Toniolo and Linder, Equation (7)

Average Error: 43.3 → 8.9

Time: 29.8s

Precision: binary64

Cost: 28172

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{x}{\ell}}\\ t_2 := 2 + \left(\frac{2}{x} + \frac{2}{x}\right)\\ t_3 := \ell \cdot \frac{\ell}{x}\\ \mathbf{if}\;t \leq -1.3 \cdot 10^{+109}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq -1.35 \cdot 10^{-177}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{t_1 + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + t_3\right)}{2}}}\\ \mathbf{elif}\;t \leq -7.6 \cdot 10^{-273}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{t_2}, \sqrt{\frac{1}{t_2}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{+94}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{t_1 + \left(t_3 + 2 \cdot \left(t \cdot t\right)\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ l (/ x l)))
        (t_2 (+ 2.0 (+ (/ 2.0 x) (/ 2.0 x))))
        (t_3 (* l (/ l x))))
   (if (<= t -1.3e+109)
     (+ -1.0 (/ 1.0 x))
     (if (<= t -1.35e-177)
       (/ t (sqrt (/ (+ t_1 (+ (* 2.0 (+ (* t t) (/ t (/ x t)))) t_3)) 2.0)))
       (if (<= t -7.6e-273)
         (*
          t
          (/
           (sqrt 2.0)
           (- (fma t (sqrt t_2) (* (sqrt (/ 1.0 t_2)) (* (/ l x) (/ l t)))))))
         (if (<= t 2.4e+94)
           (/ t (sqrt (/ (+ t_1 (+ t_3 (* 2.0 (* t t)))) 2.0)))
           (+ 1.0 (/ -1.0 x))))))))

C

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

↓

double code(double x, double l, double t) {
	double t_1 = l / (x / l);
	double t_2 = 2.0 + ((2.0 / x) + (2.0 / x));
	double t_3 = l * (l / x);
	double tmp;
	if (t <= -1.3e+109) {
		tmp = -1.0 + (1.0 / x);
	} else if (t <= -1.35e-177) {
		tmp = t / sqrt(((t_1 + ((2.0 * ((t * t) + (t / (x / t)))) + t_3)) / 2.0));
	} else if (t <= -7.6e-273) {
		tmp = t * (sqrt(2.0) / -fma(t, sqrt(t_2), (sqrt((1.0 / t_2)) * ((l / x) * (l / t)))));
	} else if (t <= 2.4e+94) {
		tmp = t / sqrt(((t_1 + (t_3 + (2.0 * (t * t)))) / 2.0));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Julia

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

↓

function code(x, l, t)
	t_1 = Float64(l / Float64(x / l))
	t_2 = Float64(2.0 + Float64(Float64(2.0 / x) + Float64(2.0 / x)))
	t_3 = Float64(l * Float64(l / x))
	tmp = 0.0
	if (t <= -1.3e+109)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	elseif (t <= -1.35e-177)
		tmp = Float64(t / sqrt(Float64(Float64(t_1 + Float64(Float64(2.0 * Float64(Float64(t * t) + Float64(t / Float64(x / t)))) + t_3)) / 2.0)));
	elseif (t <= -7.6e-273)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(-fma(t, sqrt(t_2), Float64(sqrt(Float64(1.0 / t_2)) * Float64(Float64(l / x) * Float64(l / t)))))));
	elseif (t <= 2.4e+94)
		tmp = Float64(t / sqrt(Float64(Float64(t_1 + Float64(t_3 + Float64(2.0 * Float64(t * t)))) / 2.0)));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	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[(l / N[(x / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.3e+109], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.35e-177], N[(t / N[Sqrt[N[(N[(t$95$1 + N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(t / N[(x / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -7.6e-273], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / (-N[(t * N[Sqrt[t$95$2], $MachinePrecision] + N[(N[Sqrt[N[(1.0 / t$95$2), $MachinePrecision]], $MachinePrecision] * N[(N[(l / x), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.4e+94], N[(t / N[Sqrt[N[(N[(t$95$1 + N[(t$95$3 + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -1.2999999999999999e109

   1. Initial program 53.4

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

   2. Simplified 53.4

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

      [Start] 53.4	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 53.4	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 53.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 53.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Applied egg-rr 53.3

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

   4. Taylor expanded in t around inf 63.0

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

   5. Taylor expanded in x around -inf 64.0

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

   6. Simplified 2.4

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

      [Start] 64.0	\[ {\left(\sqrt{-1}\right)}^{2} + \frac{1}{x} \]
      unpow2 [=>] 64.0	\[ \color{blue}{\sqrt{-1} \cdot \sqrt{-1}} + \frac{1}{x} \]
      rem-square-sqrt [=>] 2.4	\[ \color{blue}{-1} + \frac{1}{x} \]

   if -1.2999999999999999e109 < t < -1.3500000000000001e-177

   1. Initial program 28.5

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

   2. Simplified 28.5

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

      [Start] 28.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 28.5	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 28.5	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 28.5	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Applied egg-rr 28.3

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

   4. Taylor expanded in x around inf 11.5

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

   5. Simplified 11.5

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

      [Start] 11.5	\[ \frac{t}{\sqrt{\frac{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}{2}}} \]
      cancel-sign-sub-inv [=>] 11.5	\[ \frac{t}{\sqrt{\frac{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}{2}}} \]
      associate-+l+ [=>] 11.5	\[ \frac{t}{\sqrt{\frac{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}{2}}} \]
      unpow2 [=>] 11.5	\[ \frac{t}{\sqrt{\frac{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      associate-/l* [=>] 11.5	\[ \frac{t}{\sqrt{\frac{\color{blue}{\frac{\ell}{\frac{x}{\ell}}} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      distribute-lft-out [=>] 11.5	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      +-commutative [=>] 11.5	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \color{blue}{\left({t}^{2} + \frac{{t}^{2}}{x}\right)} + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      unpow2 [=>] 11.5	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(\color{blue}{t \cdot t} + \frac{{t}^{2}}{x}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      unpow2 [=>] 11.5	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{\color{blue}{t \cdot t}}{x}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      associate-/l* [=>] 11.5	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \color{blue}{\frac{t}{\frac{x}{t}}}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      metadata-eval [=>] 11.5	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + \color{blue}{1} \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      +-commutative [=>] 11.5	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + 1 \cdot \frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x}\right)}{2}}} \]
      unpow2 [=>] 11.5	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + 1 \cdot \frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x}\right)}{2}}} \]
      fma-def [=>] 11.5	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + 1 \cdot \frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)}{2}}} \]
      unpow2 [=>] 11.5	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + 1 \cdot \frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)}{2}}} \]

   6. Taylor expanded in l around inf 11.8

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

   7. Simplified 6.8

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

      [Start] 11.8	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + \frac{{\ell}^{2}}{x}\right)}{2}}} \]
      unpow2 [=>] 11.8	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + \frac{\color{blue}{\ell \cdot \ell}}{x}\right)}{2}}} \]
      associate-*r/ [<=] 6.8	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + \color{blue}{\ell \cdot \frac{\ell}{x}}\right)}{2}}} \]

   if -1.3500000000000001e-177 < t < -7.6000000000000007e-273

   1. Initial program 63.4

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

   2. Simplified 63.4

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

      [Start] 63.4	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 63.4	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 63.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 63.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Taylor expanded in x around inf 35.1

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

   4. Simplified 35.1

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

      [Start] 35.1	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 35.1	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 35.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 35.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 35.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 35.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 35.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      +-commutative [=>] 35.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 35.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 35.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 35.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)\right)}} \cdot t \]

   5. Taylor expanded in t around -inf 23.5

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

   6. Simplified 23.6

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

      [Start] 23.5	\[ \frac{\sqrt{2}}{-1 \cdot \left(\frac{{\ell}^{2}}{t \cdot x} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}\right) + -1 \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)} \cdot t \]
      *-commutative [=>] 23.5	\[ \frac{\sqrt{2}}{-1 \cdot \color{blue}{\left(\sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} + -1 \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)} \cdot t \]
      *-commutative [=>] 23.5	\[ \frac{\sqrt{2}}{-1 \cdot \left(\sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right) + -1 \cdot \color{blue}{\left(\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t\right)}} \cdot t \]
      distribute-lft-out [=>] 23.5	\[ \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x} + \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t\right)}} \cdot t \]
      +-commutative [<=] 23.5	\[ \frac{\sqrt{2}}{-1 \cdot \color{blue}{\left(\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)}} \cdot t \]
      mul-1-neg [=>] 23.5	\[ \frac{\sqrt{2}}{\color{blue}{-\left(\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)}} \cdot t \]

   if -7.6000000000000007e-273 < t < 2.39999999999999983e94

   1. Initial program 39.8

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

   2. Simplified 39.7

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

      [Start] 39.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 39.7	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 39.7	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 39.7	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Applied egg-rr 39.6

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

   4. Taylor expanded in x around inf 18.9

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

   5. Simplified 18.9

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

      [Start] 18.9	\[ \frac{t}{\sqrt{\frac{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}{2}}} \]
      cancel-sign-sub-inv [=>] 18.9	\[ \frac{t}{\sqrt{\frac{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}{2}}} \]
      associate-+l+ [=>] 18.9	\[ \frac{t}{\sqrt{\frac{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}{2}}} \]
      unpow2 [=>] 18.9	\[ \frac{t}{\sqrt{\frac{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      associate-/l* [=>] 18.9	\[ \frac{t}{\sqrt{\frac{\color{blue}{\frac{\ell}{\frac{x}{\ell}}} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      distribute-lft-out [=>] 18.9	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      +-commutative [=>] 18.9	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \color{blue}{\left({t}^{2} + \frac{{t}^{2}}{x}\right)} + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      unpow2 [=>] 18.9	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(\color{blue}{t \cdot t} + \frac{{t}^{2}}{x}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      unpow2 [=>] 18.9	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{\color{blue}{t \cdot t}}{x}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      associate-/l* [=>] 18.9	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \color{blue}{\frac{t}{\frac{x}{t}}}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      metadata-eval [=>] 18.9	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + \color{blue}{1} \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      +-commutative [=>] 18.9	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + 1 \cdot \frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x}\right)}{2}}} \]
      unpow2 [=>] 18.9	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + 1 \cdot \frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x}\right)}{2}}} \]
      fma-def [=>] 18.9	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + 1 \cdot \frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)}{2}}} \]
      unpow2 [=>] 18.9	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + 1 \cdot \frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)}{2}}} \]

   6. Taylor expanded in l around inf 19.2

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

   7. Simplified 15.3

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

      [Start] 19.2	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + \frac{{\ell}^{2}}{x}\right)}{2}}} \]
      unpow2 [=>] 19.2	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + \frac{\color{blue}{\ell \cdot \ell}}{x}\right)}{2}}} \]
      associate-*r/ [<=] 15.3	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + \color{blue}{\ell \cdot \frac{\ell}{x}}\right)}{2}}} \]

   8. Taylor expanded in x around inf 15.3

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

   9. Simplified 15.3

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

      [Start] 15.3	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot {t}^{2} + \ell \cdot \frac{\ell}{x}\right)}{2}}} \]
      unpow2 [=>] 15.3	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \frac{\ell}{x}\right)}{2}}} \]

   if 2.39999999999999983e94 < t

   1. Initial program 49.6

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

   2. Simplified 49.6

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

      [Start] 49.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 49.6	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 49.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 49.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Applied egg-rr 49.5

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

   4. Taylor expanded in t around inf 2.6

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

   5. Taylor expanded in x around inf 3.1

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

4. Final simplification 8.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{+109}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq -1.35 \cdot 10^{-177}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + \ell \cdot \frac{\ell}{x}\right)}{2}}}\\ \mathbf{elif}\;t \leq -7.6 \cdot 10^{-273}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{+94}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(\ell \cdot \frac{\ell}{x} + 2 \cdot \left(t \cdot t\right)\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.6
Cost	8392

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{x}{\ell}}\\ t_2 := \ell \cdot \frac{\ell}{x}\\ \mathbf{if}\;t \leq -1.55 \cdot 10^{+113}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-177}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{t_1 + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + t_2\right)}{2}}}\\ \mathbf{elif}\;t \leq -2.55 \cdot 10^{-273}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+91}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{t_1 + \left(t_2 + 2 \cdot \left(t \cdot t\right)\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 2
Error	9.6
Cost	8272

\[\begin{array}{l} t_1 := \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(\ell \cdot \frac{\ell}{x} + 2 \cdot \left(t \cdot t\right)\right)}{2}}}\\ \mathbf{if}\;t \leq -4.6 \cdot 10^{+108}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq -2.7 \cdot 10^{-177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4.7 \cdot 10^{-272}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+92}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 3
Error	15.0
Cost	7880

\[\begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-273}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 1.16 \cdot 10^{-116}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + \frac{\ell}{x} \cdot \frac{\ell}{x}\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 4
Error	15.0
Cost	7368

\[\begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{-273}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-119}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 5
Error	14.7
Cost	7048

\[\begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{-274}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-240}:\\ \;\;\;\;\frac{\sqrt{x}}{\ell} \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 6
Error	15.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	15.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 8
Error	15.8
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Error	39.2
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023060 
(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)))))