;
	} else {
		tmp = sqrt(2.0) / (sqrt(2.0) * t_2);
	}
	return tmp;
}

Java

public static double code(double x, double l, double t) {
	return (Math.sqrt(2.0) * t) / Math.sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
}

↓

public static double code(double x, double l, double t) {
	double t_1 = t * (Math.sqrt(2.0) / Math.sqrt((2.0 * ((l * (l / x)) + (t * (t + (t / x)))))));
	double t_2 = Math.sqrt(((x + 1.0) / (x + -1.0)));
	double t_3 = l / Math.sqrt(x);
	double tmp;
	if (t <= -3.55e+121) {
		tmp = Math.sqrt(2.0) / (Math.sqrt(2.0) * -t_2);
	} else if (t <= -3.3e-139) {
		tmp = t_1;
	} else if (t <= 2.15e-144) {
		tmp = t * (Math.sqrt(2.0) / Math.hypot(Math.hypot((Math.sqrt(2.0) * Math.hypot(t, (t / Math.sqrt(x)))), t_3), t_3));
	} else if (t <= 8.8e+16) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt(2.0) / (Math.sqrt(2.0) * 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 = t * (math.sqrt(2.0) / math.sqrt((2.0 * ((l * (l / x)) + (t * (t + (t / x)))))))
	t_2 = math.sqrt(((x + 1.0) / (x + -1.0)))
	t_3 = l / math.sqrt(x)
	tmp = 0
	if t <= -3.55e+121:
		tmp = math.sqrt(2.0) / (math.sqrt(2.0) * -t_2)
	elif t <= -3.3e-139:
		tmp = t_1
	elif t <= 2.15e-144:
		tmp = t * (math.sqrt(2.0) / math.hypot(math.hypot((math.sqrt(2.0) * math.hypot(t, (t / math.sqrt(x)))), t_3), t_3))
	elif t <= 8.8e+16:
		tmp = t_1
	else:
		tmp = math.sqrt(2.0) / (math.sqrt(2.0) * 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(t * Float64(sqrt(2.0) / sqrt(Float64(2.0 * Float64(Float64(l * Float64(l / x)) + Float64(t * Float64(t + Float64(t / x))))))))
	t_2 = sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0)))
	t_3 = Float64(l / sqrt(x))
	tmp = 0.0
	if (t <= -3.55e+121)
		tmp = Float64(sqrt(2.0) / Float64(sqrt(2.0) * Float64(-t_2)));
	elseif (t <= -3.3e-139)
		tmp = t_1;
	elseif (t <= 2.15e-144)
		tmp = Float64(t * Float64(sqrt(2.0) / hypot(hypot(Float64(sqrt(2.0) * hypot(t, Float64(t / sqrt(x)))), t_3), t_3)));
	elseif (t <= 8.8e+16)
		tmp = t_1;
	else
		tmp = Float64(sqrt(2.0) / Float64(sqrt(2.0) * 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 = t * (sqrt(2.0) / sqrt((2.0 * ((l * (l / x)) + (t * (t + (t / x)))))));
	t_2 = sqrt(((x + 1.0) / (x + -1.0)));
	t_3 = l / sqrt(x);
	tmp = 0.0;
	if (t <= -3.55e+121)
		tmp = sqrt(2.0) / (sqrt(2.0) * -t_2);
	elseif (t <= -3.3e-139)
		tmp = t_1;
	elseif (t <= 2.15e-144)
		tmp = t * (sqrt(2.0) / hypot(hypot((sqrt(2.0) * hypot(t, (t / sqrt(x)))), t_3), t_3));
	elseif (t <= 8.8e+16)
		tmp = t_1;
	else
		tmp = sqrt(2.0) / (sqrt(2.0) * 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[(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]}, Block[{t$95$2 = N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(l / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3.55e+121], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * (-t$95$2)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -3.3e-139], t$95$1, If[LessEqual[t, 2.15e-144], N[(t * N[(N[Sqrt[2.0], $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$3 ^ 2], $MachinePrecision] ^ 2 + t$95$3 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.8e+16], t$95$1, N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -3.55000000000000012e121

   1. Initial program 18.9%

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

   2. Simplified 18.9%

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

      [Start] 18.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-/l* [=>] 18.9	\[ \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]
      fma-neg [=>] 18.9	\[ \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}}{t}} \]
      remove-double-neg [<=] 18.9	\[ \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}}{t}} \]
      fma-neg [<=] 18.9	\[ \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}}{t}} \]
      sub-neg [=>] 18.9	\[ \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]
      metadata-eval [=>] 18.9	\[ \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]
      remove-double-neg [=>] 18.9	\[ \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]
      fma-def [=>] 18.9	\[ \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

   3. Taylor expanded in t around -inf 98.3%

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

   4. Simplified 98.3%

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

      [Start] 98.3	\[ \frac{\sqrt{2}}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}\right)} \]
      mul-1-neg [=>] 98.3	\[ \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
      *-commutative [=>] 98.3	\[ \frac{\sqrt{2}}{-\color{blue}{\sqrt{\frac{1 + x}{x - 1}} \cdot \sqrt{2}}} \]
      distribute-rgt-neg-in [=>] 98.3	\[ \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{1 + x}{x - 1}} \cdot \left(-\sqrt{2}\right)}} \]
      +-commutative [=>] 98.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{x + 1}}{x - 1}} \cdot \left(-\sqrt{2}\right)} \]
      sub-neg [=>] 98.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}} \cdot \left(-\sqrt{2}\right)} \]
      metadata-eval [=>] 98.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}}} \cdot \left(-\sqrt{2}\right)} \]
      +-commutative [=>] 98.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}} \cdot \left(-\sqrt{2}\right)} \]

   if -3.55000000000000012e121 < t < -3.3e-139 or 2.14999999999999995e-144 < t < 8.8e16

   1. Initial program 55.3%

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

   2. Simplified 55.4%

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

      [Start] 55.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 55.4	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Taylor expanded in x around inf 82.8%

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

   4. Simplified 82.8%

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

      [Start] 82.8	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 82.8	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 82.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 82.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 82.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 82.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 82.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      unpow2 [=>] 82.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}}{x}\right)\right)}} \cdot t \]
      +-commutative [=>] 82.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + \ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 82.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 82.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]

   5. Taylor expanded in t around 0 82.6%

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

   6. Simplified 82.6%

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

      [Start] 82.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{{\ell}^{2}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 82.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]

   7. Applied egg-rr 90.7%

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

      [Start] 82.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)\right)}} \cdot t \]
      pow1/2 [=>] 82.6	\[ \frac{\sqrt{2}}{\color{blue}{{\left(\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)\right)\right)}^{0.5}}} \cdot t \]
      sqr-pow [=>] 82.4	\[ \frac{\sqrt{2}}{\color{blue}{{\left(\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}}} \cdot t \]

   8. Simplified 90.8%

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

      [Start] 90.7	\[ \frac{\sqrt{2}}{{\left(\mathsf{fma}\left(2, \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right), \mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right)\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(2, \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right), \mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right)\right)\right)}^{0.25}} \cdot t \]
      pow-sqr [=>] 90.8	\[ \frac{\sqrt{2}}{\color{blue}{{\left(\mathsf{fma}\left(2, \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right), \mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right)\right)\right)}^{\left(2 \cdot 0.25\right)}}} \cdot t \]
      metadata-eval [=>] 90.8	\[ \frac{\sqrt{2}}{{\left(\mathsf{fma}\left(2, \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right), \mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right)\right)\right)}^{\color{blue}{0.5}}} \cdot t \]
      unpow1/2 [=>] 90.8	\[ \frac{\sqrt{2}}{\color{blue}{\sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right), \mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right)\right)}}} \cdot t \]
      fma-udef [=>] 90.8	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right) + \mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right)}}} \cdot t \]
      +-commutative [=>] 90.8	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right) + 2 \cdot \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right)}}} \cdot t \]
      fma-udef [=>] 90.8	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\frac{\ell}{x} \cdot \ell + \frac{\ell}{x} \cdot \ell\right)} + 2 \cdot \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right)}} \cdot t \]
      count-2 [=>] 90.8	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \left(\frac{\ell}{x} \cdot \ell\right)} + 2 \cdot \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right)}} \cdot t \]
      distribute-lft-out [=>] 90.8	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right)\right)}}} \cdot t \]
      *-commutative [=>] 90.8	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\color{blue}{\ell \cdot \frac{\ell}{x}} + \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right)\right)}} \cdot t \]
      fma-udef [=>] 90.8	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + \color{blue}{\left(\frac{t}{x} \cdot t + t \cdot t\right)}\right)}} \cdot t \]
      +-commutative [=>] 90.8	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + \color{blue}{\left(t \cdot t + \frac{t}{x} \cdot t\right)}\right)}} \cdot t \]
      distribute-rgt-out [=>] 90.8	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + \color{blue}{t \cdot \left(t + \frac{t}{x}\right)}\right)}} \cdot t \]

   if -3.3e-139 < t < 2.14999999999999995e-144

   1. Initial program 11.0%

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

   2. Simplified 11.1%

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

      [Start] 11.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 11.1	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Taylor expanded in x around inf 65.6%

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

   4. Simplified 65.6%

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

      [Start] 65.6	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 65.6	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 65.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 65.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 65.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 65.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 65.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      unpow2 [=>] 65.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}}{x}\right)\right)}} \cdot t \]
      +-commutative [=>] 65.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + \ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 65.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 65.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]

   5. Taylor expanded in t around 0 65.6%

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

   6. Simplified 65.6%

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

      [Start] 65.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{{\ell}^{2}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 65.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]

   7. Applied egg-rr 93.7%

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

      [Start] 65.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)\right)}} \cdot t \]
      +-commutative [=>] 65.6	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)\right) + \frac{\ell \cdot \ell}{x}}}} \cdot t \]
      add-sqr-sqrt [=>] 65.5	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\sqrt{2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)} \cdot \sqrt{2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)}} + \frac{\ell \cdot \ell}{x}}} \cdot t \]
      add-sqr-sqrt [=>] 65.5	\[ \frac{\sqrt{2}}{\sqrt{\sqrt{2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)} \cdot \sqrt{2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)} + \color{blue}{\sqrt{\frac{\ell \cdot \ell}{x}} \cdot \sqrt{\frac{\ell \cdot \ell}{x}}}}} \cdot t \]
      hypot-def [=>] 65.5	\[ \frac{\sqrt{2}}{\color{blue}{\mathsf{hypot}\left(\sqrt{2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)}, \sqrt{\frac{\ell \cdot \ell}{x}}\right)}} \cdot t \]

   if 8.8e16 < t

   1. Initial program 37.7%

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

   2. Simplified 37.6%

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

      [Start] 37.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-/l* [=>] 37.6	\[ \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]
      fma-neg [=>] 37.6	\[ \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}}{t}} \]
      remove-double-neg [<=] 37.6	\[ \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}}{t}} \]
      fma-neg [<=] 37.6	\[ \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}}{t}} \]
      sub-neg [=>] 37.6	\[ \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]
      metadata-eval [=>] 37.6	\[ \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]
      remove-double-neg [=>] 37.6	\[ \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]
      fma-def [=>] 37.6	\[ \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

   3. Taylor expanded in l around 0 94.6%

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

   4. Simplified 94.6%

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

      [Start] 94.6	\[ \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]
      +-commutative [=>] 94.6	\[ \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]
      sub-neg [=>] 94.6	\[ \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]
      metadata-eval [=>] 94.6	\[ \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]
      +-commutative [=>] 94.6	\[ \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.55 \cdot 10^{+121}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq -3.3 \cdot 10^{-139}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{-144}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{hypot}\left(\mathsf{hypot}\left(\sqrt{2} \cdot \mathsf{hypot}\left(t, \frac{t}{\sqrt{x}}\right), \frac{\ell}{\sqrt{x}}\right), \frac{\ell}{\sqrt{x}}\right)}\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{+16}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	90.4%
Cost	33164

\[\begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ t_2 := \sqrt{\frac{x + 1}{x + -1}}\\ \mathbf{if}\;t \leq -1.6 \cdot 10^{+121}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2} \cdot \left(-t_2\right)}\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{-144}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\mathsf{hypot}\left(\sqrt{2} \cdot \frac{\ell}{\sqrt{x}}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2} \cdot t_2}\\ \end{array} \]

Alternative 2
Accuracy	85.3%
Cost	20168

\[\begin{array}{l} \mathbf{if}\;t \leq -1.95 \cdot 10^{+121}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 + \frac{4}{x}} \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+16}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \]

Alternative 3
Accuracy	85.5%
Cost	20168

\[\begin{array}{l} t_1 := \sqrt{\frac{x + 1}{x + -1}}\\ \mathbf{if}\;t \leq -3 \cdot 10^{+121}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2} \cdot \left(-t_1\right)}\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{+16}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2} \cdot t_1}\\ \end{array} \]

Alternative 4
Accuracy	85.0%
Cost	14280

\[\begin{array}{l} t_1 := \sqrt{2 + \frac{4}{x}}\\ \mathbf{if}\;t \leq -1.76 \cdot 10^{+121}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{t_1 \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{+16}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{t \cdot t_1}\\ \end{array} \]

Alternative 5
Accuracy	84.9%
Cost	14024

\[\begin{array}{l} t_1 := \sqrt{2 + \frac{4}{x}}\\ \mathbf{if}\;t \leq -7.6 \cdot 10^{+120}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{t_1 \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{+16}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{t \cdot t_1}\\ \end{array} \]

Alternative 6
Accuracy	77.2%
Cost	13768

\[\begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{-203}:\\ \;\;\;\;\frac{\sqrt{2}}{-\sqrt{2}}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-193}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Accuracy	77.2%
Cost	13768

\[\begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{-203}:\\ \;\;\;\;\frac{\sqrt{2}}{-\sqrt{2}}\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{-215}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\ell \cdot \left(\ell \cdot \frac{2}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Accuracy	77.5%
Cost	13768

\[\begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{-203}:\\ \;\;\;\;\frac{\sqrt{2}}{-\sqrt{2}}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-187}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\ell \cdot \left(\ell \cdot \frac{2}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{t \cdot \sqrt{2 + \frac{4}{x}}}\\ \end{array} \]

Alternative 9
Accuracy	77.7%
Cost	13768

\[\begin{array}{l} t_1 := \sqrt{2 + \frac{4}{x}}\\ \mathbf{if}\;t \leq -1.8 \cdot 10^{-212}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{t_1 \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-207}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\ell \cdot \left(\ell \cdot \frac{2}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{t \cdot t_1}\\ \end{array} \]

Alternative 10
Accuracy	76.6%
Cost	13704

\[\begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{-211}:\\ \;\;\;\;\frac{\sqrt{2}}{-\sqrt{2}}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-211}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\left(-\ell\right) \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Accuracy	76.7%
Cost	13188

\[\begin{array}{l} \mathbf{if}\;t \leq -1.32 \cdot 10^{-213}:\\ \;\;\;\;\frac{\sqrt{2}}{-\sqrt{2}}\\ \mathbf{elif}\;t \leq 4.9 \cdot 10^{-216}:\\ \;\;\;\;\frac{\frac{t}{{x}^{-0.5}}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12
Accuracy	45.3%
Cost	7116

\[\begin{array}{l} t_1 := \sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{if}\;t \leq -3.4 \cdot 10^{-35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.55 \cdot 10^{-185}:\\ \;\;\;\;\sqrt{x} \cdot \frac{-t}{\ell}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-217}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13
Accuracy	44.7%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.7 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{x} \cdot \frac{-t}{\ell}\\ \mathbf{elif}\;\ell \leq 1.38 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t}{{x}^{-0.5}}}{\ell}\\ \end{array} \]

Alternative 14
Accuracy	61.9%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-155}:\\ \;\;\;\;t \cdot \sqrt{\frac{1}{t \cdot t}}\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-216}:\\ \;\;\;\;\frac{\frac{t}{{x}^{-0.5}}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 15
Accuracy	45.5%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;t \leq 1.22 \cdot 10^{-216}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 16
Accuracy	41.1%
Cost	6724

\[\begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{-217}:\\ \;\;\;\;t \cdot \left|t\right|\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 17
Accuracy	40.6%
Cost	580

\[\begin{array}{l} \mathbf{if}\;t \leq 10^{-216}:\\ \;\;\;\;t \cdot \left(-1 + \left(t + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 18
Accuracy	40.6%
Cost	580

\[\begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{-217}:\\ \;\;\;\;-1 + \left(1 + t \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 19
Accuracy	40.2%
Cost	324

\[\begin{array}{l} \mathbf{if}\;t \leq 7.4 \cdot 10^{-217}:\\ \;\;\;\;t \cdot t\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 20
Accuracy	38.8%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023153 
(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)))))