Toniolo and Linder, Equation (7)

Percentage Accurate: 33.4% → 80.7%

Time: 23.9s

Alternatives: 9

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: 33.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x l t)
 :precision binary64
 (/
  (* (sqrt 2.0) t)
  (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))

C

double code(double x, double l, double t) {
	return (sqrt(2.0) * t) / sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
}

Fortran

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

Java

public static double code(double x, double l, double t) {
	return (Math.sqrt(2.0) * t) / Math.sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
}

Python

def code(x, l, t):
	return (math.sqrt(2.0) * t) / math.sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)))

Julia

function code(x, l, t)
	return Float64(Float64(sqrt(2.0) * t) / sqrt(Float64(Float64(Float64(Float64(x + 1.0) / Float64(x - 1.0)) * Float64(Float64(l * l) + Float64(2.0 * Float64(t * t)))) - Float64(l * l))))
end

MATLAB

function tmp = code(x, l, t)
	tmp = (sqrt(2.0) * t) / sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
end

Wolfram

code[x_, l_, t_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 80.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \frac{{t}^{2}}{x}\\ t_2 := 2 \cdot {t}^{2}\\ t_3 := t_2 + {\ell}^{2}\\ t_4 := \frac{t_3}{x}\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{+36}:\\ \;\;\;\;\frac{\left(--1\right) - x}{x + -1}\\ \mathbf{elif}\;t \leq -1.26 \cdot 10^{-179}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\left(t_1 + \left(t_2 + \ell \cdot \frac{\ell}{x}\right)\right) + t_4}}{t}}\\ \mathbf{elif}\;t \leq -6 \cdot 10^{-275}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{x \cdot 0.5 - 0.5} \cdot \frac{t}{\ell}\right)\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-164}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 950000:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\left(\frac{t_3 + t_3}{{x}^{2}} + \left(t_1 + \left(2 \cdot \frac{{t}^{2}}{{x}^{3}} + \left(t_2 + \left(\frac{{\ell}^{2}}{x} + \frac{{\ell}^{2}}{{x}^{3}}\right)\right)\right)\right)\right) + \left(\frac{t_3}{{x}^{3}} + t_4\right)}}{t}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* 2.0 (/ (pow t 2.0) x)))
        (t_2 (* 2.0 (pow t 2.0)))
        (t_3 (+ t_2 (pow l 2.0)))
        (t_4 (/ t_3 x)))
   (if (<= t -1.4e+36)
     (/ (- (- -1.0) x) (+ x -1.0))
     (if (<= t -1.26e-179)
       (/ (sqrt 2.0) (/ (sqrt (+ (+ t_1 (+ t_2 (* l (/ l x)))) t_4)) t))
       (if (<= t -6e-275)
         (- (sqrt (/ (+ x -1.0) (+ x 1.0))))
         (if (<= t 2.2e-305)
           (* (sqrt 2.0) (* (sqrt (- (* x 0.5) 0.5)) (/ t l)))
           (if (<= t 2.3e-164)
             1.0
             (if (<= t 950000.0)
               (/
                (sqrt 2.0)
                (/
                 (sqrt
                  (+
                   (+
                    (/ (+ t_3 t_3) (pow x 2.0))
                    (+
                     t_1
                     (+
                      (* 2.0 (/ (pow t 2.0) (pow x 3.0)))
                      (+
                       t_2
                       (+ (/ (pow l 2.0) x) (/ (pow l 2.0) (pow x 3.0)))))))
                   (+ (/ t_3 (pow x 3.0)) t_4)))
                 t))
               (+ 1.0 (/ -1.0 x))))))))))

C

double code(double x, double l, double t) {
	double t_1 = 2.0 * (pow(t, 2.0) / x);
	double t_2 = 2.0 * pow(t, 2.0);
	double t_3 = t_2 + pow(l, 2.0);
	double t_4 = t_3 / x;
	double tmp;
	if (t <= -1.4e+36) {
		tmp = (-(-1.0) - x) / (x + -1.0);
	} else if (t <= -1.26e-179) {
		tmp = sqrt(2.0) / (sqrt(((t_1 + (t_2 + (l * (l / x)))) + t_4)) / t);
	} else if (t <= -6e-275) {
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= 2.2e-305) {
		tmp = sqrt(2.0) * (sqrt(((x * 0.5) - 0.5)) * (t / l));
	} else if (t <= 2.3e-164) {
		tmp = 1.0;
	} else if (t <= 950000.0) {
		tmp = sqrt(2.0) / (sqrt(((((t_3 + t_3) / pow(x, 2.0)) + (t_1 + ((2.0 * (pow(t, 2.0) / pow(x, 3.0))) + (t_2 + ((pow(l, 2.0) / x) + (pow(l, 2.0) / pow(x, 3.0))))))) + ((t_3 / pow(x, 3.0)) + t_4))) / t);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = 2.0d0 * ((t ** 2.0d0) / x)
    t_2 = 2.0d0 * (t ** 2.0d0)
    t_3 = t_2 + (l ** 2.0d0)
    t_4 = t_3 / x
    if (t <= (-1.4d+36)) then
        tmp = (-(-1.0d0) - x) / (x + (-1.0d0))
    else if (t <= (-1.26d-179)) then
        tmp = sqrt(2.0d0) / (sqrt(((t_1 + (t_2 + (l * (l / x)))) + t_4)) / t)
    else if (t <= (-6d-275)) then
        tmp = -sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else if (t <= 2.2d-305) then
        tmp = sqrt(2.0d0) * (sqrt(((x * 0.5d0) - 0.5d0)) * (t / l))
    else if (t <= 2.3d-164) then
        tmp = 1.0d0
    else if (t <= 950000.0d0) then
        tmp = sqrt(2.0d0) / (sqrt(((((t_3 + t_3) / (x ** 2.0d0)) + (t_1 + ((2.0d0 * ((t ** 2.0d0) / (x ** 3.0d0))) + (t_2 + (((l ** 2.0d0) / x) + ((l ** 2.0d0) / (x ** 3.0d0))))))) + ((t_3 / (x ** 3.0d0)) + t_4))) / t)
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double t_1 = 2.0 * (Math.pow(t, 2.0) / x);
	double t_2 = 2.0 * Math.pow(t, 2.0);
	double t_3 = t_2 + Math.pow(l, 2.0);
	double t_4 = t_3 / x;
	double tmp;
	if (t <= -1.4e+36) {
		tmp = (-(-1.0) - x) / (x + -1.0);
	} else if (t <= -1.26e-179) {
		tmp = Math.sqrt(2.0) / (Math.sqrt(((t_1 + (t_2 + (l * (l / x)))) + t_4)) / t);
	} else if (t <= -6e-275) {
		tmp = -Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= 2.2e-305) {
		tmp = Math.sqrt(2.0) * (Math.sqrt(((x * 0.5) - 0.5)) * (t / l));
	} else if (t <= 2.3e-164) {
		tmp = 1.0;
	} else if (t <= 950000.0) {
		tmp = Math.sqrt(2.0) / (Math.sqrt(((((t_3 + t_3) / Math.pow(x, 2.0)) + (t_1 + ((2.0 * (Math.pow(t, 2.0) / Math.pow(x, 3.0))) + (t_2 + ((Math.pow(l, 2.0) / x) + (Math.pow(l, 2.0) / Math.pow(x, 3.0))))))) + ((t_3 / Math.pow(x, 3.0)) + t_4))) / t);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = 2.0 * (math.pow(t, 2.0) / x)
	t_2 = 2.0 * math.pow(t, 2.0)
	t_3 = t_2 + math.pow(l, 2.0)
	t_4 = t_3 / x
	tmp = 0
	if t <= -1.4e+36:
		tmp = (-(-1.0) - x) / (x + -1.0)
	elif t <= -1.26e-179:
		tmp = math.sqrt(2.0) / (math.sqrt(((t_1 + (t_2 + (l * (l / x)))) + t_4)) / t)
	elif t <= -6e-275:
		tmp = -math.sqrt(((x + -1.0) / (x + 1.0)))
	elif t <= 2.2e-305:
		tmp = math.sqrt(2.0) * (math.sqrt(((x * 0.5) - 0.5)) * (t / l))
	elif t <= 2.3e-164:
		tmp = 1.0
	elif t <= 950000.0:
		tmp = math.sqrt(2.0) / (math.sqrt(((((t_3 + t_3) / math.pow(x, 2.0)) + (t_1 + ((2.0 * (math.pow(t, 2.0) / math.pow(x, 3.0))) + (t_2 + ((math.pow(l, 2.0) / x) + (math.pow(l, 2.0) / math.pow(x, 3.0))))))) + ((t_3 / math.pow(x, 3.0)) + t_4))) / t)
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

function code(x, l, t)
	t_1 = Float64(2.0 * Float64((t ^ 2.0) / x))
	t_2 = Float64(2.0 * (t ^ 2.0))
	t_3 = Float64(t_2 + (l ^ 2.0))
	t_4 = Float64(t_3 / x)
	tmp = 0.0
	if (t <= -1.4e+36)
		tmp = Float64(Float64(Float64(-(-1.0)) - x) / Float64(x + -1.0));
	elseif (t <= -1.26e-179)
		tmp = Float64(sqrt(2.0) / Float64(sqrt(Float64(Float64(t_1 + Float64(t_2 + Float64(l * Float64(l / x)))) + t_4)) / t));
	elseif (t <= -6e-275)
		tmp = Float64(-sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0))));
	elseif (t <= 2.2e-305)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(Float64(x * 0.5) - 0.5)) * Float64(t / l)));
	elseif (t <= 2.3e-164)
		tmp = 1.0;
	elseif (t <= 950000.0)
		tmp = Float64(sqrt(2.0) / Float64(sqrt(Float64(Float64(Float64(Float64(t_3 + t_3) / (x ^ 2.0)) + Float64(t_1 + Float64(Float64(2.0 * Float64((t ^ 2.0) / (x ^ 3.0))) + Float64(t_2 + Float64(Float64((l ^ 2.0) / x) + Float64((l ^ 2.0) / (x ^ 3.0))))))) + Float64(Float64(t_3 / (x ^ 3.0)) + t_4))) / t));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = 2.0 * ((t ^ 2.0) / x);
	t_2 = 2.0 * (t ^ 2.0);
	t_3 = t_2 + (l ^ 2.0);
	t_4 = t_3 / x;
	tmp = 0.0;
	if (t <= -1.4e+36)
		tmp = (-(-1.0) - x) / (x + -1.0);
	elseif (t <= -1.26e-179)
		tmp = sqrt(2.0) / (sqrt(((t_1 + (t_2 + (l * (l / x)))) + t_4)) / t);
	elseif (t <= -6e-275)
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	elseif (t <= 2.2e-305)
		tmp = sqrt(2.0) * (sqrt(((x * 0.5) - 0.5)) * (t / l));
	elseif (t <= 2.3e-164)
		tmp = 1.0;
	elseif (t <= 950000.0)
		tmp = sqrt(2.0) / (sqrt(((((t_3 + t_3) / (x ^ 2.0)) + (t_1 + ((2.0 * ((t ^ 2.0) / (x ^ 3.0))) + (t_2 + (((l ^ 2.0) / x) + ((l ^ 2.0) / (x ^ 3.0))))))) + ((t_3 / (x ^ 3.0)) + t_4))) / t);
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(2.0 * N[(N[Power[t, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / x), $MachinePrecision]}, If[LessEqual[t, -1.4e+36], N[(N[((--1.0) - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.26e-179], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[N[(N[(t$95$1 + N[(t$95$2 + N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -6e-275], (-N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, 2.2e-305], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(N[(x * 0.5), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.3e-164], 1.0, If[LessEqual[t, 950000.0], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[N[(N[(N[(N[(t$95$3 + t$95$3), $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(N[(2.0 * N[(N[Power[t, 2.0], $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision] + N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if t < -1.4e36

   1. Initial program 24.9%

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

      1. *-commutative 24.9%

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

      2. associate-*l/ 24.8%

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

      3. fma-neg 24.8%

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

      4. sub-neg 24.8%

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

      5. metadata-eval 24.8%

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

      6. +-commutative 24.8%

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

      7. fma-def 24.8%

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

      8. distribute-rgt-neg-in 24.8%

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

   3. Simplified 24.8%

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

   4. Taylor expanded in t around -inf 91.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)\right)} \cdot \sqrt{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 91.4%

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

      2. *-commutative 91.4%

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

      3. sub-neg 91.4%

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

      4. metadata-eval 91.4%

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

      5. +-commutative 91.4%

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

      6. +-commutative 91.4%

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

   6. Simplified 91.4%

      \[\leadsto \color{blue}{\left(-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right)} \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 1.6%

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

      3. sqr-neg 1.6%

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

      4. sqrt-unprod 1.6%

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

      5. add-sqr-sqrt 1.6%

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

      6. associate-*l* 1.6%

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

      7. *-commutative 1.6%

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

      8. sqrt-unprod 1.6%

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

      9. metadata-eval 1.6%

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

      10. metadata-eval 1.6%

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

      11. *-commutative 1.6%

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

      12. pow1/2 1.6%

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

   8. Applied egg-rr 1.6%

      \[\leadsto \color{blue}{{\left(\frac{x + -1}{x + 1}\right)}^{0.5}} \]
   9. Step-by-step derivation

      1. unpow1/2 1.6%

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

   10. Simplified 1.6%

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

   12. Applied egg-rr 92.9%

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

   if -1.4e36 < t < -1.2599999999999999e-179

   1. Initial program 59.3%

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

   3. Simplified 59.4%

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

   4. Taylor expanded in x around inf 90.6%

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

      1. unpow2 90.6%

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

      2. *-un-lft-identity 90.6%

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

      3. times-frac 90.7%

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

   6. Applied egg-rr 90.7%

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

   if -1.2599999999999999e-179 < t < -6.000000000000001e-275

   1. Initial program 2.6%

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

      1. *-commutative 2.6%

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

      2. associate-*l/ 2.6%

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

      3. fma-neg 2.6%

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

      4. sub-neg 2.6%

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

      5. metadata-eval 2.6%

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

      6. +-commutative 2.6%

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

      7. fma-def 2.6%

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

      8. distribute-rgt-neg-in 2.6%

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

   3. Simplified 2.6%

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

   4. Taylor expanded in t around -inf 58.5%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)\right)} \cdot \sqrt{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 58.5%

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

      2. *-commutative 58.5%

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

      3. sub-neg 58.5%

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

      4. metadata-eval 58.5%

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

      5. +-commutative 58.5%

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

      6. +-commutative 58.5%

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

   6. Simplified 58.5%

      \[\leadsto \color{blue}{\left(-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right)} \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. distribute-lft-neg-out 58.5%

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

      2. associate-*l* 58.5%

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

      3. *-commutative 58.5%

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

      4. sqrt-unprod 59.4%

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

      5. metadata-eval 59.4%

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

      6. metadata-eval 59.4%

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

      7. *-commutative 59.4%

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

      8. *-un-lft-identity 59.4%

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

      9. neg-sub0 59.4%

         \[\leadsto \color{blue}{0 - \sqrt{\frac{-1 + x}{x + 1}}} \]

      10. +-commutative 59.4%

          \[\leadsto 0 - \sqrt{\frac{\color{blue}{x + -1}}{x + 1}} \]

   8. Applied egg-rr 59.4%

      \[\leadsto \color{blue}{0 - \sqrt{\frac{x + -1}{x + 1}}} \]
   9. Step-by-step derivation

      1. sub0-neg 59.4%

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

   10. Simplified 59.4%

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

   if -6.000000000000001e-275 < t < 2.19999999999999997e-305

   1. Initial program 6.5%

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

      1. *-commutative 6.5%

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

      2. associate-*l/ 6.5%

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

      3. fma-neg 5.9%

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

      4. sub-neg 5.9%

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

      5. metadata-eval 5.9%

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

      6. +-commutative 5.9%

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

      7. fma-def 5.9%

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

      8. distribute-rgt-neg-in 5.9%

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

   3. Simplified 5.9%

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

   4. Taylor expanded in l around inf 25.9%

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

      1. *-commutative 25.9%

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

      2. associate--l+ 69.8%

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

      3. sub-neg 69.8%

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

      4. metadata-eval 69.8%

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

      5. +-commutative 69.8%

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

      6. sub-neg 69.8%

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

      7. metadata-eval 69.8%

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

      8. +-commutative 69.8%

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

   6. Simplified 69.8%

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

   7. Taylor expanded in x around 0 99.7%

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

   if 2.19999999999999997e-305 < t < 2.29999999999999985e-164

   1. Initial program 2.4%

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

      1. *-commutative 2.4%

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

      2. associate-*l/ 2.4%

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

      3. fma-neg 2.4%

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

      4. sub-neg 2.4%

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

      5. metadata-eval 2.4%

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

      6. +-commutative 2.4%

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

      7. fma-def 2.4%

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

      8. distribute-rgt-neg-in 2.4%

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

   3. Simplified 2.4%

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

   4. Taylor expanded in t around -inf 1.9%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)\right)} \cdot \sqrt{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 1.9%

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

      2. *-commutative 1.9%

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

      3. sub-neg 1.9%

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

      4. metadata-eval 1.9%

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

      5. +-commutative 1.9%

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

      6. +-commutative 1.9%

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

   6. Simplified 1.9%

      \[\leadsto \color{blue}{\left(-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right)} \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 64.1%

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

      3. sqr-neg 64.1%

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

      4. sqrt-unprod 64.1%

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

      5. add-sqr-sqrt 64.1%

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

      6. associate-*l* 64.1%

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

      7. *-commutative 64.1%

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

      8. sqrt-unprod 65.0%

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

      9. metadata-eval 65.0%

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

      10. metadata-eval 65.0%

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

      11. *-commutative 65.0%

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

      12. pow1/2 65.0%

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

   8. Applied egg-rr 65.0%

      \[\leadsto \color{blue}{{\left(\frac{x + -1}{x + 1}\right)}^{0.5}} \]
   9. Step-by-step derivation

      1. unpow1/2 65.0%

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

   10. Simplified 65.0%

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

   11. Taylor expanded in x around inf 65.0%

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

   if 2.29999999999999985e-164 < t < 9.5e5

   1. Initial program 60.9%

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

   3. Simplified 61.0%

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

   4. Taylor expanded in x around -inf 88.8%

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

   if 9.5e5 < t

   1. Initial program 27.6%

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

      1. *-commutative 27.6%

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

      2. associate-*l/ 27.6%

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

      3. fma-neg 27.6%

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

      4. sub-neg 27.6%

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

      5. metadata-eval 27.6%

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

      6. +-commutative 27.6%

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

      7. fma-def 27.6%

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

      8. distribute-rgt-neg-in 27.6%

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

   3. Simplified 27.6%

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

   4. Taylor expanded in t around -inf 1.6%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)\right)} \cdot \sqrt{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 1.6%

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

      2. *-commutative 1.6%

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

      3. sub-neg 1.6%

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

      4. metadata-eval 1.6%

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

      5. +-commutative 1.6%

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

      6. +-commutative 1.6%

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

   6. Simplified 1.6%

      \[\leadsto \color{blue}{\left(-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right)} \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 93.4%

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

      3. sqr-neg 93.4%

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

      4. sqrt-unprod 93.5%

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

      5. add-sqr-sqrt 93.4%

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

      6. associate-*l* 93.4%

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

      7. *-commutative 93.4%

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

      8. sqrt-unprod 94.9%

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

      9. metadata-eval 94.9%

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

      10. metadata-eval 94.9%

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

      11. *-commutative 94.9%

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

      12. pow1/2 94.9%

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

   8. Applied egg-rr 94.9%

      \[\leadsto \color{blue}{{\left(\frac{x + -1}{x + 1}\right)}^{0.5}} \]
   9. Step-by-step derivation

      1. unpow1/2 94.9%

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

   10. Simplified 94.9%

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

   11. Taylor expanded in x around inf 94.9%

       \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{+36}:\\ \;\;\;\;\frac{\left(--1\right) - x}{x + -1}\\ \mathbf{elif}\;t \leq -1.26 \cdot 10^{-179}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \ell \cdot \frac{\ell}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}{t}}\\ \mathbf{elif}\;t \leq -6 \cdot 10^{-275}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{x \cdot 0.5 - 0.5} \cdot \frac{t}{\ell}\right)\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-164}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 950000:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\left(\frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{{x}^{2}} + \left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{{x}^{3}} + \left(2 \cdot {t}^{2} + \left(\frac{{\ell}^{2}}{x} + \frac{{\ell}^{2}}{{x}^{3}}\right)\right)\right)\right)\right) + \left(\frac{2 \cdot {t}^{2} + {\ell}^{2}}{{x}^{3}} + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}}{t}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 2: 80.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \frac{{t}^{2}}{x}\\ t_2 := 2 \cdot {t}^{2}\\ t_3 := t_2 + {\ell}^{2}\\ t_4 := \frac{t_3}{x}\\ \mathbf{if}\;t \leq -8.6 \cdot 10^{+34}:\\ \;\;\;\;\frac{\left(--1\right) - x}{x + -1}\\ \mathbf{elif}\;t \leq -5.5 \cdot 10^{-177}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\left(t_1 + \left(t_2 + \ell \cdot \frac{\ell}{x}\right)\right) + t_4}}{t}}\\ \mathbf{elif}\;t \leq -1.02 \cdot 10^{-274}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{x \cdot 0.5 - 0.5} \cdot \frac{t}{\ell}\right)\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-163}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 480000:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\left(\frac{t_3 + t_3}{{x}^{2}} + \left(t_1 + \left(t_2 + \frac{{\ell}^{2}}{x}\right)\right)\right) + t_4}}{t}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* 2.0 (/ (pow t 2.0) x)))
        (t_2 (* 2.0 (pow t 2.0)))
        (t_3 (+ t_2 (pow l 2.0)))
        (t_4 (/ t_3 x)))
   (if (<= t -8.6e+34)
     (/ (- (- -1.0) x) (+ x -1.0))
     (if (<= t -5.5e-177)
       (/ (sqrt 2.0) (/ (sqrt (+ (+ t_1 (+ t_2 (* l (/ l x)))) t_4)) t))
       (if (<= t -1.02e-274)
         (- (sqrt (/ (+ x -1.0) (+ x 1.0))))
         (if (<= t 2.3e-305)
           (* (sqrt 2.0) (* (sqrt (- (* x 0.5) 0.5)) (/ t l)))
           (if (<= t 7.5e-163)
             1.0
             (if (<= t 480000.0)
               (/
                (sqrt 2.0)
                (/
                 (sqrt
                  (+
                   (+
                    (/ (+ t_3 t_3) (pow x 2.0))
                    (+ t_1 (+ t_2 (/ (pow l 2.0) x))))
                   t_4))
                 t))
               (+ 1.0 (/ -1.0 x))))))))))

C

double code(double x, double l, double t) {
	double t_1 = 2.0 * (pow(t, 2.0) / x);
	double t_2 = 2.0 * pow(t, 2.0);
	double t_3 = t_2 + pow(l, 2.0);
	double t_4 = t_3 / x;
	double tmp;
	if (t <= -8.6e+34) {
		tmp = (-(-1.0) - x) / (x + -1.0);
	} else if (t <= -5.5e-177) {
		tmp = sqrt(2.0) / (sqrt(((t_1 + (t_2 + (l * (l / x)))) + t_4)) / t);
	} else if (t <= -1.02e-274) {
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= 2.3e-305) {
		tmp = sqrt(2.0) * (sqrt(((x * 0.5) - 0.5)) * (t / l));
	} else if (t <= 7.5e-163) {
		tmp = 1.0;
	} else if (t <= 480000.0) {
		tmp = sqrt(2.0) / (sqrt(((((t_3 + t_3) / pow(x, 2.0)) + (t_1 + (t_2 + (pow(l, 2.0) / x)))) + t_4)) / t);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = 2.0d0 * ((t ** 2.0d0) / x)
    t_2 = 2.0d0 * (t ** 2.0d0)
    t_3 = t_2 + (l ** 2.0d0)
    t_4 = t_3 / x
    if (t <= (-8.6d+34)) then
        tmp = (-(-1.0d0) - x) / (x + (-1.0d0))
    else if (t <= (-5.5d-177)) then
        tmp = sqrt(2.0d0) / (sqrt(((t_1 + (t_2 + (l * (l / x)))) + t_4)) / t)
    else if (t <= (-1.02d-274)) then
        tmp = -sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else if (t <= 2.3d-305) then
        tmp = sqrt(2.0d0) * (sqrt(((x * 0.5d0) - 0.5d0)) * (t / l))
    else if (t <= 7.5d-163) then
        tmp = 1.0d0
    else if (t <= 480000.0d0) then
        tmp = sqrt(2.0d0) / (sqrt(((((t_3 + t_3) / (x ** 2.0d0)) + (t_1 + (t_2 + ((l ** 2.0d0) / x)))) + t_4)) / t)
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double t_1 = 2.0 * (Math.pow(t, 2.0) / x);
	double t_2 = 2.0 * Math.pow(t, 2.0);
	double t_3 = t_2 + Math.pow(l, 2.0);
	double t_4 = t_3 / x;
	double tmp;
	if (t <= -8.6e+34) {
		tmp = (-(-1.0) - x) / (x + -1.0);
	} else if (t <= -5.5e-177) {
		tmp = Math.sqrt(2.0) / (Math.sqrt(((t_1 + (t_2 + (l * (l / x)))) + t_4)) / t);
	} else if (t <= -1.02e-274) {
		tmp = -Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= 2.3e-305) {
		tmp = Math.sqrt(2.0) * (Math.sqrt(((x * 0.5) - 0.5)) * (t / l));
	} else if (t <= 7.5e-163) {
		tmp = 1.0;
	} else if (t <= 480000.0) {
		tmp = Math.sqrt(2.0) / (Math.sqrt(((((t_3 + t_3) / Math.pow(x, 2.0)) + (t_1 + (t_2 + (Math.pow(l, 2.0) / x)))) + t_4)) / t);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = 2.0 * (math.pow(t, 2.0) / x)
	t_2 = 2.0 * math.pow(t, 2.0)
	t_3 = t_2 + math.pow(l, 2.0)
	t_4 = t_3 / x
	tmp = 0
	if t <= -8.6e+34:
		tmp = (-(-1.0) - x) / (x + -1.0)
	elif t <= -5.5e-177:
		tmp = math.sqrt(2.0) / (math.sqrt(((t_1 + (t_2 + (l * (l / x)))) + t_4)) / t)
	elif t <= -1.02e-274:
		tmp = -math.sqrt(((x + -1.0) / (x + 1.0)))
	elif t <= 2.3e-305:
		tmp = math.sqrt(2.0) * (math.sqrt(((x * 0.5) - 0.5)) * (t / l))
	elif t <= 7.5e-163:
		tmp = 1.0
	elif t <= 480000.0:
		tmp = math.sqrt(2.0) / (math.sqrt(((((t_3 + t_3) / math.pow(x, 2.0)) + (t_1 + (t_2 + (math.pow(l, 2.0) / x)))) + t_4)) / t)
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

function code(x, l, t)
	t_1 = Float64(2.0 * Float64((t ^ 2.0) / x))
	t_2 = Float64(2.0 * (t ^ 2.0))
	t_3 = Float64(t_2 + (l ^ 2.0))
	t_4 = Float64(t_3 / x)
	tmp = 0.0
	if (t <= -8.6e+34)
		tmp = Float64(Float64(Float64(-(-1.0)) - x) / Float64(x + -1.0));
	elseif (t <= -5.5e-177)
		tmp = Float64(sqrt(2.0) / Float64(sqrt(Float64(Float64(t_1 + Float64(t_2 + Float64(l * Float64(l / x)))) + t_4)) / t));
	elseif (t <= -1.02e-274)
		tmp = Float64(-sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0))));
	elseif (t <= 2.3e-305)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(Float64(x * 0.5) - 0.5)) * Float64(t / l)));
	elseif (t <= 7.5e-163)
		tmp = 1.0;
	elseif (t <= 480000.0)
		tmp = Float64(sqrt(2.0) / Float64(sqrt(Float64(Float64(Float64(Float64(t_3 + t_3) / (x ^ 2.0)) + Float64(t_1 + Float64(t_2 + Float64((l ^ 2.0) / x)))) + t_4)) / t));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = 2.0 * ((t ^ 2.0) / x);
	t_2 = 2.0 * (t ^ 2.0);
	t_3 = t_2 + (l ^ 2.0);
	t_4 = t_3 / x;
	tmp = 0.0;
	if (t <= -8.6e+34)
		tmp = (-(-1.0) - x) / (x + -1.0);
	elseif (t <= -5.5e-177)
		tmp = sqrt(2.0) / (sqrt(((t_1 + (t_2 + (l * (l / x)))) + t_4)) / t);
	elseif (t <= -1.02e-274)
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	elseif (t <= 2.3e-305)
		tmp = sqrt(2.0) * (sqrt(((x * 0.5) - 0.5)) * (t / l));
	elseif (t <= 7.5e-163)
		tmp = 1.0;
	elseif (t <= 480000.0)
		tmp = sqrt(2.0) / (sqrt(((((t_3 + t_3) / (x ^ 2.0)) + (t_1 + (t_2 + ((l ^ 2.0) / x)))) + t_4)) / t);
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(2.0 * N[(N[Power[t, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / x), $MachinePrecision]}, If[LessEqual[t, -8.6e+34], N[(N[((--1.0) - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -5.5e-177], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[N[(N[(t$95$1 + N[(t$95$2 + N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.02e-274], (-N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, 2.3e-305], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(N[(x * 0.5), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.5e-163], 1.0, If[LessEqual[t, 480000.0], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[N[(N[(N[(N[(t$95$3 + t$95$3), $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(t$95$2 + N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if t < -8.59999999999999988e34

   1. Initial program 24.9%

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

      1. *-commutative 24.9%

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

      2. associate-*l/ 24.8%

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

      3. fma-neg 24.8%

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

      4. sub-neg 24.8%

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

      5. metadata-eval 24.8%

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

      6. +-commutative 24.8%

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

      7. fma-def 24.8%

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

      8. distribute-rgt-neg-in 24.8%

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

   3. Simplified 24.8%

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

   4. Taylor expanded in t around -inf 91.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)\right)} \cdot \sqrt{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 91.4%

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

      2. *-commutative 91.4%

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

      3. sub-neg 91.4%

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

      4. metadata-eval 91.4%

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

      5. +-commutative 91.4%

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

      6. +-commutative 91.4%

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

   6. Simplified 91.4%

      \[\leadsto \color{blue}{\left(-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right)} \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 1.6%

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

      3. sqr-neg 1.6%

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

      4. sqrt-unprod 1.6%

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

      5. add-sqr-sqrt 1.6%

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

      6. associate-*l* 1.6%

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

      7. *-commutative 1.6%

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

      8. sqrt-unprod 1.6%

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

      9. metadata-eval 1.6%

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

      10. metadata-eval 1.6%

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

      11. *-commutative 1.6%

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

      12. pow1/2 1.6%

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

   8. Applied egg-rr 1.6%

      \[\leadsto \color{blue}{{\left(\frac{x + -1}{x + 1}\right)}^{0.5}} \]
   9. Step-by-step derivation

      1. unpow1/2 1.6%

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

   10. Simplified 1.6%

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

   12. Applied egg-rr 92.9%

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

   if -8.59999999999999988e34 < t < -5.4999999999999996e-177

   1. Initial program 59.3%

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

   3. Simplified 59.4%

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

   4. Taylor expanded in x around inf 90.6%

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

      1. unpow2 90.6%

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

      2. *-un-lft-identity 90.6%

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

      3. times-frac 90.7%

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

   6. Applied egg-rr 90.7%

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

   if -5.4999999999999996e-177 < t < -1.01999999999999997e-274

   1. Initial program 2.6%

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

      1. *-commutative 2.6%

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

      2. associate-*l/ 2.6%

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

      3. fma-neg 2.6%

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

      4. sub-neg 2.6%

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

      5. metadata-eval 2.6%

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

      6. +-commutative 2.6%

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

      7. fma-def 2.6%

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

      8. distribute-rgt-neg-in 2.6%

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

   3. Simplified 2.6%

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

   4. Taylor expanded in t around -inf 58.5%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)\right)} \cdot \sqrt{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 58.5%

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

      2. *-commutative 58.5%

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

      3. sub-neg 58.5%

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

      4. metadata-eval 58.5%

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

      5. +-commutative 58.5%

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

      6. +-commutative 58.5%

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

   6. Simplified 58.5%

      \[\leadsto \color{blue}{\left(-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right)} \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. distribute-lft-neg-out 58.5%

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

      2. associate-*l* 58.5%

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

      3. *-commutative 58.5%

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

      4. sqrt-unprod 59.4%

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

      5. metadata-eval 59.4%

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

      6. metadata-eval 59.4%

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

      7. *-commutative 59.4%

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

      8. *-un-lft-identity 59.4%

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

      9. neg-sub0 59.4%

         \[\leadsto \color{blue}{0 - \sqrt{\frac{-1 + x}{x + 1}}} \]

      10. +-commutative 59.4%

          \[\leadsto 0 - \sqrt{\frac{\color{blue}{x + -1}}{x + 1}} \]

   8. Applied egg-rr 59.4%

      \[\leadsto \color{blue}{0 - \sqrt{\frac{x + -1}{x + 1}}} \]
   9. Step-by-step derivation

      1. sub0-neg 59.4%

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

   10. Simplified 59.4%

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

   if -1.01999999999999997e-274 < t < 2.3e-305

   1. Initial program 6.5%

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

      1. *-commutative 6.5%

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

      2. associate-*l/ 6.5%

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

      3. fma-neg 5.9%

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

      4. sub-neg 5.9%

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

      5. metadata-eval 5.9%

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

      6. +-commutative 5.9%

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

      7. fma-def 5.9%

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

      8. distribute-rgt-neg-in 5.9%

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

   3. Simplified 5.9%

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

   4. Taylor expanded in l around inf 25.9%

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

      1. *-commutative 25.9%

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

      2. associate--l+ 69.8%

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

      3. sub-neg 69.8%

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

      4. metadata-eval 69.8%

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

      5. +-commutative 69.8%

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

      6. sub-neg 69.8%

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

      7. metadata-eval 69.8%

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

      8. +-commutative 69.8%

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

   6. Simplified 69.8%

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

   7. Taylor expanded in x around 0 99.7%

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

   if 2.3e-305 < t < 7.49999999999999996e-163

   1. Initial program 2.4%

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

      1. *-commutative 2.4%

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

      2. associate-*l/ 2.4%

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

      3. fma-neg 2.4%

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

      4. sub-neg 2.4%

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

      5. metadata-eval 2.4%

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

      6. +-commutative 2.4%

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

      7. fma-def 2.4%

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

      8. distribute-rgt-neg-in 2.4%

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

   3. Simplified 2.4%

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

   4. Taylor expanded in t around -inf 1.9%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)\right)} \cdot \sqrt{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 1.9%

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

      2. *-commutative 1.9%

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

      3. sub-neg 1.9%

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

      4. metadata-eval 1.9%

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

      5. +-commutative 1.9%

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

      6. +-commutative 1.9%

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

   6. Simplified 1.9%

      \[\leadsto \color{blue}{\left(-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right)} \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 64.1%

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

      3. sqr-neg 64.1%

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

      4. sqrt-unprod 64.1%

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

      5. add-sqr-sqrt 64.1%

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

      6. associate-*l* 64.1%

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

      7. *-commutative 64.1%

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

      8. sqrt-unprod 65.0%

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

      9. metadata-eval 65.0%

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

      10. metadata-eval 65.0%

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

      11. *-commutative 65.0%

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

      12. pow1/2 65.0%

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

   8. Applied egg-rr 65.0%

      \[\leadsto \color{blue}{{\left(\frac{x + -1}{x + 1}\right)}^{0.5}} \]
   9. Step-by-step derivation

      1. unpow1/2 65.0%

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

   10. Simplified 65.0%

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

   11. Taylor expanded in x around inf 65.0%

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

   if 7.49999999999999996e-163 < t < 4.8e5

   1. Initial program 60.9%

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

   3. Simplified 61.0%

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

   4. Taylor expanded in x around -inf 88.6%

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

   if 4.8e5 < t

   1. Initial program 27.6%

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

      1. *-commutative 27.6%

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

      2. associate-*l/ 27.6%

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

      3. fma-neg 27.6%

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

      4. sub-neg 27.6%

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

      5. metadata-eval 27.6%

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

      6. +-commutative 27.6%

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

      7. fma-def 27.6%

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

      8. distribute-rgt-neg-in 27.6%

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

   3. Simplified 27.6%

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

   4. Taylor expanded in t around -inf 1.6%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)\right)} \cdot \sqrt{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 1.6%

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

      2. *-commutative 1.6%

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

      3. sub-neg 1.6%

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

      4. metadata-eval 1.6%

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

      5. +-commutative 1.6%

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

      6. +-commutative 1.6%

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

   6. Simplified 1.6%

      \[\leadsto \color{blue}{\left(-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right)} \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 93.4%

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

      3. sqr-neg 93.4%

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

      4. sqrt-unprod 93.5%

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

      5. add-sqr-sqrt 93.4%

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

      6. associate-*l* 93.4%

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

      7. *-commutative 93.4%

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

      8. sqrt-unprod 94.9%

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

      9. metadata-eval 94.9%

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

      10. metadata-eval 94.9%

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

      11. *-commutative 94.9%

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

      12. pow1/2 94.9%

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

   8. Applied egg-rr 94.9%

      \[\leadsto \color{blue}{{\left(\frac{x + -1}{x + 1}\right)}^{0.5}} \]
   9. Step-by-step derivation

      1. unpow1/2 94.9%

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

   10. Simplified 94.9%

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

   11. Taylor expanded in x around inf 94.9%

       \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.6 \cdot 10^{+34}:\\ \;\;\;\;\frac{\left(--1\right) - x}{x + -1}\\ \mathbf{elif}\;t \leq -5.5 \cdot 10^{-177}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \ell \cdot \frac{\ell}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}{t}}\\ \mathbf{elif}\;t \leq -1.02 \cdot 10^{-274}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{x \cdot 0.5 - 0.5} \cdot \frac{t}{\ell}\right)\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-163}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 480000:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\left(\frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{{x}^{2}} + \left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}{t}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 3: 80.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot {t}^{2}\\ t_2 := \frac{\sqrt{2}}{\frac{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(t_1 + \ell \cdot \frac{\ell}{x}\right)\right) + \frac{t_1 + {\ell}^{2}}{x}}}{t}}\\ \mathbf{if}\;t \leq -5.2 \cdot 10^{+35}:\\ \;\;\;\;\frac{\left(--1\right) - x}{x + -1}\\ \mathbf{elif}\;t \leq -1.2 \cdot 10^{-176}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -4.3 \cdot 10^{-274}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{x \cdot 0.5 - 0.5} \cdot \frac{t}{\ell}\right)\\ \mathbf{elif}\;t \leq 10^{-164}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 950000:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* 2.0 (pow t 2.0)))
        (t_2
         (/
          (sqrt 2.0)
          (/
           (sqrt
            (+
             (+ (* 2.0 (/ (pow t 2.0) x)) (+ t_1 (* l (/ l x))))
             (/ (+ t_1 (pow l 2.0)) x)))
           t))))
   (if (<= t -5.2e+35)
     (/ (- (- -1.0) x) (+ x -1.0))
     (if (<= t -1.2e-176)
       t_2
       (if (<= t -4.3e-274)
         (- (sqrt (/ (+ x -1.0) (+ x 1.0))))
         (if (<= t 2.3e-305)
           (* (sqrt 2.0) (* (sqrt (- (* x 0.5) 0.5)) (/ t l)))
           (if (<= t 1e-164)
             1.0
             (if (<= t 950000.0) t_2 (+ 1.0 (/ -1.0 x))))))))))

C

double code(double x, double l, double t) {
	double t_1 = 2.0 * pow(t, 2.0);
	double t_2 = sqrt(2.0) / (sqrt((((2.0 * (pow(t, 2.0) / x)) + (t_1 + (l * (l / x)))) + ((t_1 + pow(l, 2.0)) / x))) / t);
	double tmp;
	if (t <= -5.2e+35) {
		tmp = (-(-1.0) - x) / (x + -1.0);
	} else if (t <= -1.2e-176) {
		tmp = t_2;
	} else if (t <= -4.3e-274) {
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= 2.3e-305) {
		tmp = sqrt(2.0) * (sqrt(((x * 0.5) - 0.5)) * (t / l));
	} else if (t <= 1e-164) {
		tmp = 1.0;
	} else if (t <= 950000.0) {
		tmp = t_2;
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 2.0d0 * (t ** 2.0d0)
    t_2 = sqrt(2.0d0) / (sqrt((((2.0d0 * ((t ** 2.0d0) / x)) + (t_1 + (l * (l / x)))) + ((t_1 + (l ** 2.0d0)) / x))) / t)
    if (t <= (-5.2d+35)) then
        tmp = (-(-1.0d0) - x) / (x + (-1.0d0))
    else if (t <= (-1.2d-176)) then
        tmp = t_2
    else if (t <= (-4.3d-274)) then
        tmp = -sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else if (t <= 2.3d-305) then
        tmp = sqrt(2.0d0) * (sqrt(((x * 0.5d0) - 0.5d0)) * (t / l))
    else if (t <= 1d-164) then
        tmp = 1.0d0
    else if (t <= 950000.0d0) then
        tmp = t_2
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double t_1 = 2.0 * Math.pow(t, 2.0);
	double t_2 = Math.sqrt(2.0) / (Math.sqrt((((2.0 * (Math.pow(t, 2.0) / x)) + (t_1 + (l * (l / x)))) + ((t_1 + Math.pow(l, 2.0)) / x))) / t);
	double tmp;
	if (t <= -5.2e+35) {
		tmp = (-(-1.0) - x) / (x + -1.0);
	} else if (t <= -1.2e-176) {
		tmp = t_2;
	} else if (t <= -4.3e-274) {
		tmp = -Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= 2.3e-305) {
		tmp = Math.sqrt(2.0) * (Math.sqrt(((x * 0.5) - 0.5)) * (t / l));
	} else if (t <= 1e-164) {
		tmp = 1.0;
	} else if (t <= 950000.0) {
		tmp = t_2;
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = 2.0 * math.pow(t, 2.0)
	t_2 = math.sqrt(2.0) / (math.sqrt((((2.0 * (math.pow(t, 2.0) / x)) + (t_1 + (l * (l / x)))) + ((t_1 + math.pow(l, 2.0)) / x))) / t)
	tmp = 0
	if t <= -5.2e+35:
		tmp = (-(-1.0) - x) / (x + -1.0)
	elif t <= -1.2e-176:
		tmp = t_2
	elif t <= -4.3e-274:
		tmp = -math.sqrt(((x + -1.0) / (x + 1.0)))
	elif t <= 2.3e-305:
		tmp = math.sqrt(2.0) * (math.sqrt(((x * 0.5) - 0.5)) * (t / l))
	elif t <= 1e-164:
		tmp = 1.0
	elif t <= 950000.0:
		tmp = t_2
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

function code(x, l, t)
	t_1 = Float64(2.0 * (t ^ 2.0))
	t_2 = Float64(sqrt(2.0) / Float64(sqrt(Float64(Float64(Float64(2.0 * Float64((t ^ 2.0) / x)) + Float64(t_1 + Float64(l * Float64(l / x)))) + Float64(Float64(t_1 + (l ^ 2.0)) / x))) / t))
	tmp = 0.0
	if (t <= -5.2e+35)
		tmp = Float64(Float64(Float64(-(-1.0)) - x) / Float64(x + -1.0));
	elseif (t <= -1.2e-176)
		tmp = t_2;
	elseif (t <= -4.3e-274)
		tmp = Float64(-sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0))));
	elseif (t <= 2.3e-305)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(Float64(x * 0.5) - 0.5)) * Float64(t / l)));
	elseif (t <= 1e-164)
		tmp = 1.0;
	elseif (t <= 950000.0)
		tmp = t_2;
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = 2.0 * (t ^ 2.0);
	t_2 = sqrt(2.0) / (sqrt((((2.0 * ((t ^ 2.0) / x)) + (t_1 + (l * (l / x)))) + ((t_1 + (l ^ 2.0)) / x))) / t);
	tmp = 0.0;
	if (t <= -5.2e+35)
		tmp = (-(-1.0) - x) / (x + -1.0);
	elseif (t <= -1.2e-176)
		tmp = t_2;
	elseif (t <= -4.3e-274)
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	elseif (t <= 2.3e-305)
		tmp = sqrt(2.0) * (sqrt(((x * 0.5) - 0.5)) * (t / l));
	elseif (t <= 1e-164)
		tmp = 1.0;
	elseif (t <= 950000.0)
		tmp = t_2;
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(2.0 * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[N[(N[(N[(2.0 * N[(N[Power[t, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5.2e+35], N[(N[((--1.0) - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.2e-176], t$95$2, If[LessEqual[t, -4.3e-274], (-N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, 2.3e-305], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(N[(x * 0.5), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1e-164], 1.0, If[LessEqual[t, 950000.0], t$95$2, N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -5.20000000000000013e35

   1. Initial program 24.9%

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

      1. *-commutative 24.9%

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

      2. associate-*l/ 24.8%

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

      3. fma-neg 24.8%

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

      4. sub-neg 24.8%

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

      5. metadata-eval 24.8%

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

      6. +-commutative 24.8%

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

      7. fma-def 24.8%

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

      8. distribute-rgt-neg-in 24.8%

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

   3. Simplified 24.8%

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

   4. Taylor expanded in t around -inf 91.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)\right)} \cdot \sqrt{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 91.4%

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

      2. *-commutative 91.4%

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

      3. sub-neg 91.4%

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

      4. metadata-eval 91.4%

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

      5. +-commutative 91.4%

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

      6. +-commutative 91.4%

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

   6. Simplified 91.4%

      \[\leadsto \color{blue}{\left(-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right)} \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 1.6%

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

      3. sqr-neg 1.6%

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

      4. sqrt-unprod 1.6%

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

      5. add-sqr-sqrt 1.6%

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

      6. associate-*l* 1.6%

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

      7. *-commutative 1.6%

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

      8. sqrt-unprod 1.6%

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

      9. metadata-eval 1.6%

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

      10. metadata-eval 1.6%

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

      11. *-commutative 1.6%

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

      12. pow1/2 1.6%

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

   8. Applied egg-rr 1.6%

      \[\leadsto \color{blue}{{\left(\frac{x + -1}{x + 1}\right)}^{0.5}} \]
   9. Step-by-step derivation

      1. unpow1/2 1.6%

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

   10. Simplified 1.6%

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

   12. Applied egg-rr 92.9%

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

   if -5.20000000000000013e35 < t < -1.20000000000000003e-176 or 9.99999999999999962e-165 < t < 9.5e5

   1. Initial program 60.0%

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

   3. Simplified 60.0%

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

   4. Taylor expanded in x around inf 89.5%

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

      1. unpow2 89.5%

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

      2. *-un-lft-identity 89.5%

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

      3. times-frac 89.5%

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

   6. Applied egg-rr 89.5%

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

   if -1.20000000000000003e-176 < t < -4.29999999999999989e-274

   1. Initial program 2.6%

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

      1. *-commutative 2.6%

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

      2. associate-*l/ 2.6%

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

      3. fma-neg 2.6%

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

      4. sub-neg 2.6%

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

      5. metadata-eval 2.6%

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

      6. +-commutative 2.6%

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

      7. fma-def 2.6%

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

      8. distribute-rgt-neg-in 2.6%

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

   3. Simplified 2.6%

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

   4. Taylor expanded in t around -inf 58.5%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)\right)} \cdot \sqrt{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 58.5%

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

      2. *-commutative 58.5%

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

      3. sub-neg 58.5%

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

      4. metadata-eval 58.5%

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

      5. +-commutative 58.5%

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

      6. +-commutative 58.5%

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

   6. Simplified 58.5%

      \[\leadsto \color{blue}{\left(-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right)} \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. distribute-lft-neg-out 58.5%

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

      2. associate-*l* 58.5%

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

      3. *-commutative 58.5%

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

      4. sqrt-unprod 59.4%

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

      5. metadata-eval 59.4%

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

      6. metadata-eval 59.4%

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

      7. *-commutative 59.4%

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

      8. *-un-lft-identity 59.4%

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

      9. neg-sub0 59.4%

         \[\leadsto \color{blue}{0 - \sqrt{\frac{-1 + x}{x + 1}}} \]

      10. +-commutative 59.4%

          \[\leadsto 0 - \sqrt{\frac{\color{blue}{x + -1}}{x + 1}} \]

   8. Applied egg-rr 59.4%

      \[\leadsto \color{blue}{0 - \sqrt{\frac{x + -1}{x + 1}}} \]
   9. Step-by-step derivation

      1. sub0-neg 59.4%

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

   10. Simplified 59.4%

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

   if -4.29999999999999989e-274 < t < 2.3e-305

   1. Initial program 6.5%

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

      1. *-commutative 6.5%

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

      2. associate-*l/ 6.5%

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

      3. fma-neg 5.9%

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

      4. sub-neg 5.9%

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

      5. metadata-eval 5.9%

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

      6. +-commutative 5.9%

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

      7. fma-def 5.9%

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

      8. distribute-rgt-neg-in 5.9%

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

   3. Simplified 5.9%

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

   4. Taylor expanded in l around inf 25.9%

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

      1. *-commutative 25.9%

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

      2. associate--l+ 69.8%

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

      3. sub-neg 69.8%

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

      4. metadata-eval 69.8%

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

      5. +-commutative 69.8%

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

      6. sub-neg 69.8%

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

      7. metadata-eval 69.8%

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

      8. +-commutative 69.8%

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

   6. Simplified 69.8%

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

   7. Taylor expanded in x around 0 99.7%

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

   if 2.3e-305 < t < 9.99999999999999962e-165

   1. Initial program 2.4%

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

      1. *-commutative 2.4%

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

      2. associate-*l/ 2.4%

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

      3. fma-neg 2.4%

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

      4. sub-neg 2.4%

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

      5. metadata-eval 2.4%

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

      6. +-commutative 2.4%

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

      7. fma-def 2.4%

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

      8. distribute-rgt-neg-in 2.4%

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

   3. Simplified 2.4%

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

   4. Taylor expanded in t around -inf 1.9%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)\right)} \cdot \sqrt{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 1.9%

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

      2. *-commutative 1.9%

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

      3. sub-neg 1.9%

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

      4. metadata-eval 1.9%

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

      5. +-commutative 1.9%

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

      6. +-commutative 1.9%

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

   6. Simplified 1.9%

      \[\leadsto \color{blue}{\left(-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right)} \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 64.1%

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

      3. sqr-neg 64.1%

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

      4. sqrt-unprod 64.1%

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

      5. add-sqr-sqrt 64.1%

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

      6. associate-*l* 64.1%

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

      7. *-commutative 64.1%

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

      8. sqrt-unprod 65.0%

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

      9. metadata-eval 65.0%

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

      10. metadata-eval 65.0%

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

      11. *-commutative 65.0%

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

      12. pow1/2 65.0%

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

   8. Applied egg-rr 65.0%

      \[\leadsto \color{blue}{{\left(\frac{x + -1}{x + 1}\right)}^{0.5}} \]
   9. Step-by-step derivation

      1. unpow1/2 65.0%

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

   10. Simplified 65.0%

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

   11. Taylor expanded in x around inf 65.0%

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

   if 9.5e5 < t

   1. Initial program 27.6%

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

      1. *-commutative 27.6%

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

      2. associate-*l/ 27.6%

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

      3. fma-neg 27.6%

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

      4. sub-neg 27.6%

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

      5. metadata-eval 27.6%

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

      6. +-commutative 27.6%

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

      7. fma-def 27.6%

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

      8. distribute-rgt-neg-in 27.6%

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

   3. Simplified 27.6%

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

   4. Taylor expanded in t around -inf 1.6%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)\right)} \cdot \sqrt{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 1.6%

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

      2. *-commutative 1.6%

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

      3. sub-neg 1.6%

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

      4. metadata-eval 1.6%

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

      5. +-commutative 1.6%

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

      6. +-commutative 1.6%

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

   6. Simplified 1.6%

      \[\leadsto \color{blue}{\left(-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right)} \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 93.4%

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

      3. sqr-neg 93.4%

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

      4. sqrt-unprod 93.5%

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

      5. add-sqr-sqrt 93.4%

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

      6. associate-*l* 93.4%

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

      7. *-commutative 93.4%

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

      8. sqrt-unprod 94.9%

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

      9. metadata-eval 94.9%

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

      10. metadata-eval 94.9%

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

      11. *-commutative 94.9%

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

      12. pow1/2 94.9%

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

   8. Applied egg-rr 94.9%

      \[\leadsto \color{blue}{{\left(\frac{x + -1}{x + 1}\right)}^{0.5}} \]
   9. Step-by-step derivation

      1. unpow1/2 94.9%

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

   10. Simplified 94.9%

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

   11. Taylor expanded in x around inf 94.9%

       \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{+35}:\\ \;\;\;\;\frac{\left(--1\right) - x}{x + -1}\\ \mathbf{elif}\;t \leq -1.2 \cdot 10^{-176}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \ell \cdot \frac{\ell}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}{t}}\\ \mathbf{elif}\;t \leq -4.3 \cdot 10^{-274}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{x \cdot 0.5 - 0.5} \cdot \frac{t}{\ell}\right)\\ \mathbf{elif}\;t \leq 10^{-164}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 950000:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \ell \cdot \frac{\ell}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}{t}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 4: 76.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-157}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-305}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -1.15e-157)
   (- (sqrt (/ (+ x -1.0) (+ x 1.0))))
   (if (<= t 2.3e-305) (/ t (/ l (sqrt x))) (+ 1.0 (/ -1.0 x)))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.15e-157) {
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= 2.3e-305) {
		tmp = t / (l / sqrt(x));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.15d-157)) then
        tmp = -sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else if (t <= 2.3d-305) then
        tmp = t / (l / sqrt(x))
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.15e-157) {
		tmp = -Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= 2.3e-305) {
		tmp = t / (l / Math.sqrt(x));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -1.15e-157:
		tmp = -math.sqrt(((x + -1.0) / (x + 1.0)))
	elif t <= 2.3e-305:
		tmp = t / (l / math.sqrt(x))
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -1.15e-157)
		tmp = Float64(-sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0))));
	elseif (t <= 2.3e-305)
		tmp = Float64(t / Float64(l / sqrt(x)));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1.15e-157)
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	elseif (t <= 2.3e-305)
		tmp = t / (l / sqrt(x));
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -1.15e-157], (-N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, 2.3e-305], N[(t / N[(l / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.14999999999999994e-157

   1. Initial program 39.3%

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

      1. *-commutative 39.3%

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

      2. associate-*l/ 39.1%

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

      3. fma-neg 39.1%

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

      4. sub-neg 39.1%

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

      5. metadata-eval 39.1%

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

      6. +-commutative 39.1%

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

      7. fma-def 39.1%

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

      8. distribute-rgt-neg-in 39.1%

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

   3. Simplified 39.1%

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

   4. Taylor expanded in t around -inf 85.7%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)\right)} \cdot \sqrt{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 85.7%

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

      2. *-commutative 85.7%

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

      3. sub-neg 85.7%

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

      4. metadata-eval 85.7%

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

      5. +-commutative 85.7%

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

      6. +-commutative 85.7%

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

   6. Simplified 85.7%

      \[\leadsto \color{blue}{\left(-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right)} \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. distribute-lft-neg-out 85.7%

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

      2. associate-*l* 85.7%

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

      3. *-commutative 85.7%

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

      4. sqrt-unprod 87.0%

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

      5. metadata-eval 87.0%

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

      6. metadata-eval 87.0%

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

      7. *-commutative 87.0%

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

      8. *-un-lft-identity 87.0%

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

      9. neg-sub0 87.0%

         \[\leadsto \color{blue}{0 - \sqrt{\frac{-1 + x}{x + 1}}} \]

      10. +-commutative 87.0%

          \[\leadsto 0 - \sqrt{\frac{\color{blue}{x + -1}}{x + 1}} \]

   8. Applied egg-rr 87.0%

      \[\leadsto \color{blue}{0 - \sqrt{\frac{x + -1}{x + 1}}} \]
   9. Step-by-step derivation

      1. sub0-neg 87.0%

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

   10. Simplified 87.0%

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

   if -1.14999999999999994e-157 < t < 2.3e-305

   1. Initial program 3.2%

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

      1. *-commutative 3.2%

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

      2. associate-*l/ 3.2%

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

      3. fma-neg 3.1%

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

      4. sub-neg 3.1%

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

      5. metadata-eval 3.1%

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

      6. +-commutative 3.1%

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

      7. fma-def 3.1%

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

      8. distribute-rgt-neg-in 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in l around inf 10.2%

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

      1. sub-neg 10.2%

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

      2. +-commutative 10.2%

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

      3. metadata-eval 10.2%

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

      4. associate-+l+ 10.2%

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

      5. sub-neg 10.2%

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

      6. metadata-eval 10.2%

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

      7. +-commutative 10.2%

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

      8. sub-neg 10.2%

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

      9. metadata-eval 10.2%

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

      10. +-commutative 10.2%

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

   6. Simplified 10.2%

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

   7. Taylor expanded in x around inf 49.4%

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

   8. Taylor expanded in t around 0 48.4%

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

      1. associate-*l/ 49.7%

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

      2. associate-/l* 49.7%

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

   10. Applied egg-rr 49.7%

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

   if 2.3e-305 < t

   1. Initial program 29.8%

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

      1. *-commutative 29.8%

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

      2. associate-*l/ 29.8%

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

      3. fma-neg 29.8%

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

      4. sub-neg 29.8%

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

      5. metadata-eval 29.8%

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

      6. +-commutative 29.8%

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

      7. fma-def 29.8%

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

      8. distribute-rgt-neg-in 29.8%

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

   3. Simplified 29.8%

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

   4. Taylor expanded in t around -inf 1.7%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)\right)} \cdot \sqrt{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 1.7%

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

      2. *-commutative 1.7%

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

      3. sub-neg 1.7%

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

      4. metadata-eval 1.7%

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

      5. +-commutative 1.7%

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

      6. +-commutative 1.7%

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

   6. Simplified 1.7%

      \[\leadsto \color{blue}{\left(-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right)} \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 81.8%

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

      3. sqr-neg 81.8%

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

      4. sqrt-unprod 81.8%

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

      5. add-sqr-sqrt 81.8%

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

      6. associate-*l* 81.7%

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

      7. *-commutative 81.7%

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

      8. sqrt-unprod 83.0%

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

      9. metadata-eval 83.0%

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

      10. metadata-eval 83.0%

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

      11. *-commutative 83.0%

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

      12. pow1/2 83.0%

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

   8. Applied egg-rr 83.0%

      \[\leadsto \color{blue}{{\left(\frac{x + -1}{x + 1}\right)}^{0.5}} \]
   9. Step-by-step derivation

      1. unpow1/2 83.0%

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

   10. Simplified 83.0%

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

   11. Taylor expanded in x around inf 83.0%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-157}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-305}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 5: 76.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{-274}:\\ \;\;\;\;\frac{\left(--1\right) - x}{x + -1}\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{-305}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -4.2e-274)
   (/ (- (- -1.0) x) (+ x -1.0))
   (if (<= t 2.15e-305) (* (/ t l) (sqrt x)) (+ 1.0 (/ -1.0 x)))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -4.2e-274) {
		tmp = (-(-1.0) - x) / (x + -1.0);
	} else if (t <= 2.15e-305) {
		tmp = (t / l) * sqrt(x);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-4.2d-274)) then
        tmp = (-(-1.0d0) - x) / (x + (-1.0d0))
    else if (t <= 2.15d-305) then
        tmp = (t / l) * sqrt(x)
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -4.2e-274) {
		tmp = (-(-1.0) - x) / (x + -1.0);
	} else if (t <= 2.15e-305) {
		tmp = (t / l) * Math.sqrt(x);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -4.2e-274:
		tmp = (-(-1.0) - x) / (x + -1.0)
	elif t <= 2.15e-305:
		tmp = (t / l) * math.sqrt(x)
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -4.2e-274)
		tmp = Float64(Float64(Float64(-(-1.0)) - x) / Float64(x + -1.0));
	elseif (t <= 2.15e-305)
		tmp = Float64(Float64(t / l) * sqrt(x));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -4.2e-274)
		tmp = (-(-1.0) - x) / (x + -1.0);
	elseif (t <= 2.15e-305)
		tmp = (t / l) * sqrt(x);
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -4.2e-274], N[(N[((--1.0) - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.15e-305], N[(N[(t / l), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.19999999999999988e-274

   1. Initial program 32.8%

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

      1. *-commutative 32.8%

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

      2. associate-*l/ 32.6%

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

      3. fma-neg 32.6%

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

      4. sub-neg 32.6%

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

      5. metadata-eval 32.6%

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

      6. +-commutative 32.6%

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

      7. fma-def 32.6%

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

      8. distribute-rgt-neg-in 32.6%

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

   3. Simplified 32.6%

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

   4. Taylor expanded in t around -inf 79.2%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)\right)} \cdot \sqrt{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 79.2%

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

      2. *-commutative 79.2%

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

      3. sub-neg 79.2%

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

      4. metadata-eval 79.2%

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

      5. +-commutative 79.2%

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

      6. +-commutative 79.2%

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

   6. Simplified 79.2%

      \[\leadsto \color{blue}{\left(-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right)} \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 1.7%

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

      3. sqr-neg 1.7%

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

      4. sqrt-unprod 1.7%

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

      5. add-sqr-sqrt 1.7%

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

      6. associate-*l* 1.7%

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

      7. *-commutative 1.7%

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

      8. sqrt-unprod 1.7%

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

      9. metadata-eval 1.7%

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

      10. metadata-eval 1.7%

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

      11. *-commutative 1.7%

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

      12. pow1/2 1.7%

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

   8. Applied egg-rr 1.7%

      \[\leadsto \color{blue}{{\left(\frac{x + -1}{x + 1}\right)}^{0.5}} \]
   9. Step-by-step derivation

      1. unpow1/2 1.7%

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

   10. Simplified 1.7%

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

   12. Applied egg-rr 80.1%

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

   if -4.19999999999999988e-274 < t < 2.1500000000000001e-305

   1. Initial program 6.5%

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

      1. *-commutative 6.5%

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

      2. associate-*l/ 6.5%

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

      3. fma-neg 5.9%

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

      4. sub-neg 5.9%

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

      5. metadata-eval 5.9%

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

      6. +-commutative 5.9%

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

      7. fma-def 5.9%

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

      8. distribute-rgt-neg-in 5.9%

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

   3. Simplified 5.9%

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

   4. Taylor expanded in l around inf 26.5%

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

      1. sub-neg 26.5%

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

      2. +-commutative 26.5%

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

      3. metadata-eval 26.5%

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

      4. associate-+l+ 26.5%

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

      5. sub-neg 26.5%

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

      6. metadata-eval 26.5%

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

      7. +-commutative 26.5%

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

      8. sub-neg 26.5%

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

      9. metadata-eval 26.5%

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

      10. +-commutative 26.5%

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

   6. Simplified 26.5%

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

   7. Taylor expanded in x around inf 98.9%

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

   8. Taylor expanded in t around 0 99.0%

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

   if 2.1500000000000001e-305 < t

   1. Initial program 29.8%

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

      1. *-commutative 29.8%

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

      2. associate-*l/ 29.8%

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

      3. fma-neg 29.8%

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

      4. sub-neg 29.8%

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

      5. metadata-eval 29.8%

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

      6. +-commutative 29.8%

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

      7. fma-def 29.8%

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

      8. distribute-rgt-neg-in 29.8%

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

   3. Simplified 29.8%

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

   4. Taylor expanded in t around -inf 1.7%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)\right)} \cdot \sqrt{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 1.7%

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

      2. *-commutative 1.7%

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

      3. sub-neg 1.7%

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

      4. metadata-eval 1.7%

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

      5. +-commutative 1.7%

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

      6. +-commutative 1.7%

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

   6. Simplified 1.7%

      \[\leadsto \color{blue}{\left(-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right)} \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 81.8%

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

      3. sqr-neg 81.8%

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

      4. sqrt-unprod 81.8%

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

      5. add-sqr-sqrt 81.8%

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

      6. associate-*l* 81.7%

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

      7. *-commutative 81.7%

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

      8. sqrt-unprod 83.0%

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

      9. metadata-eval 83.0%

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

      10. metadata-eval 83.0%

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

      11. *-commutative 83.0%

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

      12. pow1/2 83.0%

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

   8. Applied egg-rr 83.0%

      \[\leadsto \color{blue}{{\left(\frac{x + -1}{x + 1}\right)}^{0.5}} \]
   9. Step-by-step derivation

      1. unpow1/2 83.0%

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

   10. Simplified 83.0%

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

   11. Taylor expanded in x around inf 83.0%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{-274}:\\ \;\;\;\;\frac{\left(--1\right) - x}{x + -1}\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{-305}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 6: 75.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{-158}:\\ \;\;\;\;\frac{\left(--1\right) - x}{x + -1}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-305}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -2.8e-158)
   (/ (- (- -1.0) x) (+ x -1.0))
   (if (<= t 2.3e-305) (/ t (/ l (sqrt x))) (+ 1.0 (/ -1.0 x)))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -2.8e-158) {
		tmp = (-(-1.0) - x) / (x + -1.0);
	} else if (t <= 2.3e-305) {
		tmp = t / (l / sqrt(x));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-2.8d-158)) then
        tmp = (-(-1.0d0) - x) / (x + (-1.0d0))
    else if (t <= 2.3d-305) then
        tmp = t / (l / sqrt(x))
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -2.8e-158) {
		tmp = (-(-1.0) - x) / (x + -1.0);
	} else if (t <= 2.3e-305) {
		tmp = t / (l / Math.sqrt(x));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -2.8e-158:
		tmp = (-(-1.0) - x) / (x + -1.0)
	elif t <= 2.3e-305:
		tmp = t / (l / math.sqrt(x))
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -2.8e-158)
		tmp = Float64(Float64(Float64(-(-1.0)) - x) / Float64(x + -1.0));
	elseif (t <= 2.3e-305)
		tmp = Float64(t / Float64(l / sqrt(x)));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -2.8e-158)
		tmp = (-(-1.0) - x) / (x + -1.0);
	elseif (t <= 2.3e-305)
		tmp = t / (l / sqrt(x));
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -2.8e-158], N[(N[((--1.0) - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.3e-305], N[(t / N[(l / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.80000000000000002e-158

   1. Initial program 39.3%

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

      1. *-commutative 39.3%

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

      2. associate-*l/ 39.1%

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

      3. fma-neg 39.1%

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

      4. sub-neg 39.1%

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

      5. metadata-eval 39.1%

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

      6. +-commutative 39.1%

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

      7. fma-def 39.1%

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

      8. distribute-rgt-neg-in 39.1%

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

   3. Simplified 39.1%

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

   4. Taylor expanded in t around -inf 85.7%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)\right)} \cdot \sqrt{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 85.7%

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

      2. *-commutative 85.7%

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

      3. sub-neg 85.7%

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

      4. metadata-eval 85.7%

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

      5. +-commutative 85.7%

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

      6. +-commutative 85.7%

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

   6. Simplified 85.7%

      \[\leadsto \color{blue}{\left(-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right)} \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 1.6%

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

      3. sqr-neg 1.6%

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

      4. sqrt-unprod 1.6%

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

      5. add-sqr-sqrt 1.6%

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

      6. associate-*l* 1.6%

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

      7. *-commutative 1.6%

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

      8. sqrt-unprod 1.6%

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

      9. metadata-eval 1.6%

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

      10. metadata-eval 1.6%

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

      11. *-commutative 1.6%

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

      12. pow1/2 1.6%

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

   8. Applied egg-rr 1.6%

      \[\leadsto \color{blue}{{\left(\frac{x + -1}{x + 1}\right)}^{0.5}} \]
   9. Step-by-step derivation

      1. unpow1/2 1.6%

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

   10. Simplified 1.6%

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

   12. Applied egg-rr 86.7%

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

   if -2.80000000000000002e-158 < t < 2.3e-305

   1. Initial program 3.2%

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

      1. *-commutative 3.2%

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

      2. associate-*l/ 3.2%

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

      3. fma-neg 3.1%

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

      4. sub-neg 3.1%

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

      5. metadata-eval 3.1%

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

      6. +-commutative 3.1%

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

      7. fma-def 3.1%

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

      8. distribute-rgt-neg-in 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in l around inf 10.2%

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

      1. sub-neg 10.2%

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

      2. +-commutative 10.2%

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

      3. metadata-eval 10.2%

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

      4. associate-+l+ 10.2%

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

      5. sub-neg 10.2%

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

      6. metadata-eval 10.2%

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

      7. +-commutative 10.2%

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

      8. sub-neg 10.2%

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

      9. metadata-eval 10.2%

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

      10. +-commutative 10.2%

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

   6. Simplified 10.2%

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

   7. Taylor expanded in x around inf 49.4%

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

   8. Taylor expanded in t around 0 48.4%

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

      1. associate-*l/ 49.7%

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

      2. associate-/l* 49.7%

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

   10. Applied egg-rr 49.7%

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

   if 2.3e-305 < t

   1. Initial program 29.8%

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

      1. *-commutative 29.8%

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

      2. associate-*l/ 29.8%

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

      3. fma-neg 29.8%

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

      4. sub-neg 29.8%

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

      5. metadata-eval 29.8%

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

      6. +-commutative 29.8%

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

      7. fma-def 29.8%

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

      8. distribute-rgt-neg-in 29.8%

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

   3. Simplified 29.8%

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

   4. Taylor expanded in t around -inf 1.7%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)\right)} \cdot \sqrt{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 1.7%

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

      2. *-commutative 1.7%

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

      3. sub-neg 1.7%

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

      4. metadata-eval 1.7%

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

      5. +-commutative 1.7%

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

      6. +-commutative 1.7%

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

   6. Simplified 1.7%

      \[\leadsto \color{blue}{\left(-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right)} \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 81.8%

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

      3. sqr-neg 81.8%

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

      4. sqrt-unprod 81.8%

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

      5. add-sqr-sqrt 81.8%

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

      6. associate-*l* 81.7%

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

      7. *-commutative 81.7%

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

      8. sqrt-unprod 83.0%

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

      9. metadata-eval 83.0%

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

      10. metadata-eval 83.0%

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

      11. *-commutative 83.0%

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

      12. pow1/2 83.0%

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

   8. Applied egg-rr 83.0%

      \[\leadsto \color{blue}{{\left(\frac{x + -1}{x + 1}\right)}^{0.5}} \]
   9. Step-by-step derivation

      1. unpow1/2 83.0%

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

   10. Simplified 83.0%

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

   11. Taylor expanded in x around inf 83.0%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{-158}:\\ \;\;\;\;\frac{\left(--1\right) - x}{x + -1}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-305}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 7: 75.7% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(--1\right) - x}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -2e-310) (/ (- (- -1.0) x) (+ x -1.0)) (+ 1.0 (/ -1.0 x))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -2e-310) {
		tmp = (-(-1.0) - x) / (x + -1.0);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-2d-310)) then
        tmp = (-(-1.0d0) - x) / (x + (-1.0d0))
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -2e-310) {
		tmp = (-(-1.0) - x) / (x + -1.0);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -2e-310:
		tmp = (-(-1.0) - x) / (x + -1.0)
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -2e-310)
		tmp = Float64(Float64(Float64(-(-1.0)) - x) / Float64(x + -1.0));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -2e-310)
		tmp = (-(-1.0) - x) / (x + -1.0);
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -2e-310], N[(N[((--1.0) - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.999999999999994e-310

   1. Initial program 32.0%

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

      1. *-commutative 32.0%

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

      2. associate-*l/ 31.9%

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

      3. fma-neg 31.8%

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

      4. sub-neg 31.8%

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

      5. metadata-eval 31.8%

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

      6. +-commutative 31.8%

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

      7. fma-def 31.8%

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

      8. distribute-rgt-neg-in 31.8%

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

   3. Simplified 31.8%

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

   4. Taylor expanded in t around -inf 76.8%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)\right)} \cdot \sqrt{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 76.8%

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

      2. *-commutative 76.8%

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

      3. sub-neg 76.8%

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

      4. metadata-eval 76.8%

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

      5. +-commutative 76.8%

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

      6. +-commutative 76.8%

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

   6. Simplified 76.8%

      \[\leadsto \color{blue}{\left(-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right)} \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 1.7%

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

      3. sqr-neg 1.7%

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

      4. sqrt-unprod 1.7%

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

      5. add-sqr-sqrt 1.7%

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

      6. associate-*l* 1.7%

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

      7. *-commutative 1.7%

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

      8. sqrt-unprod 1.7%

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

      9. metadata-eval 1.7%

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

      10. metadata-eval 1.7%

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

      11. *-commutative 1.7%

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

      12. pow1/2 1.7%

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

   8. Applied egg-rr 1.7%

      \[\leadsto \color{blue}{{\left(\frac{x + -1}{x + 1}\right)}^{0.5}} \]
   9. Step-by-step derivation

      1. unpow1/2 1.7%

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

   10. Simplified 1.7%

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

   12. Applied egg-rr 77.7%

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

   if -1.999999999999994e-310 < t

   1. Initial program 29.6%

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

      1. *-commutative 29.6%

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

      2. associate-*l/ 29.6%

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

      3. fma-neg 29.6%

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

      4. sub-neg 29.6%

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

      5. metadata-eval 29.6%

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

      6. +-commutative 29.6%

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

      7. fma-def 29.6%

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

      8. distribute-rgt-neg-in 29.6%

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

   3. Simplified 29.6%

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

   4. Taylor expanded in t around -inf 1.7%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)\right)} \cdot \sqrt{2} \]
   5. Step-by-step derivation

      1. mul-1-neg 1.7%

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

      2. *-commutative 1.7%

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

      3. sub-neg 1.7%

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

      4. metadata-eval 1.7%

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

      5. +-commutative 1.7%

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

      6. +-commutative 1.7%

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

   6. Simplified 1.7%

      \[\leadsto \color{blue}{\left(-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right)} \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 81.1%

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

      3. sqr-neg 81.1%

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

      4. sqrt-unprod 81.1%

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

      5. add-sqr-sqrt 81.1%

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

      6. associate-*l* 81.1%

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

      7. *-commutative 81.1%

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

      8. sqrt-unprod 82.4%

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

      9. metadata-eval 82.4%

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

      10. metadata-eval 82.4%

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

      11. *-commutative 82.4%

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

      12. pow1/2 82.4%

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

   8. Applied egg-rr 82.4%

      \[\leadsto \color{blue}{{\left(\frac{x + -1}{x + 1}\right)}^{0.5}} \]
   9. Step-by-step derivation

      1. unpow1/2 82.4%

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

   10. Simplified 82.4%

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

   11. Taylor expanded in x around inf 82.4%

       \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(--1\right) - x}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 8: 39.0% accurate, 45.0× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{-1}{x} \end{array} \]

FPCore

(FPCore (x l t) :precision binary64 (+ 1.0 (/ -1.0 x)))

C

double code(double x, double l, double t) {
	return 1.0 + (-1.0 / x);
}

Fortran

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

Java

public static double code(double x, double l, double t) {
	return 1.0 + (-1.0 / x);
}

Python

def code(x, l, t):
	return 1.0 + (-1.0 / x)

Julia

function code(x, l, t)
	return Float64(1.0 + Float64(-1.0 / x))
end

MATLAB

function tmp = code(x, l, t)
	tmp = 1.0 + (-1.0 / x);
end

Wolfram

code[x_, l_, t_] := N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.8%

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

   1. *-commutative 30.8%

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

   2. associate-*l/ 30.7%

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

   3. fma-neg 30.7%

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

   4. sub-neg 30.7%

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

   5. metadata-eval 30.7%

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

   6. +-commutative 30.7%

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

   7. fma-def 30.7%

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

   8. distribute-rgt-neg-in 30.7%

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

3. Simplified 30.7%

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

4. Taylor expanded in t around -inf 39.3%

   \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)\right)} \cdot \sqrt{2} \]
5. Step-by-step derivation

   1. mul-1-neg 39.3%

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

   2. *-commutative 39.3%

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

   3. sub-neg 39.3%

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

   4. metadata-eval 39.3%

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

   5. +-commutative 39.3%

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

   6. +-commutative 39.3%

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

6. Simplified 39.3%

   \[\leadsto \color{blue}{\left(-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right)} \cdot \sqrt{2} \]
7. Step-by-step derivation

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 41.4%

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

   3. sqr-neg 41.4%

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

   4. sqrt-unprod 41.4%

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

   5. add-sqr-sqrt 41.4%

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

   6. associate-*l* 41.4%

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

   7. *-commutative 41.4%

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

   8. sqrt-unprod 42.1%

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

   9. metadata-eval 42.1%

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

   10. metadata-eval 42.1%

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

   11. *-commutative 42.1%

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

   12. pow1/2 42.1%

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

8. Applied egg-rr 42.1%

   \[\leadsto \color{blue}{{\left(\frac{x + -1}{x + 1}\right)}^{0.5}} \]
9. Step-by-step derivation

   1. unpow1/2 42.1%

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

10. Simplified 42.1%

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

11. Taylor expanded in x around inf 42.1%

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

12. Final simplification 42.1%

    \[\leadsto 1 + \frac{-1}{x} \]

Alternative 9: 38.7% accurate, 225.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x l t) :precision binary64 1.0)

C

double code(double x, double l, double t) {
	return 1.0;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    code = 1.0d0
end function

Java

public static double code(double x, double l, double t) {
	return 1.0;
}

Python

def code(x, l, t):
	return 1.0

Julia

function code(x, l, t)
	return 1.0
end

MATLAB

function tmp = code(x, l, t)
	tmp = 1.0;
end

Wolfram

code[x_, l_, t_] := 1.0

Derivation

1. Initial program 30.8%

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

   1. *-commutative 30.8%

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

   2. associate-*l/ 30.7%

      \[\leadsto \color{blue}{\frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot \sqrt{2}} \]

   3. fma-neg 30.7%

      \[\leadsto \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \cdot \sqrt{2} \]

   4. sub-neg 30.7%

      \[\leadsto \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{\color{blue}{x + \left(-1\right)}}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}} \cdot \sqrt{2} \]

   5. metadata-eval 30.7%

      \[\leadsto \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + \color{blue}{-1}}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}} \cdot \sqrt{2} \]

   6. +-commutative 30.7%

      \[\leadsto \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \color{blue}{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}, -\ell \cdot \ell\right)}} \cdot \sqrt{2} \]

   7. fma-def 30.7%

      \[\leadsto \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}, -\ell \cdot \ell\right)}} \cdot \sqrt{2} \]

   8. distribute-rgt-neg-in 30.7%

      \[\leadsto \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \color{blue}{\ell \cdot \left(-\ell\right)}\right)}} \cdot \sqrt{2} \]

3. Simplified 30.7%

   \[\leadsto \color{blue}{\frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}} \cdot \sqrt{2}} \]

4. Taylor expanded in t around -inf 39.3%

   \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)\right)} \cdot \sqrt{2} \]
5. Step-by-step derivation

   1. mul-1-neg 39.3%

      \[\leadsto \color{blue}{\left(-\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)} \cdot \sqrt{2} \]

   2. *-commutative 39.3%

      \[\leadsto \left(-\color{blue}{\sqrt{\frac{x - 1}{1 + x}} \cdot \sqrt{0.5}}\right) \cdot \sqrt{2} \]

   3. sub-neg 39.3%

      \[\leadsto \left(-\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \cdot \sqrt{0.5}\right) \cdot \sqrt{2} \]

   4. metadata-eval 39.3%

      \[\leadsto \left(-\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \cdot \sqrt{0.5}\right) \cdot \sqrt{2} \]

   5. +-commutative 39.3%

      \[\leadsto \left(-\sqrt{\frac{\color{blue}{-1 + x}}{1 + x}} \cdot \sqrt{0.5}\right) \cdot \sqrt{2} \]

   6. +-commutative 39.3%

      \[\leadsto \left(-\sqrt{\frac{-1 + x}{\color{blue}{x + 1}}} \cdot \sqrt{0.5}\right) \cdot \sqrt{2} \]

6. Simplified 39.3%

   \[\leadsto \color{blue}{\left(-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right)} \cdot \sqrt{2} \]
7. Step-by-step derivation

   1. add-sqr-sqrt 0.0%

      \[\leadsto \color{blue}{\left(\sqrt{-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}} \cdot \sqrt{-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}}\right)} \cdot \sqrt{2} \]

   2. sqrt-unprod 41.4%

      \[\leadsto \color{blue}{\sqrt{\left(-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right) \cdot \left(-\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right)}} \cdot \sqrt{2} \]

   3. sqr-neg 41.4%

      \[\leadsto \sqrt{\color{blue}{\left(\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right) \cdot \left(\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right)}} \cdot \sqrt{2} \]

   4. sqrt-unprod 41.4%

      \[\leadsto \color{blue}{\left(\sqrt{\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}} \cdot \sqrt{\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}}\right)} \cdot \sqrt{2} \]

   5. add-sqr-sqrt 41.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{0.5}\right)} \cdot \sqrt{2} \]

   6. associate-*l* 41.4%

      \[\leadsto \color{blue}{\sqrt{\frac{-1 + x}{x + 1}} \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)} \]

   7. *-commutative 41.4%

      \[\leadsto \sqrt{\frac{-1 + x}{x + 1}} \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{0.5}\right)} \]

   8. sqrt-unprod 42.1%

      \[\leadsto \sqrt{\frac{-1 + x}{x + 1}} \cdot \color{blue}{\sqrt{2 \cdot 0.5}} \]

   9. metadata-eval 42.1%

      \[\leadsto \sqrt{\frac{-1 + x}{x + 1}} \cdot \sqrt{\color{blue}{1}} \]

   10. metadata-eval 42.1%

       \[\leadsto \sqrt{\frac{-1 + x}{x + 1}} \cdot \color{blue}{1} \]

   11. *-commutative 42.1%

       \[\leadsto \color{blue}{1 \cdot \sqrt{\frac{-1 + x}{x + 1}}} \]

   12. pow1/2 42.1%

       \[\leadsto 1 \cdot \color{blue}{{\left(\frac{-1 + x}{x + 1}\right)}^{0.5}} \]

8. Applied egg-rr 42.1%

   \[\leadsto \color{blue}{{\left(\frac{x + -1}{x + 1}\right)}^{0.5}} \]
9. Step-by-step derivation

   1. unpow1/2 42.1%

      \[\leadsto \color{blue}{\sqrt{\frac{x + -1}{x + 1}}} \]

10. Simplified 42.1%

    \[\leadsto \color{blue}{\sqrt{\frac{x + -1}{x + 1}}} \]

11. Taylor expanded in x around inf 42.0%

    \[\leadsto \color{blue}{1} \]

12. Final simplification 42.0%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023301 
(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)))))