Toniolo and Linder, Equation (7)

Average Error: 43.6 → 11.9

Time: 18.7s

Precision: binary64

Cost: 47120

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\sqrt{2} \cdot t}{\sqrt{\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + \left(\frac{{\ell}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{{t}^{2}}{x}\right)\right)}}\\ \mathbf{if}\;t \leq -7.9 \cdot 10^{+31}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq -1.3 \cdot 10^{-248}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -8.6 \cdot 10^{-277}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+74}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

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
 (let* ((t_1
         (/
          (* (sqrt 2.0) t)
          (sqrt
           (+
            (/ (+ (pow l 2.0) (* 2.0 (pow t 2.0))) x)
            (+
             (/ (pow l 2.0) x)
             (* 2.0 (+ (pow t 2.0) (/ (pow t 2.0) x)))))))))
   (if (<= t -7.9e+31)
     (- (sqrt (/ (+ -1.0 x) (+ x 1.0))))
     (if (<= t -1.3e-248)
       t_1
       (if (<= t -8.6e-277) -1.0 (if (<= t 8e+74) t_1 (- 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 t_1 = (sqrt(2.0) * t) / sqrt((((pow(l, 2.0) + (2.0 * pow(t, 2.0))) / x) + ((pow(l, 2.0) / x) + (2.0 * (pow(t, 2.0) + (pow(t, 2.0) / x))))));
	double tmp;
	if (t <= -7.9e+31) {
		tmp = -sqrt(((-1.0 + x) / (x + 1.0)));
	} else if (t <= -1.3e-248) {
		tmp = t_1;
	} else if (t <= -8.6e-277) {
		tmp = -1.0;
	} else if (t <= 8e+74) {
		tmp = t_1;
	} else {
		tmp = 1.0 - (1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    code = (sqrt(2.0d0) * t) / sqrt(((((x + 1.0d0) / (x - 1.0d0)) * ((l * l) + (2.0d0 * (t * t)))) - (l * l)))
end function

↓

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (sqrt(2.0d0) * t) / sqrt(((((l ** 2.0d0) + (2.0d0 * (t ** 2.0d0))) / x) + (((l ** 2.0d0) / x) + (2.0d0 * ((t ** 2.0d0) + ((t ** 2.0d0) / x))))))
    if (t <= (-7.9d+31)) then
        tmp = -sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    else if (t <= (-1.3d-248)) then
        tmp = t_1
    else if (t <= (-8.6d-277)) then
        tmp = -1.0d0
    else if (t <= 8d+74) then
        tmp = t_1
    else
        tmp = 1.0d0 - (1.0d0 / x)
    end if
    code = tmp
end function

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 = (Math.sqrt(2.0) * t) / Math.sqrt((((Math.pow(l, 2.0) + (2.0 * Math.pow(t, 2.0))) / x) + ((Math.pow(l, 2.0) / x) + (2.0 * (Math.pow(t, 2.0) + (Math.pow(t, 2.0) / x))))));
	double tmp;
	if (t <= -7.9e+31) {
		tmp = -Math.sqrt(((-1.0 + x) / (x + 1.0)));
	} else if (t <= -1.3e-248) {
		tmp = t_1;
	} else if (t <= -8.6e-277) {
		tmp = -1.0;
	} else if (t <= 8e+74) {
		tmp = t_1;
	} else {
		tmp = 1.0 - (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):
	t_1 = (math.sqrt(2.0) * t) / math.sqrt((((math.pow(l, 2.0) + (2.0 * math.pow(t, 2.0))) / x) + ((math.pow(l, 2.0) / x) + (2.0 * (math.pow(t, 2.0) + (math.pow(t, 2.0) / x))))))
	tmp = 0
	if t <= -7.9e+31:
		tmp = -math.sqrt(((-1.0 + x) / (x + 1.0)))
	elif t <= -1.3e-248:
		tmp = t_1
	elif t <= -8.6e-277:
		tmp = -1.0
	elif t <= 8e+74:
		tmp = t_1
	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)
	t_1 = Float64(Float64(sqrt(2.0) * t) / sqrt(Float64(Float64(Float64((l ^ 2.0) + Float64(2.0 * (t ^ 2.0))) / x) + Float64(Float64((l ^ 2.0) / x) + Float64(2.0 * Float64((t ^ 2.0) + Float64((t ^ 2.0) / x)))))))
	tmp = 0.0
	if (t <= -7.9e+31)
		tmp = Float64(-sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0))));
	elseif (t <= -1.3e-248)
		tmp = t_1;
	elseif (t <= -8.6e-277)
		tmp = -1.0;
	elseif (t <= 8e+74)
		tmp = t_1;
	else
		tmp = Float64(1.0 - 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)
	t_1 = (sqrt(2.0) * t) / sqrt(((((l ^ 2.0) + (2.0 * (t ^ 2.0))) / x) + (((l ^ 2.0) / x) + (2.0 * ((t ^ 2.0) + ((t ^ 2.0) / x))))));
	tmp = 0.0;
	if (t <= -7.9e+31)
		tmp = -sqrt(((-1.0 + x) / (x + 1.0)));
	elseif (t <= -1.3e-248)
		tmp = t_1;
	elseif (t <= -8.6e-277)
		tmp = -1.0;
	elseif (t <= 8e+74)
		tmp = t_1;
	else
		tmp = 1.0 - (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_] := Block[{t$95$1 = N[(N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] + N[(2.0 * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(N[Power[t, 2.0], $MachinePrecision] + N[(N[Power[t, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -7.9e+31], (-N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, -1.3e-248], t$95$1, If[LessEqual[t, -8.6e-277], -1.0, If[LessEqual[t, 8e+74], t$95$1, N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -7.9000000000000003e31

   1. Initial program 43.4

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

   2. Taylor expanded in t around -inf 5.4

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

   3. Simplified 4.5

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

      [Start] 5.4	\[ -1 \cdot \left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{x - 1}{1 + x}}\right) \]
      rational_best-simplify-2 [=>] 5.4	\[ \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{x - 1}{1 + x}}\right) \cdot -1} \]
      rational_best-simplify-12 [=>] 5.4	\[ \color{blue}{-\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
      rational_best-simplify-2 [=>] 5.4	\[ -\color{blue}{\sqrt{\frac{x - 1}{1 + x}} \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)} \]
      exponential-simplify-19 [=>] 4.5	\[ -\sqrt{\frac{x - 1}{1 + x}} \cdot \color{blue}{\sqrt{0.5 \cdot 2}} \]
      metadata-eval [=>] 4.5	\[ -\sqrt{\frac{x - 1}{1 + x}} \cdot \sqrt{\color{blue}{1}} \]
      metadata-eval [=>] 4.5	\[ -\sqrt{\frac{x - 1}{1 + x}} \cdot \color{blue}{1} \]
      rational_best-simplify-5 [=>] 4.5	\[ -\color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
      rational_best-simplify-18 [<=] 4.5	\[ -\sqrt{\frac{\color{blue}{x + -1}}{1 + x}} \]
      rational_best-simplify-1 [=>] 4.5	\[ -\sqrt{\frac{\color{blue}{-1 + x}}{1 + x}} \]
      rational_best-simplify-1 [=>] 4.5	\[ -\sqrt{\frac{-1 + x}{\color{blue}{x + 1}}} \]

   if -7.9000000000000003e31 < t < -1.30000000000000003e-248 or -8.59999999999999981e-277 < t < 7.99999999999999961e74

   1. Initial program 41.3

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

   2. Taylor expanded in x around inf 18.9

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

   3. Simplified 18.9

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

      [Start] 18.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      rational_best-simplify-2 [=>] 18.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - \color{blue}{\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} \cdot -1}}} \]
      rational_best-simplify-12 [=>] 18.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]
      rational_best-simplify-11 [=>] 18.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - \color{blue}{\left(0 - \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]
      rational_best-simplify-46 [=>] 18.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + \left(\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - 0\right)}}} \]
      rational_best-simplify-6 [=>] 18.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + \color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right)}}} \]
      rational_best-simplify-47 [=>] 18.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + \left(\frac{{\ell}^{2}}{x} + \color{blue}{2 \cdot \left({t}^{2} + \frac{{t}^{2}}{x}\right)}\right)}} \]

   if -1.30000000000000003e-248 < t < -8.59999999999999981e-277

   1. Initial program 63.4

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

   2. Taylor expanded in t around -inf 35.6

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

   3. Simplified 35.6

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

      [Start] 35.6	\[ \frac{\sqrt{2} \cdot t}{-1 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)} \]
      rational_best-simplify-2 [=>] 35.6	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{\left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right) \cdot -1}} \]
      rational_best-simplify-12 [=>] 35.6	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{-\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
      rational_best-simplify-2 [=>] 35.6	\[ \frac{\sqrt{2} \cdot t}{-\color{blue}{\sqrt{\frac{1 + x}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}} \]
      rational_best-simplify-44 [=>] 35.6	\[ \frac{\sqrt{2} \cdot t}{-\color{blue}{\sqrt{2} \cdot \left(\sqrt{\frac{1 + x}{x - 1}} \cdot t\right)}} \]
      rational_best-simplify-1 [=>] 35.6	\[ \frac{\sqrt{2} \cdot t}{-\sqrt{2} \cdot \left(\sqrt{\frac{\color{blue}{x + 1}}{x - 1}} \cdot t\right)} \]
      rational_best-simplify-18 [<=] 35.6	\[ \frac{\sqrt{2} \cdot t}{-\sqrt{2} \cdot \left(\sqrt{\frac{x + 1}{\color{blue}{x + -1}}} \cdot t\right)} \]
      rational_best-simplify-1 [=>] 35.6	\[ \frac{\sqrt{2} \cdot t}{-\sqrt{2} \cdot \left(\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}} \cdot t\right)} \]

   4. Taylor expanded in x around inf 35.6

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

   if 7.99999999999999961e74 < t

   1. Initial program 47.5

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

   2. Taylor expanded in t around inf 4.2

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

   3. Simplified 3.2

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

      [Start] 4.2	\[ \left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{x - 1}{1 + x}} \]
      rational_best-simplify-2 [=>] 4.2	\[ \color{blue}{\sqrt{\frac{x - 1}{1 + x}} \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)} \]
      exponential-simplify-19 [=>] 3.2	\[ \sqrt{\frac{x - 1}{1 + x}} \cdot \color{blue}{\sqrt{0.5 \cdot 2}} \]
      metadata-eval [=>] 3.2	\[ \sqrt{\frac{x - 1}{1 + x}} \cdot \sqrt{\color{blue}{1}} \]
      metadata-eval [=>] 3.2	\[ \sqrt{\frac{x - 1}{1 + x}} \cdot \color{blue}{1} \]
      rational_best-simplify-5 [=>] 3.2	\[ \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
      rational_best-simplify-18 [<=] 3.2	\[ \sqrt{\frac{\color{blue}{x + -1}}{1 + x}} \]
      rational_best-simplify-1 [=>] 3.2	\[ \sqrt{\frac{\color{blue}{-1 + x}}{1 + x}} \]
      rational_best-simplify-1 [=>] 3.2	\[ \sqrt{\frac{-1 + x}{\color{blue}{x + 1}}} \]

   4. Taylor expanded in x around inf 3.9

      \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 11.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.9 \cdot 10^{+31}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq -1.3 \cdot 10^{-248}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + \left(\frac{{\ell}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{{t}^{2}}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq -8.6 \cdot 10^{-277}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+74}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + \left(\frac{{\ell}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{{t}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	15.1
Cost	20560

\[\begin{array}{l} t_1 := t \cdot \sqrt{2 \cdot \frac{1}{\frac{{\ell}^{2} + {\ell}^{2}}{x}}}\\ t_2 := \sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{if}\;t \leq -8.6 \cdot 10^{-277}:\\ \;\;\;\;-t_2\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.26 \cdot 10^{-129}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-113}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 2
Error	14.8
Cost	13640

\[\begin{array}{l} t_1 := \sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{if}\;t \leq -8.6 \cdot 10^{-277}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-159}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{2}{x}} \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	15.3
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq -8.6 \cdot 10^{-277}:\\ \;\;\;\;\frac{1}{x} - 1\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{-156}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]

Alternative 4
Error	15.1
Cost	7112

\[\begin{array}{l} t_1 := \sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{if}\;t \leq -4.3 \cdot 10^{-280}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-156}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	15.5
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{-278}:\\ \;\;\;\;\frac{1}{x} - 1\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-159}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 6
Error	15.8
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 7
Error	15.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{-308}:\\ \;\;\;\;\frac{1}{x} - 1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 8
Error	16.0
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Error	39.8
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023094 
(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)))))