Toniolo and Linder, Equation (7)

Average Accuracy: 34.0% → 97.2%

Time: 37.2s

Precision: binary64

Cost: 52356

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\ell}{\sqrt{x}}\\ t_2 := t \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -5000000:\\ \;\;\;\;t_2 \cdot \left|\frac{1}{t_2}\right|\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{hypot}\left(\mathsf{hypot}\left(\sqrt{2} \cdot \mathsf{hypot}\left(t, \frac{t}{\sqrt{x}}\right), t_1\right), t_1\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ l (sqrt x))) (t_2 (* t (sqrt 2.0))))
   (if (<= x -5000000.0)
     (* t_2 (fabs (/ 1.0 t_2)))
     (*
      t
      (/
       (sqrt 2.0)
       (hypot (hypot (* (sqrt 2.0) (hypot t (/ t (sqrt x)))) t_1) t_1))))))

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 = l / sqrt(x);
	double t_2 = t * sqrt(2.0);
	double tmp;
	if (x <= -5000000.0) {
		tmp = t_2 * fabs((1.0 / t_2));
	} else {
		tmp = t * (sqrt(2.0) / hypot(hypot((sqrt(2.0) * hypot(t, (t / sqrt(x)))), t_1), t_1));
	}
	return tmp;
}

Java

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

↓

public static double code(double x, double l, double t) {
	double t_1 = l / Math.sqrt(x);
	double t_2 = t * Math.sqrt(2.0);
	double tmp;
	if (x <= -5000000.0) {
		tmp = t_2 * Math.abs((1.0 / t_2));
	} else {
		tmp = t * (Math.sqrt(2.0) / Math.hypot(Math.hypot((Math.sqrt(2.0) * Math.hypot(t, (t / Math.sqrt(x)))), t_1), t_1));
	}
	return tmp;
}

Python

def code(x, l, t):
	return (math.sqrt(2.0) * t) / math.sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)))

↓

def code(x, l, t):
	t_1 = l / math.sqrt(x)
	t_2 = t * math.sqrt(2.0)
	tmp = 0
	if x <= -5000000.0:
		tmp = t_2 * math.fabs((1.0 / t_2))
	else:
		tmp = t * (math.sqrt(2.0) / math.hypot(math.hypot((math.sqrt(2.0) * math.hypot(t, (t / math.sqrt(x)))), t_1), t_1))
	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(l / sqrt(x))
	t_2 = Float64(t * sqrt(2.0))
	tmp = 0.0
	if (x <= -5000000.0)
		tmp = Float64(t_2 * abs(Float64(1.0 / t_2)));
	else
		tmp = Float64(t * Float64(sqrt(2.0) / hypot(hypot(Float64(sqrt(2.0) * hypot(t, Float64(t / sqrt(x)))), t_1), t_1)));
	end
	return tmp
end

MATLAB

function tmp = code(x, l, t)
	tmp = (sqrt(2.0) * t) / sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
end

↓

function tmp_2 = code(x, l, t)
	t_1 = l / sqrt(x);
	t_2 = t * sqrt(2.0);
	tmp = 0.0;
	if (x <= -5000000.0)
		tmp = t_2 * abs((1.0 / t_2));
	else
		tmp = t * (sqrt(2.0) / hypot(hypot((sqrt(2.0) * hypot(t, (t / sqrt(x)))), t_1), t_1));
	end
	tmp_2 = 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[(l / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5000000.0], N[(t$95$2 * N[Abs[N[(1.0 / t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[Sqrt[N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[t ^ 2 + N[(t / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -5e6

   1. Initial program 36.5%

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

   2. Simplified 36.4%

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

      [Start] 36.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 36.4	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      fma-neg [=>] 36.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]
      sub-neg [=>] 36.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{\color{blue}{x + \left(-1\right)}}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}} \]
      metadata-eval [=>] 36.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + \color{blue}{-1}}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}} \]
      +-commutative [=>] 36.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \color{blue}{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}, -\ell \cdot \ell\right)}} \]
      fma-def [=>] 36.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}, -\ell \cdot \ell\right)}} \]
      distribute-rgt-neg-in [=>] 36.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \color{blue}{\ell \cdot \left(-\ell\right)}\right)}} \]

   3. Taylor expanded in x around inf 35.9%

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

   4. Simplified 45.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{0 \cdot \left(\ell \cdot \ell\right) + 2 \cdot \left(t \cdot t\right)}} \cdot \left(t \cdot \sqrt{2}\right)} \]
      Proof

      [Start] 35.9	\[ \left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1}{{\ell}^{2} + \left(-1 \cdot {\ell}^{2} + 2 \cdot {t}^{2}\right)}} \]
      *-commutative [=>] 35.9	\[ \color{blue}{\sqrt{\frac{1}{{\ell}^{2} + \left(-1 \cdot {\ell}^{2} + 2 \cdot {t}^{2}\right)}} \cdot \left(\sqrt{2} \cdot t\right)} \]
      unpow2 [=>] 35.9	\[ \sqrt{\frac{1}{\color{blue}{\ell \cdot \ell} + \left(-1 \cdot {\ell}^{2} + 2 \cdot {t}^{2}\right)}} \cdot \left(\sqrt{2} \cdot t\right) \]
      associate-+r+ [=>] 45.0	\[ \sqrt{\frac{1}{\color{blue}{\left(\ell \cdot \ell + -1 \cdot {\ell}^{2}\right) + 2 \cdot {t}^{2}}}} \cdot \left(\sqrt{2} \cdot t\right) \]
      unpow2 [=>] 45.0	\[ \sqrt{\frac{1}{\left(\ell \cdot \ell + -1 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) + 2 \cdot {t}^{2}}} \cdot \left(\sqrt{2} \cdot t\right) \]
      distribute-rgt1-in [=>] 45.0	\[ \sqrt{\frac{1}{\color{blue}{\left(-1 + 1\right) \cdot \left(\ell \cdot \ell\right)} + 2 \cdot {t}^{2}}} \cdot \left(\sqrt{2} \cdot t\right) \]
      metadata-eval [=>] 45.0	\[ \sqrt{\frac{1}{\color{blue}{0} \cdot \left(\ell \cdot \ell\right) + 2 \cdot {t}^{2}}} \cdot \left(\sqrt{2} \cdot t\right) \]
      unpow2 [=>] 45.0	\[ \sqrt{\frac{1}{0 \cdot \left(\ell \cdot \ell\right) + 2 \cdot \color{blue}{\left(t \cdot t\right)}}} \cdot \left(\sqrt{2} \cdot t\right) \]
      *-commutative [=>] 45.0	\[ \sqrt{\frac{1}{0 \cdot \left(\ell \cdot \ell\right) + 2 \cdot \left(t \cdot t\right)}} \cdot \color{blue}{\left(t \cdot \sqrt{2}\right)} \]

   5. Applied egg-rr 96.0%

      \[\leadsto \color{blue}{\left|\frac{1}{t \cdot \sqrt{2}}\right|} \cdot \left(t \cdot \sqrt{2}\right) \]
      Proof

      [Start] 45.0	\[ \sqrt{\frac{1}{0 \cdot \left(\ell \cdot \ell\right) + 2 \cdot \left(t \cdot t\right)}} \cdot \left(t \cdot \sqrt{2}\right) \]
      add-sqr-sqrt [=>] 45.0	\[ \sqrt{\color{blue}{\sqrt{\frac{1}{0 \cdot \left(\ell \cdot \ell\right) + 2 \cdot \left(t \cdot t\right)}} \cdot \sqrt{\frac{1}{0 \cdot \left(\ell \cdot \ell\right) + 2 \cdot \left(t \cdot t\right)}}}} \cdot \left(t \cdot \sqrt{2}\right) \]
      rem-sqrt-square [=>] 45.0	\[ \color{blue}{\left|\sqrt{\frac{1}{0 \cdot \left(\ell \cdot \ell\right) + 2 \cdot \left(t \cdot t\right)}}\right|} \cdot \left(t \cdot \sqrt{2}\right) \]
      inv-pow [=>] 45.0	\[ \left|\sqrt{\color{blue}{{\left(0 \cdot \left(\ell \cdot \ell\right) + 2 \cdot \left(t \cdot t\right)\right)}^{-1}}}\right| \cdot \left(t \cdot \sqrt{2}\right) \]
      sqrt-pow1 [=>] 45.3	\[ \left|\color{blue}{{\left(0 \cdot \left(\ell \cdot \ell\right) + 2 \cdot \left(t \cdot t\right)\right)}^{\left(\frac{-1}{2}\right)}}\right| \cdot \left(t \cdot \sqrt{2}\right) \]
      mul0-lft [=>] 46.4	\[ \left|{\left(\color{blue}{0} + 2 \cdot \left(t \cdot t\right)\right)}^{\left(\frac{-1}{2}\right)}\right| \cdot \left(t \cdot \sqrt{2}\right) \]
      +-lft-identity [=>] 46.4	\[ \left|{\color{blue}{\left(2 \cdot \left(t \cdot t\right)\right)}}^{\left(\frac{-1}{2}\right)}\right| \cdot \left(t \cdot \sqrt{2}\right) \]
      metadata-eval [=>] 46.4	\[ \left|{\left(2 \cdot \left(t \cdot t\right)\right)}^{\color{blue}{-0.5}}\right| \cdot \left(t \cdot \sqrt{2}\right) \]
      metadata-eval [<=] 46.4	\[ \left|{\left(2 \cdot \left(t \cdot t\right)\right)}^{\color{blue}{\left(-0.5\right)}}\right| \cdot \left(t \cdot \sqrt{2}\right) \]
      pow-flip [<=] 46.3	\[ \left|\color{blue}{\frac{1}{{\left(2 \cdot \left(t \cdot t\right)\right)}^{0.5}}}\right| \cdot \left(t \cdot \sqrt{2}\right) \]
      pow1/2 [<=] 46.3	\[ \left|\frac{1}{\color{blue}{\sqrt{2 \cdot \left(t \cdot t\right)}}}\right| \cdot \left(t \cdot \sqrt{2}\right) \]
      sqrt-prod [=>] 46.4	\[ \left|\frac{1}{\color{blue}{\sqrt{2} \cdot \sqrt{t \cdot t}}}\right| \cdot \left(t \cdot \sqrt{2}\right) \]
      sqrt-prod [=>] 52.6	\[ \left|\frac{1}{\sqrt{2} \cdot \color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)}}\right| \cdot \left(t \cdot \sqrt{2}\right) \]
      add-sqr-sqrt [<=] 96.0	\[ \left|\frac{1}{\sqrt{2} \cdot \color{blue}{t}}\right| \cdot \left(t \cdot \sqrt{2}\right) \]
      *-commutative [<=] 96.0	\[ \left|\frac{1}{\color{blue}{t \cdot \sqrt{2}}}\right| \cdot \left(t \cdot \sqrt{2}\right) \]

   if -5e6 < x

   1. Initial program 22.1%

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

   2. Simplified 22.2%

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

      [Start] 22.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 22.2	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Taylor expanded in x around inf 51.2%

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

   4. Simplified 51.2%

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

      [Start] 51.2	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 51.2	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 51.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 51.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 51.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 51.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 51.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      unpow2 [=>] 51.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}}{x}\right)\right)}} \cdot t \]
      +-commutative [=>] 51.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + \ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 51.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 51.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]

   5. Taylor expanded in t around 0 51.1%

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

   6. Simplified 51.1%

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

      [Start] 51.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{{\ell}^{2}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 51.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]

   7. Applied egg-rr 98.1%

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

      [Start] 51.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)\right)}} \cdot t \]
      +-commutative [=>] 51.1	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)\right) + \frac{\ell \cdot \ell}{x}}}} \cdot t \]
      add-sqr-sqrt [=>] 51.1	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\sqrt{2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)} \cdot \sqrt{2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)}} + \frac{\ell \cdot \ell}{x}}} \cdot t \]
      add-sqr-sqrt [=>] 51.1	\[ \frac{\sqrt{2}}{\sqrt{\sqrt{2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)} \cdot \sqrt{2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)} + \color{blue}{\sqrt{\frac{\ell \cdot \ell}{x}} \cdot \sqrt{\frac{\ell \cdot \ell}{x}}}}} \cdot t \]
      hypot-def [=>] 51.1	\[ \frac{\sqrt{2}}{\color{blue}{\mathsf{hypot}\left(\sqrt{2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)}, \sqrt{\frac{\ell \cdot \ell}{x}}\right)}} \cdot t \]

3. Recombined 2 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5000000:\\ \;\;\;\;\left(t \cdot \sqrt{2}\right) \cdot \left|\frac{1}{t \cdot \sqrt{2}}\right|\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{hypot}\left(\mathsf{hypot}\left(\sqrt{2} \cdot \mathsf{hypot}\left(t, \frac{t}{\sqrt{x}}\right), \frac{\ell}{\sqrt{x}}\right), \frac{\ell}{\sqrt{x}}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	85.6%
Cost	40652

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{x}\\ t_2 := t \cdot \sqrt{2}\\ t_3 := \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)\\ \mathbf{if}\;t \leq -2.36 \cdot 10^{+86}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(-\sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{-284}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{{\left({\left(\mathsf{fma}\left(2, \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right), \mathsf{fma}\left(\frac{\ell}{x}, \ell, t_1\right)\right)\right)}^{0.25}\right)}^{2}}\\ \mathbf{elif}\;t \leq 1.52 \cdot 10^{-156}:\\ \;\;\;\;\frac{t_2}{\mathsf{fma}\left(0.5, \frac{t_3 + t_3}{\sqrt{2} \cdot \left(x \cdot t\right)}, t_2\right)}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+131}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(t_1 + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 2
Accuracy	85.7%
Cost	40136

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{x}\\ \mathbf{if}\;t \leq -2.85 \cdot 10^{+87}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(-\sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-286}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{{\left({\left(\mathsf{fma}\left(2, \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right), \mathsf{fma}\left(\frac{\ell}{x}, \ell, t_1\right)\right)\right)}^{0.25}\right)}^{2}}\\ \mathbf{elif}\;t \leq 1.52 \cdot 10^{-156}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\mathsf{fma}\left(t, \sqrt{2}, \frac{0.5 \cdot \left(2 \cdot \mathsf{fma}\left(\ell, \ell, t \cdot \left(t \cdot 2\right)\right)\right)}{\sqrt{2} \cdot \left(x \cdot t\right)}\right)}{t}}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+131}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(t_1 + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 3
Accuracy	85.6%
Cost	33868

\[\begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ \mathbf{if}\;t \leq -2.2 \cdot 10^{+83}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(-\sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-280}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{-156}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\mathsf{fma}\left(t, \sqrt{2}, \frac{0.5 \cdot \left(2 \cdot \mathsf{fma}\left(\ell, \ell, t \cdot \left(t \cdot 2\right)\right)\right)}{\sqrt{2} \cdot \left(x \cdot t\right)}\right)}{t}}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+131}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 4
Accuracy	85.7%
Cost	27468

\[\begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ \mathbf{if}\;t \leq -9.6 \cdot 10^{+84}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(-\sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-285}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.52 \cdot 10^{-156}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot \left(t \cdot t + \ell \cdot \ell\right)}{\sqrt{2} \cdot \left(x \cdot t\right)}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+131}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 5
Accuracy	84.3%
Cost	20100

\[\begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ \mathbf{if}\;t \leq -1.9 \cdot 10^{+82}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(-\sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-277}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.52 \cdot 10^{-156}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+130}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 6
Accuracy	84.4%
Cost	14544

\[\begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ \mathbf{if}\;t \leq -1.45 \cdot 10^{+85}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{-274}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.52 \cdot 10^{-156}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 10^{+131}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 7
Accuracy	81.3%
Cost	9168

\[\begin{array}{l} t_1 := \frac{t}{\sqrt{\frac{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \left(\ell \cdot \frac{\ell}{x} + 2 \cdot \left(t \cdot \frac{t}{x}\right)\right)\right)}{2}}}\\ \mathbf{if}\;t \leq -3.5 \cdot 10^{+48}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-277}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.52 \cdot 10^{-156}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 10^{+130}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 8
Accuracy	81.0%
Cost	8656

\[\begin{array}{l} t_1 := \frac{t}{\sqrt{\frac{\frac{\ell \cdot \ell}{x} + \left(\ell \cdot \frac{\ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}{2}}}\\ \mathbf{if}\;t \leq -8.5 \cdot 10^{+50}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 9.6 \cdot 10^{-284}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-156}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+129}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 9
Accuracy	76.9%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;t \leq -6.8 \cdot 10^{-222}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{1}{x}\right) + \frac{0.5}{x \cdot x}\\ \end{array} \]

Alternative 10
Accuracy	76.7%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-222}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) + \frac{-0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{1}{x}\right) + \frac{0.5}{x \cdot x}\\ \end{array} \]

Alternative 11
Accuracy	75.7%
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{-285}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) + \frac{-0.5}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 12
Accuracy	75.8%
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{-285}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) + \frac{-0.5}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{1}{x}\right) + \frac{0.5}{x \cdot x}\\ \end{array} \]

Alternative 13
Accuracy	75.2%
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{-285}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14
Accuracy	75.6%
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{-285}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 15
Accuracy	74.9%
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{-285}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 16
Accuracy	38.6%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023157 
(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)))))