Toniolo and Linder, Equation (7)

Average Accuracy: 32.3% → 86.2%

Time: 32.4s

Precision: binary64

Cost: 28108

Math

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

↓

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{x}\\ t_2 := t \cdot \frac{\sqrt{2}}{\sqrt{{\left(\sqrt[3]{t_1}\right)}^{3} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + t_1\right)}}\\ t_3 := 2 + \left(\frac{2}{x} + \frac{2}{x}\right)\\ t_4 := \mathsf{fma}\left(t, \sqrt{t_3}, \sqrt{\frac{1}{t_3}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)\\ \mathbf{if}\;t \leq -1 \cdot 10^{+108}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-t_4}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-246}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-169}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t_4}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+110}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{\frac{x + -1}{x + 1}}}{t}\\ \end{array} \]

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 (/ l x)))
        (t_2
         (*
          t
          (/
           (sqrt 2.0)
           (sqrt
            (+
             (pow (cbrt t_1) 3.0)
             (+ (* 2.0 (+ (* t t) (/ (* t t) x))) t_1))))))
        (t_3 (+ 2.0 (+ (/ 2.0 x) (/ 2.0 x))))
        (t_4 (fma t (sqrt t_3) (* (sqrt (/ 1.0 t_3)) (* (/ l x) (/ l t))))))
   (if (<= t -1e+108)
     (* t (/ (sqrt 2.0) (- t_4)))
     (if (<= t 4.6e-246)
       t_2
       (if (<= t 3.5e-169)
         (* t (/ (sqrt 2.0) t_4))
         (if (<= t 6.8e+110)
           t_2
           (* t (/ (sqrt (/ (+ x -1.0) (+ x 1.0))) t))))))))

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 * (l / x);
	double t_2 = t * (sqrt(2.0) / sqrt((pow(cbrt(t_1), 3.0) + ((2.0 * ((t * t) + ((t * t) / x))) + t_1))));
	double t_3 = 2.0 + ((2.0 / x) + (2.0 / x));
	double t_4 = fma(t, sqrt(t_3), (sqrt((1.0 / t_3)) * ((l / x) * (l / t))));
	double tmp;
	if (t <= -1e+108) {
		tmp = t * (sqrt(2.0) / -t_4);
	} else if (t <= 4.6e-246) {
		tmp = t_2;
	} else if (t <= 3.5e-169) {
		tmp = t * (sqrt(2.0) / t_4);
	} else if (t <= 6.8e+110) {
		tmp = t_2;
	} else {
		tmp = t * (sqrt(((x + -1.0) / (x + 1.0))) / t);
	}
	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(l / x))
	t_2 = Float64(t * Float64(sqrt(2.0) / sqrt(Float64((cbrt(t_1) ^ 3.0) + Float64(Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))) + t_1)))))
	t_3 = Float64(2.0 + Float64(Float64(2.0 / x) + Float64(2.0 / x)))
	t_4 = fma(t, sqrt(t_3), Float64(sqrt(Float64(1.0 / t_3)) * Float64(Float64(l / x) * Float64(l / t))))
	tmp = 0.0
	if (t <= -1e+108)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(-t_4)));
	elseif (t <= 4.6e-246)
		tmp = t_2;
	elseif (t <= 3.5e-169)
		tmp = Float64(t * Float64(sqrt(2.0) / t_4));
	elseif (t <= 6.8e+110)
		tmp = t_2;
	else
		tmp = Float64(t * Float64(sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0))) / t));
	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[(l / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 3.0], $MachinePrecision] + N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t * N[Sqrt[t$95$3], $MachinePrecision] + N[(N[Sqrt[N[(1.0 / t$95$3), $MachinePrecision]], $MachinePrecision] * N[(N[(l / x), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1e+108], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / (-t$95$4)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.6e-246], t$95$2, If[LessEqual[t, 3.5e-169], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.8e+110], t$95$2, N[(t * N[(N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1e108

   1. Initial program 17.6%

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

   2. Simplified 17.7%

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

      [Start] 17.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 17.7	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Taylor expanded in x around inf 17.9%

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

   4. Simplified 17.9%

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

      [Start] 17.9	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 17.9	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 17.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 17.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 17.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 17.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 17.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      unpow2 [=>] 17.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}}{x}\right)\right)}} \cdot t \]
      +-commutative [=>] 17.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + \ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 17.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 17.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]

   5. Taylor expanded in t around -inf 74.7%

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

   6. Simplified 95.0%

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

      [Start] 74.7	\[ \frac{\sqrt{2}}{-1 \cdot \left(\frac{{\ell}^{2}}{t \cdot x} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}\right) + -1 \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)} \cdot t \]
      distribute-lft-out [=>] 74.7	\[ \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\frac{{\ell}^{2}}{t \cdot x} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} + t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)}} \cdot t \]
      mul-1-neg [=>] 74.7	\[ \frac{\sqrt{2}}{\color{blue}{-\left(\frac{{\ell}^{2}}{t \cdot x} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} + t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)}} \cdot t \]
      *-commutative [=>] 74.7	\[ \frac{\sqrt{2}}{-\left(\color{blue}{\sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}} + t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)} \cdot t \]
      *-commutative [=>] 74.7	\[ \frac{\sqrt{2}}{-\left(\sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x} + \color{blue}{\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t}\right)} \cdot t \]
      +-commutative [<=] 74.7	\[ \frac{\sqrt{2}}{-\color{blue}{\left(\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)}} \cdot t \]
      *-commutative [<=] 74.7	\[ \frac{\sqrt{2}}{-\left(\color{blue}{t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]

   if -1e108 < t < 4.5999999999999995e-246 or 3.5000000000000003e-169 < t < 6.8000000000000003e110

   1. Initial program 45.1%

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

   2. Simplified 45.1%

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

      [Start] 45.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 45.1	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Taylor expanded in x around inf 75.8%

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

   4. Simplified 75.8%

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

      [Start] 75.8	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 75.8	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 75.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 75.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 75.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 75.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 75.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      unpow2 [=>] 75.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}}{x}\right)\right)}} \cdot t \]
      +-commutative [=>] 75.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + \ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 75.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 75.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]

   5. Applied egg-rr 75.8%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{{\left(\sqrt[3]{\frac{\ell}{x} \cdot \ell}\right)}^{3}} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)\right)}} \cdot t \]
      Proof

      [Start] 75.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)\right)}} \cdot t \]
      add-cube-cbrt [=>] 75.8	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\sqrt[3]{\frac{\ell \cdot \ell}{x}} \cdot \sqrt[3]{\frac{\ell \cdot \ell}{x}}\right) \cdot \sqrt[3]{\frac{\ell \cdot \ell}{x}}} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)\right)}} \cdot t \]
      pow3 [=>] 75.8	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{{\left(\sqrt[3]{\frac{\ell \cdot \ell}{x}}\right)}^{3}} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)\right)}} \cdot t \]
      associate-/l* [=>] 75.8	\[ \frac{\sqrt{2}}{\sqrt{{\left(\sqrt[3]{\color{blue}{\frac{\ell}{\frac{x}{\ell}}}}\right)}^{3} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)\right)}} \cdot t \]
      associate-/r/ [=>] 75.8	\[ \frac{\sqrt{2}}{\sqrt{{\left(\sqrt[3]{\color{blue}{\frac{\ell}{x} \cdot \ell}}\right)}^{3} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)\right)}} \cdot t \]

   6. Taylor expanded in t around 0 75.3%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{{\left(\sqrt[3]{\frac{\ell}{x} \cdot \ell}\right)}^{3} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\color{blue}{\frac{{\ell}^{2}}{x}}\right)\right)}} \cdot t \]

   7. Simplified 82.1%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{{\left(\sqrt[3]{\frac{\ell}{x} \cdot \ell}\right)}^{3} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\color{blue}{\ell \cdot \frac{\ell}{x}}\right)\right)}} \cdot t \]
      Proof

      [Start] 75.3	\[ \frac{\sqrt{2}}{\sqrt{{\left(\sqrt[3]{\frac{\ell}{x} \cdot \ell}\right)}^{3} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{{\ell}^{2}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 75.3	\[ \frac{\sqrt{2}}{\sqrt{{\left(\sqrt[3]{\frac{\ell}{x} \cdot \ell}\right)}^{3} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      associate-*r/ [<=] 82.1	\[ \frac{\sqrt{2}}{\sqrt{{\left(\sqrt[3]{\frac{\ell}{x} \cdot \ell}\right)}^{3} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\color{blue}{\ell \cdot \frac{\ell}{x}}\right)\right)}} \cdot t \]

   if 4.5999999999999995e-246 < t < 3.5000000000000003e-169

   1. Initial program 0.9%

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

   2. Simplified 0.9%

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

      [Start] 0.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 0.9	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Taylor expanded in x around inf 45.8%

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

   4. Simplified 45.8%

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

      [Start] 45.8	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 45.8	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 45.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 45.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 45.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 45.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 45.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      unpow2 [=>] 45.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}}{x}\right)\right)}} \cdot t \]
      +-commutative [=>] 45.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + \ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 45.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 45.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]

   5. Taylor expanded in t around inf 60.7%

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

   6. Simplified 60.6%

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

      [Start] 60.7	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}} \cdot t \]
      *-commutative [<=] 60.7	\[ \frac{\sqrt{2}}{\color{blue}{t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}} \cdot t \]
      fma-def [=>] 60.7	\[ \frac{\sqrt{2}}{\color{blue}{\mathsf{fma}\left(t, \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)}} \cdot t \]
      distribute-lft-in [=>] 60.7	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{\color{blue}{\left(2 \cdot 1 + 2 \cdot \frac{1}{x}\right)} + 2 \cdot \frac{1}{x}}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      metadata-eval [=>] 60.7	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{\left(\color{blue}{2} + 2 \cdot \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      associate-+l+ [=>] 60.7	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{\color{blue}{2 + \left(2 \cdot \frac{1}{x} + 2 \cdot \frac{1}{x}\right)}}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      associate-*r/ [=>] 60.7	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\color{blue}{\frac{2 \cdot 1}{x}} + 2 \cdot \frac{1}{x}\right)}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      metadata-eval [=>] 60.7	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{\color{blue}{2}}{x} + 2 \cdot \frac{1}{x}\right)}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      associate-*r/ [=>] 60.7	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \color{blue}{\frac{2 \cdot 1}{x}}\right)}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      metadata-eval [=>] 60.7	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{\color{blue}{2}}{x}\right)}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]

   if 6.8000000000000003e110 < t

   1. Initial program 17.9%

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

   2. Simplified 17.9%

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

      [Start] 17.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 17.9	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Taylor expanded in t around inf 5.4%

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

   4. Simplified 5.4%

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

      [Start] 5.4	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}} \cdot t \]
      *-commutative [=>] 5.4	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{{t}^{2} \cdot \left(1 + x\right)}}{x - 1}}} \cdot t \]
      unpow2 [=>] 5.4	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{\left(t \cdot t\right)} \cdot \left(1 + x\right)}{x - 1}}} \cdot t \]
      +-commutative [<=] 5.4	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\left(t \cdot t\right) \cdot \color{blue}{\left(x + 1\right)}}{x - 1}}} \cdot t \]
      sub-neg [=>] 5.4	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{\color{blue}{x + \left(-1\right)}}}} \cdot t \]
      metadata-eval [=>] 5.4	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{x + \color{blue}{-1}}}} \cdot t \]
      +-commutative [=>] 5.4	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{\color{blue}{-1 + x}}}} \cdot t \]

   5. Taylor expanded in t around 0 96.6%

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

   6. Simplified 96.6%

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

      [Start] 96.6	\[ \left(\sqrt{\frac{x - 1}{1 + x}} \cdot \frac{1}{t}\right) \cdot t \]
      associate-*r/ [=>] 96.6	\[ \color{blue}{\frac{\sqrt{\frac{x - 1}{1 + x}} \cdot 1}{t}} \cdot t \]
      sub-neg [=>] 96.6	\[ \frac{\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \cdot 1}{t} \cdot t \]
      metadata-eval [=>] 96.6	\[ \frac{\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \cdot 1}{t} \cdot t \]
      +-commutative [<=] 96.6	\[ \frac{\sqrt{\frac{x + -1}{\color{blue}{x + 1}}} \cdot 1}{t} \cdot t \]
      *-rgt-identity [=>] 96.6	\[ \frac{\color{blue}{\sqrt{\frac{x + -1}{x + 1}}}}{t} \cdot t \]
      metadata-eval [<=] 96.6	\[ \frac{\sqrt{\frac{x + \color{blue}{\left(-1\right)}}{x + 1}}}{t} \cdot t \]
      sub-neg [<=] 96.6	\[ \frac{\sqrt{\frac{\color{blue}{x - 1}}{x + 1}}}{t} \cdot t \]
      +-commutative [=>] 96.6	\[ \frac{\sqrt{\frac{x - 1}{\color{blue}{1 + x}}}}{t} \cdot t \]
      sub-neg [=>] 96.6	\[ \frac{\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}}}{t} \cdot t \]
      metadata-eval [=>] 96.6	\[ \frac{\sqrt{\frac{x + \color{blue}{-1}}{1 + x}}}{t} \cdot t \]

3. Recombined 4 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+108}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-246}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{{\left(\sqrt[3]{\ell \cdot \frac{\ell}{x}}\right)}^{3} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-169}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+110}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{{\left(\sqrt[3]{\ell \cdot \frac{\ell}{x}}\right)}^{3} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{\frac{x + -1}{x + 1}}}{t}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	86.4%
Cost	28108

\[\begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ t_2 := \ell \cdot \frac{\ell}{x}\\ t_3 := t \cdot \frac{\sqrt{2}}{\sqrt{{\left(\sqrt[3]{t_2}\right)}^{3} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + t_2\right)}}\\ t_4 := 2 + \left(\frac{2}{x} + \frac{2}{x}\right)\\ \mathbf{if}\;t \leq -1 \cdot 10^{+109}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-252}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-174}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{t_4}, \sqrt{\frac{1}{t_4}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+111}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{t_1}{t}\\ \end{array} \]

Alternative 2
Accuracy	85.4%
Cost	27920

\[\begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ t_2 := \ell \cdot \frac{\ell}{x}\\ t_3 := t \cdot \frac{\sqrt{2}}{\sqrt{{\left(\sqrt[3]{t_2}\right)}^{3} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + t_2\right)}}\\ \mathbf{if}\;t \leq -2.25 \cdot 10^{+109}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-251}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-172}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+108}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{t_1}{t}\\ \end{array} \]

Alternative 3
Accuracy	76.1%
Cost	21448

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.6 \cdot 10^{+153}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{elif}\;\ell \leq -1.08 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(\ell, \ell, t \cdot \left(t \cdot 2\right)\right)}{x}\right)}}{t}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{hypot}\left(t, t\right)}\\ \end{array} \]

Alternative 4
Accuracy	76.2%
Cost	14792

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{x}\\ \mathbf{if}\;\ell \leq -4.6 \cdot 10^{+153}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{elif}\;\ell \leq -7 \cdot 10^{-10}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_1 + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + t_1\right)}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{hypot}\left(t, t\right)}\\ \end{array} \]

Alternative 5
Accuracy	76.1%
Cost	13316

\[\begin{array}{l} \mathbf{if}\;\ell \leq -7.6 \cdot 10^{+173}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{hypot}\left(t, t\right)}\\ \end{array} \]

Alternative 6
Accuracy	75.5%
Cost	7632

\[\begin{array}{l} t_1 := t \cdot \left(\sqrt{x} \cdot \frac{-1}{\ell}\right)\\ t_2 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -2.4 \cdot 10^{-41}:\\ \;\;\;\;-t_2\\ \mathbf{elif}\;t \leq -5.8 \cdot 10^{-87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{-218}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-250}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{t_2}{t}\\ \end{array} \]

Alternative 7
Accuracy	74.9%
Cost	7376

\[\begin{array}{l} t_1 := t \cdot \left(\sqrt{x} \cdot \frac{-1}{\ell}\right)\\ \mathbf{if}\;t \leq -2.4 \cdot 10^{-41}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq -5.8 \cdot 10^{-87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.9 \cdot 10^{-219}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-249}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{1}{t} + \left(\frac{\frac{0.5}{t}}{x \cdot x} + \frac{\frac{-1}{x}}{t}\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	75.4%
Cost	7376

\[\begin{array}{l} t_1 := t \cdot \left(\sqrt{x} \cdot \frac{-1}{\ell}\right)\\ \mathbf{if}\;t \leq -2.4 \cdot 10^{-41}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -5.8 \cdot 10^{-87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4.7 \cdot 10^{-219}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-251}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{1}{t} + \left(\frac{\frac{0.5}{t}}{x \cdot x} + \frac{\frac{-1}{x}}{t}\right)\right)\\ \end{array} \]

Alternative 9
Accuracy	74.4%
Cost	7312

\[\begin{array}{l} t_1 := \sqrt{x} \cdot \frac{-t}{\ell}\\ \mathbf{if}\;t \leq -2.4 \cdot 10^{-41}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq -5.8 \cdot 10^{-87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.7 \cdot 10^{-219}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-245}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{1}{t} + \left(\frac{\frac{0.5}{t}}{x \cdot x} + \frac{\frac{-1}{x}}{t}\right)\right)\\ \end{array} \]

Alternative 10
Accuracy	74.8%
Cost	7312

\[\begin{array}{l} t_1 := t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{if}\;t \leq -2.4 \cdot 10^{-41}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq -9.2 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.9 \cdot 10^{-218}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-245}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{1}{t} + \left(\frac{\frac{0.5}{t}}{x \cdot x} + \frac{\frac{-1}{x}}{t}\right)\right)\\ \end{array} \]

Alternative 11
Accuracy	75.2%
Cost	1348

\[\begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{1}{t} + \left(\frac{\frac{0.5}{t}}{x \cdot x} + \frac{\frac{-1}{x}}{t}\right)\right)\\ \end{array} \]

Alternative 12
Accuracy	75.1%
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{1}{t} - \frac{\frac{1}{x}}{t}\right)\\ \end{array} \]

Alternative 13
Accuracy	74.8%
Cost	196

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

Alternative 14
Accuracy	37.4%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023147 
(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)))))