Toniolo and Linder, Equation (7)

Average Error: 31.9% → 90.4%

Time: 47.8s

Precision: binary64

Cost: 13576.00

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-27}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-31}:\\ \;\;\;\;\frac{t}{\mathsf{hypot}\left(t, \sqrt{\ell \cdot \frac{\ell}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{-1 - x}{1 - x}}}\\ \end{array} \]

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
 (if (<= t -5.5e-27)
   (- (sqrt (/ (+ -1.0 x) (+ x 1.0))))
   (if (<= t 3.8e-31)
     (/ t (hypot t (sqrt (* l (/ l x)))))
     (/ 1.0 (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 tmp;
	if (t <= -5.5e-27) {
		tmp = -sqrt(((-1.0 + x) / (x + 1.0)));
	} else if (t <= 3.8e-31) {
		tmp = t / hypot(t, sqrt((l * (l / x))));
	} else {
		tmp = 1.0 / sqrt(((-1.0 - x) / (1.0 - x)));
	}
	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 tmp;
	if (t <= -5.5e-27) {
		tmp = -Math.sqrt(((-1.0 + x) / (x + 1.0)));
	} else if (t <= 3.8e-31) {
		tmp = t / Math.hypot(t, Math.sqrt((l * (l / x))));
	} else {
		tmp = 1.0 / Math.sqrt(((-1.0 - x) / (1.0 - x)));
	}
	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):
	tmp = 0
	if t <= -5.5e-27:
		tmp = -math.sqrt(((-1.0 + x) / (x + 1.0)))
	elif t <= 3.8e-31:
		tmp = t / math.hypot(t, math.sqrt((l * (l / x))))
	else:
		tmp = 1.0 / math.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)
	tmp = 0.0
	if (t <= -5.5e-27)
		tmp = Float64(-sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0))));
	elseif (t <= 3.8e-31)
		tmp = Float64(t / hypot(t, sqrt(Float64(l * Float64(l / x)))));
	else
		tmp = Float64(1.0 / sqrt(Float64(Float64(-1.0 - x) / Float64(1.0 - x))));
	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)
	tmp = 0.0;
	if (t <= -5.5e-27)
		tmp = -sqrt(((-1.0 + x) / (x + 1.0)));
	elseif (t <= 3.8e-31)
		tmp = t / hypot(t, sqrt((l * (l / x))));
	else
		tmp = 1.0 / sqrt(((-1.0 - x) / (1.0 - x)));
	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_] := If[LessEqual[t, -5.5e-27], (-N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, 3.8e-31], N[(t / N[Sqrt[t ^ 2 + N[Sqrt[N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(N[(-1.0 - x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.5000000000000002e-27

   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. Taylor expanded in t around -inf 91.3

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

   3. Simplified 91.3

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

      [Start] 91.3	\[ \frac{\sqrt{2} \cdot t}{-1 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)} \]
      mul-1-neg [=>] 91.3	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{-\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
      *-commutative [<=] 91.3	\[ \frac{\sqrt{2} \cdot t}{-\color{blue}{\left(t \cdot \sqrt{2}\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]
      sub-neg [=>] 91.3	\[ \frac{\sqrt{2} \cdot t}{-\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{\color{blue}{x + \left(-1\right)}}}} \]
      metadata-eval [=>] 91.3	\[ \frac{\sqrt{2} \cdot t}{-\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x + \color{blue}{-1}}}} \]
      +-commutative [=>] 91.3	\[ \frac{\sqrt{2} \cdot t}{-\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x + -1}}} \]
      +-commutative [=>] 91.3	\[ \frac{\sqrt{2} \cdot t}{-\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

   4. Taylor expanded in t around 0 91.3

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

   5. Simplified 91.3

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

      [Start] 91.3	\[ -1 \cdot \sqrt{\frac{x - 1}{1 + x}} \]
      mul-1-neg [=>] 91.3	\[ \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]
      sub-neg [=>] 91.3	\[ -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]
      metadata-eval [=>] 91.3	\[ -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]
      +-commutative [=>] 91.3	\[ -\sqrt{\frac{\color{blue}{-1 + x}}{1 + x}} \]
      +-commutative [<=] 91.3	\[ -\sqrt{\frac{-1 + x}{\color{blue}{x + 1}}} \]

   if -5.5000000000000002e-27 < t < 3.8e-31

   1. Initial program 24.5

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

   2. Simplified 13.5

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

      [Start] 24.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 24.5	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      associate-*l/ [=>] 25.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\left(x + 1\right) \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x - 1}} - \ell \cdot \ell}} \]
      associate-*r/ [<=] 12.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}} - \ell \cdot \ell}} \]
      *-lft-identity [<=] 12.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \left(x + 1\right)\right)} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} - \ell \cdot \ell}} \]
      associate-*r* [<=] 12.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{1 \cdot \left(\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 12.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{1 \cdot \color{blue}{\left(\frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} \cdot \left(x + 1\right)\right)} - \ell \cdot \ell}} \]
      associate-*r* [=>] 12.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right) \cdot \left(x + 1\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 12.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      fma-neg [=>] 13.5	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(x + 1, 1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}, -\ell \cdot \ell\right)}}} \]

   3. Taylor expanded in x around -inf 65.9

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

   4. Simplified 65.9

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

      [Start] 65.9	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + 2 \cdot {t}^{2}}} \]
      distribute-lft-out [=>] 65.9	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{2 \cdot \left(\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + {t}^{2}\right)}}} \]
      +-commutative [=>] 65.9	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 65.9	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x} + {t}^{2}\right)}} \]
      fma-udef [<=] 65.9	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 65.9	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 65.9	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} + \color{blue}{t \cdot t}\right)}} \]

   5. Taylor expanded in t around 0 65.7

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

   6. Simplified 72.2

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

      [Start] 65.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{{\ell}^{2}}{x} + t \cdot t\right)}} \]
      unpow2 [=>] 65.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{x} + t \cdot t\right)}} \]
      associate-/l* [=>] 72.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\color{blue}{\frac{\ell}{\frac{x}{\ell}}} + t \cdot t\right)}} \]

   7. Applied egg-rr 87.7

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

   8. Simplified 87.8

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

      [Start] 87.7	\[ \frac{\frac{t \cdot \sqrt{2}}{\sqrt{2}}}{\mathsf{hypot}\left(t, \sqrt{\ell \cdot \frac{\ell}{x}}\right)} \]
      associate-/l* [=>] 87.8	\[ \frac{\color{blue}{\frac{t}{\frac{\sqrt{2}}{\sqrt{2}}}}}{\mathsf{hypot}\left(t, \sqrt{\ell \cdot \frac{\ell}{x}}\right)} \]
      *-inverses [=>] 87.8	\[ \frac{\frac{t}{\color{blue}{1}}}{\mathsf{hypot}\left(t, \sqrt{\ell \cdot \frac{\ell}{x}}\right)} \]

   if 3.8e-31 < t

   1. Initial program 36.3

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

   2. Taylor expanded in t around -inf 1.6

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

   3. Simplified 1.6

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

      [Start] 1.6	\[ \frac{\sqrt{2} \cdot t}{-1 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)} \]
      mul-1-neg [=>] 1.6	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{-\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
      *-commutative [<=] 1.6	\[ \frac{\sqrt{2} \cdot t}{-\color{blue}{\left(t \cdot \sqrt{2}\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]
      sub-neg [=>] 1.6	\[ \frac{\sqrt{2} \cdot t}{-\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{\color{blue}{x + \left(-1\right)}}}} \]
      metadata-eval [=>] 1.6	\[ \frac{\sqrt{2} \cdot t}{-\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x + \color{blue}{-1}}}} \]
      +-commutative [=>] 1.6	\[ \frac{\sqrt{2} \cdot t}{-\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x + -1}}} \]
      +-commutative [=>] 1.6	\[ \frac{\sqrt{2} \cdot t}{-\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

   4. Applied egg-rr 92.2

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

   5. Simplified 92.5

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

      [Start] 92.2	\[ \frac{-1}{t \cdot \sqrt{2}} \cdot \frac{t \cdot \sqrt{2}}{-\sqrt{\frac{-1 - x}{1 - x}}} \]
      associate-*r/ [=>] 92.2	\[ \color{blue}{\frac{\frac{-1}{t \cdot \sqrt{2}} \cdot \left(t \cdot \sqrt{2}\right)}{-\sqrt{\frac{-1 - x}{1 - x}}}} \]
      neg-mul-1 [=>] 92.2	\[ \frac{\frac{-1}{t \cdot \sqrt{2}} \cdot \left(t \cdot \sqrt{2}\right)}{\color{blue}{-1 \cdot \sqrt{\frac{-1 - x}{1 - x}}}} \]
      associate-/r* [=>] 92.2	\[ \color{blue}{\frac{\frac{\frac{-1}{t \cdot \sqrt{2}} \cdot \left(t \cdot \sqrt{2}\right)}{-1}}{\sqrt{\frac{-1 - x}{1 - x}}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 90.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-27}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-31}:\\ \;\;\;\;\frac{t}{\mathsf{hypot}\left(t, \sqrt{\ell \cdot \frac{\ell}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{-1 - x}{1 - x}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	82.3%
Cost	13713.00

\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-207}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 10^{-213} \lor \neg \left(t \leq 6.2 \cdot 10^{-73}\right) \land t \leq 5.6 \cdot 10^{-31}:\\ \;\;\;\;\frac{t}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{-1 - x}{1 - x}}}\\ \end{array} \]

Alternative 2
Error	77.7%
Cost	7505.00

\[\begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-266}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 1.48 \cdot 10^{-238}:\\ \;\;\;\;\frac{1}{-\ell} \cdot \left(t \cdot \sqrt{x}\right)\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-66} \lor \neg \left(t \leq 2.35 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{1}{\sqrt{\frac{-1 - x}{1 - x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \end{array} \]

Alternative 3
Error	77.0%
Cost	7377.00

\[\begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{-266}:\\ \;\;\;\;\frac{1}{x} + \left(-1 + \frac{-0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-232}:\\ \;\;\;\;\sqrt{x} \cdot \left(-\frac{t}{\ell}\right)\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-66} \lor \neg \left(t \leq 2.35 \cdot 10^{-49}\right):\\ \;\;\;\;\sqrt{\frac{1}{1 + \frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \end{array} \]

Alternative 4
Error	77.2%
Cost	7377.00

\[\begin{array}{l} t_1 := \frac{\ell}{\sqrt{x}}\\ \mathbf{if}\;t \leq -1.15 \cdot 10^{-266}:\\ \;\;\;\;\frac{1}{x} + \left(-1 + \frac{-0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 2.75 \cdot 10^{-238}:\\ \;\;\;\;t \cdot \frac{-1}{t_1}\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-66} \lor \neg \left(t \leq 2.35 \cdot 10^{-49}\right):\\ \;\;\;\;\sqrt{\frac{1}{1 + \frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{t_1}\\ \end{array} \]

Alternative 5
Error	77.4%
Cost	7377.00

\[\begin{array}{l} t_1 := \frac{\ell}{\sqrt{x}}\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{-266}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-236}:\\ \;\;\;\;t \cdot \frac{-1}{t_1}\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-66} \lor \neg \left(t \leq 2.35 \cdot 10^{-49}\right):\\ \;\;\;\;\sqrt{\frac{1}{1 + \frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{t_1}\\ \end{array} \]

Alternative 6
Error	77.5%
Cost	7377.00

\[\begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{-266}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{-232}:\\ \;\;\;\;\frac{1}{-\ell} \cdot \left(t \cdot \sqrt{x}\right)\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-66} \lor \neg \left(t \leq 2.35 \cdot 10^{-49}\right):\\ \;\;\;\;\sqrt{\frac{1}{1 + \frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \end{array} \]

Alternative 7
Error	76.8%
Cost	7249.00

\[\begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-275}:\\ \;\;\;\;\frac{1}{x} + \left(-1 + \frac{-0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-214} \lor \neg \left(t \leq 1.65 \cdot 10^{-66}\right) \land t \leq 2.35 \cdot 10^{-49}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Error	76.8%
Cost	7248.00

\[\begin{array}{l} \mathbf{if}\;t \leq -1.02 \cdot 10^{-276}:\\ \;\;\;\;\frac{1}{x} + \left(-1 + \frac{-0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-214}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-66}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{-49}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Error	76.9%
Cost	7248.00

\[\begin{array}{l} \mathbf{if}\;t \leq -9.2 \cdot 10^{-276}:\\ \;\;\;\;\frac{1}{x} + \left(-1 + \frac{-0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-214}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-66}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{-49}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	76.7%
Cost	7248.00

\[\begin{array}{l} \mathbf{if}\;t \leq -2.35 \cdot 10^{-266}:\\ \;\;\;\;\frac{1}{x} + \left(-1 + \frac{-0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-233}:\\ \;\;\;\;\sqrt{x} \cdot \left(-\frac{t}{\ell}\right)\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-66}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{-49}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Error	76.3%
Cost	836.00

\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{x} + \left(-1 + \frac{-0.5}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12
Error	76.2%
Cost	452.00

\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13
Error	75.8%
Cost	196.00

\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14
Error	38.9%
Cost	64.00

\[-1 \]

Reproduce

herbie shell --seed 2023093 
(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)))))