Toniolo and Linder, Equation (7)

Percentage Accurate: 33.0% → 84.7%

Time: 41.6s

Alternatives: 6

Speedup: 225.0×

Specification

Math

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

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)))))

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)));
}

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

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)));
}

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)))

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

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

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]

Initial Program: 33.0% accurate, 1.0× speedup

Math

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

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)))))

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)));
}

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

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)));
}

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)))

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

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

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]

Alternative 1: 84.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t_3 := t\_2 + {\ell}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.8 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{0.5 \cdot \frac{t\_3 - -1 \cdot t\_3}{t\_m \cdot \left(x \cdot \sqrt{2}\right)} + t\_m \cdot \sqrt{2}}\\ \mathbf{elif}\;t\_m \leq 2.8 \cdot 10^{+72}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\sqrt{\left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_2 + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{t\_3}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow t_m 2.0))) (t_3 (+ t_2 (pow l 2.0))))
   (*
    t_s
    (if (<= t_m 2.8e-162)
      (*
       (sqrt 2.0)
       (/
        t_m
        (+
         (* 0.5 (/ (- t_3 (* -1.0 t_3)) (* t_m (* x (sqrt 2.0)))))
         (* t_m (sqrt 2.0)))))
      (if (<= t_m 2.8e+72)
        (/
         (* (sqrt 2.0) t_m)
         (sqrt
          (-
           (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l 2.0) x)))
           (* -1.0 (/ t_3 x)))))
        (sqrt (/ (- x 1.0) (+ 1.0 x))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double t_3 = t_2 + pow(l, 2.0);
	double tmp;
	if (t_m <= 2.8e-162) {
		tmp = sqrt(2.0) * (t_m / ((0.5 * ((t_3 - (-1.0 * t_3)) / (t_m * (x * sqrt(2.0))))) + (t_m * sqrt(2.0))));
	} else if (t_m <= 2.8e+72) {
		tmp = (sqrt(2.0) * t_m) / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l, 2.0) / x))) - (-1.0 * (t_3 / x))));
	} else {
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    t_3 = t_2 + (l ** 2.0d0)
    if (t_m <= 2.8d-162) then
        tmp = sqrt(2.0d0) * (t_m / ((0.5d0 * ((t_3 - ((-1.0d0) * t_3)) / (t_m * (x * sqrt(2.0d0))))) + (t_m * sqrt(2.0d0))))
    else if (t_m <= 2.8d+72) then
        tmp = (sqrt(2.0d0) * t_m) / sqrt((((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_2 + ((l ** 2.0d0) / x))) - ((-1.0d0) * (t_3 / x))))
    else
        tmp = sqrt(((x - 1.0d0) / (1.0d0 + x)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double t_2 = 2.0 * Math.pow(t_m, 2.0);
	double t_3 = t_2 + Math.pow(l, 2.0);
	double tmp;
	if (t_m <= 2.8e-162) {
		tmp = Math.sqrt(2.0) * (t_m / ((0.5 * ((t_3 - (-1.0 * t_3)) / (t_m * (x * Math.sqrt(2.0))))) + (t_m * Math.sqrt(2.0))));
	} else if (t_m <= 2.8e+72) {
		tmp = (Math.sqrt(2.0) * t_m) / Math.sqrt((((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_2 + (Math.pow(l, 2.0) / x))) - (-1.0 * (t_3 / x))));
	} else {
		tmp = Math.sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	t_2 = 2.0 * math.pow(t_m, 2.0)
	t_3 = t_2 + math.pow(l, 2.0)
	tmp = 0
	if t_m <= 2.8e-162:
		tmp = math.sqrt(2.0) * (t_m / ((0.5 * ((t_3 - (-1.0 * t_3)) / (t_m * (x * math.sqrt(2.0))))) + (t_m * math.sqrt(2.0))))
	elif t_m <= 2.8e+72:
		tmp = (math.sqrt(2.0) * t_m) / math.sqrt((((2.0 * (math.pow(t_m, 2.0) / x)) + (t_2 + (math.pow(l, 2.0) / x))) - (-1.0 * (t_3 / x))))
	else:
		tmp = math.sqrt(((x - 1.0) / (1.0 + x)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(2.0 * (t_m ^ 2.0))
	t_3 = Float64(t_2 + (l ^ 2.0))
	tmp = 0.0
	if (t_m <= 2.8e-162)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(Float64(0.5 * Float64(Float64(t_3 - Float64(-1.0 * t_3)) / Float64(t_m * Float64(x * sqrt(2.0))))) + Float64(t_m * sqrt(2.0)))));
	elseif (t_m <= 2.8e+72)
		tmp = Float64(Float64(sqrt(2.0) * t_m) / sqrt(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l ^ 2.0) / x))) - Float64(-1.0 * Float64(t_3 / x)))));
	else
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	t_2 = 2.0 * (t_m ^ 2.0);
	t_3 = t_2 + (l ^ 2.0);
	tmp = 0.0;
	if (t_m <= 2.8e-162)
		tmp = sqrt(2.0) * (t_m / ((0.5 * ((t_3 - (-1.0 * t_3)) / (t_m * (x * sqrt(2.0))))) + (t_m * sqrt(2.0))));
	elseif (t_m <= 2.8e+72)
		tmp = (sqrt(2.0) * t_m) / sqrt((((2.0 * ((t_m ^ 2.0) / x)) + (t_2 + ((l ^ 2.0) / x))) - (-1.0 * (t_3 / x))));
	else
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.8e-162], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(N[(0.5 * N[(N[(t$95$3 - N[(-1.0 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.8e+72], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Sqrt[N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.80000000000000022e-162

   1. Initial program 28.2%

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

   3. Simplified 22.7%

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

   5. Taylor expanded in x around inf 19.6%

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

   if 2.80000000000000022e-162 < t < 2.7999999999999999e72

   1. Initial program 69.5%

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

   3. Taylor expanded in x around inf 85.9%

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

   if 2.7999999999999999e72 < t

   1. Initial program 34.2%

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

   3. Taylor expanded in l around 0 97.3%

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

   4. Taylor expanded in t around 0 97.3%

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

Alternative 2: 82.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t_3 := \sqrt{2} \cdot \frac{t\_m}{\sqrt{\left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_2 + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{t\_2 + {\ell}^{2}}{x}}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.9 \cdot 10^{-225}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_m \leq 1.42 \cdot 10^{-187}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 4.9 \cdot 10^{+73}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow t_m 2.0)))
        (t_3
         (*
          (sqrt 2.0)
          (/
           t_m
           (sqrt
            (-
             (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l 2.0) x)))
             (* -1.0 (/ (+ t_2 (pow l 2.0)) x))))))))
   (*
    t_s
    (if (<= t_m 1.9e-225)
      t_3
      (if (<= t_m 1.42e-187)
        1.0
        (if (<= t_m 4.9e+73) t_3 (sqrt (/ (- x 1.0) (+ 1.0 x)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double t_3 = sqrt(2.0) * (t_m / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l, 2.0) / x))) - (-1.0 * ((t_2 + pow(l, 2.0)) / x)))));
	double tmp;
	if (t_m <= 1.9e-225) {
		tmp = t_3;
	} else if (t_m <= 1.42e-187) {
		tmp = 1.0;
	} else if (t_m <= 4.9e+73) {
		tmp = t_3;
	} else {
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    t_3 = sqrt(2.0d0) * (t_m / sqrt((((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_2 + ((l ** 2.0d0) / x))) - ((-1.0d0) * ((t_2 + (l ** 2.0d0)) / x)))))
    if (t_m <= 1.9d-225) then
        tmp = t_3
    else if (t_m <= 1.42d-187) then
        tmp = 1.0d0
    else if (t_m <= 4.9d+73) then
        tmp = t_3
    else
        tmp = sqrt(((x - 1.0d0) / (1.0d0 + x)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double t_2 = 2.0 * Math.pow(t_m, 2.0);
	double t_3 = Math.sqrt(2.0) * (t_m / Math.sqrt((((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_2 + (Math.pow(l, 2.0) / x))) - (-1.0 * ((t_2 + Math.pow(l, 2.0)) / x)))));
	double tmp;
	if (t_m <= 1.9e-225) {
		tmp = t_3;
	} else if (t_m <= 1.42e-187) {
		tmp = 1.0;
	} else if (t_m <= 4.9e+73) {
		tmp = t_3;
	} else {
		tmp = Math.sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	t_2 = 2.0 * math.pow(t_m, 2.0)
	t_3 = math.sqrt(2.0) * (t_m / math.sqrt((((2.0 * (math.pow(t_m, 2.0) / x)) + (t_2 + (math.pow(l, 2.0) / x))) - (-1.0 * ((t_2 + math.pow(l, 2.0)) / x)))))
	tmp = 0
	if t_m <= 1.9e-225:
		tmp = t_3
	elif t_m <= 1.42e-187:
		tmp = 1.0
	elif t_m <= 4.9e+73:
		tmp = t_3
	else:
		tmp = math.sqrt(((x - 1.0) / (1.0 + x)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(2.0 * (t_m ^ 2.0))
	t_3 = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l ^ 2.0) / x))) - Float64(-1.0 * Float64(Float64(t_2 + (l ^ 2.0)) / x))))))
	tmp = 0.0
	if (t_m <= 1.9e-225)
		tmp = t_3;
	elseif (t_m <= 1.42e-187)
		tmp = 1.0;
	elseif (t_m <= 4.9e+73)
		tmp = t_3;
	else
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	t_2 = 2.0 * (t_m ^ 2.0);
	t_3 = sqrt(2.0) * (t_m / sqrt((((2.0 * ((t_m ^ 2.0) / x)) + (t_2 + ((l ^ 2.0) / x))) - (-1.0 * ((t_2 + (l ^ 2.0)) / x)))));
	tmp = 0.0;
	if (t_m <= 1.9e-225)
		tmp = t_3;
	elseif (t_m <= 1.42e-187)
		tmp = 1.0;
	elseif (t_m <= 4.9e+73)
		tmp = t_3;
	else
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(N[(t$95$2 + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.9e-225], t$95$3, If[LessEqual[t$95$m, 1.42e-187], 1.0, If[LessEqual[t$95$m, 4.9e+73], t$95$3, N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.9000000000000001e-225 or 1.42e-187 < t < 4.8999999999999999e73

   1. Initial program 40.0%

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

   3. Simplified 31.9%

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

   5. Taylor expanded in x around inf 60.8%

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

   if 1.9000000000000001e-225 < t < 1.42e-187

   1. Initial program 3.0%

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

   3. Taylor expanded in l around 0 79.0%

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

   4. Taylor expanded in x around inf 79.0%

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

   if 4.8999999999999999e73 < t

   1. Initial program 34.2%

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

   3. Taylor expanded in l around 0 97.3%

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

   4. Taylor expanded in t around 0 97.3%

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

Alternative 3: 82.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t_3 := \frac{\sqrt{2} \cdot t\_m}{\sqrt{\left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_2 + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{t\_2 + {\ell}^{2}}{x}}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.5 \cdot 10^{-228}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_m \leq 1.15 \cdot 10^{-187}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 3.7 \cdot 10^{+73}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow t_m 2.0)))
        (t_3
         (/
          (* (sqrt 2.0) t_m)
          (sqrt
           (-
            (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l 2.0) x)))
            (* -1.0 (/ (+ t_2 (pow l 2.0)) x)))))))
   (*
    t_s
    (if (<= t_m 2.5e-228)
      t_3
      (if (<= t_m 1.15e-187)
        1.0
        (if (<= t_m 3.7e+73) t_3 (sqrt (/ (- x 1.0) (+ 1.0 x)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double t_3 = (sqrt(2.0) * t_m) / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l, 2.0) / x))) - (-1.0 * ((t_2 + pow(l, 2.0)) / x))));
	double tmp;
	if (t_m <= 2.5e-228) {
		tmp = t_3;
	} else if (t_m <= 1.15e-187) {
		tmp = 1.0;
	} else if (t_m <= 3.7e+73) {
		tmp = t_3;
	} else {
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    t_3 = (sqrt(2.0d0) * t_m) / sqrt((((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_2 + ((l ** 2.0d0) / x))) - ((-1.0d0) * ((t_2 + (l ** 2.0d0)) / x))))
    if (t_m <= 2.5d-228) then
        tmp = t_3
    else if (t_m <= 1.15d-187) then
        tmp = 1.0d0
    else if (t_m <= 3.7d+73) then
        tmp = t_3
    else
        tmp = sqrt(((x - 1.0d0) / (1.0d0 + x)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double t_2 = 2.0 * Math.pow(t_m, 2.0);
	double t_3 = (Math.sqrt(2.0) * t_m) / Math.sqrt((((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_2 + (Math.pow(l, 2.0) / x))) - (-1.0 * ((t_2 + Math.pow(l, 2.0)) / x))));
	double tmp;
	if (t_m <= 2.5e-228) {
		tmp = t_3;
	} else if (t_m <= 1.15e-187) {
		tmp = 1.0;
	} else if (t_m <= 3.7e+73) {
		tmp = t_3;
	} else {
		tmp = Math.sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	t_2 = 2.0 * math.pow(t_m, 2.0)
	t_3 = (math.sqrt(2.0) * t_m) / math.sqrt((((2.0 * (math.pow(t_m, 2.0) / x)) + (t_2 + (math.pow(l, 2.0) / x))) - (-1.0 * ((t_2 + math.pow(l, 2.0)) / x))))
	tmp = 0
	if t_m <= 2.5e-228:
		tmp = t_3
	elif t_m <= 1.15e-187:
		tmp = 1.0
	elif t_m <= 3.7e+73:
		tmp = t_3
	else:
		tmp = math.sqrt(((x - 1.0) / (1.0 + x)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(2.0 * (t_m ^ 2.0))
	t_3 = Float64(Float64(sqrt(2.0) * t_m) / sqrt(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l ^ 2.0) / x))) - Float64(-1.0 * Float64(Float64(t_2 + (l ^ 2.0)) / x)))))
	tmp = 0.0
	if (t_m <= 2.5e-228)
		tmp = t_3;
	elseif (t_m <= 1.15e-187)
		tmp = 1.0;
	elseif (t_m <= 3.7e+73)
		tmp = t_3;
	else
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	t_2 = 2.0 * (t_m ^ 2.0);
	t_3 = (sqrt(2.0) * t_m) / sqrt((((2.0 * ((t_m ^ 2.0) / x)) + (t_2 + ((l ^ 2.0) / x))) - (-1.0 * ((t_2 + (l ^ 2.0)) / x))));
	tmp = 0.0;
	if (t_m <= 2.5e-228)
		tmp = t_3;
	elseif (t_m <= 1.15e-187)
		tmp = 1.0;
	elseif (t_m <= 3.7e+73)
		tmp = t_3;
	else
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Sqrt[N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(N[(t$95$2 + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.5e-228], t$95$3, If[LessEqual[t$95$m, 1.15e-187], 1.0, If[LessEqual[t$95$m, 3.7e+73], t$95$3, N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.49999999999999986e-228 or 1.14999999999999999e-187 < t < 3.69999999999999973e73

   1. Initial program 40.0%

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

   3. Taylor expanded in x around inf 60.9%

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

   if 2.49999999999999986e-228 < t < 1.14999999999999999e-187

   1. Initial program 3.0%

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

   3. Taylor expanded in l around 0 79.0%

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

   4. Taylor expanded in x around inf 79.0%

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

   if 3.69999999999999973e73 < t

   1. Initial program 34.2%

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

   3. Taylor expanded in l around 0 97.3%

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

   4. Taylor expanded in t around 0 97.3%

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

Alternative 4: 76.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \sqrt{\frac{x - 1}{1 + x}} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (* t_s (sqrt (/ (- x 1.0) (+ 1.0 x)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	return t_s * sqrt(((x - 1.0) / (1.0 + x)));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    code = t_s * sqrt(((x - 1.0d0) / (1.0d0 + x)))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	return t_s * Math.sqrt(((x - 1.0) / (1.0 + x)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	return t_s * math.sqrt(((x - 1.0) / (1.0 + x)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	return Float64(t_s * sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l, t_m)
	tmp = t_s * sqrt(((x - 1.0) / (1.0 + x)));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.2%

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

3. Taylor expanded in l around 0 46.4%

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

4. Taylor expanded in t around 0 46.4%

   \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
5. Add Preprocessing

Alternative 5: 75.7% accurate, 45.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(1 - \frac{1}{x}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m) :precision binary64 (* t_s (- 1.0 (/ 1.0 x))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	return t_s * (1.0 - (1.0 / x));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    code = t_s * (1.0d0 - (1.0d0 / x))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	return t_s * (1.0 - (1.0 / x));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	return t_s * (1.0 - (1.0 / x))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	return Float64(t_s * Float64(1.0 - Float64(1.0 / x)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l, t_m)
	tmp = t_s * (1.0 - (1.0 / x));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.2%

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

3. Taylor expanded in l around 0 46.4%

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

4. Taylor expanded in x around inf 46.2%

   \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
5. Add Preprocessing

Alternative 6: 75.0% accurate, 225.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot 1 \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m) :precision binary64 (* t_s 1.0))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	return t_s * 1.0;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    code = t_s * 1.0d0
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	return t_s * 1.0;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	return t_s * 1.0

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	return Float64(t_s * 1.0)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l, t_m)
	tmp = t_s * 1.0;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * 1.0), $MachinePrecision]

Derivation

1. Initial program 37.2%

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

3. Taylor expanded in l around 0 46.4%

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

4. Taylor expanded in x around inf 45.8%

   \[\leadsto \color{blue}{1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024076 -o generate:simplify
(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)))))