Toniolo and Linder, Equation (7)

Percentage Accurate: 34.4% → 84.9%

Time: 21.9s

Alternatives: 13

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: 34.4% 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: 84.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t_3 := t\_2 + {\ell}^{2}\\ t_4 := t\_3 + t\_3\\ t_5 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.7 \cdot 10^{-170}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{0.5 \cdot \frac{t\_4}{t\_m \cdot \left(\sqrt{2} \cdot x\right)} + t\_5}\\ \mathbf{elif}\;t\_m \leq 6.2 \cdot 10^{+46}:\\ \;\;\;\;\frac{t\_5}{\sqrt{t\_2 + \frac{\left(2 \cdot \frac{{t\_m}^{2}}{x} + \frac{{\ell}^{2}}{x}\right) + \left(t\_4 + \frac{t\_3}{x}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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_4 (+ t_3 t_3))
        (t_5 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 1.7e-170)
      (* (sqrt 2.0) (/ t_m (+ (* 0.5 (/ t_4 (* t_m (* (sqrt 2.0) x)))) t_5)))
      (if (<= t_m 6.2e+46)
        (/
         t_5
         (sqrt
          (+
           t_2
           (/
            (+
             (+ (* 2.0 (/ (pow t_m 2.0) x)) (/ (pow l 2.0) x))
             (+ t_4 (/ t_3 x)))
            x))))
        (sqrt (/ (+ -1.0 x) (+ x 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) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double t_3 = t_2 + pow(l, 2.0);
	double t_4 = t_3 + t_3;
	double t_5 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 1.7e-170) {
		tmp = sqrt(2.0) * (t_m / ((0.5 * (t_4 / (t_m * (sqrt(2.0) * x)))) + t_5));
	} else if (t_m <= 6.2e+46) {
		tmp = t_5 / sqrt((t_2 + ((((2.0 * (pow(t_m, 2.0) / x)) + (pow(l, 2.0) / x)) + (t_4 + (t_3 / x))) / x)));
	} else {
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	}
	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) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    t_3 = t_2 + (l ** 2.0d0)
    t_4 = t_3 + t_3
    t_5 = t_m * sqrt(2.0d0)
    if (t_m <= 1.7d-170) then
        tmp = sqrt(2.0d0) * (t_m / ((0.5d0 * (t_4 / (t_m * (sqrt(2.0d0) * x)))) + t_5))
    else if (t_m <= 6.2d+46) then
        tmp = t_5 / sqrt((t_2 + ((((2.0d0 * ((t_m ** 2.0d0) / x)) + ((l ** 2.0d0) / x)) + (t_4 + (t_3 / x))) / x)))
    else
        tmp = sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    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 t_4 = t_3 + t_3;
	double t_5 = t_m * Math.sqrt(2.0);
	double tmp;
	if (t_m <= 1.7e-170) {
		tmp = Math.sqrt(2.0) * (t_m / ((0.5 * (t_4 / (t_m * (Math.sqrt(2.0) * x)))) + t_5));
	} else if (t_m <= 6.2e+46) {
		tmp = t_5 / Math.sqrt((t_2 + ((((2.0 * (Math.pow(t_m, 2.0) / x)) + (Math.pow(l, 2.0) / x)) + (t_4 + (t_3 / x))) / x)));
	} else {
		tmp = Math.sqrt(((-1.0 + x) / (x + 1.0)));
	}
	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)
	t_4 = t_3 + t_3
	t_5 = t_m * math.sqrt(2.0)
	tmp = 0
	if t_m <= 1.7e-170:
		tmp = math.sqrt(2.0) * (t_m / ((0.5 * (t_4 / (t_m * (math.sqrt(2.0) * x)))) + t_5))
	elif t_m <= 6.2e+46:
		tmp = t_5 / math.sqrt((t_2 + ((((2.0 * (math.pow(t_m, 2.0) / x)) + (math.pow(l, 2.0) / x)) + (t_4 + (t_3 / x))) / x)))
	else:
		tmp = math.sqrt(((-1.0 + x) / (x + 1.0)))
	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))
	t_4 = Float64(t_3 + t_3)
	t_5 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 1.7e-170)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(Float64(0.5 * Float64(t_4 / Float64(t_m * Float64(sqrt(2.0) * x)))) + t_5)));
	elseif (t_m <= 6.2e+46)
		tmp = Float64(t_5 / sqrt(Float64(t_2 + Float64(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64((l ^ 2.0) / x)) + Float64(t_4 + Float64(t_3 / x))) / x))));
	else
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)));
	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);
	t_4 = t_3 + t_3;
	t_5 = t_m * sqrt(2.0);
	tmp = 0.0;
	if (t_m <= 1.7e-170)
		tmp = sqrt(2.0) * (t_m / ((0.5 * (t_4 / (t_m * (sqrt(2.0) * x)))) + t_5));
	elseif (t_m <= 6.2e+46)
		tmp = t_5 / sqrt((t_2 + ((((2.0 * ((t_m ^ 2.0) / x)) + ((l ^ 2.0) / x)) + (t_4 + (t_3 / x))) / x)));
	else
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	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]}, Block[{t$95$4 = N[(t$95$3 + t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.7e-170], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(N[(0.5 * N[(t$95$4 / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.2e+46], N[(t$95$5 / N[Sqrt[N[(t$95$2 + N[(N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.70000000000000006e-170

   1. Initial program 32.9%

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

   3. Simplified 29.2%

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

   5. Taylor expanded in x around inf 11.7%

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

   if 1.70000000000000006e-170 < t < 6.1999999999999995e46

   1. Initial program 55.5%

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

      1. add-cube-cbrt 55.9%

         \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

      2. pow3 55.9%

         \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{{\left(\sqrt[3]{x + 1}\right)}^{3}}}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   4. Applied egg-rr 55.9%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{{\left(\sqrt[3]{x + 1}\right)}^{3}}}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   5. Taylor expanded in x around -inf 80.6%

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

   if 6.1999999999999995e46 < t

   1. Initial program 29.8%

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

   3. Simplified 29.8%

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

   5. Taylor expanded in t around inf 96.8%

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

   6. Taylor expanded in t around 0 97.1%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{-170}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(\sqrt{2} \cdot x\right)} + t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{+46}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot {t}^{2} + \frac{\left(2 \cdot \frac{{t}^{2}}{x} + \frac{{\ell}^{2}}{x}\right) + \left(\left(\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t_3 := t\_2 + {\ell}^{2}\\ t_4 := t\_3 + t\_3\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.3 \cdot 10^{-170}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{0.5 \cdot \frac{t\_4}{t\_m \cdot \left(\sqrt{2} \cdot x\right)} + t\_m \cdot \sqrt{2}}\\ \mathbf{elif}\;t\_m \leq 2.9 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{t\_2 + \frac{\left(2 \cdot \frac{{t\_m}^{2}}{x} + \frac{{\ell}^{2}}{x}\right) + \left(t\_4 + \frac{t\_3}{x}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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_4 (+ t_3 t_3)))
   (*
    t_s
    (if (<= t_m 1.3e-170)
      (*
       (sqrt 2.0)
       (/ t_m (+ (* 0.5 (/ t_4 (* t_m (* (sqrt 2.0) x)))) (* t_m (sqrt 2.0)))))
      (if (<= t_m 2.9e+44)
        (*
         (sqrt 2.0)
         (/
          t_m
          (sqrt
           (+
            t_2
            (/
             (+
              (+ (* 2.0 (/ (pow t_m 2.0) x)) (/ (pow l 2.0) x))
              (+ t_4 (/ t_3 x)))
             x)))))
        (sqrt (/ (+ -1.0 x) (+ x 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) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double t_3 = t_2 + pow(l, 2.0);
	double t_4 = t_3 + t_3;
	double tmp;
	if (t_m <= 1.3e-170) {
		tmp = sqrt(2.0) * (t_m / ((0.5 * (t_4 / (t_m * (sqrt(2.0) * x)))) + (t_m * sqrt(2.0))));
	} else if (t_m <= 2.9e+44) {
		tmp = sqrt(2.0) * (t_m / sqrt((t_2 + ((((2.0 * (pow(t_m, 2.0) / x)) + (pow(l, 2.0) / x)) + (t_4 + (t_3 / x))) / x))));
	} else {
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	}
	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) :: t_4
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    t_3 = t_2 + (l ** 2.0d0)
    t_4 = t_3 + t_3
    if (t_m <= 1.3d-170) then
        tmp = sqrt(2.0d0) * (t_m / ((0.5d0 * (t_4 / (t_m * (sqrt(2.0d0) * x)))) + (t_m * sqrt(2.0d0))))
    else if (t_m <= 2.9d+44) then
        tmp = sqrt(2.0d0) * (t_m / sqrt((t_2 + ((((2.0d0 * ((t_m ** 2.0d0) / x)) + ((l ** 2.0d0) / x)) + (t_4 + (t_3 / x))) / x))))
    else
        tmp = sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    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 t_4 = t_3 + t_3;
	double tmp;
	if (t_m <= 1.3e-170) {
		tmp = Math.sqrt(2.0) * (t_m / ((0.5 * (t_4 / (t_m * (Math.sqrt(2.0) * x)))) + (t_m * Math.sqrt(2.0))));
	} else if (t_m <= 2.9e+44) {
		tmp = Math.sqrt(2.0) * (t_m / Math.sqrt((t_2 + ((((2.0 * (Math.pow(t_m, 2.0) / x)) + (Math.pow(l, 2.0) / x)) + (t_4 + (t_3 / x))) / x))));
	} else {
		tmp = Math.sqrt(((-1.0 + x) / (x + 1.0)));
	}
	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)
	t_4 = t_3 + t_3
	tmp = 0
	if t_m <= 1.3e-170:
		tmp = math.sqrt(2.0) * (t_m / ((0.5 * (t_4 / (t_m * (math.sqrt(2.0) * x)))) + (t_m * math.sqrt(2.0))))
	elif t_m <= 2.9e+44:
		tmp = math.sqrt(2.0) * (t_m / math.sqrt((t_2 + ((((2.0 * (math.pow(t_m, 2.0) / x)) + (math.pow(l, 2.0) / x)) + (t_4 + (t_3 / x))) / x))))
	else:
		tmp = math.sqrt(((-1.0 + x) / (x + 1.0)))
	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))
	t_4 = Float64(t_3 + t_3)
	tmp = 0.0
	if (t_m <= 1.3e-170)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(Float64(0.5 * Float64(t_4 / Float64(t_m * Float64(sqrt(2.0) * x)))) + Float64(t_m * sqrt(2.0)))));
	elseif (t_m <= 2.9e+44)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(t_2 + Float64(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64((l ^ 2.0) / x)) + Float64(t_4 + Float64(t_3 / x))) / x)))));
	else
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)));
	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);
	t_4 = t_3 + t_3;
	tmp = 0.0;
	if (t_m <= 1.3e-170)
		tmp = sqrt(2.0) * (t_m / ((0.5 * (t_4 / (t_m * (sqrt(2.0) * x)))) + (t_m * sqrt(2.0))));
	elseif (t_m <= 2.9e+44)
		tmp = sqrt(2.0) * (t_m / sqrt((t_2 + ((((2.0 * ((t_m ^ 2.0) / x)) + ((l ^ 2.0) / x)) + (t_4 + (t_3 / x))) / x))));
	else
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	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]}, Block[{t$95$4 = N[(t$95$3 + t$95$3), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.3e-170], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(N[(0.5 * N[(t$95$4 / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.9e+44], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(t$95$2 + N[(N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.3000000000000001e-170

   1. Initial program 32.9%

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

   3. Simplified 29.2%

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

   5. Taylor expanded in x around inf 11.7%

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

   if 1.3000000000000001e-170 < t < 2.9000000000000002e44

   1. Initial program 55.5%

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

   3. Simplified 55.5%

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

   5. Taylor expanded in x around -inf 80.6%

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

   if 2.9000000000000002e44 < t

   1. Initial program 29.8%

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

   3. Simplified 29.8%

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

   5. Taylor expanded in t around inf 96.8%

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

   6. Taylor expanded in t around 0 97.1%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{-170}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(\sqrt{2} \cdot x\right)} + t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot {t}^{2} + \frac{\left(2 \cdot \frac{{t}^{2}}{x} + \frac{{\ell}^{2}}{x}\right) + \left(\left(\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t_3 := t\_2 + {\ell}^{2}\\ t_4 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.9 \cdot 10^{-170}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{0.5 \cdot \frac{t\_3 + t\_3}{t\_m \cdot \left(\sqrt{2} \cdot x\right)} + t\_4}\\ \mathbf{elif}\;t\_m \leq 1.7 \cdot 10^{+47}:\\ \;\;\;\;\frac{t\_4}{\sqrt{\frac{t\_3}{x} + \left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_2 + \frac{{\ell}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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_4 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 1.9e-170)
      (*
       (sqrt 2.0)
       (/ t_m (+ (* 0.5 (/ (+ t_3 t_3) (* t_m (* (sqrt 2.0) x)))) t_4)))
      (if (<= t_m 1.7e+47)
        (/
         t_4
         (sqrt
          (+
           (/ t_3 x)
           (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l 2.0) x))))))
        (sqrt (/ (+ -1.0 x) (+ x 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) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double t_3 = t_2 + pow(l, 2.0);
	double t_4 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 1.9e-170) {
		tmp = sqrt(2.0) * (t_m / ((0.5 * ((t_3 + t_3) / (t_m * (sqrt(2.0) * x)))) + t_4));
	} else if (t_m <= 1.7e+47) {
		tmp = t_4 / sqrt(((t_3 / x) + ((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l, 2.0) / x)))));
	} else {
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	}
	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) :: t_4
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    t_3 = t_2 + (l ** 2.0d0)
    t_4 = t_m * sqrt(2.0d0)
    if (t_m <= 1.9d-170) then
        tmp = sqrt(2.0d0) * (t_m / ((0.5d0 * ((t_3 + t_3) / (t_m * (sqrt(2.0d0) * x)))) + t_4))
    else if (t_m <= 1.7d+47) then
        tmp = t_4 / sqrt(((t_3 / x) + ((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_2 + ((l ** 2.0d0) / x)))))
    else
        tmp = sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    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 t_4 = t_m * Math.sqrt(2.0);
	double tmp;
	if (t_m <= 1.9e-170) {
		tmp = Math.sqrt(2.0) * (t_m / ((0.5 * ((t_3 + t_3) / (t_m * (Math.sqrt(2.0) * x)))) + t_4));
	} else if (t_m <= 1.7e+47) {
		tmp = t_4 / Math.sqrt(((t_3 / x) + ((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_2 + (Math.pow(l, 2.0) / x)))));
	} else {
		tmp = Math.sqrt(((-1.0 + x) / (x + 1.0)));
	}
	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)
	t_4 = t_m * math.sqrt(2.0)
	tmp = 0
	if t_m <= 1.9e-170:
		tmp = math.sqrt(2.0) * (t_m / ((0.5 * ((t_3 + t_3) / (t_m * (math.sqrt(2.0) * x)))) + t_4))
	elif t_m <= 1.7e+47:
		tmp = t_4 / math.sqrt(((t_3 / x) + ((2.0 * (math.pow(t_m, 2.0) / x)) + (t_2 + (math.pow(l, 2.0) / x)))))
	else:
		tmp = math.sqrt(((-1.0 + x) / (x + 1.0)))
	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))
	t_4 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 1.9e-170)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(Float64(0.5 * Float64(Float64(t_3 + t_3) / Float64(t_m * Float64(sqrt(2.0) * x)))) + t_4)));
	elseif (t_m <= 1.7e+47)
		tmp = Float64(t_4 / sqrt(Float64(Float64(t_3 / x) + Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l ^ 2.0) / x))))));
	else
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)));
	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);
	t_4 = t_m * sqrt(2.0);
	tmp = 0.0;
	if (t_m <= 1.9e-170)
		tmp = sqrt(2.0) * (t_m / ((0.5 * ((t_3 + t_3) / (t_m * (sqrt(2.0) * x)))) + t_4));
	elseif (t_m <= 1.7e+47)
		tmp = t_4 / sqrt(((t_3 / x) + ((2.0 * ((t_m ^ 2.0) / x)) + (t_2 + ((l ^ 2.0) / x)))));
	else
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	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]}, Block[{t$95$4 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.9e-170], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(N[(0.5 * N[(N[(t$95$3 + t$95$3), $MachinePrecision] / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.7e+47], N[(t$95$4 / N[Sqrt[N[(N[(t$95$3 / x), $MachinePrecision] + 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]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.8999999999999999e-170

   1. Initial program 32.9%

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

   3. Simplified 29.2%

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

   5. Taylor expanded in x around inf 11.7%

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

   if 1.8999999999999999e-170 < t < 1.6999999999999999e47

   1. Initial program 55.5%

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

      1. add-cube-cbrt 55.9%

         \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

      2. pow3 55.9%

         \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{{\left(\sqrt[3]{x + 1}\right)}^{3}}}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   4. Applied egg-rr 55.9%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{{\left(\sqrt[3]{x + 1}\right)}^{3}}}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   5. Taylor expanded in x around inf 80.2%

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

   if 1.6999999999999999e47 < t

   1. Initial program 29.8%

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

   3. Simplified 29.8%

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

   5. Taylor expanded in t around inf 96.8%

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

   6. Taylor expanded in t around 0 97.1%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.9 \cdot 10^{-170}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(\sqrt{2} \cdot x\right)} + t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+47}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{2 \cdot {t}^{2} + {\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t_3 := 2 \cdot {t\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.26 \cdot 10^{-135}:\\ \;\;\;\;\frac{t\_2}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{elif}\;t\_m \leq 10^{+41}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{t\_3 + {\ell}^{2}}{x} + \left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_3 + \frac{{\ell}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 (* t_m (sqrt 2.0))) (t_3 (* 2.0 (pow t_m 2.0))))
   (*
    t_s
    (if (<= t_m 1.26e-135)
      (/ t_2 (* l (sqrt (/ 2.0 x))))
      (if (<= t_m 1e+41)
        (/
         t_2
         (sqrt
          (+
           (/ (+ t_3 (pow l 2.0)) x)
           (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_3 (/ (pow l 2.0) x))))))
        (sqrt (/ (+ -1.0 x) (+ x 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) {
	double t_2 = t_m * sqrt(2.0);
	double t_3 = 2.0 * pow(t_m, 2.0);
	double tmp;
	if (t_m <= 1.26e-135) {
		tmp = t_2 / (l * sqrt((2.0 / x)));
	} else if (t_m <= 1e+41) {
		tmp = t_2 / sqrt((((t_3 + pow(l, 2.0)) / x) + ((2.0 * (pow(t_m, 2.0) / x)) + (t_3 + (pow(l, 2.0) / x)))));
	} else {
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	}
	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 = t_m * sqrt(2.0d0)
    t_3 = 2.0d0 * (t_m ** 2.0d0)
    if (t_m <= 1.26d-135) then
        tmp = t_2 / (l * sqrt((2.0d0 / x)))
    else if (t_m <= 1d+41) then
        tmp = t_2 / sqrt((((t_3 + (l ** 2.0d0)) / x) + ((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_3 + ((l ** 2.0d0) / x)))))
    else
        tmp = sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    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 = t_m * Math.sqrt(2.0);
	double t_3 = 2.0 * Math.pow(t_m, 2.0);
	double tmp;
	if (t_m <= 1.26e-135) {
		tmp = t_2 / (l * Math.sqrt((2.0 / x)));
	} else if (t_m <= 1e+41) {
		tmp = t_2 / Math.sqrt((((t_3 + Math.pow(l, 2.0)) / x) + ((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_3 + (Math.pow(l, 2.0) / x)))));
	} else {
		tmp = Math.sqrt(((-1.0 + x) / (x + 1.0)));
	}
	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 = t_m * math.sqrt(2.0)
	t_3 = 2.0 * math.pow(t_m, 2.0)
	tmp = 0
	if t_m <= 1.26e-135:
		tmp = t_2 / (l * math.sqrt((2.0 / x)))
	elif t_m <= 1e+41:
		tmp = t_2 / math.sqrt((((t_3 + math.pow(l, 2.0)) / x) + ((2.0 * (math.pow(t_m, 2.0) / x)) + (t_3 + (math.pow(l, 2.0) / x)))))
	else:
		tmp = math.sqrt(((-1.0 + x) / (x + 1.0)))
	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(t_m * sqrt(2.0))
	t_3 = Float64(2.0 * (t_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 1.26e-135)
		tmp = Float64(t_2 / Float64(l * sqrt(Float64(2.0 / x))));
	elseif (t_m <= 1e+41)
		tmp = Float64(t_2 / sqrt(Float64(Float64(Float64(t_3 + (l ^ 2.0)) / x) + Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_3 + Float64((l ^ 2.0) / x))))));
	else
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)));
	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 = t_m * sqrt(2.0);
	t_3 = 2.0 * (t_m ^ 2.0);
	tmp = 0.0;
	if (t_m <= 1.26e-135)
		tmp = t_2 / (l * sqrt((2.0 / x)));
	elseif (t_m <= 1e+41)
		tmp = t_2 / sqrt((((t_3 + (l ^ 2.0)) / x) + ((2.0 * ((t_m ^ 2.0) / x)) + (t_3 + ((l ^ 2.0) / x)))));
	else
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	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[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.26e-135], N[(t$95$2 / N[(l * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1e+41], N[(t$95$2 / N[Sqrt[N[(N[(N[(t$95$3 + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 + N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.2600000000000001e-135

   1. Initial program 32.5%

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

      1. add-cube-cbrt 33.0%

         \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

      2. pow3 33.0%

         \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{{\left(\sqrt[3]{x + 1}\right)}^{3}}}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   4. Applied egg-rr 33.0%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{{\left(\sqrt[3]{x + 1}\right)}^{3}}}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   5. Taylor expanded in l around inf 3.4%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\ell \cdot \sqrt{\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1}}} \]
   6. Step-by-step derivation

      1. associate--l+ 8.7%

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

      2. sub-neg 8.7%

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

      3. metadata-eval 8.7%

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

      4. sub-neg 8.7%

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

      5. metadata-eval 8.7%

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

   7. Simplified 8.7%

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

   8. Taylor expanded in x around inf 17.2%

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

   if 1.2600000000000001e-135 < t < 1.00000000000000001e41

   1. Initial program 60.8%

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

      1. add-cube-cbrt 61.4%

         \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

      2. pow3 61.4%

         \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{{\left(\sqrt[3]{x + 1}\right)}^{3}}}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   4. Applied egg-rr 61.4%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{{\left(\sqrt[3]{x + 1}\right)}^{3}}}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   5. Taylor expanded in x around inf 83.0%

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

   if 1.00000000000000001e41 < t

   1. Initial program 29.8%

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

   3. Simplified 29.8%

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

   5. Taylor expanded in t around inf 96.8%

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

   6. Taylor expanded in t around 0 97.1%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.26 \cdot 10^{-135}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{elif}\;t \leq 10^{+41}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{2 \cdot {t}^{2} + {\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.26 \cdot 10^{-135}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{elif}\;t\_m \leq 1.35 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{{t\_m}^{2} \cdot \left(2 + \frac{4}{x}\right) + \frac{{\ell}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.26e-135)
    (/ (* t_m (sqrt 2.0)) (* l (sqrt (/ 2.0 x))))
    (if (<= t_m 1.35e+49)
      (*
       (sqrt 2.0)
       (/
        t_m
        (sqrt
         (+
          (* (pow t_m 2.0) (+ 2.0 (/ 4.0 x)))
          (/ (+ (pow l 2.0) (pow l 2.0)) x)))))
      (sqrt (/ (+ -1.0 x) (+ x 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) {
	double tmp;
	if (t_m <= 1.26e-135) {
		tmp = (t_m * sqrt(2.0)) / (l * sqrt((2.0 / x)));
	} else if (t_m <= 1.35e+49) {
		tmp = sqrt(2.0) * (t_m / sqrt(((pow(t_m, 2.0) * (2.0 + (4.0 / x))) + ((pow(l, 2.0) + pow(l, 2.0)) / x))));
	} else {
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	}
	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) :: tmp
    if (t_m <= 1.26d-135) then
        tmp = (t_m * sqrt(2.0d0)) / (l * sqrt((2.0d0 / x)))
    else if (t_m <= 1.35d+49) then
        tmp = sqrt(2.0d0) * (t_m / sqrt((((t_m ** 2.0d0) * (2.0d0 + (4.0d0 / x))) + (((l ** 2.0d0) + (l ** 2.0d0)) / x))))
    else
        tmp = sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    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 tmp;
	if (t_m <= 1.26e-135) {
		tmp = (t_m * Math.sqrt(2.0)) / (l * Math.sqrt((2.0 / x)));
	} else if (t_m <= 1.35e+49) {
		tmp = Math.sqrt(2.0) * (t_m / Math.sqrt(((Math.pow(t_m, 2.0) * (2.0 + (4.0 / x))) + ((Math.pow(l, 2.0) + Math.pow(l, 2.0)) / x))));
	} else {
		tmp = Math.sqrt(((-1.0 + x) / (x + 1.0)));
	}
	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):
	tmp = 0
	if t_m <= 1.26e-135:
		tmp = (t_m * math.sqrt(2.0)) / (l * math.sqrt((2.0 / x)))
	elif t_m <= 1.35e+49:
		tmp = math.sqrt(2.0) * (t_m / math.sqrt(((math.pow(t_m, 2.0) * (2.0 + (4.0 / x))) + ((math.pow(l, 2.0) + math.pow(l, 2.0)) / x))))
	else:
		tmp = math.sqrt(((-1.0 + x) / (x + 1.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 1.26e-135)
		tmp = Float64(Float64(t_m * sqrt(2.0)) / Float64(l * sqrt(Float64(2.0 / x))));
	elseif (t_m <= 1.35e+49)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(Float64((t_m ^ 2.0) * Float64(2.0 + Float64(4.0 / x))) + Float64(Float64((l ^ 2.0) + (l ^ 2.0)) / x)))));
	else
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)));
	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)
	tmp = 0.0;
	if (t_m <= 1.26e-135)
		tmp = (t_m * sqrt(2.0)) / (l * sqrt((2.0 / x)));
	elseif (t_m <= 1.35e+49)
		tmp = sqrt(2.0) * (t_m / sqrt((((t_m ^ 2.0) * (2.0 + (4.0 / x))) + (((l ^ 2.0) + (l ^ 2.0)) / x))));
	else
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	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_] := N[(t$95$s * If[LessEqual[t$95$m, 1.26e-135], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(l * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.35e+49], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[l, 2.0], $MachinePrecision] + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.2600000000000001e-135

   1. Initial program 32.5%

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

      1. add-cube-cbrt 33.0%

         \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

      2. pow3 33.0%

         \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{{\left(\sqrt[3]{x + 1}\right)}^{3}}}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   4. Applied egg-rr 33.0%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{{\left(\sqrt[3]{x + 1}\right)}^{3}}}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   5. Taylor expanded in l around inf 3.4%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\ell \cdot \sqrt{\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1}}} \]
   6. Step-by-step derivation

      1. associate--l+ 8.7%

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

      2. sub-neg 8.7%

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

      3. metadata-eval 8.7%

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

      4. sub-neg 8.7%

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

      5. metadata-eval 8.7%

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

   7. Simplified 8.7%

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

   8. Taylor expanded in x around inf 17.2%

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

   if 1.2600000000000001e-135 < t < 1.35000000000000005e49

   1. Initial program 60.8%

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

   3. Simplified 60.7%

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

   5. Taylor expanded in x around inf 82.9%

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

   6. Taylor expanded in t around 0 82.9%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left({t}^{2} \cdot \left(2 + 4 \cdot \frac{1}{x}\right) + \frac{{\ell}^{2}}{x}\right) - -1 \cdot \frac{{\ell}^{2}}{x}}}} \]
   7. Step-by-step derivation

      1. associate--l+ 82.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{{t}^{2} \cdot \left(2 + 4 \cdot \frac{1}{x}\right) + \left(\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}\right)}}} \]

      2. associate-*r/ 82.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{{t}^{2} \cdot \left(2 + \color{blue}{\frac{4 \cdot 1}{x}}\right) + \left(\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}\right)}} \]

      3. metadata-eval 82.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{{t}^{2} \cdot \left(2 + \frac{\color{blue}{4}}{x}\right) + \left(\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}\right)}} \]

      4. associate-*r/ 82.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{{t}^{2} \cdot \left(2 + \frac{4}{x}\right) + \left(\frac{{\ell}^{2}}{x} - \color{blue}{\frac{-1 \cdot {\ell}^{2}}{x}}\right)}} \]

      5. div-sub 82.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{{t}^{2} \cdot \left(2 + \frac{4}{x}\right) + \color{blue}{\frac{{\ell}^{2} - -1 \cdot {\ell}^{2}}{x}}}} \]

      6. neg-mul-1 82.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{{t}^{2} \cdot \left(2 + \frac{4}{x}\right) + \frac{{\ell}^{2} - \color{blue}{\left(-{\ell}^{2}\right)}}{x}}} \]

   8. Simplified 82.9%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{{t}^{2} \cdot \left(2 + \frac{4}{x}\right) + \frac{{\ell}^{2} - \left(-{\ell}^{2}\right)}{x}}}} \]

   if 1.35000000000000005e49 < t

   1. Initial program 29.8%

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

   3. Simplified 29.8%

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

   5. Taylor expanded in t around inf 96.8%

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

   6. Taylor expanded in t around 0 97.1%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.26 \cdot 10^{-135}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{{t}^{2} \cdot \left(2 + \frac{4}{x}\right) + \frac{{\ell}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 160000000000:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+73}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2 + \frac{2}{x}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= x 160000000000.0)
    (sqrt (/ (+ -1.0 x) (+ x 1.0)))
    (if (<= x 4.1e+73)
      (/ (* t_m (sqrt 2.0)) (* l (sqrt (/ (+ 2.0 (/ 2.0 x)) 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 tmp;
	if (x <= 160000000000.0) {
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	} else if (x <= 4.1e+73) {
		tmp = (t_m * sqrt(2.0)) / (l * sqrt(((2.0 + (2.0 / x)) / x)));
	} else {
		tmp = 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) :: tmp
    if (x <= 160000000000.0d0) then
        tmp = sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    else if (x <= 4.1d+73) then
        tmp = (t_m * sqrt(2.0d0)) / (l * sqrt(((2.0d0 + (2.0d0 / x)) / x)))
    else
        tmp = 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 tmp;
	if (x <= 160000000000.0) {
		tmp = Math.sqrt(((-1.0 + x) / (x + 1.0)));
	} else if (x <= 4.1e+73) {
		tmp = (t_m * Math.sqrt(2.0)) / (l * Math.sqrt(((2.0 + (2.0 / x)) / x)));
	} else {
		tmp = 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):
	tmp = 0
	if x <= 160000000000.0:
		tmp = math.sqrt(((-1.0 + x) / (x + 1.0)))
	elif x <= 4.1e+73:
		tmp = (t_m * math.sqrt(2.0)) / (l * math.sqrt(((2.0 + (2.0 / x)) / x)))
	else:
		tmp = 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)
	tmp = 0.0
	if (x <= 160000000000.0)
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)));
	elseif (x <= 4.1e+73)
		tmp = Float64(Float64(t_m * sqrt(2.0)) / Float64(l * sqrt(Float64(Float64(2.0 + Float64(2.0 / x)) / x))));
	else
		tmp = Float64(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)
	tmp = 0.0;
	if (x <= 160000000000.0)
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	elseif (x <= 4.1e+73)
		tmp = (t_m * sqrt(2.0)) / (l * sqrt(((2.0 + (2.0 / x)) / x)));
	else
		tmp = 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_] := N[(t$95$s * If[LessEqual[x, 160000000000.0], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4.1e+73], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(l * N[Sqrt[N[(N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 1.6e11

   1. Initial program 44.6%

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

   3. Simplified 38.3%

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

   5. Taylor expanded in t around inf 44.7%

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

   6. Taylor expanded in t around 0 44.8%

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

   if 1.6e11 < x < 4.0999999999999998e73

   1. Initial program 26.9%

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

      1. add-cube-cbrt 27.8%

         \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

      2. pow3 27.9%

         \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{{\left(\sqrt[3]{x + 1}\right)}^{3}}}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   4. Applied egg-rr 27.9%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{{\left(\sqrt[3]{x + 1}\right)}^{3}}}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   5. Taylor expanded in l around inf 10.6%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\ell \cdot \sqrt{\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1}}} \]
   6. Step-by-step derivation

      1. associate--l+ 28.5%

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

      2. sub-neg 28.5%

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

      3. metadata-eval 28.5%

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

      4. sub-neg 28.5%

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

      5. metadata-eval 28.5%

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

   7. Simplified 28.5%

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

   8. Taylor expanded in x around inf 56.5%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\color{blue}{\frac{2 + 2 \cdot \frac{1}{x}}{x}}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 56.5%

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

      2. metadata-eval 56.5%

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

   10. Simplified 56.5%

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

   if 4.0999999999999998e73 < x

   1. Initial program 28.5%

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

   3. Simplified 25.5%

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

   5. Taylor expanded in t around inf 37.4%

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

   6. Taylor expanded in x around inf 37.5%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 160000000000:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+73}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2 + \frac{2}{x}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 132000000000:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+73}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= x 132000000000.0)
    (sqrt (/ (+ -1.0 x) (+ x 1.0)))
    (if (<= x 3.7e+73)
      (/ (* t_m (sqrt 2.0)) (* l (sqrt (/ 2.0 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 tmp;
	if (x <= 132000000000.0) {
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	} else if (x <= 3.7e+73) {
		tmp = (t_m * sqrt(2.0)) / (l * sqrt((2.0 / x)));
	} else {
		tmp = 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) :: tmp
    if (x <= 132000000000.0d0) then
        tmp = sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    else if (x <= 3.7d+73) then
        tmp = (t_m * sqrt(2.0d0)) / (l * sqrt((2.0d0 / x)))
    else
        tmp = 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 tmp;
	if (x <= 132000000000.0) {
		tmp = Math.sqrt(((-1.0 + x) / (x + 1.0)));
	} else if (x <= 3.7e+73) {
		tmp = (t_m * Math.sqrt(2.0)) / (l * Math.sqrt((2.0 / x)));
	} else {
		tmp = 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):
	tmp = 0
	if x <= 132000000000.0:
		tmp = math.sqrt(((-1.0 + x) / (x + 1.0)))
	elif x <= 3.7e+73:
		tmp = (t_m * math.sqrt(2.0)) / (l * math.sqrt((2.0 / x)))
	else:
		tmp = 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)
	tmp = 0.0
	if (x <= 132000000000.0)
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)));
	elseif (x <= 3.7e+73)
		tmp = Float64(Float64(t_m * sqrt(2.0)) / Float64(l * sqrt(Float64(2.0 / x))));
	else
		tmp = Float64(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)
	tmp = 0.0;
	if (x <= 132000000000.0)
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	elseif (x <= 3.7e+73)
		tmp = (t_m * sqrt(2.0)) / (l * sqrt((2.0 / x)));
	else
		tmp = 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_] := N[(t$95$s * If[LessEqual[x, 132000000000.0], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 3.7e+73], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(l * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 1.32e11

   1. Initial program 44.6%

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

   3. Simplified 38.3%

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

   5. Taylor expanded in t around inf 44.7%

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

   6. Taylor expanded in t around 0 44.8%

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

   if 1.32e11 < x < 3.69999999999999973e73

   1. Initial program 26.9%

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

      1. add-cube-cbrt 27.8%

         \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

      2. pow3 27.9%

         \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{{\left(\sqrt[3]{x + 1}\right)}^{3}}}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   4. Applied egg-rr 27.9%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{{\left(\sqrt[3]{x + 1}\right)}^{3}}}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   5. Taylor expanded in l around inf 10.6%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\ell \cdot \sqrt{\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1}}} \]
   6. Step-by-step derivation

      1. associate--l+ 28.5%

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

      2. sub-neg 28.5%

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

      3. metadata-eval 28.5%

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

      4. sub-neg 28.5%

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

      5. metadata-eval 28.5%

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

   7. Simplified 28.5%

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

   8. Taylor expanded in x around inf 55.7%

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

   if 3.69999999999999973e73 < x

   1. Initial program 28.5%

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

   3. Simplified 25.5%

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

   5. Taylor expanded in t around inf 37.4%

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

   6. Taylor expanded in x around inf 37.5%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 132000000000:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+73}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 132000000000:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+73}:\\ \;\;\;\;\frac{t\_m}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= x 132000000000.0)
    (sqrt (/ (+ -1.0 x) (+ x 1.0)))
    (if (<= x 3.7e+73) (* (/ t_m l) (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 tmp;
	if (x <= 132000000000.0) {
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	} else if (x <= 3.7e+73) {
		tmp = (t_m / l) * sqrt(x);
	} else {
		tmp = 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) :: tmp
    if (x <= 132000000000.0d0) then
        tmp = sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    else if (x <= 3.7d+73) then
        tmp = (t_m / l) * sqrt(x)
    else
        tmp = 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 tmp;
	if (x <= 132000000000.0) {
		tmp = Math.sqrt(((-1.0 + x) / (x + 1.0)));
	} else if (x <= 3.7e+73) {
		tmp = (t_m / l) * Math.sqrt(x);
	} else {
		tmp = 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):
	tmp = 0
	if x <= 132000000000.0:
		tmp = math.sqrt(((-1.0 + x) / (x + 1.0)))
	elif x <= 3.7e+73:
		tmp = (t_m / l) * math.sqrt(x)
	else:
		tmp = 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)
	tmp = 0.0
	if (x <= 132000000000.0)
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)));
	elseif (x <= 3.7e+73)
		tmp = Float64(Float64(t_m / l) * sqrt(x));
	else
		tmp = Float64(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)
	tmp = 0.0;
	if (x <= 132000000000.0)
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	elseif (x <= 3.7e+73)
		tmp = (t_m / l) * sqrt(x);
	else
		tmp = 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_] := N[(t$95$s * If[LessEqual[x, 132000000000.0], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 3.7e+73], N[(N[(t$95$m / l), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 1.32e11

   1. Initial program 44.6%

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

   3. Simplified 38.3%

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

   5. Taylor expanded in t around inf 44.7%

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

   6. Taylor expanded in t around 0 44.8%

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

   if 1.32e11 < x < 3.69999999999999973e73

   1. Initial program 26.9%

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

   3. Simplified 26.9%

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

   5. Taylor expanded in x around inf 66.4%

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

   6. Taylor expanded in l around inf 42.9%

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

   7. Taylor expanded in t around 0 53.4%

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

   if 3.69999999999999973e73 < x

   1. Initial program 28.5%

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

   3. Simplified 25.5%

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

   5. Taylor expanded in t around inf 37.4%

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

   6. Taylor expanded in x around inf 37.5%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 132000000000:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+73}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 160000000000:\\ \;\;\;\;1 + \frac{-1 + \frac{0.5 + \frac{-0.5}{x}}{x}}{x}\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+73}:\\ \;\;\;\;\frac{t\_m}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= x 160000000000.0)
    (+ 1.0 (/ (+ -1.0 (/ (+ 0.5 (/ -0.5 x)) x)) x))
    (if (<= x 4.1e+73) (* (/ t_m l) (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 tmp;
	if (x <= 160000000000.0) {
		tmp = 1.0 + ((-1.0 + ((0.5 + (-0.5 / x)) / x)) / x);
	} else if (x <= 4.1e+73) {
		tmp = (t_m / l) * sqrt(x);
	} else {
		tmp = 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) :: tmp
    if (x <= 160000000000.0d0) then
        tmp = 1.0d0 + (((-1.0d0) + ((0.5d0 + ((-0.5d0) / x)) / x)) / x)
    else if (x <= 4.1d+73) then
        tmp = (t_m / l) * sqrt(x)
    else
        tmp = 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 tmp;
	if (x <= 160000000000.0) {
		tmp = 1.0 + ((-1.0 + ((0.5 + (-0.5 / x)) / x)) / x);
	} else if (x <= 4.1e+73) {
		tmp = (t_m / l) * Math.sqrt(x);
	} else {
		tmp = 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):
	tmp = 0
	if x <= 160000000000.0:
		tmp = 1.0 + ((-1.0 + ((0.5 + (-0.5 / x)) / x)) / x)
	elif x <= 4.1e+73:
		tmp = (t_m / l) * math.sqrt(x)
	else:
		tmp = 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)
	tmp = 0.0
	if (x <= 160000000000.0)
		tmp = Float64(1.0 + Float64(Float64(-1.0 + Float64(Float64(0.5 + Float64(-0.5 / x)) / x)) / x));
	elseif (x <= 4.1e+73)
		tmp = Float64(Float64(t_m / l) * sqrt(x));
	else
		tmp = Float64(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)
	tmp = 0.0;
	if (x <= 160000000000.0)
		tmp = 1.0 + ((-1.0 + ((0.5 + (-0.5 / x)) / x)) / x);
	elseif (x <= 4.1e+73)
		tmp = (t_m / l) * sqrt(x);
	else
		tmp = 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_] := N[(t$95$s * If[LessEqual[x, 160000000000.0], N[(1.0 + N[(N[(-1.0 + N[(N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.1e+73], N[(N[(t$95$m / l), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 1.6e11

   1. Initial program 44.6%

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

   3. Simplified 38.3%

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

   5. Taylor expanded in t around inf 44.7%

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

   6. Taylor expanded in t around 0 44.8%

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

   7. Taylor expanded in x around inf 44.8%

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

   8. Taylor expanded in x around -inf 44.6%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{1 + -1 \cdot \frac{0.5 - 0.5 \cdot \frac{1}{x}}{x}}{x}} \]
   9. Step-by-step derivation

      1. mul-1-neg 44.6%

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

      2. unsub-neg 44.6%

         \[\leadsto \color{blue}{1 - \frac{1 + -1 \cdot \frac{0.5 - 0.5 \cdot \frac{1}{x}}{x}}{x}} \]

      3. mul-1-neg 44.6%

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

      4. unsub-neg 44.6%

         \[\leadsto 1 - \frac{\color{blue}{1 - \frac{0.5 - 0.5 \cdot \frac{1}{x}}{x}}}{x} \]

      5. sub-neg 44.6%

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

      6. associate-*r/ 44.6%

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

      7. metadata-eval 44.6%

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

      8. distribute-neg-frac 44.6%

         \[\leadsto 1 - \frac{1 - \frac{0.5 + \color{blue}{\frac{-0.5}{x}}}{x}}{x} \]

      9. metadata-eval 44.6%

         \[\leadsto 1 - \frac{1 - \frac{0.5 + \frac{\color{blue}{-0.5}}{x}}{x}}{x} \]

   10. Simplified 44.6%

       \[\leadsto \color{blue}{1 - \frac{1 - \frac{0.5 + \frac{-0.5}{x}}{x}}{x}} \]

   if 1.6e11 < x < 4.0999999999999998e73

   1. Initial program 26.9%

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

   3. Simplified 26.9%

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

   5. Taylor expanded in x around inf 66.4%

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

   6. Taylor expanded in l around inf 42.9%

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

   7. Taylor expanded in t around 0 53.4%

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

   if 4.0999999999999998e73 < x

   1. Initial program 28.5%

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

   3. Simplified 25.5%

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

   5. Taylor expanded in t around inf 37.4%

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

   6. Taylor expanded in x around inf 37.5%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 160000000000:\\ \;\;\;\;1 + \frac{-1 + \frac{0.5 + \frac{-0.5}{x}}{x}}{x}\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+73}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.6% accurate, 17.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(1 + \frac{-1 + \frac{0.5 + \frac{-0.5}{x}}{x}}{x}\right) \end{array} \]

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 (/ (+ 0.5 (/ -0.5 x)) x)) 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 + ((0.5 + (-0.5 / x)) / x)) / 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) + ((0.5d0 + ((-0.5d0) / x)) / x)) / 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 + ((0.5 + (-0.5 / x)) / x)) / 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 + ((0.5 + (-0.5 / x)) / x)) / 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(Float64(-1.0 + Float64(Float64(0.5 + Float64(-0.5 / x)) / x)) / 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 + ((0.5 + (-0.5 / x)) / x)) / 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[(N[(-1.0 + N[(N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.5%

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

3. Simplified 31.4%

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

5. Taylor expanded in t around inf 37.2%

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

6. Taylor expanded in t around 0 37.3%

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

7. Taylor expanded in x around inf 37.3%

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

8. Taylor expanded in x around -inf 37.2%

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{1 + -1 \cdot \frac{0.5 - 0.5 \cdot \frac{1}{x}}{x}}{x}} \]
9. Step-by-step derivation

   1. mul-1-neg 37.2%

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

   2. unsub-neg 37.2%

      \[\leadsto \color{blue}{1 - \frac{1 + -1 \cdot \frac{0.5 - 0.5 \cdot \frac{1}{x}}{x}}{x}} \]

   3. mul-1-neg 37.2%

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

   4. unsub-neg 37.2%

      \[\leadsto 1 - \frac{\color{blue}{1 - \frac{0.5 - 0.5 \cdot \frac{1}{x}}{x}}}{x} \]

   5. sub-neg 37.2%

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

   6. associate-*r/ 37.2%

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

   7. metadata-eval 37.2%

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

   8. distribute-neg-frac 37.2%

      \[\leadsto 1 - \frac{1 - \frac{0.5 + \color{blue}{\frac{-0.5}{x}}}{x}}{x} \]

   9. metadata-eval 37.2%

      \[\leadsto 1 - \frac{1 - \frac{0.5 + \frac{\color{blue}{-0.5}}{x}}{x}}{x} \]

10. Simplified 37.2%

    \[\leadsto \color{blue}{1 - \frac{1 - \frac{0.5 + \frac{-0.5}{x}}{x}}{x}} \]

11. Final simplification 37.2%

    \[\leadsto 1 + \frac{-1 + \frac{0.5 + \frac{-0.5}{x}}{x}}{x} \]
12. Add Preprocessing

Alternative 11: 76.5% accurate, 25.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(1 + \frac{-1 - \frac{-0.5}{x}}{x}\right) \end{array} \]

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 (/ -0.5 x)) 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 - (-0.5 / x)) / 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) - ((-0.5d0) / x)) / 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 - (-0.5 / x)) / 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 - (-0.5 / x)) / 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(Float64(-1.0 - Float64(-0.5 / x)) / 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 - (-0.5 / x)) / 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[(N[(-1.0 - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.5%

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

3. Simplified 31.4%

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

5. Taylor expanded in t around inf 37.2%

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

6. Taylor expanded in t around 0 37.3%

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

7. Taylor expanded in x around inf 37.3%

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

8. Taylor expanded in x around -inf 37.2%

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{1 - 0.5 \cdot \frac{1}{x}}{x}} \]
9. Step-by-step derivation

   1. mul-1-neg 37.2%

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

   2. unsub-neg 37.2%

      \[\leadsto \color{blue}{1 - \frac{1 - 0.5 \cdot \frac{1}{x}}{x}} \]

   3. sub-neg 37.2%

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

   4. associate-*r/ 37.2%

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

   5. metadata-eval 37.2%

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

   6. distribute-neg-frac 37.2%

      \[\leadsto 1 - \frac{1 + \color{blue}{\frac{-0.5}{x}}}{x} \]

   7. metadata-eval 37.2%

      \[\leadsto 1 - \frac{1 + \frac{\color{blue}{-0.5}}{x}}{x} \]

10. Simplified 37.2%

    \[\leadsto \color{blue}{1 - \frac{1 + \frac{-0.5}{x}}{x}} \]

11. Final simplification 37.2%

    \[\leadsto 1 + \frac{-1 - \frac{-0.5}{x}}{x} \]
12. Add Preprocessing

Alternative 12: 76.3% 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 35.5%

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

3. Simplified 31.4%

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

5. Taylor expanded in t around inf 37.2%

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

6. Taylor expanded in x around inf 37.1%

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

7. Final simplification 37.1%

   \[\leadsto 1 + \frac{-1}{x} \]
8. Add Preprocessing

Alternative 13: 75.5% 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 35.5%

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

3. Simplified 31.4%

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

5. Taylor expanded in t around inf 37.2%

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

6. Taylor expanded in x around inf 36.7%

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

Reproduce

herbie shell --seed 2024136 
(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)))))