Toniolo and Linder, Equation (7)

Percentage Accurate: 33.7% → 85.6%

Time: 20.3s

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: 33.7% 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: 85.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.4 \cdot 10^{-241}:\\ \;\;\;\;\frac{t\_m \cdot \left(\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{x}\right)}{l\_m}\\ \mathbf{elif}\;t\_m \leq 1.8 \cdot 10^{-165}:\\ \;\;\;\;1 - \frac{1 - \frac{0.5 + \frac{-0.5}{x}}{x}}{x}\\ \mathbf{elif}\;t\_m \leq 3.25 \cdot 10^{+18}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2}{l\_m \cdot \left(l\_m \cdot \frac{2}{x}\right) + \frac{2 \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \left(x + 1\right)\right)}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x + 1}{x + -1}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.4e-241)
    (/ (* t_m (* (* (sqrt 0.5) (sqrt 2.0)) (sqrt x))) l_m)
    (if (<= t_m 1.8e-165)
      (- 1.0 (/ (- 1.0 (/ (+ 0.5 (/ -0.5 x)) x)) x))
      (if (<= t_m 3.25e+18)
        (*
         t_m
         (sqrt
          (/
           2.0
           (+
            (* l_m (* l_m (/ 2.0 x)))
            (/ (* 2.0 (* (* t_m t_m) (+ x 1.0))) (+ x -1.0))))))
        (pow (/ (+ x 1.0) (+ x -1.0)) -0.5))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 2.4e-241) {
		tmp = (t_m * ((sqrt(0.5) * sqrt(2.0)) * sqrt(x))) / l_m;
	} else if (t_m <= 1.8e-165) {
		tmp = 1.0 - ((1.0 - ((0.5 + (-0.5 / x)) / x)) / x);
	} else if (t_m <= 3.25e+18) {
		tmp = t_m * sqrt((2.0 / ((l_m * (l_m * (2.0 / x))) + ((2.0 * ((t_m * t_m) * (x + 1.0))) / (x + -1.0)))));
	} else {
		tmp = pow(((x + 1.0) / (x + -1.0)), -0.5);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 2.4d-241) then
        tmp = (t_m * ((sqrt(0.5d0) * sqrt(2.0d0)) * sqrt(x))) / l_m
    else if (t_m <= 1.8d-165) then
        tmp = 1.0d0 - ((1.0d0 - ((0.5d0 + ((-0.5d0) / x)) / x)) / x)
    else if (t_m <= 3.25d+18) then
        tmp = t_m * sqrt((2.0d0 / ((l_m * (l_m * (2.0d0 / x))) + ((2.0d0 * ((t_m * t_m) * (x + 1.0d0))) / (x + (-1.0d0))))))
    else
        tmp = ((x + 1.0d0) / (x + (-1.0d0))) ** (-0.5d0)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 2.4e-241) {
		tmp = (t_m * ((Math.sqrt(0.5) * Math.sqrt(2.0)) * Math.sqrt(x))) / l_m;
	} else if (t_m <= 1.8e-165) {
		tmp = 1.0 - ((1.0 - ((0.5 + (-0.5 / x)) / x)) / x);
	} else if (t_m <= 3.25e+18) {
		tmp = t_m * Math.sqrt((2.0 / ((l_m * (l_m * (2.0 / x))) + ((2.0 * ((t_m * t_m) * (x + 1.0))) / (x + -1.0)))));
	} else {
		tmp = Math.pow(((x + 1.0) / (x + -1.0)), -0.5);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 2.4e-241:
		tmp = (t_m * ((math.sqrt(0.5) * math.sqrt(2.0)) * math.sqrt(x))) / l_m
	elif t_m <= 1.8e-165:
		tmp = 1.0 - ((1.0 - ((0.5 + (-0.5 / x)) / x)) / x)
	elif t_m <= 3.25e+18:
		tmp = t_m * math.sqrt((2.0 / ((l_m * (l_m * (2.0 / x))) + ((2.0 * ((t_m * t_m) * (x + 1.0))) / (x + -1.0)))))
	else:
		tmp = math.pow(((x + 1.0) / (x + -1.0)), -0.5)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 2.4e-241)
		tmp = Float64(Float64(t_m * Float64(Float64(sqrt(0.5) * sqrt(2.0)) * sqrt(x))) / l_m);
	elseif (t_m <= 1.8e-165)
		tmp = Float64(1.0 - Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.5 / x)) / x)) / x));
	elseif (t_m <= 3.25e+18)
		tmp = Float64(t_m * sqrt(Float64(2.0 / Float64(Float64(l_m * Float64(l_m * Float64(2.0 / x))) + Float64(Float64(2.0 * Float64(Float64(t_m * t_m) * Float64(x + 1.0))) / Float64(x + -1.0))))));
	else
		tmp = Float64(Float64(x + 1.0) / Float64(x + -1.0)) ^ -0.5;
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 2.4e-241)
		tmp = (t_m * ((sqrt(0.5) * sqrt(2.0)) * sqrt(x))) / l_m;
	elseif (t_m <= 1.8e-165)
		tmp = 1.0 - ((1.0 - ((0.5 + (-0.5 / x)) / x)) / x);
	elseif (t_m <= 3.25e+18)
		tmp = t_m * sqrt((2.0 / ((l_m * (l_m * (2.0 / x))) + ((2.0 * ((t_m * t_m) * (x + 1.0))) / (x + -1.0)))));
	else
		tmp = ((x + 1.0) / (x + -1.0)) ^ -0.5;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2.4e-241], N[(N[(t$95$m * N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 1.8e-165], 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[t$95$m, 3.25e+18], N[(t$95$m * N[Sqrt[N[(2.0 / N[(N[(l$95$m * N[(l$95$m * N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 2.4e-241

   1. Initial program 25.1%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)}\right) \]

      4. sqrt-undiv N/A

         \[\leadsto \mathsf{*.f64}\left(t, \left(\sqrt{\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 25.2%

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

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1} + {\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left({\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right) + 2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left({\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{x - 1} + \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\left(\frac{1}{x - 1}\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x - 1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x + -1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\left(\frac{x}{x - 1}\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x - 1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x + -1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, -1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

   7. Simplified 31.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \color{blue}{\left(\frac{2}{x}\right)}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 40.5%

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{/.f64}\left(2, x\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]

   10. Simplified 40.5%

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

   11. Taylor expanded in t around 0

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

       1. associate-*l/ N/A

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

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(t \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{x}\right), \color{blue}{\ell}\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(t \cdot \left(\left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right) \cdot \sqrt{x}\right)\right), \ell\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(t \cdot \left(\sqrt{x} \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right)\right), \ell\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{x} \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right)\right), \ell\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(\left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right) \cdot \sqrt{x}\right)\right), \ell\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right), \left(\sqrt{x}\right)\right)\right), \ell\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}}\right), \left(\sqrt{2}\right)\right), \left(\sqrt{x}\right)\right)\right), \ell\right) \]

       9. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\sqrt{2}\right)\right), \left(\sqrt{x}\right)\right)\right), \ell\right) \]

       10. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{x}\right)\right)\right), \ell\right) \]

       11. sqrt-lowering-sqrt.f64 19.5%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right), \ell\right) \]

   13. Simplified 19.5%

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

   if 2.4e-241 < t < 1.79999999999999992e-165

   1. Initial program 8.7%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 68.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

   5. Simplified 68.5%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. *-inverses N/A

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

      4. pow1/2 N/A

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

      5. pow-flip N/A

         \[\leadsto {\left(\frac{1 + x}{x + -1}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{1 + x}{x + -1}\right)}^{\frac{-1}{2}} \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{1 + x}{x + -1}\right), \color{blue}{\frac{-1}{2}}\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{x + 1}{x + -1}\right), \frac{-1}{2}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(x + 1\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

      12. +-lowering-+.f64 68.5%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right), \frac{-1}{2}\right) \]

   7. Applied egg-rr 68.5%

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

   8. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

         \[\leadsto 1 + \left(\mathsf{neg}\left(\frac{1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}}{x}\right)\right) \]

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}}{x}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right), \color{blue}{x}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right)\right), x\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(1 - \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right), x\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right), x\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}\right), x\right)\right), x\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right)\right), x\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right)\right), x\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right)\right), x\right)\right), x\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{x}\right)\right)\right), x\right)\right), x\right)\right) \]

      13. distribute-neg-frac N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}\right)\right), x\right)\right), x\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{-1}{2}}{x}\right)\right), x\right)\right), x\right)\right) \]

      15. /-lowering-/.f64 68.5%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), x\right)\right), x\right)\right) \]

   10. Simplified 68.5%

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

   if 1.79999999999999992e-165 < t < 3.25e18

   1. Initial program 58.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)}\right) \]

      4. sqrt-undiv N/A

         \[\leadsto \mathsf{*.f64}\left(t, \left(\sqrt{\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 58.4%

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

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1} + {\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left({\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right) + 2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left({\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{x - 1} + \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\left(\frac{1}{x - 1}\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x - 1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x + -1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\left(\frac{x}{x - 1}\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x - 1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x + -1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, -1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

   7. Simplified 75.2%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \color{blue}{\left(\frac{2}{x}\right)}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 89.1%

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{/.f64}\left(2, x\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]

   10. Simplified 89.1%

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

       1. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(\ell \cdot \left(\ell \cdot \frac{2}{x}\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(\left(\ell \cdot \frac{2}{x}\right) \cdot \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \frac{2}{x}\right), \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{2}{x}\right)\right), \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]

       5. /-lowering-/.f64 95.8%

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(2, x\right)\right), \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 95.8%

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

   if 3.25e18 < t

   1. Initial program 34.7%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 98.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

   5. Simplified 98.7%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. *-inverses N/A

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

      4. pow1/2 N/A

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

      5. pow-flip N/A

         \[\leadsto {\left(\frac{1 + x}{x + -1}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{1 + x}{x + -1}\right)}^{\frac{-1}{2}} \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{1 + x}{x + -1}\right), \color{blue}{\frac{-1}{2}}\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{x + 1}{x + -1}\right), \frac{-1}{2}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(x + 1\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

      12. +-lowering-+.f64 98.7%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right), \frac{-1}{2}\right) \]

   7. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{{\left(\frac{1 + x}{x + -1}\right)}^{-0.5}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-241}:\\ \;\;\;\;\frac{t \cdot \left(\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{x}\right)}{\ell}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-165}:\\ \;\;\;\;1 - \frac{1 - \frac{0.5 + \frac{-0.5}{x}}{x}}{x}\\ \mathbf{elif}\;t \leq 3.25 \cdot 10^{+18}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\ell \cdot \left(\ell \cdot \frac{2}{x}\right) + \frac{2 \cdot \left(\left(t \cdot t\right) \cdot \left(x + 1\right)\right)}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x + 1}{x + -1}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.1 \cdot 10^{-239}:\\ \;\;\;\;t\_m \cdot \frac{\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{x}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 8 \cdot 10^{-165}:\\ \;\;\;\;1 - \frac{1 - \frac{0.5 + \frac{-0.5}{x}}{x}}{x}\\ \mathbf{elif}\;t\_m \leq 3.2 \cdot 10^{+18}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2}{l\_m \cdot \left(l\_m \cdot \frac{2}{x}\right) + \frac{2 \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \left(x + 1\right)\right)}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x + 1}{x + -1}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.1e-239)
    (* t_m (/ (* (* (sqrt 0.5) (sqrt 2.0)) (sqrt x)) l_m))
    (if (<= t_m 8e-165)
      (- 1.0 (/ (- 1.0 (/ (+ 0.5 (/ -0.5 x)) x)) x))
      (if (<= t_m 3.2e+18)
        (*
         t_m
         (sqrt
          (/
           2.0
           (+
            (* l_m (* l_m (/ 2.0 x)))
            (/ (* 2.0 (* (* t_m t_m) (+ x 1.0))) (+ x -1.0))))))
        (pow (/ (+ x 1.0) (+ x -1.0)) -0.5))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.1e-239) {
		tmp = t_m * (((sqrt(0.5) * sqrt(2.0)) * sqrt(x)) / l_m);
	} else if (t_m <= 8e-165) {
		tmp = 1.0 - ((1.0 - ((0.5 + (-0.5 / x)) / x)) / x);
	} else if (t_m <= 3.2e+18) {
		tmp = t_m * sqrt((2.0 / ((l_m * (l_m * (2.0 / x))) + ((2.0 * ((t_m * t_m) * (x + 1.0))) / (x + -1.0)))));
	} else {
		tmp = pow(((x + 1.0) / (x + -1.0)), -0.5);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1.1d-239) then
        tmp = t_m * (((sqrt(0.5d0) * sqrt(2.0d0)) * sqrt(x)) / l_m)
    else if (t_m <= 8d-165) then
        tmp = 1.0d0 - ((1.0d0 - ((0.5d0 + ((-0.5d0) / x)) / x)) / x)
    else if (t_m <= 3.2d+18) then
        tmp = t_m * sqrt((2.0d0 / ((l_m * (l_m * (2.0d0 / x))) + ((2.0d0 * ((t_m * t_m) * (x + 1.0d0))) / (x + (-1.0d0))))))
    else
        tmp = ((x + 1.0d0) / (x + (-1.0d0))) ** (-0.5d0)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.1e-239) {
		tmp = t_m * (((Math.sqrt(0.5) * Math.sqrt(2.0)) * Math.sqrt(x)) / l_m);
	} else if (t_m <= 8e-165) {
		tmp = 1.0 - ((1.0 - ((0.5 + (-0.5 / x)) / x)) / x);
	} else if (t_m <= 3.2e+18) {
		tmp = t_m * Math.sqrt((2.0 / ((l_m * (l_m * (2.0 / x))) + ((2.0 * ((t_m * t_m) * (x + 1.0))) / (x + -1.0)))));
	} else {
		tmp = Math.pow(((x + 1.0) / (x + -1.0)), -0.5);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 1.1e-239:
		tmp = t_m * (((math.sqrt(0.5) * math.sqrt(2.0)) * math.sqrt(x)) / l_m)
	elif t_m <= 8e-165:
		tmp = 1.0 - ((1.0 - ((0.5 + (-0.5 / x)) / x)) / x)
	elif t_m <= 3.2e+18:
		tmp = t_m * math.sqrt((2.0 / ((l_m * (l_m * (2.0 / x))) + ((2.0 * ((t_m * t_m) * (x + 1.0))) / (x + -1.0)))))
	else:
		tmp = math.pow(((x + 1.0) / (x + -1.0)), -0.5)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 1.1e-239)
		tmp = Float64(t_m * Float64(Float64(Float64(sqrt(0.5) * sqrt(2.0)) * sqrt(x)) / l_m));
	elseif (t_m <= 8e-165)
		tmp = Float64(1.0 - Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.5 / x)) / x)) / x));
	elseif (t_m <= 3.2e+18)
		tmp = Float64(t_m * sqrt(Float64(2.0 / Float64(Float64(l_m * Float64(l_m * Float64(2.0 / x))) + Float64(Float64(2.0 * Float64(Float64(t_m * t_m) * Float64(x + 1.0))) / Float64(x + -1.0))))));
	else
		tmp = Float64(Float64(x + 1.0) / Float64(x + -1.0)) ^ -0.5;
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 1.1e-239)
		tmp = t_m * (((sqrt(0.5) * sqrt(2.0)) * sqrt(x)) / l_m);
	elseif (t_m <= 8e-165)
		tmp = 1.0 - ((1.0 - ((0.5 + (-0.5 / x)) / x)) / x);
	elseif (t_m <= 3.2e+18)
		tmp = t_m * sqrt((2.0 / ((l_m * (l_m * (2.0 / x))) + ((2.0 * ((t_m * t_m) * (x + 1.0))) / (x + -1.0)))));
	else
		tmp = ((x + 1.0) / (x + -1.0)) ^ -0.5;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.1e-239], N[(t$95$m * N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8e-165], 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[t$95$m, 3.2e+18], N[(t$95$m * N[Sqrt[N[(2.0 / N[(N[(l$95$m * N[(l$95$m * N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 1.09999999999999991e-239

   1. Initial program 25.1%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)}\right) \]

      4. sqrt-undiv N/A

         \[\leadsto \mathsf{*.f64}\left(t, \left(\sqrt{\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 25.2%

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

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1} + {\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left({\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right) + 2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left({\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{x - 1} + \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\left(\frac{1}{x - 1}\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x - 1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x + -1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\left(\frac{x}{x - 1}\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x - 1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x + -1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, -1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

   7. Simplified 31.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \color{blue}{\left(\frac{2}{x}\right)}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 40.5%

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{/.f64}\left(2, x\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]

   10. Simplified 40.5%

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

   11. Taylor expanded in l around inf

       \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{\sqrt{\frac{1}{2}} \cdot \sqrt{2}}{\ell} \cdot \sqrt{x}\right)}\right) \]
   12. Step-by-step derivation

       1. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(t, \left(\frac{\left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right) \cdot \sqrt{x}}{\color{blue}{\ell}}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(t, \left(\frac{\sqrt{x} \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}{\ell}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\left(\sqrt{x} \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right), \color{blue}{\ell}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\left(\left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right) \cdot \sqrt{x}\right), \ell\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right), \left(\sqrt{x}\right)\right), \ell\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}}\right), \left(\sqrt{2}\right)\right), \left(\sqrt{x}\right)\right), \ell\right)\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\sqrt{2}\right)\right), \left(\sqrt{x}\right)\right), \ell\right)\right) \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{x}\right)\right), \ell\right)\right) \]

       9. sqrt-lowering-sqrt.f64 19.5%

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(x\right)\right), \ell\right)\right) \]

   13. Simplified 19.5%

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

   if 1.09999999999999991e-239 < t < 8.0000000000000001e-165

   1. Initial program 8.7%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 68.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

   5. Simplified 68.5%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. *-inverses N/A

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

      4. pow1/2 N/A

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

      5. pow-flip N/A

         \[\leadsto {\left(\frac{1 + x}{x + -1}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{1 + x}{x + -1}\right)}^{\frac{-1}{2}} \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{1 + x}{x + -1}\right), \color{blue}{\frac{-1}{2}}\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{x + 1}{x + -1}\right), \frac{-1}{2}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(x + 1\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

      12. +-lowering-+.f64 68.5%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right), \frac{-1}{2}\right) \]

   7. Applied egg-rr 68.5%

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

   8. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

         \[\leadsto 1 + \left(\mathsf{neg}\left(\frac{1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}}{x}\right)\right) \]

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}}{x}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right), \color{blue}{x}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right)\right), x\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(1 - \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right), x\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right), x\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}\right), x\right)\right), x\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right)\right), x\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right)\right), x\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right)\right), x\right)\right), x\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{x}\right)\right)\right), x\right)\right), x\right)\right) \]

      13. distribute-neg-frac N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}\right)\right), x\right)\right), x\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{-1}{2}}{x}\right)\right), x\right)\right), x\right)\right) \]

      15. /-lowering-/.f64 68.5%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), x\right)\right), x\right)\right) \]

   10. Simplified 68.5%

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

   if 8.0000000000000001e-165 < t < 3.2e18

   1. Initial program 58.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)}\right) \]

      4. sqrt-undiv N/A

         \[\leadsto \mathsf{*.f64}\left(t, \left(\sqrt{\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 58.4%

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

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1} + {\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left({\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right) + 2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left({\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{x - 1} + \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\left(\frac{1}{x - 1}\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x - 1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x + -1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\left(\frac{x}{x - 1}\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x - 1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x + -1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, -1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

   7. Simplified 75.2%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \color{blue}{\left(\frac{2}{x}\right)}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 89.1%

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{/.f64}\left(2, x\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]

   10. Simplified 89.1%

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

       1. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(\ell \cdot \left(\ell \cdot \frac{2}{x}\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(\left(\ell \cdot \frac{2}{x}\right) \cdot \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \frac{2}{x}\right), \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{2}{x}\right)\right), \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]

       5. /-lowering-/.f64 95.8%

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(2, x\right)\right), \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 95.8%

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

   if 3.2e18 < t

   1. Initial program 34.7%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 98.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

   5. Simplified 98.7%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. *-inverses N/A

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

      4. pow1/2 N/A

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

      5. pow-flip N/A

         \[\leadsto {\left(\frac{1 + x}{x + -1}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{1 + x}{x + -1}\right)}^{\frac{-1}{2}} \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{1 + x}{x + -1}\right), \color{blue}{\frac{-1}{2}}\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{x + 1}{x + -1}\right), \frac{-1}{2}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(x + 1\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

      12. +-lowering-+.f64 98.7%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right), \frac{-1}{2}\right) \]

   7. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{{\left(\frac{1 + x}{x + -1}\right)}^{-0.5}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-239}:\\ \;\;\;\;t \cdot \frac{\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-165}:\\ \;\;\;\;1 - \frac{1 - \frac{0.5 + \frac{-0.5}{x}}{x}}{x}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+18}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\ell \cdot \left(\ell \cdot \frac{2}{x}\right) + \frac{2 \cdot \left(\left(t \cdot t\right) \cdot \left(x + 1\right)\right)}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x + 1}{x + -1}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot \left(t\_m \cdot t\_m\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8 \cdot 10^{-288}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{x}{l\_m \cdot l\_m}}\\ \mathbf{elif}\;t\_m \leq 2.8 \cdot 10^{-165}:\\ \;\;\;\;1 - \frac{1 - \frac{0.5 + \frac{-0.5}{x}}{x}}{x}\\ \mathbf{elif}\;t\_m \leq 4.6 \cdot 10^{+57}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2}{t\_2 - \frac{-2 \cdot \left(l\_m \cdot l\_m + t\_2\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x + 1}{x + -1}\right)}^{-0.5}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (* t_m t_m))))
   (*
    t_s
    (if (<= t_m 8e-288)
      (* t_m (sqrt (/ x (* l_m l_m))))
      (if (<= t_m 2.8e-165)
        (- 1.0 (/ (- 1.0 (/ (+ 0.5 (/ -0.5 x)) x)) x))
        (if (<= t_m 4.6e+57)
          (* t_m (sqrt (/ 2.0 (- t_2 (/ (* -2.0 (+ (* l_m l_m) t_2)) x)))))
          (pow (/ (+ x 1.0) (+ x -1.0)) -0.5)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * (t_m * t_m);
	double tmp;
	if (t_m <= 8e-288) {
		tmp = t_m * sqrt((x / (l_m * l_m)));
	} else if (t_m <= 2.8e-165) {
		tmp = 1.0 - ((1.0 - ((0.5 + (-0.5 / x)) / x)) / x);
	} else if (t_m <= 4.6e+57) {
		tmp = t_m * sqrt((2.0 / (t_2 - ((-2.0 * ((l_m * l_m) + t_2)) / x))));
	} else {
		tmp = pow(((x + 1.0) / (x + -1.0)), -0.5);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m * t_m)
    if (t_m <= 8d-288) then
        tmp = t_m * sqrt((x / (l_m * l_m)))
    else if (t_m <= 2.8d-165) then
        tmp = 1.0d0 - ((1.0d0 - ((0.5d0 + ((-0.5d0) / x)) / x)) / x)
    else if (t_m <= 4.6d+57) then
        tmp = t_m * sqrt((2.0d0 / (t_2 - (((-2.0d0) * ((l_m * l_m) + t_2)) / x))))
    else
        tmp = ((x + 1.0d0) / (x + (-1.0d0))) ** (-0.5d0)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * (t_m * t_m);
	double tmp;
	if (t_m <= 8e-288) {
		tmp = t_m * Math.sqrt((x / (l_m * l_m)));
	} else if (t_m <= 2.8e-165) {
		tmp = 1.0 - ((1.0 - ((0.5 + (-0.5 / x)) / x)) / x);
	} else if (t_m <= 4.6e+57) {
		tmp = t_m * Math.sqrt((2.0 / (t_2 - ((-2.0 * ((l_m * l_m) + t_2)) / x))));
	} else {
		tmp = Math.pow(((x + 1.0) / (x + -1.0)), -0.5);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = 2.0 * (t_m * t_m)
	tmp = 0
	if t_m <= 8e-288:
		tmp = t_m * math.sqrt((x / (l_m * l_m)))
	elif t_m <= 2.8e-165:
		tmp = 1.0 - ((1.0 - ((0.5 + (-0.5 / x)) / x)) / x)
	elif t_m <= 4.6e+57:
		tmp = t_m * math.sqrt((2.0 / (t_2 - ((-2.0 * ((l_m * l_m) + t_2)) / x))))
	else:
		tmp = math.pow(((x + 1.0) / (x + -1.0)), -0.5)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(2.0 * Float64(t_m * t_m))
	tmp = 0.0
	if (t_m <= 8e-288)
		tmp = Float64(t_m * sqrt(Float64(x / Float64(l_m * l_m))));
	elseif (t_m <= 2.8e-165)
		tmp = Float64(1.0 - Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.5 / x)) / x)) / x));
	elseif (t_m <= 4.6e+57)
		tmp = Float64(t_m * sqrt(Float64(2.0 / Float64(t_2 - Float64(Float64(-2.0 * Float64(Float64(l_m * l_m) + t_2)) / x)))));
	else
		tmp = Float64(Float64(x + 1.0) / Float64(x + -1.0)) ^ -0.5;
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = 2.0 * (t_m * t_m);
	tmp = 0.0;
	if (t_m <= 8e-288)
		tmp = t_m * sqrt((x / (l_m * l_m)));
	elseif (t_m <= 2.8e-165)
		tmp = 1.0 - ((1.0 - ((0.5 + (-0.5 / x)) / x)) / x);
	elseif (t_m <= 4.6e+57)
		tmp = t_m * sqrt((2.0 / (t_2 - ((-2.0 * ((l_m * l_m) + t_2)) / x))));
	else
		tmp = ((x + 1.0) / (x + -1.0)) ^ -0.5;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 8e-288], N[(t$95$m * N[Sqrt[N[(x / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.8e-165], 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[t$95$m, 4.6e+57], N[(t$95$m * N[Sqrt[N[(2.0 / N[(t$95$2 - N[(N[(-2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 8.00000000000000046e-288

   1. Initial program 26.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)}\right) \]

      4. sqrt-undiv N/A

         \[\leadsto \mathsf{*.f64}\left(t, \left(\sqrt{\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 26.5%

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

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1} + {\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left({\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right) + 2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left({\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{x - 1} + \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\left(\frac{1}{x - 1}\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x - 1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x + -1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\left(\frac{x}{x - 1}\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x - 1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x + -1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, -1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

   7. Simplified 33.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \color{blue}{\left(\frac{2}{x}\right)}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 41.0%

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{/.f64}\left(2, x\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]

   10. Simplified 41.0%

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

   11. Taylor expanded in l around inf

       \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{x}{{\ell}^{2}}\right)}\right)\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(x, \left({\ell}^{2}\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(x, \left(\ell \cdot \ell\right)\right)\right)\right) \]

       3. *-lowering-*.f64 18.0%

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right) \]

   13. Simplified 18.0%

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

   if 8.00000000000000046e-288 < t < 2.7999999999999999e-165

   1. Initial program 6.3%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 55.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

   5. Simplified 55.0%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. *-inverses N/A

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

      4. pow1/2 N/A

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

      5. pow-flip N/A

         \[\leadsto {\left(\frac{1 + x}{x + -1}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{1 + x}{x + -1}\right)}^{\frac{-1}{2}} \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{1 + x}{x + -1}\right), \color{blue}{\frac{-1}{2}}\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{x + 1}{x + -1}\right), \frac{-1}{2}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(x + 1\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

      12. +-lowering-+.f64 55.0%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right), \frac{-1}{2}\right) \]

   7. Applied egg-rr 55.0%

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

   8. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

         \[\leadsto 1 + \left(\mathsf{neg}\left(\frac{1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}}{x}\right)\right) \]

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}}{x}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right), \color{blue}{x}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right)\right), x\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(1 - \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right), x\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right), x\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}\right), x\right)\right), x\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right)\right), x\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right)\right), x\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right)\right), x\right)\right), x\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{x}\right)\right)\right), x\right)\right), x\right)\right) \]

      13. distribute-neg-frac N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}\right)\right), x\right)\right), x\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{-1}{2}}{x}\right)\right), x\right)\right), x\right)\right) \]

      15. /-lowering-/.f64 55.0%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), x\right)\right), x\right)\right) \]

   10. Simplified 55.0%

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

   if 2.7999999999999999e-165 < t < 4.5999999999999998e57

   1. Initial program 62.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)}\right) \]

      4. sqrt-undiv N/A

         \[\leadsto \mathsf{*.f64}\left(t, \left(\sqrt{\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 62.7%

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

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1} + {\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left({\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right) + 2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left({\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{x - 1} + \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\left(\frac{1}{x - 1}\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x - 1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x + -1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\left(\frac{x}{x - 1}\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x - 1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x + -1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, -1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

   7. Simplified 74.3%

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

   8. Taylor expanded in x around -inf

      \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\left(-1 \cdot \frac{-2 \cdot \left({t}^{2} - -1 \cdot {t}^{2}\right) + -2 \cdot {\ell}^{2}}{x} + 2 \cdot {t}^{2}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(2 \cdot {t}^{2} + -1 \cdot \frac{-2 \cdot \left({t}^{2} - -1 \cdot {t}^{2}\right) + -2 \cdot {\ell}^{2}}{x}\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(2 \cdot {t}^{2} + \left(\mathsf{neg}\left(\frac{-2 \cdot \left({t}^{2} - -1 \cdot {t}^{2}\right) + -2 \cdot {\ell}^{2}}{x}\right)\right)\right)\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(2 \cdot {t}^{2} - \frac{-2 \cdot \left({t}^{2} - -1 \cdot {t}^{2}\right) + -2 \cdot {\ell}^{2}}{x}\right)\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\left(2 \cdot {t}^{2}\right), \left(\frac{-2 \cdot \left({t}^{2} - -1 \cdot {t}^{2}\right) + -2 \cdot {\ell}^{2}}{x}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(2, \left({t}^{2}\right)\right), \left(\frac{-2 \cdot \left({t}^{2} - -1 \cdot {t}^{2}\right) + -2 \cdot {\ell}^{2}}{x}\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(2, \left(t \cdot t\right)\right), \left(\frac{-2 \cdot \left({t}^{2} - -1 \cdot {t}^{2}\right) + -2 \cdot {\ell}^{2}}{x}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{-2 \cdot \left({t}^{2} - -1 \cdot {t}^{2}\right) + -2 \cdot {\ell}^{2}}{x}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(-2 \cdot \left({t}^{2} - -1 \cdot {t}^{2}\right) + -2 \cdot {\ell}^{2}\right), x\right)\right)\right)\right)\right) \]

   10. Simplified 89.2%

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

   if 4.5999999999999998e57 < t

   1. Initial program 27.4%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

   5. Simplified 100.0%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. *-inverses N/A

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

      4. pow1/2 N/A

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

      5. pow-flip N/A

         \[\leadsto {\left(\frac{1 + x}{x + -1}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{1 + x}{x + -1}\right)}^{\frac{-1}{2}} \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{1 + x}{x + -1}\right), \color{blue}{\frac{-1}{2}}\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{x + 1}{x + -1}\right), \frac{-1}{2}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(x + 1\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

      12. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right), \frac{-1}{2}\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(\frac{1 + x}{x + -1}\right)}^{-0.5}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{-288}:\\ \;\;\;\;t \cdot \sqrt{\frac{x}{\ell \cdot \ell}}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-165}:\\ \;\;\;\;1 - \frac{1 - \frac{0.5 + \frac{-0.5}{x}}{x}}{x}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{+57}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{2 \cdot \left(t \cdot t\right) - \frac{-2 \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x + 1}{x + -1}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.35 \cdot 10^{-287}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{x}{l\_m \cdot l\_m}}\\ \mathbf{elif}\;t\_m \leq 8 \cdot 10^{-165}:\\ \;\;\;\;1 - \frac{1 - \frac{0.5 + \frac{-0.5}{x}}{x}}{x}\\ \mathbf{elif}\;t\_m \leq 2.5 \cdot 10^{+57}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2}{2 \cdot \left(t\_m \cdot t\_m\right) + \frac{2}{x} \cdot \left(l\_m \cdot l\_m\right)}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x + 1}{x + -1}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.35e-287)
    (* t_m (sqrt (/ x (* l_m l_m))))
    (if (<= t_m 8e-165)
      (- 1.0 (/ (- 1.0 (/ (+ 0.5 (/ -0.5 x)) x)) x))
      (if (<= t_m 2.5e+57)
        (*
         t_m
         (sqrt (/ 2.0 (+ (* 2.0 (* t_m t_m)) (* (/ 2.0 x) (* l_m l_m))))))
        (pow (/ (+ x 1.0) (+ x -1.0)) -0.5))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 2.35e-287) {
		tmp = t_m * sqrt((x / (l_m * l_m)));
	} else if (t_m <= 8e-165) {
		tmp = 1.0 - ((1.0 - ((0.5 + (-0.5 / x)) / x)) / x);
	} else if (t_m <= 2.5e+57) {
		tmp = t_m * sqrt((2.0 / ((2.0 * (t_m * t_m)) + ((2.0 / x) * (l_m * l_m)))));
	} else {
		tmp = pow(((x + 1.0) / (x + -1.0)), -0.5);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 2.35d-287) then
        tmp = t_m * sqrt((x / (l_m * l_m)))
    else if (t_m <= 8d-165) then
        tmp = 1.0d0 - ((1.0d0 - ((0.5d0 + ((-0.5d0) / x)) / x)) / x)
    else if (t_m <= 2.5d+57) then
        tmp = t_m * sqrt((2.0d0 / ((2.0d0 * (t_m * t_m)) + ((2.0d0 / x) * (l_m * l_m)))))
    else
        tmp = ((x + 1.0d0) / (x + (-1.0d0))) ** (-0.5d0)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 2.35e-287) {
		tmp = t_m * Math.sqrt((x / (l_m * l_m)));
	} else if (t_m <= 8e-165) {
		tmp = 1.0 - ((1.0 - ((0.5 + (-0.5 / x)) / x)) / x);
	} else if (t_m <= 2.5e+57) {
		tmp = t_m * Math.sqrt((2.0 / ((2.0 * (t_m * t_m)) + ((2.0 / x) * (l_m * l_m)))));
	} else {
		tmp = Math.pow(((x + 1.0) / (x + -1.0)), -0.5);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 2.35e-287:
		tmp = t_m * math.sqrt((x / (l_m * l_m)))
	elif t_m <= 8e-165:
		tmp = 1.0 - ((1.0 - ((0.5 + (-0.5 / x)) / x)) / x)
	elif t_m <= 2.5e+57:
		tmp = t_m * math.sqrt((2.0 / ((2.0 * (t_m * t_m)) + ((2.0 / x) * (l_m * l_m)))))
	else:
		tmp = math.pow(((x + 1.0) / (x + -1.0)), -0.5)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 2.35e-287)
		tmp = Float64(t_m * sqrt(Float64(x / Float64(l_m * l_m))));
	elseif (t_m <= 8e-165)
		tmp = Float64(1.0 - Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.5 / x)) / x)) / x));
	elseif (t_m <= 2.5e+57)
		tmp = Float64(t_m * sqrt(Float64(2.0 / Float64(Float64(2.0 * Float64(t_m * t_m)) + Float64(Float64(2.0 / x) * Float64(l_m * l_m))))));
	else
		tmp = Float64(Float64(x + 1.0) / Float64(x + -1.0)) ^ -0.5;
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 2.35e-287)
		tmp = t_m * sqrt((x / (l_m * l_m)));
	elseif (t_m <= 8e-165)
		tmp = 1.0 - ((1.0 - ((0.5 + (-0.5 / x)) / x)) / x);
	elseif (t_m <= 2.5e+57)
		tmp = t_m * sqrt((2.0 / ((2.0 * (t_m * t_m)) + ((2.0 / x) * (l_m * l_m)))));
	else
		tmp = ((x + 1.0) / (x + -1.0)) ^ -0.5;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2.35e-287], N[(t$95$m * N[Sqrt[N[(x / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8e-165], 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[t$95$m, 2.5e+57], N[(t$95$m * N[Sqrt[N[(2.0 / N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / x), $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 2.3499999999999999e-287

   1. Initial program 26.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)}\right) \]

      4. sqrt-undiv N/A

         \[\leadsto \mathsf{*.f64}\left(t, \left(\sqrt{\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 26.5%

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

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1} + {\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left({\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right) + 2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left({\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{x - 1} + \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\left(\frac{1}{x - 1}\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x - 1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x + -1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\left(\frac{x}{x - 1}\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x - 1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x + -1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, -1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

   7. Simplified 33.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \color{blue}{\left(\frac{2}{x}\right)}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 41.0%

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{/.f64}\left(2, x\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]

   10. Simplified 41.0%

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

   11. Taylor expanded in l around inf

       \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{x}{{\ell}^{2}}\right)}\right)\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(x, \left({\ell}^{2}\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(x, \left(\ell \cdot \ell\right)\right)\right)\right) \]

       3. *-lowering-*.f64 18.0%

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right) \]

   13. Simplified 18.0%

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

   if 2.3499999999999999e-287 < t < 8.0000000000000001e-165

   1. Initial program 6.3%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 55.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

   5. Simplified 55.0%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. *-inverses N/A

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

      4. pow1/2 N/A

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

      5. pow-flip N/A

         \[\leadsto {\left(\frac{1 + x}{x + -1}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{1 + x}{x + -1}\right)}^{\frac{-1}{2}} \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{1 + x}{x + -1}\right), \color{blue}{\frac{-1}{2}}\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{x + 1}{x + -1}\right), \frac{-1}{2}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(x + 1\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

      12. +-lowering-+.f64 55.0%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right), \frac{-1}{2}\right) \]

   7. Applied egg-rr 55.0%

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

   8. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

         \[\leadsto 1 + \left(\mathsf{neg}\left(\frac{1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}}{x}\right)\right) \]

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}}{x}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right), \color{blue}{x}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right)\right), x\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(1 - \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right), x\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right), x\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}\right), x\right)\right), x\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right)\right), x\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right)\right), x\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right)\right), x\right)\right), x\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{x}\right)\right)\right), x\right)\right), x\right)\right) \]

      13. distribute-neg-frac N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}\right)\right), x\right)\right), x\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{-1}{2}}{x}\right)\right), x\right)\right), x\right)\right) \]

      15. /-lowering-/.f64 55.0%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), x\right)\right), x\right)\right) \]

   10. Simplified 55.0%

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

   if 8.0000000000000001e-165 < t < 2.49999999999999986e57

   1. Initial program 62.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)}\right) \]

      4. sqrt-undiv N/A

         \[\leadsto \mathsf{*.f64}\left(t, \left(\sqrt{\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 62.7%

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

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1} + {\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left({\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right) + 2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left({\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{x - 1} + \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\left(\frac{1}{x - 1}\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x - 1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x + -1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\left(\frac{x}{x - 1}\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x - 1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x + -1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, -1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

   7. Simplified 74.3%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \color{blue}{\left(\frac{2}{x}\right)}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 85.4%

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{/.f64}\left(2, x\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]

   10. Simplified 85.4%

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

   11. Taylor expanded in x around inf

       \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{/.f64}\left(2, x\right)\right), \color{blue}{\left(2 \cdot {t}^{2}\right)}\right)\right)\right)\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{/.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(2, \left({t}^{2}\right)\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{/.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(2, \left(t \cdot t\right)\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 89.2%

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{/.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right)\right) \]

   13. Simplified 89.2%

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

   if 2.49999999999999986e57 < t

   1. Initial program 27.4%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

   5. Simplified 100.0%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. *-inverses N/A

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

      4. pow1/2 N/A

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

      5. pow-flip N/A

         \[\leadsto {\left(\frac{1 + x}{x + -1}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{1 + x}{x + -1}\right)}^{\frac{-1}{2}} \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{1 + x}{x + -1}\right), \color{blue}{\frac{-1}{2}}\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{x + 1}{x + -1}\right), \frac{-1}{2}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(x + 1\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

      12. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right), \frac{-1}{2}\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(\frac{1 + x}{x + -1}\right)}^{-0.5}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.35 \cdot 10^{-287}:\\ \;\;\;\;t \cdot \sqrt{\frac{x}{\ell \cdot \ell}}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-165}:\\ \;\;\;\;1 - \frac{1 - \frac{0.5 + \frac{-0.5}{x}}{x}}{x}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+57}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{2 \cdot \left(t \cdot t\right) + \frac{2}{x} \cdot \left(\ell \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x + 1}{x + -1}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.7 \cdot 10^{+18}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2}{l\_m \cdot \left(l\_m \cdot \frac{2}{x}\right) + \frac{2 \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \left(x + 1\right)\right)}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x + 1}{x + -1}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.7e+18)
    (*
     t_m
     (sqrt
      (/
       2.0
       (+
        (* l_m (* l_m (/ 2.0 x)))
        (/ (* 2.0 (* (* t_m t_m) (+ x 1.0))) (+ x -1.0))))))
    (pow (/ (+ x 1.0) (+ x -1.0)) -0.5))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 6.7e+18) {
		tmp = t_m * sqrt((2.0 / ((l_m * (l_m * (2.0 / x))) + ((2.0 * ((t_m * t_m) * (x + 1.0))) / (x + -1.0)))));
	} else {
		tmp = pow(((x + 1.0) / (x + -1.0)), -0.5);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 6.7d+18) then
        tmp = t_m * sqrt((2.0d0 / ((l_m * (l_m * (2.0d0 / x))) + ((2.0d0 * ((t_m * t_m) * (x + 1.0d0))) / (x + (-1.0d0))))))
    else
        tmp = ((x + 1.0d0) / (x + (-1.0d0))) ** (-0.5d0)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 6.7e+18) {
		tmp = t_m * Math.sqrt((2.0 / ((l_m * (l_m * (2.0 / x))) + ((2.0 * ((t_m * t_m) * (x + 1.0))) / (x + -1.0)))));
	} else {
		tmp = Math.pow(((x + 1.0) / (x + -1.0)), -0.5);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 6.7e+18:
		tmp = t_m * math.sqrt((2.0 / ((l_m * (l_m * (2.0 / x))) + ((2.0 * ((t_m * t_m) * (x + 1.0))) / (x + -1.0)))))
	else:
		tmp = math.pow(((x + 1.0) / (x + -1.0)), -0.5)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 6.7e+18)
		tmp = Float64(t_m * sqrt(Float64(2.0 / Float64(Float64(l_m * Float64(l_m * Float64(2.0 / x))) + Float64(Float64(2.0 * Float64(Float64(t_m * t_m) * Float64(x + 1.0))) / Float64(x + -1.0))))));
	else
		tmp = Float64(Float64(x + 1.0) / Float64(x + -1.0)) ^ -0.5;
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 6.7e+18)
		tmp = t_m * sqrt((2.0 / ((l_m * (l_m * (2.0 / x))) + ((2.0 * ((t_m * t_m) * (x + 1.0))) / (x + -1.0)))));
	else
		tmp = ((x + 1.0) / (x + -1.0)) ^ -0.5;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 6.7e+18], N[(t$95$m * N[Sqrt[N[(2.0 / N[(N[(l$95$m * N[(l$95$m * N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.7e18

   1. Initial program 31.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)}\right) \]

      4. sqrt-undiv N/A

         \[\leadsto \mathsf{*.f64}\left(t, \left(\sqrt{\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 31.5%

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

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1} + {\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left({\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right) + 2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left({\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{x - 1} + \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\left(\frac{1}{x - 1}\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x - 1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x + -1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\left(\frac{x}{x - 1}\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x - 1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x + -1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, -1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

   7. Simplified 40.2%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \color{blue}{\left(\frac{2}{x}\right)}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 50.2%

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{/.f64}\left(2, x\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]

   10. Simplified 50.2%

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

       1. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(\ell \cdot \left(\ell \cdot \frac{2}{x}\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(\left(\ell \cdot \frac{2}{x}\right) \cdot \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \frac{2}{x}\right), \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{2}{x}\right)\right), \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]

       5. /-lowering-/.f64 54.6%

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(2, x\right)\right), \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 54.6%

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

   if 6.7e18 < t

   1. Initial program 34.7%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 98.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

   5. Simplified 98.7%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. *-inverses N/A

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

      4. pow1/2 N/A

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

      5. pow-flip N/A

         \[\leadsto {\left(\frac{1 + x}{x + -1}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{1 + x}{x + -1}\right)}^{\frac{-1}{2}} \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{1 + x}{x + -1}\right), \color{blue}{\frac{-1}{2}}\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{x + 1}{x + -1}\right), \frac{-1}{2}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(x + 1\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

      12. +-lowering-+.f64 98.7%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right), \frac{-1}{2}\right) \]

   7. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{{\left(\frac{1 + x}{x + -1}\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.7 \cdot 10^{+18}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\ell \cdot \left(\ell \cdot \frac{2}{x}\right) + \frac{2 \cdot \left(\left(t \cdot t\right) \cdot \left(x + 1\right)\right)}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x + 1}{x + -1}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.9 \cdot 10^{-286}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{x}{l\_m \cdot l\_m}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x + 1}{x + -1}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.9e-286)
    (* t_m (sqrt (/ x (* l_m l_m))))
    (pow (/ (+ x 1.0) (+ x -1.0)) -0.5))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.9e-286) {
		tmp = t_m * sqrt((x / (l_m * l_m)));
	} else {
		tmp = pow(((x + 1.0) / (x + -1.0)), -0.5);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1.9d-286) then
        tmp = t_m * sqrt((x / (l_m * l_m)))
    else
        tmp = ((x + 1.0d0) / (x + (-1.0d0))) ** (-0.5d0)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.9e-286) {
		tmp = t_m * Math.sqrt((x / (l_m * l_m)));
	} else {
		tmp = Math.pow(((x + 1.0) / (x + -1.0)), -0.5);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 1.9e-286:
		tmp = t_m * math.sqrt((x / (l_m * l_m)))
	else:
		tmp = math.pow(((x + 1.0) / (x + -1.0)), -0.5)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 1.9e-286)
		tmp = Float64(t_m * sqrt(Float64(x / Float64(l_m * l_m))));
	else
		tmp = Float64(Float64(x + 1.0) / Float64(x + -1.0)) ^ -0.5;
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 1.9e-286)
		tmp = t_m * sqrt((x / (l_m * l_m)));
	else
		tmp = ((x + 1.0) / (x + -1.0)) ^ -0.5;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.9e-286], N[(t$95$m * N[Sqrt[N[(x / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.9000000000000001e-286

   1. Initial program 26.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)}\right) \]

      4. sqrt-undiv N/A

         \[\leadsto \mathsf{*.f64}\left(t, \left(\sqrt{\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 26.5%

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

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1} + {\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left({\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right) + 2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left({\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{x - 1} + \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\left(\frac{1}{x - 1}\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x - 1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x + -1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\left(\frac{x}{x - 1}\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x - 1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x + -1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, -1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

   7. Simplified 33.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \color{blue}{\left(\frac{2}{x}\right)}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 41.0%

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{/.f64}\left(2, x\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]

   10. Simplified 41.0%

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

   11. Taylor expanded in l around inf

       \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{x}{{\ell}^{2}}\right)}\right)\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(x, \left({\ell}^{2}\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(x, \left(\ell \cdot \ell\right)\right)\right)\right) \]

       3. *-lowering-*.f64 18.0%

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right) \]

   13. Simplified 18.0%

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

   if 1.9000000000000001e-286 < t

   1. Initial program 37.9%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 82.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

   5. Simplified 82.8%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. *-inverses N/A

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

      4. pow1/2 N/A

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

      5. pow-flip N/A

         \[\leadsto {\left(\frac{1 + x}{x + -1}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{1 + x}{x + -1}\right)}^{\frac{-1}{2}} \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{1 + x}{x + -1}\right), \color{blue}{\frac{-1}{2}}\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{x + 1}{x + -1}\right), \frac{-1}{2}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(x + 1\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

      12. +-lowering-+.f64 82.8%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right), \frac{-1}{2}\right) \]

   7. Applied egg-rr 82.8%

      \[\leadsto \color{blue}{{\left(\frac{1 + x}{x + -1}\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.9 \cdot 10^{-286}:\\ \;\;\;\;t \cdot \sqrt{\frac{x}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x + 1}{x + -1}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.7 \cdot 10^{-286}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{x}{l\_m \cdot l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.7e-286)
    (* t_m (sqrt (/ x (* l_m l_m))))
    (sqrt (/ (+ x -1.0) (+ x 1.0))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 2.7e-286) {
		tmp = t_m * sqrt((x / (l_m * l_m)));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 2.7d-286) then
        tmp = t_m * sqrt((x / (l_m * l_m)))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 2.7e-286) {
		tmp = t_m * Math.sqrt((x / (l_m * l_m)));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 2.7e-286:
		tmp = t_m * math.sqrt((x / (l_m * l_m)))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 2.7e-286)
		tmp = Float64(t_m * sqrt(Float64(x / Float64(l_m * l_m))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 2.7e-286)
		tmp = t_m * sqrt((x / (l_m * l_m)));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2.7e-286], N[(t$95$m * N[Sqrt[N[(x / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.7000000000000002e-286

   1. Initial program 26.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)}\right) \]

      4. sqrt-undiv N/A

         \[\leadsto \mathsf{*.f64}\left(t, \left(\sqrt{\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 26.5%

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

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1} + {\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left({\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right) + 2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left({\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{x - 1} + \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\left(\frac{1}{x - 1}\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x - 1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x + -1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \left(\frac{x}{x - 1} - 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\left(\frac{x}{x - 1}\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x - 1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(x + -1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, -1\right)\right), 1\right)\right)\right), \left(2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}\right)\right)\right)\right)\right) \]

   7. Simplified 33.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \color{blue}{\left(\frac{2}{x}\right)}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 41.0%

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{/.f64}\left(2, x\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right)\right) \]

   10. Simplified 41.0%

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

   11. Taylor expanded in l around inf

       \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{x}{{\ell}^{2}}\right)}\right)\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(x, \left({\ell}^{2}\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(x, \left(\ell \cdot \ell\right)\right)\right)\right) \]

       3. *-lowering-*.f64 18.0%

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right) \]

   13. Simplified 18.0%

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

   if 2.7000000000000002e-286 < t

   1. Initial program 37.9%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 82.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

   5. Simplified 82.8%

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

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
   7. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{x - 1}{1 + x}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(x - 1\right), \left(1 + x\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(1 + x\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(x + -1\right), \left(1 + x\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(1 + x\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(x + 1\right)\right)\right) \]

      7. +-lowering-+.f64 82.8%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, 1\right)\right)\right) \]

   8. Simplified 82.8%

      \[\leadsto \color{blue}{\sqrt{\frac{x + -1}{x + 1}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 76.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, t$95$m_] := N[(t$95$s * N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.3%

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

3. Taylor expanded in l around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

   3. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f64 43.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

5. Simplified 43.6%

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

6. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
7. Step-by-step derivation

   1. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{x - 1}{1 + x}\right)\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(x - 1\right), \left(1 + x\right)\right)\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(1 + x\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(x + -1\right), \left(1 + x\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(1 + x\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(x + 1\right)\right)\right) \]

   7. +-lowering-+.f64 43.6%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, 1\right)\right)\right) \]

8. Simplified 43.6%

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

Alternative 9: 76.6% accurate, 17.3× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (* t_s (- 1.0 (/ (- 1.0 (/ (+ 0.5 (/ -0.5 x)) x)) x))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 - ((1.0 - ((0.5 + (-0.5 / x)) / x)) / x));
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * (1.0d0 - ((1.0d0 - ((0.5d0 + ((-0.5d0) / x)) / x)) / x))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 - ((1.0 - ((0.5 + (-0.5 / x)) / x)) / x));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * (1.0 - ((1.0 - ((0.5 + (-0.5 / x)) / x)) / x))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, 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

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * (1.0 - ((1.0 - ((0.5 + (-0.5 / x)) / x)) / x));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, 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 32.3%

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

3. Taylor expanded in l around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

   3. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f64 43.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

5. Simplified 43.6%

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. *-inverses N/A

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

   4. pow1/2 N/A

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

   5. pow-flip N/A

      \[\leadsto {\left(\frac{1 + x}{x + -1}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

   6. metadata-eval N/A

      \[\leadsto {\left(\frac{1 + x}{x + -1}\right)}^{\frac{-1}{2}} \]

   7. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\frac{1 + x}{x + -1}\right), \color{blue}{\frac{-1}{2}}\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\frac{x + 1}{x + -1}\right), \frac{-1}{2}\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(x + 1\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

   12. +-lowering-+.f64 43.6%

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right), \frac{-1}{2}\right) \]

7. Applied egg-rr 43.6%

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

8. Taylor expanded in x around -inf

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

   1. mul-1-neg N/A

      \[\leadsto 1 + \left(\mathsf{neg}\left(\frac{1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}}{x}\right)\right) \]

   2. unsub-neg N/A

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

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}}{x}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right), \color{blue}{x}\right)\right) \]

   5. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right)\right), x\right)\right) \]

   6. unsub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(1 - \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right), x\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right), x\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}\right), x\right)\right), x\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right)\right), x\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right)\right), x\right)\right) \]

   11. associate-*r/ N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right)\right), x\right)\right), x\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{x}\right)\right)\right), x\right)\right), x\right)\right) \]

   13. distribute-neg-frac N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}\right)\right), x\right)\right), x\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{-1}{2}}{x}\right)\right), x\right)\right), x\right)\right) \]

   15. /-lowering-/.f64 43.3%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), x\right)\right), x\right)\right) \]

10. Simplified 43.3%

    \[\leadsto \color{blue}{1 - \frac{1 - \frac{0.5 + \frac{-0.5}{x}}{x}}{x}} \]
11. Add Preprocessing

Alternative 10: 76.5% accurate, 20.5× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (* t_s (+ (+ 1.0 (/ 0.5 (* x x))) (/ -1.0 x))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * ((1.0 + (0.5 / (x * x))) + (-1.0 / x));
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * ((1.0d0 + (0.5d0 / (x * x))) + ((-1.0d0) / x))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * ((1.0 + (0.5 / (x * x))) + (-1.0 / x));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * ((1.0 + (0.5 / (x * x))) + (-1.0 / x))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) + Float64(-1.0 / x)))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * ((1.0 + (0.5 / (x * x))) + (-1.0 / x));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, t$95$m_] := N[(t$95$s * N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.3%

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

3. Taylor expanded in l around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

   3. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f64 43.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

5. Simplified 43.6%

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

6. Taylor expanded in x around inf

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right), \color{blue}{\left(\frac{1}{x}\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right), \left(\frac{\color{blue}{1}}{x}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{x}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{x}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{x}\right)\right) \]

   6. /-lowering-/.f64 43.2%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{x}\right)\right) \]

8. Simplified 43.2%

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

9. Final simplification 43.2%

   \[\leadsto \left(1 + \frac{0.5}{x \cdot x}\right) + \frac{-1}{x} \]
10. Add Preprocessing

Alternative 11: 76.6% accurate, 25.0× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (* t_s (+ 1.0 (/ (- -1.0 (/ -0.5 x)) x))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 + ((-1.0 - (-0.5 / x)) / x));
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * (1.0d0 + (((-1.0d0) - ((-0.5d0) / x)) / x))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 + ((-1.0 - (-0.5 / x)) / x));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * (1.0 + ((-1.0 - (-0.5 / x)) / x))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * Float64(1.0 + Float64(Float64(-1.0 - Float64(-0.5 / x)) / x)))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * (1.0 + ((-1.0 - (-0.5 / x)) / x));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, 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 32.3%

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

3. Taylor expanded in l around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

   3. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f64 43.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

5. Simplified 43.6%

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. *-inverses N/A

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

   4. pow1/2 N/A

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

   5. pow-flip N/A

      \[\leadsto {\left(\frac{1 + x}{x + -1}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

   6. metadata-eval N/A

      \[\leadsto {\left(\frac{1 + x}{x + -1}\right)}^{\frac{-1}{2}} \]

   7. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\frac{1 + x}{x + -1}\right), \color{blue}{\frac{-1}{2}}\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\frac{x + 1}{x + -1}\right), \frac{-1}{2}\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(x + 1\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right), \frac{-1}{2}\right) \]

   12. +-lowering-+.f64 43.6%

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right), \frac{-1}{2}\right) \]

7. Applied egg-rr 43.6%

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

8. Taylor expanded in x around -inf

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

   1. mul-1-neg N/A

      \[\leadsto 1 + \left(\mathsf{neg}\left(\frac{1 - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right) \]

   2. unsub-neg N/A

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

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1 - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(1 - \frac{1}{2} \cdot \frac{1}{x}\right), \color{blue}{x}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right)\right), x\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{x}\right)\right)\right), x\right)\right) \]

   9. distribute-neg-frac N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}\right)\right), x\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{2}}{x}\right)\right), x\right)\right) \]

   11. /-lowering-/.f64 43.2%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), x\right)\right) \]

10. Simplified 43.2%

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

11. Final simplification 43.2%

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

Alternative 12: 76.3% accurate, 45.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, t$95$m_] := N[(t$95$s * N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.3%

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

3. Taylor expanded in l around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

   3. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f64 43.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

5. Simplified 43.6%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
7. Step-by-step derivation

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1}{x}\right)}\right) \]

   2. /-lowering-/.f64 42.9%

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{x}\right)\right) \]

8. Simplified 42.9%

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

9. Final simplification 42.9%

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

Alternative 13: 75.7% accurate, 225.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, t$95$m_] := N[(t$95$s * 1.0), $MachinePrecision]

Derivation

1. Initial program 32.3%

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

3. Taylor expanded in l around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

   3. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f64 43.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

5. Simplified 43.6%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 42.6%

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

Reproduce

herbie shell --seed 2024288 
(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)))))