Toniolo and Linder, Equation (7)

Average Error: 42.8 → 9.1

Time: 25.4s

Precision: binary64

Cost: 33868

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \frac{t_1}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{if}\;t \leq -3.3 \cdot 10^{+52}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) + \frac{-0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{-245}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-127}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\frac{\ell \cdot \ell}{\sqrt{2} \cdot \left(t \cdot x\right)} + \mathsf{fma}\left(2, \frac{t}{x \cdot \sqrt{2}}, t_1\right)}\\ \mathbf{elif}\;t \leq 1.72 \cdot 10^{+96}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1 - x}{-1 - x}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* t (sqrt 2.0)))
        (t_2
         (/ t_1 (sqrt (+ (* 2.0 (* t (+ t (/ t x)))) (* 2.0 (* l (/ l x))))))))
   (if (<= t -3.3e+52)
     (+ (+ -1.0 (/ 1.0 x)) (/ -0.5 (* x x)))
     (if (<= t 4.3e-245)
       t_2
       (if (<= t 1.7e-127)
         (*
          (sqrt 2.0)
          (/
           t
           (+
            (/ (* l l) (* (sqrt 2.0) (* t x)))
            (fma 2.0 (/ t (* x (sqrt 2.0))) t_1))))
         (if (<= t 1.72e+96) t_2 (sqrt (/ (- 1.0 x) (- -1.0 x)))))))))

C

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

↓

double code(double x, double l, double t) {
	double t_1 = t * sqrt(2.0);
	double t_2 = t_1 / sqrt(((2.0 * (t * (t + (t / x)))) + (2.0 * (l * (l / x)))));
	double tmp;
	if (t <= -3.3e+52) {
		tmp = (-1.0 + (1.0 / x)) + (-0.5 / (x * x));
	} else if (t <= 4.3e-245) {
		tmp = t_2;
	} else if (t <= 1.7e-127) {
		tmp = sqrt(2.0) * (t / (((l * l) / (sqrt(2.0) * (t * x))) + fma(2.0, (t / (x * sqrt(2.0))), t_1)));
	} else if (t <= 1.72e+96) {
		tmp = t_2;
	} else {
		tmp = sqrt(((1.0 - x) / (-1.0 - x)));
	}
	return tmp;
}

Julia

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

↓

function code(x, l, t)
	t_1 = Float64(t * sqrt(2.0))
	t_2 = Float64(t_1 / sqrt(Float64(Float64(2.0 * Float64(t * Float64(t + Float64(t / x)))) + Float64(2.0 * Float64(l * Float64(l / x))))))
	tmp = 0.0
	if (t <= -3.3e+52)
		tmp = Float64(Float64(-1.0 + Float64(1.0 / x)) + Float64(-0.5 / Float64(x * x)));
	elseif (t <= 4.3e-245)
		tmp = t_2;
	elseif (t <= 1.7e-127)
		tmp = Float64(sqrt(2.0) * Float64(t / Float64(Float64(Float64(l * l) / Float64(sqrt(2.0) * Float64(t * x))) + fma(2.0, Float64(t / Float64(x * sqrt(2.0))), t_1))));
	elseif (t <= 1.72e+96)
		tmp = t_2;
	else
		tmp = sqrt(Float64(Float64(1.0 - x) / Float64(-1.0 - x)));
	end
	return tmp
end

Wolfram

code[x_, l_, t_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[Sqrt[N[(N[(2.0 * N[(t * N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3.3e+52], N[(N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.3e-245], t$95$2, If[LessEqual[t, 1.7e-127], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[(N[(N[(l * l), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[(t * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(t / N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.72e+96], t$95$2, N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -3.3e52

   1. Initial program 44.7

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

   2. Simplified 44.8

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

      [Start] 44.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 44.8	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      fma-neg [=>] 44.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]
      +-commutative [=>] 44.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}, -\ell \cdot \ell\right)}} \]
      fma-def [=>] 44.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}, -\ell \cdot \ell\right)}} \]
      distribute-rgt-neg-in [=>] 44.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \color{blue}{\ell \cdot \left(-\ell\right)}\right)}} \]

   3. Taylor expanded in t around inf 63.0

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

   4. Simplified 63.0

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

      [Start] 63.0	\[ \left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{x - 1}{1 + x}} \]
      associate-*l* [=>] 63.0	\[ \color{blue}{\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)} \]
      *-commutative [=>] 63.0	\[ \sqrt{2} \cdot \color{blue}{\left(\sqrt{\frac{x - 1}{1 + x}} \cdot \sqrt{0.5}\right)} \]
      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 63.0

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

   6. Simplified 63.0

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

      [Start] 63.0	\[ e^{\mathsf{log1p}\left(\sqrt{2 \cdot \frac{1 - x}{\frac{-1 - x}{0.5}}}\right)} - 1 \]
      expm1-def [=>] 63.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \frac{1 - x}{\frac{-1 - x}{0.5}}}\right)\right)} \]
      expm1-log1p [=>] 63.0	\[ \color{blue}{\sqrt{2 \cdot \frac{1 - x}{\frac{-1 - x}{0.5}}}} \]
      *-commutative [=>] 63.0	\[ \sqrt{\color{blue}{\frac{1 - x}{\frac{-1 - x}{0.5}} \cdot 2}} \]
      associate-/r/ [=>] 63.0	\[ \sqrt{\color{blue}{\left(\frac{1 - x}{-1 - x} \cdot 0.5\right)} \cdot 2} \]
      associate-*l* [=>] 63.0	\[ \sqrt{\color{blue}{\frac{1 - x}{-1 - x} \cdot \left(0.5 \cdot 2\right)}} \]
      metadata-eval [=>] 63.0	\[ \sqrt{\frac{1 - x}{-1 - x} \cdot \color{blue}{1}} \]
      *-rgt-identity [=>] 63.0	\[ \sqrt{\color{blue}{\frac{1 - x}{-1 - x}}} \]

   7. Taylor expanded in x around inf 64.0

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

   8. Simplified 3.6

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

      [Start] 64.0	\[ -0.5 \cdot \frac{2 + {\left(\frac{1}{\sqrt{-1}}\right)}^{2}}{{x}^{2}} + \left({\left(\sqrt{-1}\right)}^{2} + \frac{1}{x}\right) \]
      +-commutative [=>] 64.0	\[ \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} + \frac{1}{x}\right) + -0.5 \cdot \frac{2 + {\left(\frac{1}{\sqrt{-1}}\right)}^{2}}{{x}^{2}}} \]
      unpow2 [=>] 64.0	\[ \left(\color{blue}{\sqrt{-1} \cdot \sqrt{-1}} + \frac{1}{x}\right) + -0.5 \cdot \frac{2 + {\left(\frac{1}{\sqrt{-1}}\right)}^{2}}{{x}^{2}} \]
      rem-square-sqrt [=>] 64.0	\[ \left(\color{blue}{-1} + \frac{1}{x}\right) + -0.5 \cdot \frac{2 + {\left(\frac{1}{\sqrt{-1}}\right)}^{2}}{{x}^{2}} \]
      associate-*r/ [=>] 64.0	\[ \left(-1 + \frac{1}{x}\right) + \color{blue}{\frac{-0.5 \cdot \left(2 + {\left(\frac{1}{\sqrt{-1}}\right)}^{2}\right)}{{x}^{2}}} \]
      associate-/l* [=>] 64.0	\[ \left(-1 + \frac{1}{x}\right) + \color{blue}{\frac{-0.5}{\frac{{x}^{2}}{2 + {\left(\frac{1}{\sqrt{-1}}\right)}^{2}}}} \]
      remove-double-neg [<=] 64.0	\[ \left(-1 + \frac{1}{x}\right) + \frac{-0.5}{\frac{{x}^{2}}{\color{blue}{-\left(-\left(2 + {\left(\frac{1}{\sqrt{-1}}\right)}^{2}\right)\right)}}} \]
      mul-1-neg [<=] 64.0	\[ \left(-1 + \frac{1}{x}\right) + \frac{-0.5}{\frac{{x}^{2}}{-\color{blue}{-1 \cdot \left(2 + {\left(\frac{1}{\sqrt{-1}}\right)}^{2}\right)}}} \]
      distribute-lft-neg-in [=>] 64.0	\[ \left(-1 + \frac{1}{x}\right) + \frac{-0.5}{\frac{{x}^{2}}{\color{blue}{\left(--1\right) \cdot \left(2 + {\left(\frac{1}{\sqrt{-1}}\right)}^{2}\right)}}} \]
      metadata-eval [=>] 64.0	\[ \left(-1 + \frac{1}{x}\right) + \frac{-0.5}{\frac{{x}^{2}}{\color{blue}{1} \cdot \left(2 + {\left(\frac{1}{\sqrt{-1}}\right)}^{2}\right)}} \]
      metadata-eval [<=] 64.0	\[ \left(-1 + \frac{1}{x}\right) + \frac{-0.5}{\frac{{x}^{2}}{\color{blue}{\frac{-1}{-1}} \cdot \left(2 + {\left(\frac{1}{\sqrt{-1}}\right)}^{2}\right)}} \]
      associate-/r/ [<=] 64.0	\[ \left(-1 + \frac{1}{x}\right) + \frac{-0.5}{\frac{{x}^{2}}{\color{blue}{\frac{-1}{\frac{-1}{2 + {\left(\frac{1}{\sqrt{-1}}\right)}^{2}}}}}} \]
      rem-square-sqrt [<=] 64.0	\[ \left(-1 + \frac{1}{x}\right) + \frac{-0.5}{\frac{{x}^{2}}{\frac{-1}{\frac{\color{blue}{\sqrt{-1} \cdot \sqrt{-1}}}{2 + {\left(\frac{1}{\sqrt{-1}}\right)}^{2}}}}} \]
      unpow2 [<=] 64.0	\[ \left(-1 + \frac{1}{x}\right) + \frac{-0.5}{\frac{{x}^{2}}{\frac{-1}{\frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2}}}{2 + {\left(\frac{1}{\sqrt{-1}}\right)}^{2}}}}} \]
      associate-/l* [<=] 64.0	\[ \left(-1 + \frac{1}{x}\right) + \frac{-0.5}{\frac{{x}^{2}}{\color{blue}{\frac{-1 \cdot \left(2 + {\left(\frac{1}{\sqrt{-1}}\right)}^{2}\right)}{{\left(\sqrt{-1}\right)}^{2}}}}} \]
      associate-*r/ [<=] 64.0	\[ \left(-1 + \frac{1}{x}\right) + \frac{-0.5}{\frac{{x}^{2}}{\color{blue}{-1 \cdot \frac{2 + {\left(\frac{1}{\sqrt{-1}}\right)}^{2}}{{\left(\sqrt{-1}\right)}^{2}}}}} \]

   if -3.3e52 < t < 4.30000000000000003e-245 or 1.6999999999999999e-127 < t < 1.72e96

   1. Initial program 36.6

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

   2. Taylor expanded in x around inf 16.5

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

   3. Simplified 16.5

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

      [Start] 16.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      associate--l+ [=>] 16.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]
      unpow2 [=>] 16.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      distribute-lft-out [=>] 16.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 16.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 16.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-*r/ [=>] 16.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \]
      mul-1-neg [=>] 16.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \]
      +-commutative [=>] 16.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + {\ell}^{2}\right)}}{x}\right)}} \]
      unpow2 [=>] 16.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}\right)}{x}\right)}} \]
      fma-udef [<=] 16.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)}} \]
      unpow2 [=>] 16.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)}} \]

   4. Taylor expanded in t around 0 16.8

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

   5. Simplified 16.8

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

      [Start] 16.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - -1 \cdot \frac{{\ell}^{2}}{x}\right)}} \]
      associate-*r/ [=>] 16.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot {\ell}^{2}}{x}}\right)}} \]
      mul-1-neg [=>] 16.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-{\ell}^{2}}}{x}\right)}} \]
      unpow2 [=>] 16.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\ell \cdot \ell}}{x}\right)}} \]
      distribute-rgt-neg-out [<=] 16.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{\ell \cdot \left(-\ell\right)}}{x}\right)}} \]

   6. Applied egg-rr 12.6

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

   7. Simplified 12.5

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

      [Start] 12.6	\[ \frac{\sqrt{2} \cdot t}{{\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)\right)}^{0.25} \cdot {\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)\right)}^{0.25}} \]
      pow-sqr [=>] 12.5	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{{\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)\right)}^{\left(2 \cdot 0.25\right)}}} \]
      metadata-eval [=>] 12.5	\[ \frac{\sqrt{2} \cdot t}{{\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)\right)}^{\color{blue}{0.5}}} \]
      unpow1/2 [=>] 12.5	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)}}} \]
      +-commutative [=>] 12.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right) + \frac{\ell}{x} \cdot \ell}}} \]
      fma-udef [=>] 12.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) + \frac{\ell}{x} \cdot \ell\right)} + \frac{\ell}{x} \cdot \ell}} \]
      associate-+l+ [=>] 12.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) + \left(\frac{\ell}{x} \cdot \ell + \frac{\ell}{x} \cdot \ell\right)}}} \]
      fma-udef [=>] 12.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \color{blue}{\left(t \cdot t + \frac{t}{x} \cdot t\right)} + \left(\frac{\ell}{x} \cdot \ell + \frac{\ell}{x} \cdot \ell\right)}} \]
      distribute-rgt-out [=>] 12.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \color{blue}{\left(t \cdot \left(t + \frac{t}{x}\right)\right)} + \left(\frac{\ell}{x} \cdot \ell + \frac{\ell}{x} \cdot \ell\right)}} \]
      count-2 [=>] 12.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + \color{blue}{2 \cdot \left(\frac{\ell}{x} \cdot \ell\right)}}} \]
      *-commutative [=>] 12.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{x}\right)}}} \]

   if 4.30000000000000003e-245 < t < 1.6999999999999999e-127

   1. Initial program 56.3

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

   2. Simplified 61.3

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

      [Start] 56.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 56.3	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      associate-*l/ [=>] 55.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\left(x + 1\right) \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x - 1}} - \ell \cdot \ell}} \]
      associate-*r/ [<=] 62.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}} - \ell \cdot \ell}} \]
      *-lft-identity [<=] 62.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \left(x + 1\right)\right)} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} - \ell \cdot \ell}} \]
      associate-*r* [<=] 62.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{1 \cdot \left(\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 62.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{1 \cdot \color{blue}{\left(\frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} \cdot \left(x + 1\right)\right)} - \ell \cdot \ell}} \]
      associate-*r* [=>] 62.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right) \cdot \left(x + 1\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 62.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      fma-neg [=>] 61.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(x + 1, 1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}, -\ell \cdot \ell\right)}}} \]

   3. Taylor expanded in x around -inf 23.6

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

   4. Simplified 23.6

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

      [Start] 23.6	\[ \sqrt{2} \cdot \frac{t}{\frac{{\ell}^{2}}{\sqrt{2} \cdot \left(t \cdot x\right)} + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + \sqrt{2} \cdot t\right)} \]
      unpow2 [=>] 23.6	\[ \sqrt{2} \cdot \frac{t}{\frac{\color{blue}{\ell \cdot \ell}}{\sqrt{2} \cdot \left(t \cdot x\right)} + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + \sqrt{2} \cdot t\right)} \]
      fma-def [=>] 23.6	\[ \sqrt{2} \cdot \frac{t}{\frac{\ell \cdot \ell}{\sqrt{2} \cdot \left(t \cdot x\right)} + \color{blue}{\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, \sqrt{2} \cdot t\right)}} \]
      *-commutative [=>] 23.6	\[ \sqrt{2} \cdot \frac{t}{\frac{\ell \cdot \ell}{\sqrt{2} \cdot \left(t \cdot x\right)} + \mathsf{fma}\left(2, \frac{t}{\color{blue}{x \cdot \sqrt{2}}}, \sqrt{2} \cdot t\right)} \]
      *-commutative [=>] 23.6	\[ \sqrt{2} \cdot \frac{t}{\frac{\ell \cdot \ell}{\sqrt{2} \cdot \left(t \cdot x\right)} + \mathsf{fma}\left(2, \frac{t}{x \cdot \sqrt{2}}, \color{blue}{t \cdot \sqrt{2}}\right)} \]

   if 1.72e96 < t

   1. Initial program 49.9

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

   2. Simplified 49.9

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

      [Start] 49.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 50.0	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      fma-neg [=>] 49.9	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]
      +-commutative [=>] 49.9	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}, -\ell \cdot \ell\right)}} \]
      fma-def [=>] 49.9	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}, -\ell \cdot \ell\right)}} \]
      distribute-rgt-neg-in [=>] 49.9	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \color{blue}{\ell \cdot \left(-\ell\right)}\right)}} \]

   3. Taylor expanded in t around inf 3.5

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

   4. Simplified 3.5

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

      [Start] 3.5	\[ \left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{x - 1}{1 + x}} \]
      associate-*l* [=>] 3.5	\[ \color{blue}{\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)} \]
      *-commutative [=>] 3.5	\[ \sqrt{2} \cdot \color{blue}{\left(\sqrt{\frac{x - 1}{1 + x}} \cdot \sqrt{0.5}\right)} \]
      sub-neg [=>] 3.5	\[ \sqrt{2} \cdot \left(\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \cdot \sqrt{0.5}\right) \]
      metadata-eval [=>] 3.5	\[ \sqrt{2} \cdot \left(\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \cdot \sqrt{0.5}\right) \]
      +-commutative [=>] 3.5	\[ \sqrt{2} \cdot \left(\sqrt{\frac{\color{blue}{-1 + x}}{1 + x}} \cdot \sqrt{0.5}\right) \]
      +-commutative [=>] 3.5	\[ \sqrt{2} \cdot \left(\sqrt{\frac{-1 + x}{\color{blue}{x + 1}}} \cdot \sqrt{0.5}\right) \]

   5. Applied egg-rr 2.6

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

   6. Simplified 2.6

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

      [Start] 2.6	\[ e^{\mathsf{log1p}\left(\sqrt{2 \cdot \frac{1 - x}{\frac{-1 - x}{0.5}}}\right)} - 1 \]
      expm1-def [=>] 2.6	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \frac{1 - x}{\frac{-1 - x}{0.5}}}\right)\right)} \]
      expm1-log1p [=>] 2.6	\[ \color{blue}{\sqrt{2 \cdot \frac{1 - x}{\frac{-1 - x}{0.5}}}} \]
      *-commutative [=>] 2.6	\[ \sqrt{\color{blue}{\frac{1 - x}{\frac{-1 - x}{0.5}} \cdot 2}} \]
      associate-/r/ [=>] 2.6	\[ \sqrt{\color{blue}{\left(\frac{1 - x}{-1 - x} \cdot 0.5\right)} \cdot 2} \]
      associate-*l* [=>] 2.6	\[ \sqrt{\color{blue}{\frac{1 - x}{-1 - x} \cdot \left(0.5 \cdot 2\right)}} \]
      metadata-eval [=>] 2.6	\[ \sqrt{\frac{1 - x}{-1 - x} \cdot \color{blue}{1}} \]
      *-rgt-identity [=>] 2.6	\[ \sqrt{\color{blue}{\frac{1 - x}{-1 - x}}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 9.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{+52}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) + \frac{-0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{-245}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-127}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\frac{\ell \cdot \ell}{\sqrt{2} \cdot \left(t \cdot x\right)} + \mathsf{fma}\left(2, \frac{t}{x \cdot \sqrt{2}}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 1.72 \cdot 10^{+96}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1 - x}{-1 - x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.1
Cost	14408

\[\begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{+52}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) + \frac{-0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+96}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1 - x}{-1 - x}}\\ \end{array} \]

Alternative 2
Error	13.5
Cost	13768

\[\begin{array}{l} \mathbf{if}\;t \leq -7.8 \cdot 10^{-175}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{1 + x}}\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{-239}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell}{x} \cdot \left(\ell + \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1 - x}{-1 - x}}\\ \end{array} \]

Alternative 3
Error	14.2
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-174}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) + \frac{-0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-202}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1 - x}{-1 - x}}\\ \end{array} \]

Alternative 4
Error	14.2
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-169}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{1 + x}}\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-208}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1 - x}{-1 - x}}\\ \end{array} \]

Alternative 5
Error	14.4
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-169}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) + \frac{-0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{-209}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{0.5}{x \cdot x}\right) + \frac{-1}{x}\\ \end{array} \]

Alternative 6
Error	15.1
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{-295}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) + \frac{-0.5}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 7
Error	15.0
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{-295}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) + \frac{-0.5}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{0.5}{x \cdot x}\right) + \frac{-1}{x}\\ \end{array} \]

Alternative 8
Error	15.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{-295}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Error	15.1
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{-295}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 10
Error	15.7
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{-295}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Error	39.0
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023066 
(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)))))