Toniolo and Linder, Equation (7)

Average Accuracy: 33.1% → 87.9%

Time: 32.2s

Precision: binary64

Cost: 28372

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \sqrt{\frac{1}{\mathsf{fma}\left(t, t + \frac{t}{x}, \frac{\ell}{\frac{x}{\ell}}\right)}}\\ t_2 := 2 + \left(\frac{2}{x} + \frac{2}{x}\right)\\ t_3 := t \cdot \sqrt{2}\\ t_4 := \frac{2}{x} + \left(2 + \frac{2}{x}\right)\\ \mathbf{if}\;t \leq -2.25 \cdot 10^{+92}:\\ \;\;\;\;\frac{-0.5}{x \cdot x} + \left(-1 + \frac{1}{x}\right)\\ \mathbf{elif}\;t \leq -5.4 \cdot 10^{-158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3.7 \cdot 10^{-268}:\\ \;\;\;\;\frac{t_3}{\sqrt{\frac{1}{t_2}} \cdot \left(-0.5 \cdot \frac{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}{t}\right) - t \cdot \sqrt{t_2}}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-293}:\\ \;\;\;\;\frac{t_3}{\left|\frac{\ell \cdot \sqrt{2}}{\sqrt{x}}\right|}\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-163}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(\sqrt{t_4}, t, \sqrt{\frac{1}{t_4}} \cdot \frac{\ell \cdot \ell}{t \cdot x}\right)}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+85}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \left(\frac{0.5}{x \cdot x} + \frac{-0.5}{{x}^{3}}\right)\right)\\ \end{array} \]

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 (/ 1.0 (fma t (+ t (/ t x)) (/ l (/ x l)))))))
        (t_2 (+ 2.0 (+ (/ 2.0 x) (/ 2.0 x))))
        (t_3 (* t (sqrt 2.0)))
        (t_4 (+ (/ 2.0 x) (+ 2.0 (/ 2.0 x)))))
   (if (<= t -2.25e+92)
     (+ (/ -0.5 (* x x)) (+ -1.0 (/ 1.0 x)))
     (if (<= t -5.4e-158)
       t_1
       (if (<= t -3.7e-268)
         (/
          t_3
          (-
           (* (sqrt (/ 1.0 t_2)) (* -0.5 (/ (* 2.0 (* l (/ l x))) t)))
           (* t (sqrt t_2))))
         (if (<= t 1.25e-293)
           (/ t_3 (fabs (/ (* l (sqrt 2.0)) (sqrt x))))
           (if (<= t 2.05e-163)
             (*
              t
              (/
               (sqrt 2.0)
               (fma (sqrt t_4) t (* (sqrt (/ 1.0 t_4)) (/ (* l l) (* t x))))))
             (if (<= t 1.35e+85)
               t_1
               (+
                1.0
                (+
                 (/ -1.0 x)
                 (+ (/ 0.5 (* x x)) (/ -0.5 (pow x 3.0)))))))))))))

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((1.0 / fma(t, (t + (t / x)), (l / (x / l)))));
	double t_2 = 2.0 + ((2.0 / x) + (2.0 / x));
	double t_3 = t * sqrt(2.0);
	double t_4 = (2.0 / x) + (2.0 + (2.0 / x));
	double tmp;
	if (t <= -2.25e+92) {
		tmp = (-0.5 / (x * x)) + (-1.0 + (1.0 / x));
	} else if (t <= -5.4e-158) {
		tmp = t_1;
	} else if (t <= -3.7e-268) {
		tmp = t_3 / ((sqrt((1.0 / t_2)) * (-0.5 * ((2.0 * (l * (l / x))) / t))) - (t * sqrt(t_2)));
	} else if (t <= 1.25e-293) {
		tmp = t_3 / fabs(((l * sqrt(2.0)) / sqrt(x)));
	} else if (t <= 2.05e-163) {
		tmp = t * (sqrt(2.0) / fma(sqrt(t_4), t, (sqrt((1.0 / t_4)) * ((l * l) / (t * x)))));
	} else if (t <= 1.35e+85) {
		tmp = t_1;
	} else {
		tmp = 1.0 + ((-1.0 / x) + ((0.5 / (x * x)) + (-0.5 / pow(x, 3.0))));
	}
	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(Float64(1.0 / fma(t, Float64(t + Float64(t / x)), Float64(l / Float64(x / l))))))
	t_2 = Float64(2.0 + Float64(Float64(2.0 / x) + Float64(2.0 / x)))
	t_3 = Float64(t * sqrt(2.0))
	t_4 = Float64(Float64(2.0 / x) + Float64(2.0 + Float64(2.0 / x)))
	tmp = 0.0
	if (t <= -2.25e+92)
		tmp = Float64(Float64(-0.5 / Float64(x * x)) + Float64(-1.0 + Float64(1.0 / x)));
	elseif (t <= -5.4e-158)
		tmp = t_1;
	elseif (t <= -3.7e-268)
		tmp = Float64(t_3 / Float64(Float64(sqrt(Float64(1.0 / t_2)) * Float64(-0.5 * Float64(Float64(2.0 * Float64(l * Float64(l / x))) / t))) - Float64(t * sqrt(t_2))));
	elseif (t <= 1.25e-293)
		tmp = Float64(t_3 / abs(Float64(Float64(l * sqrt(2.0)) / sqrt(x))));
	elseif (t <= 2.05e-163)
		tmp = Float64(t * Float64(sqrt(2.0) / fma(sqrt(t_4), t, Float64(sqrt(Float64(1.0 / t_4)) * Float64(Float64(l * l) / Float64(t * x))))));
	elseif (t <= 1.35e+85)
		tmp = t_1;
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) + Float64(Float64(0.5 / Float64(x * x)) + Float64(-0.5 / (x ^ 3.0)))));
	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[N[(1.0 / N[(t * N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision] + N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.25e+92], N[(N[(-0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -5.4e-158], t$95$1, If[LessEqual[t, -3.7e-268], N[(t$95$3 / N[(N[(N[Sqrt[N[(1.0 / t$95$2), $MachinePrecision]], $MachinePrecision] * N[(-0.5 * N[(N[(2.0 * N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.25e-293], N[(t$95$3 / N[Abs[N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.05e-163], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[t$95$4], $MachinePrecision] * t + N[(N[Sqrt[N[(1.0 / t$95$4), $MachinePrecision]], $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(t * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.35e+85], t$95$1, N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -2.25e92

   1. Initial program 22.0%

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

   2. Simplified 22.0%

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

      [Start] 22.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 22.0	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 22.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 22.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Taylor expanded in t around -inf 94.0%

      \[\leadsto \color{blue}{-1 \cdot \left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{x - 1}{1 + x}}\right)} \]

   4. Simplified 94.0%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)} \]
      Proof

      [Start] 94.0	\[ -1 \cdot \left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{x - 1}{1 + x}}\right) \]
      mul-1-neg [=>] 94.0	\[ \color{blue}{-\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
      associate-*l* [=>] 94.0	\[ -\color{blue}{\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)} \]
      sub-neg [=>] 94.0	\[ -\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}}\right) \]
      metadata-eval [=>] 94.0	\[ -\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x + \color{blue}{-1}}{1 + x}}\right) \]
      +-commutative [=>] 94.0	\[ -\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{\color{blue}{-1 + x}}{1 + x}}\right) \]
      +-commutative [=>] 94.0	\[ -\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{\color{blue}{x + 1}}}\right) \]

   5. Applied egg-rr 95.4%

      \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{-1 + x}{x + 1}}\right)} - 1\right)} \]

   6. Simplified 95.5%

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

      [Start] 95.4	\[ -\left(e^{\mathsf{log1p}\left(\sqrt{\frac{-1 + x}{x + 1}}\right)} - 1\right) \]
      expm1-def [=>] 95.5	\[ -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{-1 + x}{x + 1}}\right)\right)} \]
      expm1-log1p [=>] 95.5	\[ -\color{blue}{\sqrt{\frac{-1 + x}{x + 1}}} \]

   7. Taylor expanded in x around inf 94.7%

      \[\leadsto -\color{blue}{\left(\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{x}\right)} \]

   8. Simplified 94.7%

      \[\leadsto -\color{blue}{\left(\left(1 + \frac{-1}{x}\right) + \frac{0.5}{x \cdot x}\right)} \]
      Proof

      [Start] 94.7	\[ -\left(\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{x}\right) \]
      +-commutative [=>] 94.7	\[ -\left(\color{blue}{\left(0.5 \cdot \frac{1}{{x}^{2}} + 1\right)} - \frac{1}{x}\right) \]
      associate--l+ [=>] 94.7	\[ -\color{blue}{\left(0.5 \cdot \frac{1}{{x}^{2}} + \left(1 - \frac{1}{x}\right)\right)} \]
      +-commutative [=>] 94.7	\[ -\color{blue}{\left(\left(1 - \frac{1}{x}\right) + 0.5 \cdot \frac{1}{{x}^{2}}\right)} \]
      sub-neg [=>] 94.7	\[ -\left(\color{blue}{\left(1 + \left(-\frac{1}{x}\right)\right)} + 0.5 \cdot \frac{1}{{x}^{2}}\right) \]
      distribute-neg-frac [=>] 94.7	\[ -\left(\left(1 + \color{blue}{\frac{-1}{x}}\right) + 0.5 \cdot \frac{1}{{x}^{2}}\right) \]
      metadata-eval [=>] 94.7	\[ -\left(\left(1 + \frac{\color{blue}{-1}}{x}\right) + 0.5 \cdot \frac{1}{{x}^{2}}\right) \]
      associate-*r/ [=>] 94.7	\[ -\left(\left(1 + \frac{-1}{x}\right) + \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right) \]
      metadata-eval [=>] 94.7	\[ -\left(\left(1 + \frac{-1}{x}\right) + \frac{\color{blue}{0.5}}{{x}^{2}}\right) \]
      unpow2 [=>] 94.7	\[ -\left(\left(1 + \frac{-1}{x}\right) + \frac{0.5}{\color{blue}{x \cdot x}}\right) \]

   if -2.25e92 < t < -5.3999999999999997e-158 or 2.04999999999999991e-163 < t < 1.34999999999999992e85

   1. Initial program 59.2%

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

   2. Simplified 59.3%

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

      [Start] 59.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 59.3	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 59.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 59.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Taylor expanded in x around inf 84.8%

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

   4. Simplified 84.8%

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

      [Start] 84.8	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 84.8	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 84.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 84.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 84.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 84.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 84.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      +-commutative [=>] 84.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 84.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 84.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 84.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)\right)}} \cdot t \]

   5. Taylor expanded in t around 0 84.3%

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

   6. Simplified 84.3%

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

      [Start] 84.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{{\ell}^{2}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 84.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]

   7. Applied egg-rr 92.1%

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

   8. Simplified 92.4%

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

      [Start] 92.1	\[ \frac{\sqrt{2}}{{\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)\right)}^{0.25} \cdot {\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)\right)}^{0.25}} \cdot t \]
      pow-sqr [=>] 92.4	\[ \frac{\sqrt{2}}{\color{blue}{{\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)\right)}^{\left(2 \cdot 0.25\right)}}} \cdot t \]
      metadata-eval [=>] 92.4	\[ \frac{\sqrt{2}}{{\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)\right)}^{\color{blue}{0.5}}} \cdot t \]
      unpow1/2 [=>] 92.4	\[ \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)}}} \cdot t \]
      +-commutative [=>] 92.4	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right) + \frac{\ell}{x} \cdot \ell}}} \cdot t \]
      fma-udef [=>] 92.4	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) + \frac{\ell}{x} \cdot \ell\right)} + \frac{\ell}{x} \cdot \ell}} \cdot t \]
      associate-+l+ [=>] 92.4	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) + \left(\frac{\ell}{x} \cdot \ell + \frac{\ell}{x} \cdot \ell\right)}}} \cdot t \]
      fma-udef [=>] 92.4	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\left(t \cdot t + \frac{t}{x} \cdot t\right)} + \left(\frac{\ell}{x} \cdot \ell + \frac{\ell}{x} \cdot \ell\right)}} \cdot t \]
      distribute-rgt-out [=>] 92.4	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\left(t \cdot \left(t + \frac{t}{x}\right)\right)} + \left(\frac{\ell}{x} \cdot \ell + \frac{\ell}{x} \cdot \ell\right)}} \cdot t \]
      count-2 [=>] 92.4	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + \color{blue}{2 \cdot \left(\frac{\ell}{x} \cdot \ell\right)}}} \cdot t \]
      *-commutative [=>] 92.4	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{x}\right)}}} \cdot t \]

   9. Applied egg-rr 49.1%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\mathsf{fma}\left(t, t + \frac{t}{x}, \frac{\ell}{\frac{x}{\ell}}\right)}}\right)} - 1\right)} \cdot t \]

   10. Simplified 92.1%

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

       [Start] 49.1	\[ \left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\mathsf{fma}\left(t, t + \frac{t}{x}, \frac{\ell}{\frac{x}{\ell}}\right)}}\right)} - 1\right) \cdot t \]
       expm1-def [=>] 89.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\mathsf{fma}\left(t, t + \frac{t}{x}, \frac{\ell}{\frac{x}{\ell}}\right)}}\right)\right)} \cdot t \]
       expm1-log1p [=>] 92.1	\[ \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(t, t + \frac{t}{x}, \frac{\ell}{\frac{x}{\ell}}\right)}}} \cdot t \]

   if -5.3999999999999997e-158 < t < -3.70000000000000018e-268

   1. Initial program 2.5%

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

   2. Taylor expanded in x around inf 43.1%

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

   3. Simplified 43.1%

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

      [Start] 43.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      associate--l+ [=>] 43.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]
      unpow2 [=>] 43.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      distribute-lft-out [=>] 43.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 43.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 43.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-*r/ [=>] 43.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \]
      mul-1-neg [=>] 43.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \]
      +-commutative [=>] 43.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + {\ell}^{2}\right)}}{x}\right)}} \]
      unpow2 [=>] 43.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}\right)}{x}\right)}} \]
      fma-udef [<=] 43.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)}} \]
      unpow2 [=>] 43.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)}} \]

   4. Taylor expanded in t around -inf 63.2%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{-0.5 \cdot \left(\frac{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}{t} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}\right) + -1 \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)}} \]

   5. Simplified 63.2%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{\frac{1}{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}} \cdot \left(-0.5 \cdot \frac{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}{t}\right) - t \cdot \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}}} \]
      Proof

      [Start] 63.2	\[ \frac{\sqrt{2} \cdot t}{-0.5 \cdot \left(\frac{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}{t} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}\right) + -1 \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)} \]
      *-commutative [=>] 63.2	\[ \frac{\sqrt{2} \cdot t}{-0.5 \cdot \left(\frac{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}{t} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}\right) + -1 \cdot \color{blue}{\left(\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t\right)}} \]
      mul-1-neg [=>] 63.2	\[ \frac{\sqrt{2} \cdot t}{-0.5 \cdot \left(\frac{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}{t} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}\right) + \color{blue}{\left(-\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t\right)}} \]
      unsub-neg [=>] 63.2	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{-0.5 \cdot \left(\frac{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}{t} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}\right) - \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t}} \]

   if -3.70000000000000018e-268 < t < 1.2500000000000001e-293

   1. Initial program 1.0%

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

   2. Taylor expanded in t around 0 4.6%

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

   3. Simplified 4.8%

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

      [Start] 4.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\left(1 + x\right) \cdot {\ell}^{2}}{x - 1} - {\ell}^{2}}} \]
      *-commutative [=>] 4.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{{\ell}^{2} \cdot \left(1 + x\right)}}{x - 1} - {\ell}^{2}}} \]
      unpow2 [=>] 4.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(1 + x\right)}{x - 1} - {\ell}^{2}}} \]
      associate-*l* [=>] 4.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{\ell \cdot \left(\ell \cdot \left(1 + x\right)\right)}}{x - 1} - {\ell}^{2}}} \]
      +-commutative [=>] 4.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \color{blue}{\left(x + 1\right)}\right)}{x - 1} - {\ell}^{2}}} \]
      sub-neg [=>] 4.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \left(x + 1\right)\right)}{\color{blue}{x + \left(-1\right)}} - {\ell}^{2}}} \]
      metadata-eval [=>] 4.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \left(x + 1\right)\right)}{x + \color{blue}{-1}} - {\ell}^{2}}} \]
      +-commutative [=>] 4.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \left(x + 1\right)\right)}{\color{blue}{-1 + x}} - {\ell}^{2}}} \]
      unpow2 [=>] 4.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \left(x + 1\right)\right)}{-1 + x} - \color{blue}{\ell \cdot \ell}}} \]

   4. Taylor expanded in x around inf 57.1%

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

   5. Simplified 57.1%

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

      [Start] 57.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{{\ell}^{2} - -1 \cdot {\ell}^{2}}{x}}} \]
      cancel-sign-sub-inv [=>] 57.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{{\ell}^{2} + \left(--1\right) \cdot {\ell}^{2}}}{x}}} \]
      unpow2 [=>] 57.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{\ell \cdot \ell} + \left(--1\right) \cdot {\ell}^{2}}{x}}} \]
      metadata-eval [=>] 57.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell + \color{blue}{1} \cdot {\ell}^{2}}{x}}} \]
      unpow2 [=>] 57.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell + 1 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{x}}} \]
      distribute-rgt1-in [=>] 57.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{\left(1 + 1\right) \cdot \left(\ell \cdot \ell\right)}}{x}}} \]
      metadata-eval [=>] 57.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{2} \cdot \left(\ell \cdot \ell\right)}{x}}} \]

   6. Applied egg-rr 74.9%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left|\frac{\ell \cdot \sqrt{2}}{\sqrt{x}}\right|}} \]

   if 1.2500000000000001e-293 < t < 2.04999999999999991e-163

   1. Initial program 1.4%

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

   2. Simplified 1.4%

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

      [Start] 1.4	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 1.4	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 1.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 1.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Taylor expanded in x around inf 46.1%

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

   4. Simplified 46.1%

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

      [Start] 46.1	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 46.1	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 46.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 46.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 46.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 46.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 46.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      +-commutative [=>] 46.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 46.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 46.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 46.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)\right)}} \cdot t \]

   5. Taylor expanded in t around inf 58.6%

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

   6. Simplified 58.6%

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

      [Start] 58.6	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}} \cdot t \]
      fma-def [=>] 58.6	\[ \frac{\sqrt{2}}{\color{blue}{\mathsf{fma}\left(\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}, t, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)}} \cdot t \]
      distribute-lft-in [=>] 58.6	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(\sqrt{\color{blue}{\left(2 \cdot 1 + 2 \cdot \frac{1}{x}\right)} + 2 \cdot \frac{1}{x}}, t, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      metadata-eval [=>] 58.6	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(\sqrt{\left(\color{blue}{2} + 2 \cdot \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}, t, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      associate-*r/ [=>] 58.6	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(\sqrt{\left(2 + \color{blue}{\frac{2 \cdot 1}{x}}\right) + 2 \cdot \frac{1}{x}}, t, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      metadata-eval [=>] 58.6	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(\sqrt{\left(2 + \frac{\color{blue}{2}}{x}\right) + 2 \cdot \frac{1}{x}}, t, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      associate-*r/ [=>] 58.6	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(\sqrt{\left(2 + \frac{2}{x}\right) + \color{blue}{\frac{2 \cdot 1}{x}}}, t, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      metadata-eval [=>] 58.6	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(\sqrt{\left(2 + \frac{2}{x}\right) + \frac{\color{blue}{2}}{x}}, t, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]

   if 1.34999999999999992e85 < t

   1. Initial program 24.7%

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

   2. Simplified 24.7%

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

      [Start] 24.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 24.7	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 24.7	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 24.7	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Taylor expanded in t around -inf 1.6%

      \[\leadsto \color{blue}{-1 \cdot \left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{x - 1}{1 + x}}\right)} \]

   4. Simplified 1.6%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)} \]
      Proof

      [Start] 1.6	\[ -1 \cdot \left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{x - 1}{1 + x}}\right) \]
      mul-1-neg [=>] 1.6	\[ \color{blue}{-\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
      associate-*l* [=>] 1.6	\[ -\color{blue}{\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)} \]
      sub-neg [=>] 1.6	\[ -\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}}\right) \]
      metadata-eval [=>] 1.6	\[ -\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x + \color{blue}{-1}}{1 + x}}\right) \]
      +-commutative [=>] 1.6	\[ -\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{\color{blue}{-1 + x}}{1 + x}}\right) \]
      +-commutative [=>] 1.6	\[ -\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{\color{blue}{x + 1}}}\right) \]

   5. Applied egg-rr 1.6%

      \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{-1 + x}{x + 1}}\right)} - 1\right)} \]

   6. Simplified 1.6%

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

      [Start] 1.6	\[ -\left(e^{\mathsf{log1p}\left(\sqrt{\frac{-1 + x}{x + 1}}\right)} - 1\right) \]
      expm1-def [=>] 1.6	\[ -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{-1 + x}{x + 1}}\right)\right)} \]
      expm1-log1p [=>] 1.6	\[ -\color{blue}{\sqrt{\frac{-1 + x}{x + 1}}} \]

   7. Taylor expanded in x around -inf 0.0%

      \[\leadsto -\color{blue}{\left(0.5 \cdot \frac{2 \cdot \frac{1}{{\left(\sqrt{-1}\right)}^{2}} + \left(2 + \frac{1}{{\left(\sqrt{-1}\right)}^{4}}\right)}{{x}^{3}} + \left(-0.5 \cdot \frac{2 + {\left(\frac{-1}{\sqrt{-1}}\right)}^{2}}{{x}^{2}} + \left({\left(\sqrt{-1}\right)}^{2} + \frac{1}{x}\right)\right)\right)} \]

   8. Simplified 95.2%

      \[\leadsto -\color{blue}{\left(-1 + \left(\frac{1}{x} + \left(\frac{-0.5}{x \cdot x} + \frac{0.5}{{x}^{3}}\right)\right)\right)} \]
      Proof

      [Start] 0.0	\[ -\left(0.5 \cdot \frac{2 \cdot \frac{1}{{\left(\sqrt{-1}\right)}^{2}} + \left(2 + \frac{1}{{\left(\sqrt{-1}\right)}^{4}}\right)}{{x}^{3}} + \left(-0.5 \cdot \frac{2 + {\left(\frac{-1}{\sqrt{-1}}\right)}^{2}}{{x}^{2}} + \left({\left(\sqrt{-1}\right)}^{2} + \frac{1}{x}\right)\right)\right) \]
      associate-+r+ [=>] 0.0	\[ -\color{blue}{\left(\left(0.5 \cdot \frac{2 \cdot \frac{1}{{\left(\sqrt{-1}\right)}^{2}} + \left(2 + \frac{1}{{\left(\sqrt{-1}\right)}^{4}}\right)}{{x}^{3}} + -0.5 \cdot \frac{2 + {\left(\frac{-1}{\sqrt{-1}}\right)}^{2}}{{x}^{2}}\right) + \left({\left(\sqrt{-1}\right)}^{2} + \frac{1}{x}\right)\right)} \]
      +-commutative [=>] 0.0	\[ -\color{blue}{\left(\left({\left(\sqrt{-1}\right)}^{2} + \frac{1}{x}\right) + \left(0.5 \cdot \frac{2 \cdot \frac{1}{{\left(\sqrt{-1}\right)}^{2}} + \left(2 + \frac{1}{{\left(\sqrt{-1}\right)}^{4}}\right)}{{x}^{3}} + -0.5 \cdot \frac{2 + {\left(\frac{-1}{\sqrt{-1}}\right)}^{2}}{{x}^{2}}\right)\right)} \]
      unpow2 [=>] 0.0	\[ -\left(\left(\color{blue}{\sqrt{-1} \cdot \sqrt{-1}} + \frac{1}{x}\right) + \left(0.5 \cdot \frac{2 \cdot \frac{1}{{\left(\sqrt{-1}\right)}^{2}} + \left(2 + \frac{1}{{\left(\sqrt{-1}\right)}^{4}}\right)}{{x}^{3}} + -0.5 \cdot \frac{2 + {\left(\frac{-1}{\sqrt{-1}}\right)}^{2}}{{x}^{2}}\right)\right) \]
      rem-square-sqrt [=>] 0.0	\[ -\left(\left(\color{blue}{-1} + \frac{1}{x}\right) + \left(0.5 \cdot \frac{2 \cdot \frac{1}{{\left(\sqrt{-1}\right)}^{2}} + \left(2 + \frac{1}{{\left(\sqrt{-1}\right)}^{4}}\right)}{{x}^{3}} + -0.5 \cdot \frac{2 + {\left(\frac{-1}{\sqrt{-1}}\right)}^{2}}{{x}^{2}}\right)\right) \]
      associate-+l+ [=>] 0.0	\[ -\color{blue}{\left(-1 + \left(\frac{1}{x} + \left(0.5 \cdot \frac{2 \cdot \frac{1}{{\left(\sqrt{-1}\right)}^{2}} + \left(2 + \frac{1}{{\left(\sqrt{-1}\right)}^{4}}\right)}{{x}^{3}} + -0.5 \cdot \frac{2 + {\left(\frac{-1}{\sqrt{-1}}\right)}^{2}}{{x}^{2}}\right)\right)\right)} \]

3. Recombined 6 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.25 \cdot 10^{+92}:\\ \;\;\;\;\frac{-0.5}{x \cdot x} + \left(-1 + \frac{1}{x}\right)\\ \mathbf{elif}\;t \leq -5.4 \cdot 10^{-158}:\\ \;\;\;\;t \cdot \sqrt{\frac{1}{\mathsf{fma}\left(t, t + \frac{t}{x}, \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{elif}\;t \leq -3.7 \cdot 10^{-268}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{1}{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}} \cdot \left(-0.5 \cdot \frac{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}{t}\right) - t \cdot \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-293}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left|\frac{\ell \cdot \sqrt{2}}{\sqrt{x}}\right|}\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-163}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(\sqrt{\frac{2}{x} + \left(2 + \frac{2}{x}\right)}, t, \sqrt{\frac{1}{\frac{2}{x} + \left(2 + \frac{2}{x}\right)}} \cdot \frac{\ell \cdot \ell}{t \cdot x}\right)}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+85}:\\ \;\;\;\;t \cdot \sqrt{\frac{1}{\mathsf{fma}\left(t, t + \frac{t}{x}, \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \left(\frac{0.5}{x \cdot x} + \frac{-0.5}{{x}^{3}}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	87.9%
Cost	28372

\[\begin{array}{l} t_1 := t \cdot \sqrt{\frac{1}{\mathsf{fma}\left(t, t + \frac{t}{x}, \frac{\ell}{\frac{x}{\ell}}\right)}}\\ t_2 := 2 + \left(\frac{2}{x} + \frac{2}{x}\right)\\ t_3 := \sqrt{\frac{1}{t_2}}\\ t_4 := t \cdot \sqrt{2}\\ t_5 := \sqrt{t_2}\\ \mathbf{if}\;t \leq -3.4 \cdot 10^{+92}:\\ \;\;\;\;\frac{-0.5}{x \cdot x} + \left(-1 + \frac{1}{x}\right)\\ \mathbf{elif}\;t \leq -1.1 \cdot 10^{-159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3.8 \cdot 10^{-268}:\\ \;\;\;\;\frac{t_4}{t_3 \cdot \left(-0.5 \cdot \frac{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}{t}\right) - t \cdot t_5}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-293}:\\ \;\;\;\;\frac{t_4}{\left|\frac{\ell \cdot \sqrt{2}}{\sqrt{x}}\right|}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-163}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(t, t_5, t_3 \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+87}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \left(\frac{0.5}{x \cdot x} + \frac{-0.5}{{x}^{3}}\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	87.9%
Cost	28116

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := t \cdot \sqrt{\frac{1}{\mathsf{fma}\left(t, t + \frac{t}{x}, \frac{\ell}{\frac{x}{\ell}}\right)}}\\ t_3 := 2 + \left(\frac{2}{x} + \frac{2}{x}\right)\\ \mathbf{if}\;t \leq -1.15 \cdot 10^{+92}:\\ \;\;\;\;\frac{-0.5}{x \cdot x} + \left(-1 + \frac{1}{x}\right)\\ \mathbf{elif}\;t \leq -6.6 \cdot 10^{-158}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -2.1 \cdot 10^{-268}:\\ \;\;\;\;\frac{t_1}{\sqrt{\frac{1}{t_3}} \cdot \left(-0.5 \cdot \frac{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}{t}\right) - t \cdot \sqrt{t_3}}\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{-290}:\\ \;\;\;\;\frac{t_1}{\left|\frac{\ell \cdot \sqrt{2}}{\sqrt{x}}\right|}\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-163}:\\ \;\;\;\;\frac{t_1}{\mathsf{fma}\left(0.5, \frac{2 \cdot \left(\ell \cdot \ell + t \cdot t\right) + 2 \cdot \left(t \cdot t\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, t_1\right)}\\ \mathbf{elif}\;t \leq 4.9 \cdot 10^{+85}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \left(\frac{0.5}{x \cdot x} + \frac{-0.5}{{x}^{3}}\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	87.9%
Cost	26704

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := t \cdot \sqrt{\frac{1}{\mathsf{fma}\left(t, t + \frac{t}{x}, \frac{\ell}{\frac{x}{\ell}}\right)}}\\ t_3 := 2 + \left(\frac{2}{x} + \frac{2}{x}\right)\\ \mathbf{if}\;t \leq -3.8 \cdot 10^{+92}:\\ \;\;\;\;\frac{-0.5}{x \cdot x} + \left(-1 + \frac{1}{x}\right)\\ \mathbf{elif}\;t \leq -8.5 \cdot 10^{-159}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-268}:\\ \;\;\;\;\frac{t_1}{\sqrt{\frac{1}{t_3}} \cdot \left(-0.5 \cdot \frac{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}{t}\right) - t \cdot \sqrt{t_3}}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-292}:\\ \;\;\;\;\frac{t_1}{\left|\frac{\ell \cdot \sqrt{2}}{\sqrt{x}}\right|}\\ \mathbf{elif}\;t \leq 6.7 \cdot 10^{-164}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t_1 + \frac{\ell \cdot \ell + t \cdot t}{\sqrt{2} \cdot \left(t \cdot x\right)}}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+86}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \left(\frac{0.5}{x \cdot x} + \frac{-0.5}{{x}^{3}}\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	87.2%
Cost	22092

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := t \cdot \sqrt{\frac{1}{\mathsf{fma}\left(t, t + \frac{t}{x}, \frac{\ell}{\frac{x}{\ell}}\right)}}\\ t_3 := 2 + \left(\frac{2}{x} + \frac{2}{x}\right)\\ \mathbf{if}\;t \leq -3.5 \cdot 10^{+92}:\\ \;\;\;\;\frac{-0.5}{x \cdot x} + \left(-1 + \frac{1}{x}\right)\\ \mathbf{elif}\;t \leq -1.05 \cdot 10^{-159}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-309}:\\ \;\;\;\;\frac{t_1}{\sqrt{\frac{1}{t_3}} \cdot \left(-0.5 \cdot \frac{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}{t}\right) - t \cdot \sqrt{t_3}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-164}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t_1 + \frac{\ell \cdot \ell + t \cdot t}{\sqrt{2} \cdot \left(t \cdot x\right)}}\\ \mathbf{elif}\;t \leq 4.9 \cdot 10^{+86}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \left(\frac{0.5}{x \cdot x} + \frac{-0.5}{{x}^{3}}\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	85.9%
Cost	21204

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{x}{\ell}}\\ t_2 := t \cdot \sqrt{\frac{1}{\mathsf{fma}\left(t, t + \frac{t}{x}, t_1\right)}}\\ \mathbf{if}\;t \leq -5.4 \cdot 10^{+92}:\\ \;\;\;\;\frac{-0.5}{x \cdot x} + \left(-1 + \frac{1}{x}\right)\\ \mathbf{elif}\;t \leq -9.2 \cdot 10^{-158}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -3.8 \cdot 10^{-268}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{-298}:\\ \;\;\;\;\sqrt{2} \cdot \left(t \cdot {\left(t_1 \cdot 2\right)}^{-0.5}\right)\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-164}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t \cdot \sqrt{2} + \frac{\ell \cdot \ell + t \cdot t}{\sqrt{2} \cdot \left(t \cdot x\right)}}\\ \mathbf{elif}\;t \leq 2.95 \cdot 10^{+85}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \left(\frac{0.5}{x \cdot x} + \frac{-0.5}{{x}^{3}}\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	85.3%
Cost	14552

\[\begin{array}{l} t_1 := t \cdot \sqrt{\frac{1}{\mathsf{fma}\left(t, t + \frac{t}{x}, \frac{\ell}{\frac{x}{\ell}}\right)}}\\ t_2 := \frac{0.5}{x \cdot x}\\ \mathbf{if}\;t \leq -2.25 \cdot 10^{+92}:\\ \;\;\;\;\frac{-0.5}{x \cdot x} + \left(-1 + \frac{1}{x}\right)\\ \mathbf{elif}\;t \leq -3 \cdot 10^{-158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3.7 \cdot 10^{-268}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-236}:\\ \;\;\;\;\sqrt{2} \cdot \left(t \cdot {\left(2 \cdot \frac{\ell \cdot \ell}{x}\right)}^{-0.5}\right)\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{-163}:\\ \;\;\;\;1 + \left(t_2 + \frac{-1}{x}\right)\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+87}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \left(t_2 + \frac{-0.5}{{x}^{3}}\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	78.1%
Cost	13832

\[\begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-268}:\\ \;\;\;\;\frac{-0.5}{x \cdot x} + \left(-1 + \frac{1}{x}\right)\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-235}:\\ \;\;\;\;\sqrt{2} \cdot \left(t \cdot {\left(\frac{\ell}{\frac{x}{\ell}} \cdot 2\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \left(\frac{0.5}{x \cdot x} + \frac{-0.5}{{x}^{3}}\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	78.1%
Cost	13832

\[\begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-268}:\\ \;\;\;\;\frac{-0.5}{x \cdot x} + \left(-1 + \frac{1}{x}\right)\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-235}:\\ \;\;\;\;\sqrt{2} \cdot \left(t \cdot {\left(2 \cdot \frac{\ell \cdot \ell}{x}\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \left(\frac{0.5}{x \cdot x} + \frac{-0.5}{{x}^{3}}\right)\right)\\ \end{array} \]

Alternative 9
Accuracy	77.7%
Cost	13768

\[\begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-268}:\\ \;\;\;\;\frac{-0.5}{x \cdot x} + \left(-1 + \frac{1}{x}\right)\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-272}:\\ \;\;\;\;\left(t \cdot \sqrt{2}\right) \cdot \sqrt{0.5 \cdot \frac{x}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \left(\frac{0.5}{x \cdot x} + \frac{-0.5}{{x}^{3}}\right)\right)\\ \end{array} \]

Alternative 10
Accuracy	78.1%
Cost	13768

\[\begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{-268}:\\ \;\;\;\;\frac{-0.5}{x \cdot x} + \left(-1 + \frac{1}{x}\right)\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-239}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \left(\frac{0.5}{x \cdot x} + \frac{-0.5}{{x}^{3}}\right)\right)\\ \end{array} \]

Alternative 11
Accuracy	78.1%
Cost	13768

\[\begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-268}:\\ \;\;\;\;\frac{-0.5}{x \cdot x} + \left(-1 + \frac{1}{x}\right)\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-239}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\ell \cdot \frac{\ell \cdot 2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \left(\frac{0.5}{x \cdot x} + \frac{-0.5}{{x}^{3}}\right)\right)\\ \end{array} \]

Alternative 12
Accuracy	77.4%
Cost	7688

\[\begin{array}{l} \mathbf{if}\;t \leq -3.7 \cdot 10^{-268}:\\ \;\;\;\;\frac{-0.5}{x \cdot x} + \left(-1 + \frac{1}{x}\right)\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-269}:\\ \;\;\;\;\sqrt{x} \cdot \frac{-t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \left(\frac{0.5}{x \cdot x} + \frac{-0.5}{{x}^{3}}\right)\right)\\ \end{array} \]

Alternative 13
Accuracy	77.3%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-268}:\\ \;\;\;\;\frac{-0.5}{x \cdot x} + \left(-1 + \frac{1}{x}\right)\\ \mathbf{elif}\;t \leq 5.7 \cdot 10^{-269}:\\ \;\;\;\;\sqrt{x} \cdot \frac{-t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 14
Accuracy	76.3%
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 15
Accuracy	76.4%
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-0.5}{x \cdot x} + \left(-1 + \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 16
Accuracy	75.9%
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 17
Accuracy	76.2%
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 18
Accuracy	75.5%
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 19
Accuracy	38.8%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023125 
(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)))))