Toniolo and Linder, Equation (7)

Average Error: 43.2 → 11.2

Time: 49.4s

Precision: binary64

Cost: 28172

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \frac{\ell \cdot \ell}{x}\\ t_3 := 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\\ t_4 := \sqrt{\frac{x + 1}{x + -1}}\\ t_5 := 2 + \left(\frac{2}{x} + \frac{2}{x}\right)\\ \mathbf{if}\;t \leq -5.2 \cdot 10^{+53}:\\ \;\;\;\;\frac{t_1}{t_1 \cdot \left(-t_4\right)}\\ \mathbf{elif}\;t \leq -1.28 \cdot 10^{-204}:\\ \;\;\;\;\frac{t_1}{\sqrt{t_2 + \left(t_3 + \frac{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)}{x}\right)}}\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{-268}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{t_5}, \sqrt{\frac{1}{t_5}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{+16}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_2 + \left(t_2 + t_3\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_1 \cdot t_4}\\ \end{array} \]

FPCore

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

↓

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* t (sqrt 2.0)))
        (t_2 (/ (* l l) x))
        (t_3 (* 2.0 (+ (* t t) (/ (* t t) x))))
        (t_4 (sqrt (/ (+ x 1.0) (+ x -1.0))))
        (t_5 (+ 2.0 (+ (/ 2.0 x) (/ 2.0 x)))))
   (if (<= t -5.2e+53)
     (/ t_1 (* t_1 (- t_4)))
     (if (<= t -1.28e-204)
       (/ t_1 (sqrt (+ t_2 (+ t_3 (/ (fma l l (* 2.0 (* t t))) x)))))
       (if (<= t -2.3e-268)
         (*
          t
          (/
           (sqrt 2.0)
           (- (fma t (sqrt t_5) (* (sqrt (/ 1.0 t_5)) (* (/ l x) (/ l t)))))))
         (if (<= t 1.12e+16)
           (* t (/ (sqrt 2.0) (sqrt (+ t_2 (+ t_2 t_3)))))
           (/ t_1 (* t_1 t_4))))))))

C

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

↓

double code(double x, double l, double t) {
	double t_1 = t * sqrt(2.0);
	double t_2 = (l * l) / x;
	double t_3 = 2.0 * ((t * t) + ((t * t) / x));
	double t_4 = sqrt(((x + 1.0) / (x + -1.0)));
	double t_5 = 2.0 + ((2.0 / x) + (2.0 / x));
	double tmp;
	if (t <= -5.2e+53) {
		tmp = t_1 / (t_1 * -t_4);
	} else if (t <= -1.28e-204) {
		tmp = t_1 / sqrt((t_2 + (t_3 + (fma(l, l, (2.0 * (t * t))) / x))));
	} else if (t <= -2.3e-268) {
		tmp = t * (sqrt(2.0) / -fma(t, sqrt(t_5), (sqrt((1.0 / t_5)) * ((l / x) * (l / t)))));
	} else if (t <= 1.12e+16) {
		tmp = t * (sqrt(2.0) / sqrt((t_2 + (t_2 + t_3))));
	} else {
		tmp = t_1 / (t_1 * t_4);
	}
	return tmp;
}

Julia

function code(x, l, t)
	return Float64(Float64(sqrt(2.0) * t) / sqrt(Float64(Float64(Float64(Float64(x + 1.0) / Float64(x - 1.0)) * Float64(Float64(l * l) + Float64(2.0 * Float64(t * t)))) - Float64(l * l))))
end

↓

function code(x, l, t)
	t_1 = Float64(t * sqrt(2.0))
	t_2 = Float64(Float64(l * l) / x)
	t_3 = Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x)))
	t_4 = sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0)))
	t_5 = Float64(2.0 + Float64(Float64(2.0 / x) + Float64(2.0 / x)))
	tmp = 0.0
	if (t <= -5.2e+53)
		tmp = Float64(t_1 / Float64(t_1 * Float64(-t_4)));
	elseif (t <= -1.28e-204)
		tmp = Float64(t_1 / sqrt(Float64(t_2 + Float64(t_3 + Float64(fma(l, l, Float64(2.0 * Float64(t * t))) / x)))));
	elseif (t <= -2.3e-268)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(-fma(t, sqrt(t_5), Float64(sqrt(Float64(1.0 / t_5)) * Float64(Float64(l / x) * Float64(l / t)))))));
	elseif (t <= 1.12e+16)
		tmp = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(t_2 + Float64(t_2 + t_3)))));
	else
		tmp = Float64(t_1 / Float64(t_1 * t_4));
	end
	return tmp
end

Wolfram

code[x_, l_, t_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(2.0 + N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5.2e+53], N[(t$95$1 / N[(t$95$1 * (-t$95$4)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.28e-204], N[(t$95$1 / N[Sqrt[N[(t$95$2 + N[(t$95$3 + N[(N[(l * l + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -2.3e-268], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / (-N[(t * N[Sqrt[t$95$5], $MachinePrecision] + N[(N[Sqrt[N[(1.0 / t$95$5), $MachinePrecision]], $MachinePrecision] * N[(N[(l / x), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.12e+16], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(t$95$2 + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(t$95$1 * t$95$4), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -5.19999999999999996e53

   1. Initial program 46.4

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

   2. Simplified 46.4

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

      [Start] 46.4	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      *-lft-identity [<=] 46.4	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(1 \cdot \frac{x + 1}{x - 1}\right)} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      *-lft-identity [=>] 46.4	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{x + 1}{x - 1}} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      sub-neg [=>] 46.4	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      metadata-eval [=>] 46.4	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      fma-def [=>] 46.4	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}} \]

   3. Taylor expanded in t around -inf 4.1

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

   4. Simplified 4.1

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

      [Start] 4.1	\[ \frac{\sqrt{2} \cdot t}{-1 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)} \]
      associate-*r* [=>] 4.1	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{\left(-1 \cdot \left(\sqrt{2} \cdot t\right)\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
      *-commutative [<=] 4.1	\[ \frac{\sqrt{2} \cdot t}{\left(-1 \cdot \color{blue}{\left(t \cdot \sqrt{2}\right)}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}} \]
      associate-*r* [=>] 4.1	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{\left(\left(-1 \cdot t\right) \cdot \sqrt{2}\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]
      neg-mul-1 [<=] 4.1	\[ \frac{\sqrt{2} \cdot t}{\left(\color{blue}{\left(-t\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}} \]
      *-commutative [<=] 4.1	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{\left(\sqrt{2} \cdot \left(-t\right)\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]
      sub-neg [=>] 4.1	\[ \frac{\sqrt{2} \cdot t}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{1 + x}{\color{blue}{x + \left(-1\right)}}}} \]
      metadata-eval [=>] 4.1	\[ \frac{\sqrt{2} \cdot t}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{1 + x}{x + \color{blue}{-1}}}} \]
      +-commutative [=>] 4.1	\[ \frac{\sqrt{2} \cdot t}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x + -1}}} \]
      +-commutative [=>] 4.1	\[ \frac{\sqrt{2} \cdot t}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

   if -5.19999999999999996e53 < t < -1.28000000000000004e-204

   1. Initial program 33.5

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

   2. Simplified 33.5

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

      [Start] 33.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      *-lft-identity [<=] 33.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(1 \cdot \frac{x + 1}{x - 1}\right)} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      *-lft-identity [=>] 33.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{x + 1}{x - 1}} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      sub-neg [=>] 33.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      metadata-eval [=>] 33.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      fma-def [=>] 33.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}} \]

   3. Taylor expanded in x around inf 13.7

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

   4. Simplified 13.7

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

      [Start] 13.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      associate--l+ [=>] 13.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]
      unpow2 [=>] 13.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      distribute-lft-out [=>] 13.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 13.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 13.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-*r/ [=>] 13.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \]
      mul-1-neg [=>] 13.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \]
      unpow2 [=>] 13.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}\right)}{x}\right)}} \]
      unpow2 [=>] 13.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(\ell \cdot \ell + 2 \cdot \color{blue}{\left(t \cdot t\right)}\right)}{x}\right)}} \]
      fma-udef [<=] 13.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)}}{x}\right)}} \]

   if -1.28000000000000004e-204 < t < -2.3000000000000001e-268

   1. Initial program 62.6

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

   2. Simplified 62.6

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

      [Start] 62.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 62.6	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 62.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 62.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Taylor expanded in x around inf 34.4

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

   4. Simplified 34.4

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

      [Start] 34.4	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 34.4	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 34.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 34.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 34.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 34.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 34.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      +-commutative [=>] 34.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 34.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 34.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 34.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)\right)}} \cdot t \]

   5. Taylor expanded in t around -inf 26.6

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

   6. Simplified 26.7

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

      [Start] 26.6	\[ \frac{\sqrt{2}}{-1 \cdot \left(\frac{{\ell}^{2}}{t \cdot x} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}\right) + -1 \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)} \cdot t \]
      *-commutative [=>] 26.6	\[ \frac{\sqrt{2}}{-1 \cdot \color{blue}{\left(\sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} + -1 \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)} \cdot t \]
      *-commutative [=>] 26.6	\[ \frac{\sqrt{2}}{-1 \cdot \left(\sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right) + -1 \cdot \color{blue}{\left(\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t\right)}} \cdot t \]
      distribute-lft-out [=>] 26.6	\[ \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x} + \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t\right)}} \cdot t \]
      +-commutative [<=] 26.6	\[ \frac{\sqrt{2}}{-1 \cdot \color{blue}{\left(\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)}} \cdot t \]
      mul-1-neg [=>] 26.6	\[ \frac{\sqrt{2}}{\color{blue}{-\left(\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)}} \cdot t \]

   if -2.3000000000000001e-268 < t < 1.12e16

   1. Initial program 45.3

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

   2. Simplified 45.2

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

      [Start] 45.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 45.2	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 45.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 45.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Taylor expanded in x around inf 20.8

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

   4. Simplified 20.8

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

      [Start] 20.8	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 20.8	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 20.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 20.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 20.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 20.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 20.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      +-commutative [=>] 20.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 20.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 20.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 20.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)\right)}} \cdot t \]

   5. Taylor expanded in t around 0 20.9

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

   6. Simplified 20.9

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

      [Start] 20.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{{\ell}^{2}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 20.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]

   if 1.12e16 < t

   1. Initial program 42.6

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

   2. Taylor expanded in l around 0 4.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 11.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{+53}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq -1.28 \cdot 10^{-204}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)}{x}\right)}}\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{-268}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{+16}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	11.6
Cost	21448

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \frac{\ell \cdot \ell}{x}\\ t_3 := \sqrt{\frac{x + 1}{x + -1}}\\ t_4 := \frac{t_1}{t_1 \cdot \left(-t_3\right)}\\ t_5 := 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\\ \mathbf{if}\;t \leq -5.2 \cdot 10^{+53}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq -4.2 \cdot 10^{-210}:\\ \;\;\;\;\frac{t_1}{\sqrt{t_2 + \left(t_5 + \frac{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)}{x}\right)}}\\ \mathbf{elif}\;t \leq -5.4 \cdot 10^{-265}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 10^{+16}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_2 + \left(t_2 + t_5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_1 \cdot t_3}\\ \end{array} \]

Alternative 2
Error	11.8
Cost	20688

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := t \cdot \frac{\sqrt{2}}{\left(-t\right) \cdot \sqrt{\frac{2}{x} + \left(2 + \frac{2}{x}\right)}}\\ t_3 := \frac{\ell \cdot \ell}{x}\\ \mathbf{if}\;t \leq -1.95 \cdot 10^{+53}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -3.4 \cdot 10^{-209}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_3 + \left(2 \cdot \left(t \cdot t\right) - \frac{t \cdot \left(t \cdot -4\right) - \ell \cdot \ell}{x}\right)}}\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-263}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+14}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_3 + \left(t_3 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_1 \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \]

Alternative 3
Error	11.6
Cost	20688

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \frac{\ell \cdot \ell}{x}\\ t_3 := \sqrt{\frac{x + 1}{x + -1}}\\ t_4 := \frac{t_1}{t_1 \cdot \left(-t_3\right)}\\ \mathbf{if}\;t \leq -2.8 \cdot 10^{+52}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq -8.8 \cdot 10^{-210}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_2 + \left(2 \cdot \left(t \cdot t\right) - \frac{t \cdot \left(t \cdot -4\right) - \ell \cdot \ell}{x}\right)}}\\ \mathbf{elif}\;t \leq -5.5 \cdot 10^{-265}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 6.3 \cdot 10^{+15}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_2 + \left(t_2 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_1 \cdot t_3}\\ \end{array} \]

Alternative 4
Error	12.0
Cost	15056

\[\begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t\right) - \frac{t \cdot \left(t \cdot -4\right) - \ell \cdot \ell}{x}\right)}}\\ t_2 := \sqrt{\frac{2}{x} + \left(2 + \frac{2}{x}\right)}\\ t_3 := t \cdot \frac{\sqrt{2}}{\left(-t\right) \cdot t_2}\\ \mathbf{if}\;t \leq -1.35 \cdot 10^{+54}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -3.3 \cdot 10^{-207}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-263}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t \cdot t_2}\\ \end{array} \]

Alternative 5
Error	12.0
Cost	15056

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{x}\\ t_2 := \sqrt{\frac{2}{x} + \left(2 + \frac{2}{x}\right)}\\ t_3 := t \cdot \frac{\sqrt{2}}{\left(-t\right) \cdot t_2}\\ \mathbf{if}\;t \leq -1.8 \cdot 10^{+55}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -4.6 \cdot 10^{-206}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_1 + \left(2 \cdot \left(t \cdot t\right) - \frac{t \cdot \left(t \cdot -4\right) - \ell \cdot \ell}{x}\right)}}\\ \mathbf{elif}\;t \leq -6.5 \cdot 10^{-266}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+14}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_1 + \left(t_1 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t \cdot t_2}\\ \end{array} \]

Alternative 6
Error	14.6
Cost	14288

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := t_1 \cdot \frac{-\sqrt{0.5}}{t}\\ \mathbf{if}\;t \leq -2.2 \cdot 10^{+59}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -9 \cdot 10^{-158}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t\right)}}\\ \mathbf{elif}\;t \leq -3.4 \cdot 10^{-265}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-227}:\\ \;\;\;\;\frac{t_1}{\sqrt{\frac{2 \cdot \left(\ell \cdot \ell\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t \cdot \sqrt{\frac{2}{x} + \left(2 + \frac{2}{x}\right)}}\\ \end{array} \]

Alternative 7
Error	14.4
Cost	14288

\[\begin{array}{l} t_1 := \sqrt{\frac{2}{x} + \left(2 + \frac{2}{x}\right)}\\ t_2 := t \cdot \frac{\sqrt{2}}{\left(-t\right) \cdot t_1}\\ \mathbf{if}\;t \leq -2.2 \cdot 10^{+53}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -3.7 \cdot 10^{-156}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t\right)}}\\ \mathbf{elif}\;t \leq -1.25 \cdot 10^{-268}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-227}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{2 \cdot \left(\ell \cdot \ell\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t \cdot t_1}\\ \end{array} \]

Alternative 8
Error	15.2
Cost	14032

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := t_1 \cdot \frac{-\sqrt{0.5}}{t}\\ \mathbf{if}\;t \leq -2 \cdot 10^{-121}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -4.4 \cdot 10^{-142}:\\ \;\;\;\;t \cdot \sqrt{\frac{1}{\ell} \cdot \frac{x}{\ell}}\\ \mathbf{elif}\;t \leq -1.15 \cdot 10^{-263}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-233}:\\ \;\;\;\;\frac{t_1}{\sqrt{\frac{2 \cdot \left(\ell \cdot \ell\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Error	14.7
Cost	14032

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := t_1 \cdot \frac{-\sqrt{0.5}}{t}\\ \mathbf{if}\;t \leq -1.15 \cdot 10^{+57}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-157}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t\right)}}\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-263}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-227}:\\ \;\;\;\;\frac{t_1}{\sqrt{\frac{2 \cdot \left(\ell \cdot \ell\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	15.3
Cost	13708

\[\begin{array}{l} t_1 := \left(t \cdot \sqrt{2}\right) \cdot \frac{-\sqrt{0.5}}{t}\\ \mathbf{if}\;t \leq -4.6 \cdot 10^{-120}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.95 \cdot 10^{-142}:\\ \;\;\;\;t \cdot \sqrt{\frac{1}{\ell} \cdot \frac{x}{\ell}}\\ \mathbf{elif}\;t \leq -1.3 \cdot 10^{-244}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-227}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Error	15.5
Cost	13452

\[\begin{array}{l} t_1 := \sqrt{2} \cdot \left(-\sqrt{0.5}\right)\\ \mathbf{if}\;t \leq -1.7 \cdot 10^{-121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3 \cdot 10^{-142}:\\ \;\;\;\;t \cdot \sqrt{\frac{1}{\ell} \cdot \frac{x}{\ell}}\\ \mathbf{elif}\;t \leq -3.9 \cdot 10^{-246}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{-231}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12
Error	33.8
Cost	6984

\[\begin{array}{l} \mathbf{if}\;\ell \leq -7.5 \cdot 10^{+60}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{-8}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \end{array} \]

Alternative 13
Error	33.8
Cost	6984

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{+60}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \end{array} \]

Alternative 14
Error	34.8
Cost	6852

\[\begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 15
Error	34.4
Cost	6852

\[\begin{array}{l} \mathbf{if}\;t \leq 6.2 \cdot 10^{-231}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 16
Error	36.9
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{-253}:\\ \;\;\;\;t \cdot \left(\frac{1}{\ell} + \frac{\frac{-0.5}{\ell}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 17
Error	36.9
Cost	324

\[\begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-253}:\\ \;\;\;\;\frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 18
Error	38.7
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023039 
(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)))))