Toniolo and Linder, Equation (7)

Average Error: 43.3 → 9.0

Time: 25.7s

Precision: binary64

Cost: 40520

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := 1 + \frac{2}{x}\\ \mathbf{if}\;t \leq -1.22 \cdot 10^{+122}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+126}:\\ \;\;\;\;\frac{t_1}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(\ell, \frac{\ell}{x}, \left(t \cdot 2\right) \cdot \left(\frac{t}{x} + \left(t + \frac{t}{x}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\mathsf{fma}\left(t_1, \sqrt{t_2}, \frac{\ell}{\frac{\sqrt{2} \cdot \left(t \cdot x\right)}{\ell}} \cdot \sqrt{\frac{1}{t_2}}\right)}\\ \end{array} \]

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 (+ 1.0 (/ 2.0 x))))
   (if (<= t -1.22e+122)
     (- (sqrt (/ (+ -1.0 x) (+ x 1.0))))
     (if (<= t 4e+126)
       (/
        t_1
        (sqrt
         (+
          (/ l (/ x l))
          (fma l (/ l x) (* (* t 2.0) (+ (/ t x) (+ t (/ t x))))))))
       (/
        t_1
        (fma
         t_1
         (sqrt t_2)
         (* (/ l (/ (* (sqrt 2.0) (* t x)) l)) (sqrt (/ 1.0 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);
	double t_2 = 1.0 + (2.0 / x);
	double tmp;
	if (t <= -1.22e+122) {
		tmp = -sqrt(((-1.0 + x) / (x + 1.0)));
	} else if (t <= 4e+126) {
		tmp = t_1 / sqrt(((l / (x / l)) + fma(l, (l / x), ((t * 2.0) * ((t / x) + (t + (t / x)))))));
	} else {
		tmp = t_1 / fma(t_1, sqrt(t_2), ((l / ((sqrt(2.0) * (t * x)) / l)) * sqrt((1.0 / 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 * sqrt(2.0))
	t_2 = Float64(1.0 + Float64(2.0 / x))
	tmp = 0.0
	if (t <= -1.22e+122)
		tmp = Float64(-sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0))));
	elseif (t <= 4e+126)
		tmp = Float64(t_1 / sqrt(Float64(Float64(l / Float64(x / l)) + fma(l, Float64(l / x), Float64(Float64(t * 2.0) * Float64(Float64(t / x) + Float64(t + Float64(t / x))))))));
	else
		tmp = Float64(t_1 / fma(t_1, sqrt(t_2), Float64(Float64(l / Float64(Float64(sqrt(2.0) * Float64(t * x)) / l)) * sqrt(Float64(1.0 / t_2)))));
	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[(1.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.22e+122], (-N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, 4e+126], N[(t$95$1 / N[Sqrt[N[(N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision] + N[(l * N[(l / x), $MachinePrecision] + N[(N[(t * 2.0), $MachinePrecision] * N[(N[(t / x), $MachinePrecision] + N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(t$95$1 * N[Sqrt[t$95$2], $MachinePrecision] + N[(N[(l / N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t * x), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.22e122

   1. Initial program 54.3

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

   2. Simplified 54.3

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

      [Start] 54.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 54.3	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      associate-*l/ [=>] 61.9	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\left(x + 1\right) \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x - 1}} - \ell \cdot \ell}} \]
      associate-*r/ [<=] 54.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}} - \ell \cdot \ell}} \]
      *-lft-identity [<=] 54.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \left(x + 1\right)\right)} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} - \ell \cdot \ell}} \]
      associate-*r* [<=] 54.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{1 \cdot \left(\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 54.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{1 \cdot \color{blue}{\left(\frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} \cdot \left(x + 1\right)\right)} - \ell \cdot \ell}} \]
      associate-*r* [=>] 54.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right) \cdot \left(x + 1\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 54.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      fma-neg [=>] 54.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(x + 1, 1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}, -\ell \cdot \ell\right)}}} \]

   3. Taylor expanded in t around inf 61.4

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

   4. Simplified 52.3

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

      [Start] 61.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}} \]
      associate-/l* [=>] 52.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \]
      +-commutative [=>] 52.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \]
      sub-neg [=>] 52.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}} \]
      metadata-eval [=>] 52.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \]
      +-commutative [=>] 52.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \]
      unpow2 [=>] 52.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \]

   5. Taylor expanded in t around -inf 2.1

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

   6. Simplified 2.1

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

      [Start] 2.1	\[ -1 \cdot \sqrt{\frac{x - 1}{1 + x}} \]
      mul-1-neg [=>] 2.1	\[ \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]
      sub-neg [=>] 2.1	\[ -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]
      metadata-eval [=>] 2.1	\[ -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]
      +-commutative [=>] 2.1	\[ -\sqrt{\frac{\color{blue}{-1 + x}}{1 + x}} \]
      +-commutative [=>] 2.1	\[ -\sqrt{\frac{-1 + x}{\color{blue}{x + 1}}} \]

   if -1.22e122 < t < 3.9999999999999997e126

   1. Initial program 36.7

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

   2. Taylor expanded in x around inf 17.3

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

   3. Simplified 17.3

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

      [Start] 17.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      associate--l+ [=>] 17.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]
      unpow2 [=>] 17.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-/l* [=>] 17.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}}} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      distribute-lft-out [=>] 17.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      +-commutative [=>] 17.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \color{blue}{\left({t}^{2} + \frac{{t}^{2}}{x}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 17.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(\color{blue}{t \cdot t} + \frac{{t}^{2}}{x}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 17.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{\color{blue}{t \cdot t}}{x}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-/l* [=>] 17.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \color{blue}{\frac{t}{\frac{x}{t}}}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-*r/ [=>] 17.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \]
      mul-1-neg [=>] 17.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \]
      +-commutative [=>] 17.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + {\ell}^{2}\right)}}{x}\right)}} \]
      unpow2 [=>] 17.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\left(2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}\right)}{x}\right)}} \]
      fma-udef [<=] 17.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)}} \]
      unpow2 [=>] 17.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)}} \]

   4. Taylor expanded in l around 0 17.3

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

   5. Simplified 12.9

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

      [Start] 17.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\left(\frac{{\ell}^{2}}{x} + 2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)\right) - -2 \cdot \frac{{t}^{2}}{x}\right)}} \]
      associate--l+ [=>] 17.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right) - -2 \cdot \frac{{t}^{2}}{x}\right)\right)}}} \]
      unpow2 [=>] 17.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right) - -2 \cdot \frac{{t}^{2}}{x}\right)\right)}} \]
      associate-*r/ [<=] 12.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\color{blue}{\ell \cdot \frac{\ell}{x}} + \left(2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right) - -2 \cdot \frac{{t}^{2}}{x}\right)\right)}} \]
      fma-def [=>] 12.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \color{blue}{\mathsf{fma}\left(\ell, \frac{\ell}{x}, 2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right) - -2 \cdot \frac{{t}^{2}}{x}\right)}}} \]
      cancel-sign-sub-inv [=>] 12.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(\ell, \frac{\ell}{x}, \color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right) + \left(--2\right) \cdot \frac{{t}^{2}}{x}}\right)}} \]
      +-commutative [=>] 12.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(\ell, \frac{\ell}{x}, 2 \cdot \color{blue}{\left({t}^{2} + \frac{{t}^{2}}{x}\right)} + \left(--2\right) \cdot \frac{{t}^{2}}{x}\right)}} \]
      unpow2 [=>] 12.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(\ell, \frac{\ell}{x}, 2 \cdot \left(\color{blue}{t \cdot t} + \frac{{t}^{2}}{x}\right) + \left(--2\right) \cdot \frac{{t}^{2}}{x}\right)}} \]
      unpow2 [=>] 12.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(\ell, \frac{\ell}{x}, 2 \cdot \left(t \cdot t + \frac{\color{blue}{t \cdot t}}{x}\right) + \left(--2\right) \cdot \frac{{t}^{2}}{x}\right)}} \]
      associate-*r/ [<=] 12.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(\ell, \frac{\ell}{x}, 2 \cdot \left(t \cdot t + \color{blue}{t \cdot \frac{t}{x}}\right) + \left(--2\right) \cdot \frac{{t}^{2}}{x}\right)}} \]
      distribute-lft-in [<=] 12.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(\ell, \frac{\ell}{x}, 2 \cdot \color{blue}{\left(t \cdot \left(t + \frac{t}{x}\right)\right)} + \left(--2\right) \cdot \frac{{t}^{2}}{x}\right)}} \]
      associate-*r* [=>] 12.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(\ell, \frac{\ell}{x}, \color{blue}{\left(2 \cdot t\right) \cdot \left(t + \frac{t}{x}\right)} + \left(--2\right) \cdot \frac{{t}^{2}}{x}\right)}} \]
      unpow2 [=>] 12.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(\ell, \frac{\ell}{x}, \left(2 \cdot t\right) \cdot \left(t + \frac{t}{x}\right) + \left(--2\right) \cdot \frac{\color{blue}{t \cdot t}}{x}\right)}} \]
      associate-*r/ [<=] 12.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(\ell, \frac{\ell}{x}, \left(2 \cdot t\right) \cdot \left(t + \frac{t}{x}\right) + \left(--2\right) \cdot \color{blue}{\left(t \cdot \frac{t}{x}\right)}\right)}} \]
      metadata-eval [=>] 12.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(\ell, \frac{\ell}{x}, \left(2 \cdot t\right) \cdot \left(t + \frac{t}{x}\right) + \color{blue}{2} \cdot \left(t \cdot \frac{t}{x}\right)\right)}} \]
      associate-*r* [=>] 12.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(\ell, \frac{\ell}{x}, \left(2 \cdot t\right) \cdot \left(t + \frac{t}{x}\right) + \color{blue}{\left(2 \cdot t\right) \cdot \frac{t}{x}}\right)}} \]
      distribute-lft-out [=>] 12.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(\ell, \frac{\ell}{x}, \color{blue}{\left(2 \cdot t\right) \cdot \left(\left(t + \frac{t}{x}\right) + \frac{t}{x}\right)}\right)}} \]

   if 3.9999999999999997e126 < t

   1. Initial program 55.6

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

   2. Taylor expanded in x around inf 56.1

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

   3. Simplified 56.1

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

      [Start] 56.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      associate--l+ [=>] 56.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]
      unpow2 [=>] 56.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-/l* [=>] 56.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}}} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      distribute-lft-out [=>] 56.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      +-commutative [=>] 56.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \color{blue}{\left({t}^{2} + \frac{{t}^{2}}{x}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 56.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(\color{blue}{t \cdot t} + \frac{{t}^{2}}{x}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 56.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{\color{blue}{t \cdot t}}{x}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-/l* [=>] 56.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \color{blue}{\frac{t}{\frac{x}{t}}}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-*r/ [=>] 56.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \]
      mul-1-neg [=>] 56.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \]
      +-commutative [=>] 56.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + {\ell}^{2}\right)}}{x}\right)}} \]
      unpow2 [=>] 56.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\left(2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}\right)}{x}\right)}} \]
      fma-udef [<=] 56.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)}} \]
      unpow2 [=>] 56.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)}} \]

   4. Taylor expanded in l around 0 56.1

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

   5. Simplified 54.1

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

      [Start] 56.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\left(\frac{{\ell}^{2}}{x} + 2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)\right) - -2 \cdot \frac{{t}^{2}}{x}\right)}} \]
      associate--l+ [=>] 56.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right) - -2 \cdot \frac{{t}^{2}}{x}\right)\right)}}} \]
      unpow2 [=>] 56.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right) - -2 \cdot \frac{{t}^{2}}{x}\right)\right)}} \]
      associate-*r/ [<=] 54.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\color{blue}{\ell \cdot \frac{\ell}{x}} + \left(2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right) - -2 \cdot \frac{{t}^{2}}{x}\right)\right)}} \]
      fma-def [=>] 54.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \color{blue}{\mathsf{fma}\left(\ell, \frac{\ell}{x}, 2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right) - -2 \cdot \frac{{t}^{2}}{x}\right)}}} \]
      cancel-sign-sub-inv [=>] 54.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(\ell, \frac{\ell}{x}, \color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right) + \left(--2\right) \cdot \frac{{t}^{2}}{x}}\right)}} \]
      +-commutative [=>] 54.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(\ell, \frac{\ell}{x}, 2 \cdot \color{blue}{\left({t}^{2} + \frac{{t}^{2}}{x}\right)} + \left(--2\right) \cdot \frac{{t}^{2}}{x}\right)}} \]
      unpow2 [=>] 54.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(\ell, \frac{\ell}{x}, 2 \cdot \left(\color{blue}{t \cdot t} + \frac{{t}^{2}}{x}\right) + \left(--2\right) \cdot \frac{{t}^{2}}{x}\right)}} \]
      unpow2 [=>] 54.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(\ell, \frac{\ell}{x}, 2 \cdot \left(t \cdot t + \frac{\color{blue}{t \cdot t}}{x}\right) + \left(--2\right) \cdot \frac{{t}^{2}}{x}\right)}} \]
      associate-*r/ [<=] 54.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(\ell, \frac{\ell}{x}, 2 \cdot \left(t \cdot t + \color{blue}{t \cdot \frac{t}{x}}\right) + \left(--2\right) \cdot \frac{{t}^{2}}{x}\right)}} \]
      distribute-lft-in [<=] 54.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(\ell, \frac{\ell}{x}, 2 \cdot \color{blue}{\left(t \cdot \left(t + \frac{t}{x}\right)\right)} + \left(--2\right) \cdot \frac{{t}^{2}}{x}\right)}} \]
      associate-*r* [=>] 54.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(\ell, \frac{\ell}{x}, \color{blue}{\left(2 \cdot t\right) \cdot \left(t + \frac{t}{x}\right)} + \left(--2\right) \cdot \frac{{t}^{2}}{x}\right)}} \]
      unpow2 [=>] 54.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(\ell, \frac{\ell}{x}, \left(2 \cdot t\right) \cdot \left(t + \frac{t}{x}\right) + \left(--2\right) \cdot \frac{\color{blue}{t \cdot t}}{x}\right)}} \]
      associate-*r/ [<=] 54.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(\ell, \frac{\ell}{x}, \left(2 \cdot t\right) \cdot \left(t + \frac{t}{x}\right) + \left(--2\right) \cdot \color{blue}{\left(t \cdot \frac{t}{x}\right)}\right)}} \]
      metadata-eval [=>] 54.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(\ell, \frac{\ell}{x}, \left(2 \cdot t\right) \cdot \left(t + \frac{t}{x}\right) + \color{blue}{2} \cdot \left(t \cdot \frac{t}{x}\right)\right)}} \]
      associate-*r* [=>] 54.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(\ell, \frac{\ell}{x}, \left(2 \cdot t\right) \cdot \left(t + \frac{t}{x}\right) + \color{blue}{\left(2 \cdot t\right) \cdot \frac{t}{x}}\right)}} \]
      distribute-lft-out [=>] 54.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(\ell, \frac{\ell}{x}, \color{blue}{\left(2 \cdot t\right) \cdot \left(\left(t + \frac{t}{x}\right) + \frac{t}{x}\right)}\right)}} \]

   6. Taylor expanded in t around inf 13.4

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

   7. Simplified 2.4

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

      [Start] 13.4	\[ \frac{\sqrt{2} \cdot t}{\left(\sqrt{2} \cdot t\right) \cdot \sqrt{1 + 2 \cdot \frac{1}{x}} + \frac{{\ell}^{2}}{\sqrt{2} \cdot \left(t \cdot x\right)} \cdot \sqrt{\frac{1}{1 + 2 \cdot \frac{1}{x}}}} \]
      fma-def [=>] 13.4	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot t, \sqrt{1 + 2 \cdot \frac{1}{x}}, \frac{{\ell}^{2}}{\sqrt{2} \cdot \left(t \cdot x\right)} \cdot \sqrt{\frac{1}{1 + 2 \cdot \frac{1}{x}}}\right)}} \]
      associate-*r/ [=>] 13.4	\[ \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\sqrt{2} \cdot t, \sqrt{1 + \color{blue}{\frac{2 \cdot 1}{x}}}, \frac{{\ell}^{2}}{\sqrt{2} \cdot \left(t \cdot x\right)} \cdot \sqrt{\frac{1}{1 + 2 \cdot \frac{1}{x}}}\right)} \]
      metadata-eval [=>] 13.4	\[ \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\sqrt{2} \cdot t, \sqrt{1 + \frac{\color{blue}{2}}{x}}, \frac{{\ell}^{2}}{\sqrt{2} \cdot \left(t \cdot x\right)} \cdot \sqrt{\frac{1}{1 + 2 \cdot \frac{1}{x}}}\right)} \]
      unpow2 [=>] 13.4	\[ \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\sqrt{2} \cdot t, \sqrt{1 + \frac{2}{x}}, \frac{\color{blue}{\ell \cdot \ell}}{\sqrt{2} \cdot \left(t \cdot x\right)} \cdot \sqrt{\frac{1}{1 + 2 \cdot \frac{1}{x}}}\right)} \]
      associate-/l* [=>] 2.4	\[ \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\sqrt{2} \cdot t, \sqrt{1 + \frac{2}{x}}, \color{blue}{\frac{\ell}{\frac{\sqrt{2} \cdot \left(t \cdot x\right)}{\ell}}} \cdot \sqrt{\frac{1}{1 + 2 \cdot \frac{1}{x}}}\right)} \]
      associate-*r/ [=>] 2.4	\[ \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\sqrt{2} \cdot t, \sqrt{1 + \frac{2}{x}}, \frac{\ell}{\frac{\sqrt{2} \cdot \left(t \cdot x\right)}{\ell}} \cdot \sqrt{\frac{1}{1 + \color{blue}{\frac{2 \cdot 1}{x}}}}\right)} \]
      metadata-eval [=>] 2.4	\[ \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\sqrt{2} \cdot t, \sqrt{1 + \frac{2}{x}}, \frac{\ell}{\frac{\sqrt{2} \cdot \left(t \cdot x\right)}{\ell}} \cdot \sqrt{\frac{1}{1 + \frac{\color{blue}{2}}{x}}}\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 9.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.22 \cdot 10^{+122}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+126}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(\ell, \frac{\ell}{x}, \left(t \cdot 2\right) \cdot \left(\frac{t}{x} + \left(t + \frac{t}{x}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(t \cdot \sqrt{2}, \sqrt{1 + \frac{2}{x}}, \frac{\ell}{\frac{\sqrt{2} \cdot \left(t \cdot x\right)}{\ell}} \cdot \sqrt{\frac{1}{1 + \frac{2}{x}}}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.5
Cost	27976

\[\begin{array}{l} t_1 := 2 + \left(\frac{2}{x} + \frac{2}{x}\right)\\ \mathbf{if}\;t \leq -6.4 \cdot 10^{+122}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 155000000000:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(\ell, \frac{\ell}{x}, \left(t \cdot 2\right) \cdot \left(\frac{t}{x} + \left(t + \frac{t}{x}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{t_1}, \sqrt{\frac{1}{t_1}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \end{array} \]

Alternative 2
Error	9.0
Cost	21192

\[\begin{array}{l} \mathbf{if}\;t \leq -2.7 \cdot 10^{+122}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+125}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(\ell, \frac{\ell}{x}, \left(t \cdot 2\right) \cdot \left(\frac{t}{x} + \left(t + \frac{t}{x}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 3
Error	9.0
Cost	14408

\[\begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{+122}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{+125}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + 2 \cdot \frac{t}{x}\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 4
Error	14.3
Cost	14280

\[\begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{-257}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-223}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + \frac{\ell}{x} \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-115}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{-72}:\\ \;\;\;\;t \cdot \sqrt{\frac{1}{\ell \cdot \frac{\ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 5
Error	14.4
Cost	7504

\[\begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{-250}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-224}:\\ \;\;\;\;t \cdot \sqrt{x \cdot \frac{1}{\ell \cdot \ell}}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-116}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-72}:\\ \;\;\;\;t \cdot \sqrt{\frac{1}{\ell \cdot \frac{\ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 6
Error	14.2
Cost	7504

\[\begin{array}{l} \mathbf{if}\;t \leq -5.3 \cdot 10^{-244}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-226}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\ell}{x} \cdot \left(\ell + \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-106}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{-72}:\\ \;\;\;\;t \cdot \sqrt{\frac{1}{\ell \cdot \frac{\ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 7
Error	14.7
Cost	7376

\[\begin{array}{l} t_1 := \frac{t}{\sqrt{\ell \cdot \frac{\ell}{x}}}\\ \mathbf{if}\;t \leq -1.66 \cdot 10^{-245}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-219}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-115}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{-72}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 8
Error	14.3
Cost	7376

\[\begin{array}{l} t_1 := \frac{t}{\sqrt{\ell \cdot \frac{\ell}{x}}}\\ \mathbf{if}\;t \leq -1.75 \cdot 10^{-245}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 2.75 \cdot 10^{-220}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-115}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{-72}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 9
Error	14.4
Cost	7376

\[\begin{array}{l} \mathbf{if}\;t \leq -1.75 \cdot 10^{-261}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-224}:\\ \;\;\;\;t \cdot \sqrt{x \cdot \frac{1}{\ell \cdot \ell}}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-111}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{t}{\sqrt{\ell \cdot \frac{\ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 10
Error	14.8
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{-274}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 3.35 \cdot 10^{-245}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 11
Error	14.7
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{-269}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{-245}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 12
Error	15.8
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-287}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 13
Error	16.0
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-287}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14
Error	39.0
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023083 
(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)))))