Toniolo and Linder, Equation (7)

Average Accuracy: 33.8% → 84.3%

Time: 32.5s

Precision: binary64

Cost: 14288

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \sqrt{\frac{1}{\mathsf{fma}\left(t, t + \frac{t}{x}, \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{if}\;t \leq -1.12 \cdot 10^{+39}:\\ \;\;\;\;-\sqrt{\frac{1 - x}{-1 - x}}\\ \mathbf{elif}\;t \leq -1.55 \cdot 10^{-268}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-293}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \frac{2}{x}}}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+43}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

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))))))))
   (if (<= t -1.12e+39)
     (- (sqrt (/ (- 1.0 x) (- -1.0 x))))
     (if (<= t -1.55e-268)
       t_1
       (if (<= t 6.5e-293)
         (/ (* t (sqrt 2.0)) (* l (sqrt (+ (/ 2.0 (* x x)) (/ 2.0 x)))))
         (if (<= t 6.5e+43) t_1 1.0))))))

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 tmp;
	if (t <= -1.12e+39) {
		tmp = -sqrt(((1.0 - x) / (-1.0 - x)));
	} else if (t <= -1.55e-268) {
		tmp = t_1;
	} else if (t <= 6.5e-293) {
		tmp = (t * sqrt(2.0)) / (l * sqrt(((2.0 / (x * x)) + (2.0 / x))));
	} else if (t <= 6.5e+43) {
		tmp = t_1;
	} else {
		tmp = 1.0;
	}
	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))))))
	tmp = 0.0
	if (t <= -1.12e+39)
		tmp = Float64(-sqrt(Float64(Float64(1.0 - x) / Float64(-1.0 - x))));
	elseif (t <= -1.55e-268)
		tmp = t_1;
	elseif (t <= 6.5e-293)
		tmp = Float64(Float64(t * sqrt(2.0)) / Float64(l * sqrt(Float64(Float64(2.0 / Float64(x * x)) + Float64(2.0 / x)))));
	elseif (t <= 6.5e+43)
		tmp = t_1;
	else
		tmp = 1.0;
	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]}, If[LessEqual[t, -1.12e+39], (-N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, -1.55e-268], t$95$1, If[LessEqual[t, 6.5e-293], N[(N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(l * N[Sqrt[N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.5e+43], t$95$1, 1.0]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.12e39

   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 33.1%

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

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

   3. Taylor expanded in t around -inf 92.2%

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

   4. Simplified 92.2%

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

      [Start] 92.2	\[ \sqrt{2} \cdot \left(-1 \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)\right) \]
      mul-1-neg [=>] 92.2	\[ \sqrt{2} \cdot \color{blue}{\left(-\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)} \]
      *-commutative [=>] 92.2	\[ \sqrt{2} \cdot \left(-\color{blue}{\sqrt{\frac{x - 1}{1 + x}} \cdot \sqrt{0.5}}\right) \]
      sub-neg [=>] 92.2	\[ \sqrt{2} \cdot \left(-\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \cdot \sqrt{0.5}\right) \]
      metadata-eval [=>] 92.2	\[ \sqrt{2} \cdot \left(-\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \cdot \sqrt{0.5}\right) \]
      +-commutative [=>] 92.2	\[ \sqrt{2} \cdot \left(-\sqrt{\frac{\color{blue}{-1 + x}}{1 + x}} \cdot \sqrt{0.5}\right) \]
      +-commutative [=>] 92.2	\[ \sqrt{2} \cdot \left(-\sqrt{\frac{-1 + x}{\color{blue}{x + 1}}} \cdot \sqrt{0.5}\right) \]

   5. Applied egg-rr 93.6%

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

      [Start] 92.2	\[ \sqrt{2} \cdot \left(-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right) \]
      distribute-rgt-neg-out [=>] 92.2	\[ \color{blue}{-\sqrt{2} \cdot \left(\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right)} \]
      neg-sub0 [=>] 92.2	\[ \color{blue}{0 - \sqrt{2} \cdot \left(\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right)} \]
      metadata-eval [<=] 92.2	\[ \color{blue}{\log 1} - \sqrt{2} \cdot \left(\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right) \]
      *-commutative [=>] 92.2	\[ \log 1 - \sqrt{2} \cdot \color{blue}{\left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)} \]
      *-commutative [<=] 92.2	\[ \log 1 - \sqrt{2} \cdot \color{blue}{\left(\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right)} \]
      add-sqr-sqrt [=>] 92.2	\[ \log 1 - \sqrt{2} \cdot \color{blue}{\left(\sqrt{\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}} \cdot \sqrt{\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}}\right)} \]
      sqrt-unprod [=>] 92.2	\[ \log 1 - \sqrt{2} \cdot \color{blue}{\sqrt{\left(\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right) \cdot \left(\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right)}} \]
      sqr-neg [<=] 92.2	\[ \log 1 - \sqrt{2} \cdot \sqrt{\color{blue}{\left(-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right) \cdot \left(-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right)}} \]
      sqrt-unprod [<=] 0.0	\[ \log 1 - \sqrt{2} \cdot \color{blue}{\left(\sqrt{-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}} \cdot \sqrt{-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}}\right)} \]
      add-sqr-sqrt [<=] 1.6	\[ \log 1 - \sqrt{2} \cdot \color{blue}{\left(-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right)} \]
      expm1-log1p-u [=>] 0.1	\[ \log 1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2} \cdot \left(-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right)\right)\right)} \]

   6. Simplified 93.6%

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

      [Start] 93.6	\[ \left(0 - e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(\frac{1 - x}{-1 - x} \cdot 0.5\right)}\right)}\right) + 1 \]
      associate-+l- [=>] 93.6	\[ \color{blue}{0 - \left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(\frac{1 - x}{-1 - x} \cdot 0.5\right)}\right)} - 1\right)} \]
      expm1-def [=>] 93.6	\[ 0 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \left(\frac{1 - x}{-1 - x} \cdot 0.5\right)}\right)\right)} \]
      expm1-log1p [=>] 93.6	\[ 0 - \color{blue}{\sqrt{2 \cdot \left(\frac{1 - x}{-1 - x} \cdot 0.5\right)}} \]
      sub0-neg [=>] 93.6	\[ \color{blue}{-\sqrt{2 \cdot \left(\frac{1 - x}{-1 - x} \cdot 0.5\right)}} \]
      *-commutative [=>] 93.6	\[ -\sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{1 - x}{-1 - x}\right)}} \]
      associate-*r* [=>] 93.6	\[ -\sqrt{\color{blue}{\left(2 \cdot 0.5\right) \cdot \frac{1 - x}{-1 - x}}} \]
      metadata-eval [=>] 93.6	\[ -\sqrt{\color{blue}{1} \cdot \frac{1 - x}{-1 - x}} \]
      associate-*r/ [=>] 93.6	\[ -\sqrt{\color{blue}{\frac{1 \cdot \left(1 - x\right)}{-1 - x}}} \]

   if -1.12e39 < t < -1.5499999999999999e-268 or 6.50000000000000033e-293 < t < 6.4999999999999998e43

   1. Initial program 37.5%

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

   2. Simplified 37.6%

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

      [Start] 37.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 37.6	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 37.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 37.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Taylor expanded in x around inf 72.2%

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

   4. Simplified 72.2%

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

      [Start] 72.2	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 72.2	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 72.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 72.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 72.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 72.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 72.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      +-commutative [=>] 72.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 72.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 72.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 72.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)\right)}} \cdot t \]

   5. Taylor expanded in t around 0 71.9%

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

   6. Simplified 71.9%

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

      [Start] 71.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{{\ell}^{2}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 71.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]

   7. Applied egg-rr 78.2%

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

      [Start] 71.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)\right)}} \cdot t \]
      pow1/2 [=>] 71.9	\[ \frac{\sqrt{2}}{\color{blue}{{\left(\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)\right)\right)}^{0.5}}} \cdot t \]
      sqr-pow [=>] 71.7	\[ \frac{\sqrt{2}}{\color{blue}{{\left(\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}}} \cdot t \]

   8. Simplified 78.4%

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

      [Start] 78.2	\[ \frac{\sqrt{2}}{{\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)\right)}^{0.25} \cdot {\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)\right)}^{0.25}} \cdot t \]
      pow-sqr [=>] 78.4	\[ \frac{\sqrt{2}}{\color{blue}{{\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)\right)}^{\left(2 \cdot 0.25\right)}}} \cdot t \]
      metadata-eval [=>] 78.4	\[ \frac{\sqrt{2}}{{\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)\right)}^{\color{blue}{0.5}}} \cdot t \]
      unpow1/2 [=>] 78.4	\[ \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)}}} \cdot t \]
      +-commutative [=>] 78.4	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right) + \frac{\ell}{x} \cdot \ell}}} \cdot t \]
      fma-udef [=>] 78.4	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) + \frac{\ell}{x} \cdot \ell\right)} + \frac{\ell}{x} \cdot \ell}} \cdot t \]
      associate-+l+ [=>] 78.4	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) + \left(\frac{\ell}{x} \cdot \ell + \frac{\ell}{x} \cdot \ell\right)}}} \cdot t \]
      fma-udef [=>] 78.4	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\left(t \cdot t + \frac{t}{x} \cdot t\right)} + \left(\frac{\ell}{x} \cdot \ell + \frac{\ell}{x} \cdot \ell\right)}} \cdot t \]
      distribute-rgt-out [=>] 78.4	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\left(t \cdot \left(t + \frac{t}{x}\right)\right)} + \left(\frac{\ell}{x} \cdot \ell + \frac{\ell}{x} \cdot \ell\right)}} \cdot t \]
      count-2 [=>] 78.4	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + \color{blue}{2 \cdot \left(\frac{\ell}{x} \cdot \ell\right)}}} \cdot t \]
      *-commutative [=>] 78.4	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{x}\right)}}} \cdot t \]

   9. Applied egg-rr 77.7%

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

      [Start] 78.4	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}} \cdot t \]
      sqrt-undiv [=>] 77.7	\[ \color{blue}{\sqrt{\frac{2}{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}} \cdot t \]
      distribute-lft-out [=>] 77.7	\[ \sqrt{\frac{2}{\color{blue}{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right) + \ell \cdot \frac{\ell}{x}\right)}}} \cdot t \]
      associate-/r* [=>] 77.7	\[ \sqrt{\color{blue}{\frac{\frac{2}{2}}{t \cdot \left(t + \frac{t}{x}\right) + \ell \cdot \frac{\ell}{x}}}} \cdot t \]
      metadata-eval [=>] 77.7	\[ \sqrt{\frac{\color{blue}{1}}{t \cdot \left(t + \frac{t}{x}\right) + \ell \cdot \frac{\ell}{x}}} \cdot t \]
      fma-def [=>] 77.7	\[ \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(t, t + \frac{t}{x}, \ell \cdot \frac{\ell}{x}\right)}}} \cdot t \]
      *-commutative [=>] 77.7	\[ \sqrt{\frac{1}{\mathsf{fma}\left(t, t + \frac{t}{x}, \color{blue}{\frac{\ell}{x} \cdot \ell}\right)}} \cdot t \]
      associate-/r/ [<=] 77.7	\[ \sqrt{\frac{1}{\mathsf{fma}\left(t, t + \frac{t}{x}, \color{blue}{\frac{\ell}{\frac{x}{\ell}}}\right)}} \cdot t \]

   if -1.5499999999999999e-268 < t < 6.50000000000000033e-293

   1. Initial program 2.3%

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

   2. Taylor expanded in l around inf 3.5%

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

   3. Simplified 3.5%

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

      [Start] 3.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\left(\frac{x}{x - 1} + \frac{1}{x - 1}\right) - 1\right) \cdot {\ell}^{2}}} \]
      sub-neg [=>] 3.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(\left(\frac{x}{x - 1} + \frac{1}{x - 1}\right) + \left(-1\right)\right)} \cdot {\ell}^{2}}} \]
      sub-neg [=>] 3.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\left(\frac{x}{\color{blue}{x + \left(-1\right)}} + \frac{1}{x - 1}\right) + \left(-1\right)\right) \cdot {\ell}^{2}}} \]
      metadata-eval [=>] 3.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\left(\frac{x}{x + \color{blue}{-1}} + \frac{1}{x - 1}\right) + \left(-1\right)\right) \cdot {\ell}^{2}}} \]
      +-commutative [=>] 3.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\left(\frac{x}{\color{blue}{-1 + x}} + \frac{1}{x - 1}\right) + \left(-1\right)\right) \cdot {\ell}^{2}}} \]
      sub-neg [=>] 3.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\left(\frac{x}{-1 + x} + \frac{1}{\color{blue}{x + \left(-1\right)}}\right) + \left(-1\right)\right) \cdot {\ell}^{2}}} \]
      metadata-eval [=>] 3.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\left(\frac{x}{-1 + x} + \frac{1}{x + \color{blue}{-1}}\right) + \left(-1\right)\right) \cdot {\ell}^{2}}} \]
      +-commutative [=>] 3.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\left(\frac{x}{-1 + x} + \frac{1}{\color{blue}{-1 + x}}\right) + \left(-1\right)\right) \cdot {\ell}^{2}}} \]
      metadata-eval [=>] 3.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\left(\frac{x}{-1 + x} + \frac{1}{-1 + x}\right) + \color{blue}{-1}\right) \cdot {\ell}^{2}}} \]
      unpow2 [=>] 3.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\left(\frac{x}{-1 + x} + \frac{1}{-1 + x}\right) + -1\right) \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \]

   4. Applied egg-rr 1.2%

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

      [Start] 3.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\left(\frac{x}{-1 + x} + \frac{1}{-1 + x}\right) + -1\right) \cdot \left(\ell \cdot \ell\right)}} \]
      expm1-log1p-u [=>] 3.5	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(\left(\frac{x}{-1 + x} + \frac{1}{-1 + x}\right) + -1\right) \cdot \left(\ell \cdot \ell\right)}\right)\right)}} \]
      expm1-udef [=>] 2.5	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\left(\left(\frac{x}{-1 + x} + \frac{1}{-1 + x}\right) + -1\right) \cdot \left(\ell \cdot \ell\right)}\right)} - 1}} \]

   5. Simplified 3.2%

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

      [Start] 1.2	\[ \frac{\sqrt{2} \cdot t}{e^{\mathsf{log1p}\left(\ell \cdot \sqrt{-1 + \frac{-1}{1 - x} \cdot \left(x + 1\right)}\right)} - 1} \]
      expm1-def [=>] 1.9	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot \sqrt{-1 + \frac{-1}{1 - x} \cdot \left(x + 1\right)}\right)\right)}} \]
      expm1-log1p [=>] 3.2	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{\ell \cdot \sqrt{-1 + \frac{-1}{1 - x} \cdot \left(x + 1\right)}}} \]
      associate-*l/ [=>] 3.2	\[ \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{-1 + \color{blue}{\frac{-1 \cdot \left(x + 1\right)}{1 - x}}}} \]
      +-commutative [=>] 3.2	\[ \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{-1 + \frac{-1 \cdot \color{blue}{\left(1 + x\right)}}{1 - x}}} \]
      distribute-lft-in [=>] 3.2	\[ \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{-1 + \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{1 - x}}} \]
      metadata-eval [=>] 3.2	\[ \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{-1 + \frac{\color{blue}{-1} + -1 \cdot x}{1 - x}}} \]
      neg-mul-1 [<=] 3.2	\[ \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{-1 + \frac{-1 + \color{blue}{\left(-x\right)}}{1 - x}}} \]
      neg-sub0 [=>] 3.2	\[ \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{-1 + \frac{-1 + \color{blue}{\left(0 - x\right)}}{1 - x}}} \]
      associate-+r- [=>] 3.2	\[ \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{-1 + \frac{\color{blue}{\left(-1 + 0\right) - x}}{1 - x}}} \]
      metadata-eval [=>] 3.2	\[ \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{-1 + \frac{\color{blue}{-1} - x}{1 - x}}} \]

   6. Taylor expanded in x around inf 49.2%

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

   7. Simplified 49.2%

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

      [Start] 49.2	\[ \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{2 \cdot \frac{1}{{x}^{2}} + 2 \cdot \frac{1}{x}}} \]
      associate-*r/ [=>] 49.2	\[ \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\color{blue}{\frac{2 \cdot 1}{{x}^{2}}} + 2 \cdot \frac{1}{x}}} \]
      metadata-eval [=>] 49.2	\[ \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\frac{\color{blue}{2}}{{x}^{2}} + 2 \cdot \frac{1}{x}}} \]
      unpow2 [=>] 49.2	\[ \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\frac{2}{\color{blue}{x \cdot x}} + 2 \cdot \frac{1}{x}}} \]
      associate-*r/ [=>] 49.2	\[ \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \color{blue}{\frac{2 \cdot 1}{x}}}} \]
      metadata-eval [=>] 49.2	\[ \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \frac{\color{blue}{2}}{x}}} \]

   if 6.4999999999999998e43 < t

   1. Initial program 31.3%

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

   2. Simplified 31.4%

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

      [Start] 31.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 31.3	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      associate-*l/ [=>] 12.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\left(x + 1\right) \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x - 1}} - \ell \cdot \ell}} \]
      associate-*r/ [<=] 31.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}} - \ell \cdot \ell}} \]
      *-lft-identity [<=] 31.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \left(x + 1\right)\right)} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} - \ell \cdot \ell}} \]
      associate-*r* [<=] 31.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{1 \cdot \left(\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 31.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{1 \cdot \color{blue}{\left(\frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} \cdot \left(x + 1\right)\right)} - \ell \cdot \ell}} \]
      associate-*r* [=>] 31.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right) \cdot \left(x + 1\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 31.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      fma-neg [=>] 31.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(x + 1, 1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}, -\ell \cdot \ell\right)}}} \]

   3. Taylor expanded in x around inf 89.7%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]

   4. Applied egg-rr 91.1%

      \[\leadsto \color{blue}{1} \]
      Proof

      [Start] 89.7	\[ \sqrt{2} \cdot \sqrt{0.5} \]
      sqrt-unprod [=>] 91.1	\[ \color{blue}{\sqrt{2 \cdot 0.5}} \]
      metadata-eval [=>] 91.1	\[ \sqrt{\color{blue}{1}} \]
      metadata-eval [=>] 91.1	\[ \color{blue}{1} \]

3. Recombined 4 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{+39}:\\ \;\;\;\;-\sqrt{\frac{1 - x}{-1 - x}}\\ \mathbf{elif}\;t \leq -1.55 \cdot 10^{-268}:\\ \;\;\;\;t \cdot \sqrt{\frac{1}{\mathsf{fma}\left(t, t + \frac{t}{x}, \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-293}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \frac{2}{x}}}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+43}:\\ \;\;\;\;t \cdot \sqrt{\frac{1}{\mathsf{fma}\left(t, t + \frac{t}{x}, \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	84.6%
Cost	20356

\[\begin{array}{l} \mathbf{if}\;t \leq -4.7 \cdot 10^{+39}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(t \cdot \left(-\sqrt{2}\right)\right)}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+43}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2
Accuracy	84.7%
Cost	14408

\[\begin{array}{l} \mathbf{if}\;t \leq -4.6 \cdot 10^{+39}:\\ \;\;\;\;-\sqrt{\frac{1 - x}{-1 - x}}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+43}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3
Accuracy	84.3%
Cost	14024

\[\begin{array}{l} \mathbf{if}\;t \leq -4.7 \cdot 10^{+39}:\\ \;\;\;\;-\sqrt{\frac{1 - x}{-1 - x}}\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{+43}:\\ \;\;\;\;t \cdot \sqrt{\frac{1}{\mathsf{fma}\left(t, t + \frac{t}{x}, \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Accuracy	77.2%
Cost	13768

\[\begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{-117}:\\ \;\;\;\;-\sqrt{\frac{1 - x}{-1 - x}}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-149}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Accuracy	77.3%
Cost	13768

\[\begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-117}:\\ \;\;\;\;-\sqrt{\frac{1 - x}{-1 - x}}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-148}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Accuracy	76.7%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{-180}:\\ \;\;\;\;\frac{1}{x} + \left(-1 + \frac{\frac{-0.5}{x}}{x}\right)\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-149}:\\ \;\;\;\;t \cdot \sqrt{x \cdot \frac{1}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Accuracy	77.0%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{-117}:\\ \;\;\;\;\frac{1}{x} + \left(-1 + \frac{\frac{-0.5}{x}}{x}\right)\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-147}:\\ \;\;\;\;t \cdot \sqrt{\frac{1}{\ell \cdot \frac{\ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Accuracy	77.2%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;t \leq -1.16 \cdot 10^{-116}:\\ \;\;\;\;-\sqrt{\frac{1 - x}{-1 - x}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-148}:\\ \;\;\;\;t \cdot \sqrt{\frac{1}{\ell \cdot \frac{\ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Accuracy	76.1%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-237}:\\ \;\;\;\;\frac{1}{x} + \left(-1 + \frac{\frac{-0.5}{x}}{x}\right)\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-158}:\\ \;\;\;\;\frac{t}{\ell \cdot {x}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Accuracy	76.3%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-238}:\\ \;\;\;\;\frac{1}{x} + \left(-1 + \frac{\frac{-0.5}{x}}{x}\right)\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 11
Accuracy	76.1%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{-238}:\\ \;\;\;\;\frac{1}{x} + \left(-1 + \frac{\frac{-0.5}{x}}{x}\right)\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-158}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12
Accuracy	76.0%
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{-308}:\\ \;\;\;\;\frac{1}{x} + \left(-1 + \frac{\frac{-0.5}{x}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 13
Accuracy	75.5%
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{-308}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14
Accuracy	75.9%
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{-308}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 15
Accuracy	75.2%
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{-308}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 16
Accuracy	38.4%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023140 
(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)))))