Toniolo and Linder, Equation (7)

Percentage Accurate: 32.9% → 82.0%

Time: 43.2s

Precision: binary64

Cost: 21448

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-28}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+54}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(t \cdot 2, t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

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
 (if (<= t -3.8e-28)
   (- (sqrt (/ (+ x -1.0) (+ x 1.0))))
   (if (<= t 1.05e+54)
     (*
      t
      (/
       (sqrt 2.0)
       (sqrt
        (+
         (/ (* l l) x)
         (+
          (* 2.0 (+ (* t t) (/ (* t t) x)))
          (/ (fma (* t 2.0) t (* l l)) x))))))
     (+ 1.0 (/ -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 tmp;
	if (t <= -3.8e-28) {
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= 1.05e+54) {
		tmp = t * (sqrt(2.0) / sqrt((((l * l) / x) + ((2.0 * ((t * t) + ((t * t) / x))) + (fma((t * 2.0), t, (l * l)) / x)))));
	} else {
		tmp = 1.0 + (-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)
	tmp = 0.0
	if (t <= -3.8e-28)
		tmp = Float64(-sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0))));
	elseif (t <= 1.05e+54)
		tmp = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(Float64(Float64(l * l) / x) + Float64(Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))) + Float64(fma(Float64(t * 2.0), t, Float64(l * l)) / x))))));
	else
		tmp = Float64(1.0 + 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_] := If[LessEqual[t, -3.8e-28], (-N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, 1.05e+54], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] + N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t * 2.0), $MachinePrecision] * t + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.80000000000000009e-28

   1. Initial program 33.1%

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

   2. Simplified 33.0%

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

      [Start] 33.1%	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 33.0%	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      fma-neg [=>] 33.0%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]
      sub-neg [=>] 33.0%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{\color{blue}{x + \left(-1\right)}}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}} \]
      metadata-eval [=>] 33.0%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + \color{blue}{-1}}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}} \]
      +-commutative [=>] 33.0%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \color{blue}{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}, -\ell \cdot \ell\right)}} \]
      fma-def [=>] 33.0%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}, -\ell \cdot \ell\right)}} \]
      distribute-rgt-neg-in [=>] 33.0%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \color{blue}{\ell \cdot \left(-\ell\right)}\right)}} \]

   3. Applied egg-rr 71.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + -1}}, \ell\right)}{t \cdot \sqrt{2}}}} \]
      Step-by-step derivation

      [Start] 33.0%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}} \]
      associate-*r/ [=>] 33.1%	\[ \color{blue}{\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]
      metadata-eval [<=] 33.1%	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + \color{blue}{\left(-1\right)}}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}} \]
      sub-neg [<=] 33.1%	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{\color{blue}{x - 1}}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}} \]
      fma-udef [=>] 33.1%	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}, \ell \cdot \left(-\ell\right)\right)}} \]
      +-commutative [<=] 33.1%	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}, \ell \cdot \left(-\ell\right)\right)}} \]
      distribute-rgt-neg-out [=>] 33.1%	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), \color{blue}{-\ell \cdot \ell}\right)}} \]
      fma-neg [<=] 33.1%	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      clear-num [=>] 33.0%	\[ \color{blue}{\frac{1}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{\sqrt{2} \cdot t}}} \]

   4. Taylor expanded in t around -inf 90.8%

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

   5. Simplified 90.8%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{x + 1}}} \]
      Step-by-step derivation

      [Start] 90.8%	\[ -1 \cdot \sqrt{\frac{x - 1}{1 + x}} \]
      mul-1-neg [=>] 90.8%	\[ \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]
      sub-neg [=>] 90.8%	\[ -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]
      metadata-eval [=>] 90.8%	\[ -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]
      +-commutative [=>] 90.8%	\[ -\sqrt{\frac{x + -1}{\color{blue}{x + 1}}} \]

   if -3.80000000000000009e-28 < t < 1.04999999999999993e54

   1. Initial program 35.5%

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

   2. Simplified 35.6%

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

      [Start] 35.5%	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 35.6%	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Taylor expanded in x around inf 76.4%

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

   4. Simplified 76.4%

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

      [Start] 76.4%	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 76.4%	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 76.4%	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 76.4%	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 76.4%	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 76.4%	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      associate-*r/ [=>] 76.4%	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \cdot t \]
      mul-1-neg [=>] 76.4%	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \cdot t \]
      +-commutative [=>] 76.4%	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + {\ell}^{2}\right)}}{x}\right)}} \cdot t \]
      unpow2 [=>] 76.4%	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(2 \cdot \color{blue}{\left(t \cdot t\right)} + {\ell}^{2}\right)}{x}\right)}} \cdot t \]
      associate-*l* [<=] 76.4%	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(\color{blue}{\left(2 \cdot t\right) \cdot t} + {\ell}^{2}\right)}{x}\right)}} \cdot t \]
      unpow2 [=>] 76.4%	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(\left(2 \cdot t\right) \cdot t + \color{blue}{\ell \cdot \ell}\right)}{x}\right)}} \cdot t \]
      fma-udef [<=] 76.4%	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right)}}{x}\right)}} \cdot t \]

   if 1.04999999999999993e54 < t

   1. Initial program 26.1%

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

   2. Simplified 26.1%

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

      [Start] 26.1%	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 26.1%	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      fma-neg [=>] 26.1%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]
      sub-neg [=>] 26.1%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{\color{blue}{x + \left(-1\right)}}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}} \]
      metadata-eval [=>] 26.1%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + \color{blue}{-1}}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}} \]
      +-commutative [=>] 26.1%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \color{blue}{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}, -\ell \cdot \ell\right)}} \]
      fma-def [=>] 26.1%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}, -\ell \cdot \ell\right)}} \]
      distribute-rgt-neg-in [=>] 26.1%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \color{blue}{\ell \cdot \left(-\ell\right)}\right)}} \]

   3. Applied egg-rr 85.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + -1}}, \ell\right)}{t \cdot \sqrt{2}}}} \]
      Step-by-step derivation

      [Start] 26.1%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}} \]
      associate-*r/ [=>] 26.1%	\[ \color{blue}{\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]
      metadata-eval [<=] 26.1%	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + \color{blue}{\left(-1\right)}}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}} \]
      sub-neg [<=] 26.1%	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{\color{blue}{x - 1}}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}} \]
      fma-udef [=>] 26.1%	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}, \ell \cdot \left(-\ell\right)\right)}} \]
      +-commutative [<=] 26.1%	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}, \ell \cdot \left(-\ell\right)\right)}} \]
      distribute-rgt-neg-out [=>] 26.1%	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), \color{blue}{-\ell \cdot \ell}\right)}} \]
      fma-neg [<=] 26.1%	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      clear-num [=>] 26.1%	\[ \color{blue}{\frac{1}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{\sqrt{2} \cdot t}}} \]

   4. Taylor expanded in l around 0 94.5%

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

   5. Taylor expanded in x around inf 94.5%

      \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-28}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+54}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(t \cdot 2, t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	82.0%
Cost	21448

\[\begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-28}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+54}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(t \cdot 2, t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 2
Accuracy	78.6%
Cost	13896

\[\begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -9 \cdot 10^{-224}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-155}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{\ell \cdot \ell + \ell \cdot \ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	77.5%
Cost	13640

\[\begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -8.2 \cdot 10^{-227}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 1.16 \cdot 10^{-158}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\frac{2}{x}} \cdot \frac{\ell}{t}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	76.6%
Cost	7880

\[\begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -5.3 \cdot 10^{-224}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-240}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\ell \cdot \ell + \frac{x + 1}{\frac{x + -1}{\ell \cdot \ell}}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	76.8%
Cost	7496

\[\begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -2.3 \cdot 10^{-236}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-245}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{2 \cdot \left(\ell \cdot \ell + t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	76.5%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{-233}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-244}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\ell}{t}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 7
Accuracy	76.7%
Cost	7112

\[\begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -3.45 \cdot 10^{-239}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-238}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\ell}{t}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Accuracy	76.3%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -6.6 \cdot 10^{-245}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-245}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 9
Accuracy	76.2%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-226}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-244}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\ell}{t}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 10
Accuracy	76.0%
Cost	836

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

Alternative 11
Accuracy	75.6%
Cost	452

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

Alternative 12
Accuracy	75.9%
Cost	452

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

Alternative 13
Accuracy	75.3%
Cost	196

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

Alternative 14
Accuracy	37.8%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023263 
(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)))))