Toniolo and Linder, Equation (7)

Average Error: 43.3 → 5.8

Time: 29.1s

Precision: binary64

Cost: 20496

Math

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

↓

\[\begin{array}{l} t_1 := \frac{t}{\sqrt{\mathsf{fma}\left(t, t, \ell \cdot \frac{\ell}{x}\right)}}\\ t_2 := \sqrt{\frac{1 - x}{-1 - x}}\\ \mathbf{if}\;t \leq -5 \cdot 10^{+106}:\\ \;\;\;\;-t_2\\ \mathbf{elif}\;t \leq -1.3 \cdot 10^{-134}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-304}:\\ \;\;\;\;\frac{t}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-116}:\\ \;\;\;\;\frac{t}{\mathsf{hypot}\left(t, \sqrt{\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}\right)}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+103}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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 (fma t t (* l (/ l x))))))
        (t_2 (sqrt (/ (- 1.0 x) (- -1.0 x)))))
   (if (<= t -5e+106)
     (- t_2)
     (if (<= t -1.3e-134)
       t_1
       (if (<= t 3.1e-304)
         (/ t (hypot t (/ l (sqrt x))))
         (if (<= t 1.9e-116)
           (/ t (hypot t (sqrt (/ (fma 2.0 (* t t) (* l l)) x))))
           (if (<= t 5e+103) t_1 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(fma(t, t, (l * (l / x))));
	double t_2 = sqrt(((1.0 - x) / (-1.0 - x)));
	double tmp;
	if (t <= -5e+106) {
		tmp = -t_2;
	} else if (t <= -1.3e-134) {
		tmp = t_1;
	} else if (t <= 3.1e-304) {
		tmp = t / hypot(t, (l / sqrt(x)));
	} else if (t <= 1.9e-116) {
		tmp = t / hypot(t, sqrt((fma(2.0, (t * t), (l * l)) / x)));
	} else if (t <= 5e+103) {
		tmp = t_1;
	} else {
		tmp = 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(fma(t, t, Float64(l * Float64(l / x)))))
	t_2 = sqrt(Float64(Float64(1.0 - x) / Float64(-1.0 - x)))
	tmp = 0.0
	if (t <= -5e+106)
		tmp = Float64(-t_2);
	elseif (t <= -1.3e-134)
		tmp = t_1;
	elseif (t <= 3.1e-304)
		tmp = Float64(t / hypot(t, Float64(l / sqrt(x))));
	elseif (t <= 1.9e-116)
		tmp = Float64(t / hypot(t, sqrt(Float64(fma(2.0, Float64(t * t), Float64(l * l)) / x))));
	elseif (t <= 5e+103)
		tmp = t_1;
	else
		tmp = 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[N[(t * t + N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -5e+106], (-t$95$2), If[LessEqual[t, -1.3e-134], t$95$1, If[LessEqual[t, 3.1e-304], N[(t / N[Sqrt[t ^ 2 + N[(l / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.9e-116], N[(t / N[Sqrt[t ^ 2 + N[Sqrt[N[(N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5e+103], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -4.9999999999999998e106

   1. Initial program 52.0

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

   2. Simplified 52.1

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

      [Start] 52.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 52.1	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      fma-neg [=>] 52.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]
      +-commutative [=>] 52.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}, -\ell \cdot \ell\right)}} \]
      fma-def [=>] 52.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}, -\ell \cdot \ell\right)}} \]
      distribute-rgt-neg-in [=>] 52.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \color{blue}{\ell \cdot \left(-\ell\right)}\right)}} \]

   3. Taylor expanded in t around -inf 3.6

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

   4. Simplified 3.6

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

      [Start] 3.6	\[ \sqrt{2} \cdot \left(-1 \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)\right) \]
      mul-1-neg [=>] 3.6	\[ \sqrt{2} \cdot \color{blue}{\left(-\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)} \]
      *-commutative [=>] 3.6	\[ \sqrt{2} \cdot \left(-\color{blue}{\sqrt{\frac{x - 1}{1 + x}} \cdot \sqrt{0.5}}\right) \]
      sub-neg [=>] 3.6	\[ \sqrt{2} \cdot \left(-\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \cdot \sqrt{0.5}\right) \]
      metadata-eval [=>] 3.6	\[ \sqrt{2} \cdot \left(-\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \cdot \sqrt{0.5}\right) \]
      +-commutative [=>] 3.6	\[ \sqrt{2} \cdot \left(-\sqrt{\frac{\color{blue}{-1 + x}}{1 + x}} \cdot \sqrt{0.5}\right) \]
      +-commutative [=>] 3.6	\[ \sqrt{2} \cdot \left(-\sqrt{\frac{-1 + x}{\color{blue}{x + 1}}} \cdot \sqrt{0.5}\right) \]

   5. Applied egg-rr 2.7

      \[\leadsto \color{blue}{\left(0 - e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(\frac{1 - x}{-1 - x} \cdot 0.5\right)}\right)}\right) + 1} \]

   6. Simplified 2.6

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

      [Start] 2.7	\[ \left(0 - e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(\frac{1 - x}{-1 - x} \cdot 0.5\right)}\right)}\right) + 1 \]
      *-commutative [=>] 2.7	\[ \left(0 - e^{\mathsf{log1p}\left(\sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{1 - x}{-1 - x}\right)}}\right)}\right) + 1 \]
      associate-*r* [=>] 2.7	\[ \left(0 - e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(2 \cdot 0.5\right) \cdot \frac{1 - x}{-1 - x}}}\right)}\right) + 1 \]
      metadata-eval [=>] 2.7	\[ \left(0 - e^{\mathsf{log1p}\left(\sqrt{\color{blue}{1} \cdot \frac{1 - x}{-1 - x}}\right)}\right) + 1 \]
      *-lft-identity [=>] 2.7	\[ \left(0 - e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{1 - x}{-1 - x}}}\right)}\right) + 1 \]
      associate--r- [<=] 2.7	\[ \color{blue}{0 - \left(e^{\mathsf{log1p}\left(\sqrt{\frac{1 - x}{-1 - x}}\right)} - 1\right)} \]
      expm1-def [=>] 2.7	\[ 0 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1 - x}{-1 - x}}\right)\right)} \]
      expm1-log1p [=>] 2.6	\[ 0 - \color{blue}{\sqrt{\frac{1 - x}{-1 - x}}} \]
      neg-sub0 [<=] 2.6	\[ \color{blue}{-\sqrt{\frac{1 - x}{-1 - x}}} \]

   if -4.9999999999999998e106 < t < -1.30000000000000011e-134 or 1.9000000000000001e-116 < t < 5e103

   1. Initial program 25.4

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

   2. Simplified 30.7

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

      [Start] 25.4	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 25.5	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      associate-*l/ [=>] 32.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\left(x + 1\right) \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x - 1}} - \ell \cdot \ell}} \]
      associate-*r/ [<=] 31.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}} - \ell \cdot \ell}} \]
      *-lft-identity [<=] 31.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \left(x + 1\right)\right)} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} - \ell \cdot \ell}} \]
      associate-*r* [<=] 31.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{1 \cdot \left(\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 31.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{1 \cdot \color{blue}{\left(\frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} \cdot \left(x + 1\right)\right)} - \ell \cdot \ell}} \]
      associate-*r* [=>] 31.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right) \cdot \left(x + 1\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 31.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      fma-neg [=>] 30.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(x + 1, 1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}, -\ell \cdot \ell\right)}}} \]
      associate-*r/ [=>] 30.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \color{blue}{\frac{1 \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x - 1}}, -\ell \cdot \ell\right)}} \]
      *-lft-identity [=>] 30.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \frac{\color{blue}{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}}{x - 1}, -\ell \cdot \ell\right)}} \]
      +-commutative [=>] 30.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \frac{\color{blue}{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}}{x - 1}, -\ell \cdot \ell\right)}} \]
      fma-def [=>] 30.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \frac{\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x - 1}, -\ell \cdot \ell\right)}} \]
      sub-neg [=>] 30.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\color{blue}{x + \left(-1\right)}}, -\ell \cdot \ell\right)}} \]
      metadata-eval [=>] 30.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + \color{blue}{-1}}, -\ell \cdot \ell\right)}} \]

   3. Taylor expanded in x around -inf 10.6

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

   4. Simplified 10.6

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

      [Start] 10.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + 2 \cdot {t}^{2}}} \]
      distribute-lft-out [=>] 10.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{2 \cdot \left(\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + {t}^{2}\right)}}} \]
      +-commutative [=>] 10.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 10.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x} + {t}^{2}\right)}} \]
      fma-udef [<=] 10.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 10.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 10.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} + \color{blue}{t \cdot t}\right)}} \]

   5. Taylor expanded in t around 0 10.9

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

   6. Simplified 6.0

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

      [Start] 10.9	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{{\ell}^{2}}{x} + t \cdot t\right)}} \]
      unpow2 [=>] 10.9	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{x} + t \cdot t\right)}} \]
      associate-/l* [=>] 6.0	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\color{blue}{\frac{\ell}{\frac{x}{\ell}}} + t \cdot t\right)}} \]

   7. Applied egg-rr 14.7

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

   8. Simplified 14.5

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

      [Start] 14.7	\[ \frac{\frac{t \cdot \sqrt{2}}{\sqrt{2}}}{\mathsf{hypot}\left(t, \sqrt{\ell \cdot \frac{\ell}{x}}\right)} \]
      associate-/l* [=>] 14.5	\[ \frac{\color{blue}{\frac{t}{\frac{\sqrt{2}}{\sqrt{2}}}}}{\mathsf{hypot}\left(t, \sqrt{\ell \cdot \frac{\ell}{x}}\right)} \]
      *-inverses [=>] 14.5	\[ \frac{\frac{t}{\color{blue}{1}}}{\mathsf{hypot}\left(t, \sqrt{\ell \cdot \frac{\ell}{x}}\right)} \]

   9. Applied egg-rr 5.5

      \[\leadsto \frac{\frac{t}{1}}{\color{blue}{{\left(\mathsf{fma}\left(t, t, \ell \cdot \frac{\ell}{x}\right)\right)}^{0.5}}} \]

   10. Simplified 5.5

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

       [Start] 5.5	\[ \frac{\frac{t}{1}}{{\left(\mathsf{fma}\left(t, t, \ell \cdot \frac{\ell}{x}\right)\right)}^{0.5}} \]
       unpow1/2 [=>] 5.5	\[ \frac{\frac{t}{1}}{\color{blue}{\sqrt{\mathsf{fma}\left(t, t, \ell \cdot \frac{\ell}{x}\right)}}} \]

   if -1.30000000000000011e-134 < t < 3.09999999999999985e-304

   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 61.6

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

      [Start] 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-*r/ [<=] 59.3	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      associate-*l/ [=>] 57.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\left(x + 1\right) \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x - 1}} - \ell \cdot \ell}} \]
      associate-*r/ [<=] 62.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}} - \ell \cdot \ell}} \]
      *-lft-identity [<=] 62.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \left(x + 1\right)\right)} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} - \ell \cdot \ell}} \]
      associate-*r* [<=] 62.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{1 \cdot \left(\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 62.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{1 \cdot \color{blue}{\left(\frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} \cdot \left(x + 1\right)\right)} - \ell \cdot \ell}} \]
      associate-*r* [=>] 62.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right) \cdot \left(x + 1\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 62.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      fma-neg [=>] 61.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(x + 1, 1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}, -\ell \cdot \ell\right)}}} \]
      associate-*r/ [=>] 61.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \color{blue}{\frac{1 \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x - 1}}, -\ell \cdot \ell\right)}} \]
      *-lft-identity [=>] 61.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \frac{\color{blue}{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}}{x - 1}, -\ell \cdot \ell\right)}} \]
      +-commutative [=>] 61.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \frac{\color{blue}{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}}{x - 1}, -\ell \cdot \ell\right)}} \]
      fma-def [=>] 61.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \frac{\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x - 1}, -\ell \cdot \ell\right)}} \]
      sub-neg [=>] 61.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\color{blue}{x + \left(-1\right)}}, -\ell \cdot \ell\right)}} \]
      metadata-eval [=>] 61.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + \color{blue}{-1}}, -\ell \cdot \ell\right)}} \]

   3. Taylor expanded in x around -inf 29.8

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

   4. Simplified 29.8

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

      [Start] 29.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + 2 \cdot {t}^{2}}} \]
      distribute-lft-out [=>] 29.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{2 \cdot \left(\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + {t}^{2}\right)}}} \]
      +-commutative [=>] 29.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 29.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x} + {t}^{2}\right)}} \]
      fma-udef [<=] 29.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 29.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 29.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} + \color{blue}{t \cdot t}\right)}} \]

   5. Taylor expanded in t around 0 29.8

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

   6. Simplified 27.6

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

      [Start] 29.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{{\ell}^{2}}{x} + t \cdot t\right)}} \]
      unpow2 [=>] 29.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{x} + t \cdot t\right)}} \]
      associate-/l* [=>] 27.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\color{blue}{\frac{\ell}{\frac{x}{\ell}}} + t \cdot t\right)}} \]

   7. Applied egg-rr 6.0

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

   8. Simplified 5.9

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

      [Start] 6.0	\[ \frac{\frac{t \cdot \sqrt{2}}{\sqrt{2}}}{\mathsf{hypot}\left(t, \sqrt{\ell \cdot \frac{\ell}{x}}\right)} \]
      associate-/l* [=>] 5.9	\[ \frac{\color{blue}{\frac{t}{\frac{\sqrt{2}}{\sqrt{2}}}}}{\mathsf{hypot}\left(t, \sqrt{\ell \cdot \frac{\ell}{x}}\right)} \]
      *-inverses [=>] 5.9	\[ \frac{\frac{t}{\color{blue}{1}}}{\mathsf{hypot}\left(t, \sqrt{\ell \cdot \frac{\ell}{x}}\right)} \]

   9. Applied egg-rr 37.9

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

   10. Simplified 14.7

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

       [Start] 37.9	\[ e^{\mathsf{log1p}\left(\frac{t}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\right)} - 1 \]
       expm1-def [=>] 14.7	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\right)\right)} \]
       expm1-log1p [=>] 14.7	\[ \color{blue}{\frac{t}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}} \]

   if 3.09999999999999985e-304 < t < 1.9000000000000001e-116

   1. Initial program 56.8

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

   2. Simplified 61.1

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

      [Start] 56.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 56.8	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      associate-*l/ [=>] 55.9	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\left(x + 1\right) \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x - 1}} - \ell \cdot \ell}} \]
      associate-*r/ [<=] 62.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}} - \ell \cdot \ell}} \]
      *-lft-identity [<=] 62.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \left(x + 1\right)\right)} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} - \ell \cdot \ell}} \]
      associate-*r* [<=] 62.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{1 \cdot \left(\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 62.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{1 \cdot \color{blue}{\left(\frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} \cdot \left(x + 1\right)\right)} - \ell \cdot \ell}} \]
      associate-*r* [=>] 62.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right) \cdot \left(x + 1\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 62.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      fma-neg [=>] 61.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(x + 1, 1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}, -\ell \cdot \ell\right)}}} \]
      associate-*r/ [=>] 61.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \color{blue}{\frac{1 \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x - 1}}, -\ell \cdot \ell\right)}} \]
      *-lft-identity [=>] 61.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \frac{\color{blue}{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}}{x - 1}, -\ell \cdot \ell\right)}} \]
      +-commutative [=>] 61.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \frac{\color{blue}{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}}{x - 1}, -\ell \cdot \ell\right)}} \]
      fma-def [=>] 61.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \frac{\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x - 1}, -\ell \cdot \ell\right)}} \]
      sub-neg [=>] 61.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\color{blue}{x + \left(-1\right)}}, -\ell \cdot \ell\right)}} \]
      metadata-eval [=>] 61.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + \color{blue}{-1}}, -\ell \cdot \ell\right)}} \]

   3. Taylor expanded in x around -inf 27.1

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

   4. Simplified 27.1

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

      [Start] 27.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + 2 \cdot {t}^{2}}} \]
      distribute-lft-out [=>] 27.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{2 \cdot \left(\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + {t}^{2}\right)}}} \]
      +-commutative [=>] 27.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 27.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x} + {t}^{2}\right)}} \]
      fma-udef [<=] 27.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 27.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 27.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} + \color{blue}{t \cdot t}\right)}} \]

   5. Applied egg-rr 9.4

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

   6. Simplified 9.4

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

      [Start] 9.4	\[ \frac{\frac{t}{\frac{\sqrt{2}}{\sqrt{2}}}}{\mathsf{hypot}\left(t, \sqrt{\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}\right)} \]
      associate-/l/ [=>] 9.4	\[ \color{blue}{\frac{t}{\mathsf{hypot}\left(t, \sqrt{\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}\right) \cdot \frac{\sqrt{2}}{\sqrt{2}}}} \]
      *-inverses [=>] 9.4	\[ \frac{t}{\mathsf{hypot}\left(t, \sqrt{\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}\right) \cdot \color{blue}{1}} \]

   if 5e103 < t

   1. Initial program 51.0

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

   2. Simplified 51.0

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

      [Start] 51.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 51.0	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      fma-neg [=>] 51.0	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]
      +-commutative [=>] 51.0	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}, -\ell \cdot \ell\right)}} \]
      fma-def [=>] 51.0	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}, -\ell \cdot \ell\right)}} \]
      distribute-rgt-neg-in [=>] 51.0	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \color{blue}{\ell \cdot \left(-\ell\right)}\right)}} \]

   3. Taylor expanded in t around -inf 63.0

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

   4. Simplified 63.0

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

      [Start] 63.0	\[ \sqrt{2} \cdot \left(-1 \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)\right) \]
      mul-1-neg [=>] 63.0	\[ \sqrt{2} \cdot \color{blue}{\left(-\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)} \]
      *-commutative [=>] 63.0	\[ \sqrt{2} \cdot \left(-\color{blue}{\sqrt{\frac{x - 1}{1 + x}} \cdot \sqrt{0.5}}\right) \]
      sub-neg [=>] 63.0	\[ \sqrt{2} \cdot \left(-\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \cdot \sqrt{0.5}\right) \]
      metadata-eval [=>] 63.0	\[ \sqrt{2} \cdot \left(-\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \cdot \sqrt{0.5}\right) \]
      +-commutative [=>] 63.0	\[ \sqrt{2} \cdot \left(-\sqrt{\frac{\color{blue}{-1 + x}}{1 + x}} \cdot \sqrt{0.5}\right) \]
      +-commutative [=>] 63.0	\[ \sqrt{2} \cdot \left(-\sqrt{\frac{-1 + x}{\color{blue}{x + 1}}} \cdot \sqrt{0.5}\right) \]

   5. Applied egg-rr 2.1

      \[\leadsto \color{blue}{0 + \sqrt{2 \cdot \left(\frac{1 - x}{-1 - x} \cdot 0.5\right)}} \]

   6. Simplified 2.1

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

      [Start] 2.1	\[ 0 + \sqrt{2 \cdot \left(\frac{1 - x}{-1 - x} \cdot 0.5\right)} \]
      +-lft-identity [=>] 2.1	\[ \color{blue}{\sqrt{2 \cdot \left(\frac{1 - x}{-1 - x} \cdot 0.5\right)}} \]
      *-commutative [=>] 2.1	\[ \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{1 - x}{-1 - x}\right)}} \]
      associate-*r* [=>] 2.1	\[ \sqrt{\color{blue}{\left(2 \cdot 0.5\right) \cdot \frac{1 - x}{-1 - x}}} \]
      metadata-eval [=>] 2.1	\[ \sqrt{\color{blue}{1} \cdot \frac{1 - x}{-1 - x}} \]
      *-lft-identity [=>] 2.1	\[ \sqrt{\color{blue}{\frac{1 - x}{-1 - x}}} \]

3. Recombined 5 regimes into one program.

4. Final simplification 5.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+106}:\\ \;\;\;\;-\sqrt{\frac{1 - x}{-1 - x}}\\ \mathbf{elif}\;t \leq -1.3 \cdot 10^{-134}:\\ \;\;\;\;\frac{t}{\sqrt{\mathsf{fma}\left(t, t, \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-304}:\\ \;\;\;\;\frac{t}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-116}:\\ \;\;\;\;\frac{t}{\mathsf{hypot}\left(t, \sqrt{\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}\right)}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+103}:\\ \;\;\;\;\frac{t}{\sqrt{\mathsf{fma}\left(t, t, \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1 - x}{-1 - x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	5.3
Cost	14036

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{x}\\ t_2 := \frac{t}{\sqrt{\mathsf{fma}\left(t, t, t_1\right)}}\\ t_3 := \sqrt{\frac{1 - x}{-1 - x}}\\ \mathbf{if}\;t \leq -4.8 \cdot 10^{+107}:\\ \;\;\;\;-t_3\\ \mathbf{elif}\;t \leq -1.3 \cdot 10^{-134}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-304}:\\ \;\;\;\;\frac{t}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-156}:\\ \;\;\;\;\frac{t}{\mathsf{hypot}\left(t, \sqrt{t_1}\right)}\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{+106}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 2
Error	6.6
Cost	13576

\[\begin{array}{l} t_1 := \sqrt{\frac{1 - x}{-1 - x}}\\ \mathbf{if}\;t \leq -5.2 \cdot 10^{-53}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 2.45 \cdot 10^{-18}:\\ \;\;\;\;\frac{t}{\mathsf{hypot}\left(t, \sqrt{\ell \cdot \frac{\ell}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	10.5
Cost	13448

\[\begin{array}{l} t_1 := \sqrt{\frac{1 - x}{-1 - x}}\\ \mathbf{if}\;t \leq -2.75 \cdot 10^{-132}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{-173}:\\ \;\;\;\;\frac{t}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	12.6
Cost	13384

\[\begin{array}{l} \mathbf{if}\;t \leq -1.56 \cdot 10^{-218}:\\ \;\;\;\;-\sqrt{\frac{1 - x}{-1 - x}}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-210}:\\ \;\;\;\;\frac{t}{\left|\frac{\ell}{\sqrt{x}}\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(0.5, \frac{\ell}{\frac{t \cdot x}{\ell}}, t\right)}\\ \end{array} \]

Alternative 5
Error	13.6
Cost	7368

\[\begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{-220}:\\ \;\;\;\;-\sqrt{\frac{1 - x}{-1 - x}}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-219}:\\ \;\;\;\;t \cdot \sqrt{\frac{\frac{x}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(0.5, \frac{\ell}{\frac{t \cdot x}{\ell}}, t\right)}\\ \end{array} \]

Alternative 6
Error	14.1
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{-221}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\frac{t \cdot x}{\ell}} \cdot -0.5 - t}\\ \mathbf{elif}\;t \leq 1.06 \cdot 10^{-215}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1 - x}{-1 - x}}\\ \end{array} \]

Alternative 7
Error	13.7
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{-211}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\frac{t \cdot x}{\ell}} \cdot -0.5 - t}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-218}:\\ \;\;\;\;t \cdot \sqrt{\frac{\frac{x}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1 - x}{-1 - x}}\\ \end{array} \]

Alternative 8
Error	14.1
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{-211}:\\ \;\;\;\;-\sqrt{\frac{1}{1 + \frac{2}{x}}}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-218}:\\ \;\;\;\;t \cdot \sqrt{\frac{\frac{x}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1 - x}{-1 - x}}\\ \end{array} \]

Alternative 9
Error	13.9
Cost	7112

\[\begin{array}{l} t_1 := \sqrt{\frac{1 - x}{-1 - x}}\\ \mathbf{if}\;t \leq -4.7 \cdot 10^{-214}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-218}:\\ \;\;\;\;t \cdot \sqrt{\frac{\frac{x}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	14.7
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -7.4 \cdot 10^{-216}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\frac{t \cdot x}{\ell}} \cdot -0.5 - t}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-210}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{x}\\ \end{array} \]

Alternative 11
Error	14.6
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{-220}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\frac{t \cdot x}{\ell}} \cdot -0.5 - t}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-217}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{x}\\ \end{array} \]

Alternative 12
Error	14.7
Cost	1096

\[\begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-221}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 11000000000000:\\ \;\;\;\;\frac{t}{t + 0.5 \cdot \frac{\ell \cdot \ell}{t \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{x}\\ \end{array} \]

Alternative 13
Error	13.1
Cost	1096

\[\begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{-298}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\frac{t \cdot x}{\ell}} \cdot -0.5 - t}\\ \mathbf{elif}\;t \leq 11000000000000:\\ \;\;\;\;\frac{t}{t + 0.5 \cdot \frac{\ell \cdot \ell}{t \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{x}\\ \end{array} \]

Alternative 14
Error	16.1
Cost	964

\[\begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{-300}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{x}\\ \end{array} \]

Alternative 15
Error	15.8
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq -1.22 \cdot 10^{-307}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 16
Error	38.8
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022354 
(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)))))