Toniolo and Linder, Equation (7)

Percentage Accurate: 33.6% → 84.9%

Time: 19.5s

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: 33.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t_3 := 2 \cdot \left(t\_m \cdot t\_m\right)\\ t_4 := l\_m \cdot l\_m + t\_3\\ t_5 := 2 \cdot t\_4\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.05 \cdot 10^{-225}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 7.5 \cdot 10^{-150}:\\ \;\;\;\;\frac{t\_2}{t\_2 + \frac{0.5}{t\_m} \cdot \frac{t\_5}{x \cdot \sqrt{2}}}\\ \mathbf{elif}\;t\_m \leq 5 \cdot 10^{+43}:\\ \;\;\;\;\frac{t\_2}{\sqrt{t\_3 + \frac{\frac{t\_5 + \left(\left(\frac{t\_3}{x} + \frac{l\_m \cdot l\_m}{x}\right) + \frac{t\_4}{x}\right)}{x} - t\_4 \cdot -2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{0.5 + \frac{-0.5}{x}}{x} + -1}{x}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* t_m (sqrt 2.0)))
        (t_3 (* 2.0 (* t_m t_m)))
        (t_4 (+ (* l_m l_m) t_3))
        (t_5 (* 2.0 t_4)))
   (*
    t_s
    (if (<= t_m 1.05e-225)
      (/ (* t_m (sqrt x)) l_m)
      (if (<= t_m 7.5e-150)
        (/ t_2 (+ t_2 (* (/ 0.5 t_m) (/ t_5 (* x (sqrt 2.0))))))
        (if (<= t_m 5e+43)
          (/
           t_2
           (sqrt
            (+
             t_3
             (/
              (-
               (/ (+ t_5 (+ (+ (/ t_3 x) (/ (* l_m l_m) x)) (/ t_4 x))) x)
               (* t_4 -2.0))
              x))))
          (+ 1.0 (/ (+ (/ (+ 0.5 (/ -0.5 x)) x) -1.0) x))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * sqrt(2.0);
	double t_3 = 2.0 * (t_m * t_m);
	double t_4 = (l_m * l_m) + t_3;
	double t_5 = 2.0 * t_4;
	double tmp;
	if (t_m <= 1.05e-225) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else if (t_m <= 7.5e-150) {
		tmp = t_2 / (t_2 + ((0.5 / t_m) * (t_5 / (x * sqrt(2.0)))));
	} else if (t_m <= 5e+43) {
		tmp = t_2 / sqrt((t_3 + ((((t_5 + (((t_3 / x) + ((l_m * l_m) / x)) + (t_4 / x))) / x) - (t_4 * -2.0)) / x)));
	} else {
		tmp = 1.0 + ((((0.5 + (-0.5 / x)) / x) + -1.0) / x);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_2 = t_m * sqrt(2.0d0)
    t_3 = 2.0d0 * (t_m * t_m)
    t_4 = (l_m * l_m) + t_3
    t_5 = 2.0d0 * t_4
    if (t_m <= 1.05d-225) then
        tmp = (t_m * sqrt(x)) / l_m
    else if (t_m <= 7.5d-150) then
        tmp = t_2 / (t_2 + ((0.5d0 / t_m) * (t_5 / (x * sqrt(2.0d0)))))
    else if (t_m <= 5d+43) then
        tmp = t_2 / sqrt((t_3 + ((((t_5 + (((t_3 / x) + ((l_m * l_m) / x)) + (t_4 / x))) / x) - (t_4 * (-2.0d0))) / x)))
    else
        tmp = 1.0d0 + ((((0.5d0 + ((-0.5d0) / x)) / x) + (-1.0d0)) / x)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * Math.sqrt(2.0);
	double t_3 = 2.0 * (t_m * t_m);
	double t_4 = (l_m * l_m) + t_3;
	double t_5 = 2.0 * t_4;
	double tmp;
	if (t_m <= 1.05e-225) {
		tmp = (t_m * Math.sqrt(x)) / l_m;
	} else if (t_m <= 7.5e-150) {
		tmp = t_2 / (t_2 + ((0.5 / t_m) * (t_5 / (x * Math.sqrt(2.0)))));
	} else if (t_m <= 5e+43) {
		tmp = t_2 / Math.sqrt((t_3 + ((((t_5 + (((t_3 / x) + ((l_m * l_m) / x)) + (t_4 / x))) / x) - (t_4 * -2.0)) / x)));
	} else {
		tmp = 1.0 + ((((0.5 + (-0.5 / x)) / x) + -1.0) / x);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = t_m * math.sqrt(2.0)
	t_3 = 2.0 * (t_m * t_m)
	t_4 = (l_m * l_m) + t_3
	t_5 = 2.0 * t_4
	tmp = 0
	if t_m <= 1.05e-225:
		tmp = (t_m * math.sqrt(x)) / l_m
	elif t_m <= 7.5e-150:
		tmp = t_2 / (t_2 + ((0.5 / t_m) * (t_5 / (x * math.sqrt(2.0)))))
	elif t_m <= 5e+43:
		tmp = t_2 / math.sqrt((t_3 + ((((t_5 + (((t_3 / x) + ((l_m * l_m) / x)) + (t_4 / x))) / x) - (t_4 * -2.0)) / x)))
	else:
		tmp = 1.0 + ((((0.5 + (-0.5 / x)) / x) + -1.0) / x)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(t_m * sqrt(2.0))
	t_3 = Float64(2.0 * Float64(t_m * t_m))
	t_4 = Float64(Float64(l_m * l_m) + t_3)
	t_5 = Float64(2.0 * t_4)
	tmp = 0.0
	if (t_m <= 1.05e-225)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	elseif (t_m <= 7.5e-150)
		tmp = Float64(t_2 / Float64(t_2 + Float64(Float64(0.5 / t_m) * Float64(t_5 / Float64(x * sqrt(2.0))))));
	elseif (t_m <= 5e+43)
		tmp = Float64(t_2 / sqrt(Float64(t_3 + Float64(Float64(Float64(Float64(t_5 + Float64(Float64(Float64(t_3 / x) + Float64(Float64(l_m * l_m) / x)) + Float64(t_4 / x))) / x) - Float64(t_4 * -2.0)) / x))));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(0.5 + Float64(-0.5 / x)) / x) + -1.0) / x));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = t_m * sqrt(2.0);
	t_3 = 2.0 * (t_m * t_m);
	t_4 = (l_m * l_m) + t_3;
	t_5 = 2.0 * t_4;
	tmp = 0.0;
	if (t_m <= 1.05e-225)
		tmp = (t_m * sqrt(x)) / l_m;
	elseif (t_m <= 7.5e-150)
		tmp = t_2 / (t_2 + ((0.5 / t_m) * (t_5 / (x * sqrt(2.0)))));
	elseif (t_m <= 5e+43)
		tmp = t_2 / sqrt((t_3 + ((((t_5 + (((t_3 / x) + ((l_m * l_m) / x)) + (t_4 / x))) / x) - (t_4 * -2.0)) / x)));
	else
		tmp = 1.0 + ((((0.5 + (-0.5 / x)) / x) + -1.0) / x);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(l$95$m * l$95$m), $MachinePrecision] + t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(2.0 * t$95$4), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.05e-225], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 7.5e-150], N[(t$95$2 / N[(t$95$2 + N[(N[(0.5 / t$95$m), $MachinePrecision] * N[(t$95$5 / N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5e+43], N[(t$95$2 / N[Sqrt[N[(t$95$3 + N[(N[(N[(N[(t$95$5 + N[(N[(N[(t$95$3 / x), $MachinePrecision] + N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(t$95$4 * -2.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.05e-225

   1. Initial program 31.9%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. sqrt-undiv N/A

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

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

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

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

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

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

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

   4. Applied egg-rr 31.4%

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

   5. Taylor expanded in t around 0

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

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

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. +-commutative N/A

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 3.0%

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

   7. Simplified 3.0%

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

   8. Taylor expanded in x around inf

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

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

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

      2. cancel-sign-sub-inv N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 18.1%

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

   10. Simplified 18.1%

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

   11. Taylor expanded in t around 0

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

       1. associate-*l/ N/A

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

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

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

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

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

       4. sqrt-lowering-sqrt.f64 19.4%

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

   13. Simplified 19.4%

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

   if 1.05e-225 < t < 7.5000000000000004e-150

   1. Initial program 2.5%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

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

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

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

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

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

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

      5. associate-*r/ N/A

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

      6. times-frac N/A

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

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

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

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

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

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

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

   5. Simplified 78.6%

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

   if 7.5000000000000004e-150 < t < 5.0000000000000004e43

   1. Initial program 51.4%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 87.4%

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

   if 5.0000000000000004e43 < t

   1. Initial program 40.0%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 97.0%

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

   5. Simplified 97.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right) - \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right), \color{blue}{\left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right), \left(\color{blue}{\frac{1}{x}} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{\color{blue}{x}} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\frac{1}{2} \cdot 1}{\color{blue}{{x}^{3}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\frac{1}{2}}{{\color{blue}{x}}^{3}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{3}\right)}\right)\right)\right) \]

      11. cube-mult N/A

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 97.1%

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

   8. Simplified 97.1%

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

   9. Taylor expanded in x around -inf

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

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

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

       2. mul-1-neg N/A

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

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

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

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

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

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

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

       6. mul-1-neg N/A

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

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

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

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

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

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

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

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

       12. /-lowering-/.f64 97.1%

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

   11. Simplified 97.1%

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

       1. unsub-neg N/A

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

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

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

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

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

       4. unsub-neg N/A

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

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

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

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

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

       7. sub-neg N/A

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

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

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

       9. distribute-neg-frac N/A

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

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

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

       11. metadata-eval 97.1%

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

   13. Applied egg-rr 97.1%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-225}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-150}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + \frac{0.5}{t} \cdot \frac{2 \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x \cdot \sqrt{2}}}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+43}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot t\right) + \frac{\frac{2 \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) + \left(\left(\frac{2 \cdot \left(t \cdot t\right)}{x} + \frac{\ell \cdot \ell}{x}\right) + \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x}\right)}{x} - \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) \cdot -2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{0.5 + \frac{-0.5}{x}}{x} + -1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot \left(t\_m \cdot t\_m\right)\\ t_3 := l\_m \cdot l\_m + t\_2\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-215}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 7.5 \cdot 10^{-150}:\\ \;\;\;\;1 + \frac{-1 - \frac{-0.5}{x}}{x}\\ \mathbf{elif}\;t\_m \leq 1.5 \cdot 10^{+44}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{\sqrt{t\_2 + \frac{\frac{2 \cdot t\_3 + \left(\left(\frac{t\_2}{x} + \frac{l\_m \cdot l\_m}{x}\right) + \frac{t\_3}{x}\right)}{x} - t\_3 \cdot -2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{0.5 + \frac{-0.5}{x}}{x} + -1}{x}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (* t_m t_m))) (t_3 (+ (* l_m l_m) t_2)))
   (*
    t_s
    (if (<= t_m 9e-215)
      (/ (* t_m (sqrt x)) l_m)
      (if (<= t_m 7.5e-150)
        (+ 1.0 (/ (- -1.0 (/ -0.5 x)) x))
        (if (<= t_m 1.5e+44)
          (/
           (* t_m (sqrt 2.0))
           (sqrt
            (+
             t_2
             (/
              (-
               (/
                (+ (* 2.0 t_3) (+ (+ (/ t_2 x) (/ (* l_m l_m) x)) (/ t_3 x)))
                x)
               (* t_3 -2.0))
              x))))
          (+ 1.0 (/ (+ (/ (+ 0.5 (/ -0.5 x)) x) -1.0) x))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * (t_m * t_m);
	double t_3 = (l_m * l_m) + t_2;
	double tmp;
	if (t_m <= 9e-215) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else if (t_m <= 7.5e-150) {
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	} else if (t_m <= 1.5e+44) {
		tmp = (t_m * sqrt(2.0)) / sqrt((t_2 + (((((2.0 * t_3) + (((t_2 / x) + ((l_m * l_m) / x)) + (t_3 / x))) / x) - (t_3 * -2.0)) / x)));
	} else {
		tmp = 1.0 + ((((0.5 + (-0.5 / x)) / x) + -1.0) / x);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m * t_m)
    t_3 = (l_m * l_m) + t_2
    if (t_m <= 9d-215) then
        tmp = (t_m * sqrt(x)) / l_m
    else if (t_m <= 7.5d-150) then
        tmp = 1.0d0 + (((-1.0d0) - ((-0.5d0) / x)) / x)
    else if (t_m <= 1.5d+44) then
        tmp = (t_m * sqrt(2.0d0)) / sqrt((t_2 + (((((2.0d0 * t_3) + (((t_2 / x) + ((l_m * l_m) / x)) + (t_3 / x))) / x) - (t_3 * (-2.0d0))) / x)))
    else
        tmp = 1.0d0 + ((((0.5d0 + ((-0.5d0) / x)) / x) + (-1.0d0)) / x)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * (t_m * t_m);
	double t_3 = (l_m * l_m) + t_2;
	double tmp;
	if (t_m <= 9e-215) {
		tmp = (t_m * Math.sqrt(x)) / l_m;
	} else if (t_m <= 7.5e-150) {
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	} else if (t_m <= 1.5e+44) {
		tmp = (t_m * Math.sqrt(2.0)) / Math.sqrt((t_2 + (((((2.0 * t_3) + (((t_2 / x) + ((l_m * l_m) / x)) + (t_3 / x))) / x) - (t_3 * -2.0)) / x)));
	} else {
		tmp = 1.0 + ((((0.5 + (-0.5 / x)) / x) + -1.0) / x);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = 2.0 * (t_m * t_m)
	t_3 = (l_m * l_m) + t_2
	tmp = 0
	if t_m <= 9e-215:
		tmp = (t_m * math.sqrt(x)) / l_m
	elif t_m <= 7.5e-150:
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x)
	elif t_m <= 1.5e+44:
		tmp = (t_m * math.sqrt(2.0)) / math.sqrt((t_2 + (((((2.0 * t_3) + (((t_2 / x) + ((l_m * l_m) / x)) + (t_3 / x))) / x) - (t_3 * -2.0)) / x)))
	else:
		tmp = 1.0 + ((((0.5 + (-0.5 / x)) / x) + -1.0) / x)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(2.0 * Float64(t_m * t_m))
	t_3 = Float64(Float64(l_m * l_m) + t_2)
	tmp = 0.0
	if (t_m <= 9e-215)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	elseif (t_m <= 7.5e-150)
		tmp = Float64(1.0 + Float64(Float64(-1.0 - Float64(-0.5 / x)) / x));
	elseif (t_m <= 1.5e+44)
		tmp = Float64(Float64(t_m * sqrt(2.0)) / sqrt(Float64(t_2 + Float64(Float64(Float64(Float64(Float64(2.0 * t_3) + Float64(Float64(Float64(t_2 / x) + Float64(Float64(l_m * l_m) / x)) + Float64(t_3 / x))) / x) - Float64(t_3 * -2.0)) / x))));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(0.5 + Float64(-0.5 / x)) / x) + -1.0) / x));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = 2.0 * (t_m * t_m);
	t_3 = (l_m * l_m) + t_2;
	tmp = 0.0;
	if (t_m <= 9e-215)
		tmp = (t_m * sqrt(x)) / l_m;
	elseif (t_m <= 7.5e-150)
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	elseif (t_m <= 1.5e+44)
		tmp = (t_m * sqrt(2.0)) / sqrt((t_2 + (((((2.0 * t_3) + (((t_2 / x) + ((l_m * l_m) / x)) + (t_3 / x))) / x) - (t_3 * -2.0)) / x)));
	else
		tmp = 1.0 + ((((0.5 + (-0.5 / x)) / x) + -1.0) / x);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(l$95$m * l$95$m), $MachinePrecision] + t$95$2), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 9e-215], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 7.5e-150], N[(1.0 + N[(N[(-1.0 - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.5e+44], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(N[(N[(N[(N[(2.0 * t$95$3), $MachinePrecision] + N[(N[(N[(t$95$2 / x), $MachinePrecision] + N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(t$95$3 * -2.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 9e-215

   1. Initial program 31.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. sqrt-undiv N/A

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

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

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

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

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

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

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

   4. Applied egg-rr 31.0%

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

   5. Taylor expanded in t around 0

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

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

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. +-commutative N/A

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 3.0%

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

   7. Simplified 3.0%

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

   8. Taylor expanded in x around inf

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

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

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

      2. cancel-sign-sub-inv N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 18.5%

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

   10. Simplified 18.5%

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

   11. Taylor expanded in t around 0

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

       1. associate-*l/ N/A

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

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

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

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

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

       4. sqrt-lowering-sqrt.f64 19.9%

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

   13. Simplified 19.9%

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

   if 9e-215 < t < 7.5000000000000004e-150

   1. Initial program 2.7%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 78.5%

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

   5. Simplified 78.5%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right) - \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right), \color{blue}{\left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right), \left(\color{blue}{\frac{1}{x}} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{\color{blue}{x}} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\frac{1}{2} \cdot 1}{\color{blue}{{x}^{3}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\frac{1}{2}}{{\color{blue}{x}}^{3}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{3}\right)}\right)\right)\right) \]

      11. cube-mult N/A

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 78.7%

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

   8. Simplified 78.7%

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

   9. Taylor expanded in x around -inf

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

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

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

       2. mul-1-neg N/A

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

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

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

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

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

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

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

       6. mul-1-neg N/A

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

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

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

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

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

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

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

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

       12. /-lowering-/.f64 78.7%

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

   11. Simplified 78.7%

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

   12. Taylor expanded in x around -inf

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

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

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

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

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

       5. sub-neg N/A

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

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

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

       7. associate-*r/ N/A

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

       8. metadata-eval N/A

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

       9. distribute-neg-frac N/A

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

       10. metadata-eval N/A

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

       11. /-lowering-/.f64 78.7%

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

   14. Simplified 78.7%

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

   if 7.5000000000000004e-150 < t < 1.49999999999999993e44

   1. Initial program 51.4%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 87.4%

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

   if 1.49999999999999993e44 < t

   1. Initial program 40.0%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 97.0%

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

   5. Simplified 97.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right) - \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right), \color{blue}{\left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right), \left(\color{blue}{\frac{1}{x}} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{\color{blue}{x}} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\frac{1}{2} \cdot 1}{\color{blue}{{x}^{3}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\frac{1}{2}}{{\color{blue}{x}}^{3}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{3}\right)}\right)\right)\right) \]

      11. cube-mult N/A

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 97.1%

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

   8. Simplified 97.1%

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

   9. Taylor expanded in x around -inf

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

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

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

       2. mul-1-neg N/A

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

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

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

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

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

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

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

       6. mul-1-neg N/A

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

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

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

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

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

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

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

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

       12. /-lowering-/.f64 97.1%

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

   11. Simplified 97.1%

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

       1. unsub-neg N/A

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

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

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

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

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

       4. unsub-neg N/A

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

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

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

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

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

       7. sub-neg N/A

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

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

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

       9. distribute-neg-frac N/A

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

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

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

       11. metadata-eval 97.1%

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

   13. Applied egg-rr 97.1%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-215}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-150}:\\ \;\;\;\;1 + \frac{-1 - \frac{-0.5}{x}}{x}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+44}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot t\right) + \frac{\frac{2 \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) + \left(\left(\frac{2 \cdot \left(t \cdot t\right)}{x} + \frac{\ell \cdot \ell}{x}\right) + \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x}\right)}{x} - \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) \cdot -2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{0.5 + \frac{-0.5}{x}}{x} + -1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot \left(t\_m \cdot t\_m\right)\\ t_3 := l\_m \cdot l\_m + t\_2\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-215}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;1 + \frac{-1 - \frac{-0.5}{x}}{x}\\ \mathbf{elif}\;t\_m \leq 2.15 \cdot 10^{+43}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2}{t\_2 + \frac{\left(\frac{t\_3}{x} + \left(t\_3 + t\_3\right)\right) + \left(\frac{l\_m \cdot l\_m}{x} + 2 \cdot \frac{t\_m \cdot t\_m}{x}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{0.5 + \frac{-0.5}{x}}{x} + -1}{x}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (* t_m t_m))) (t_3 (+ (* l_m l_m) t_2)))
   (*
    t_s
    (if (<= t_m 4.2e-215)
      (/ (* t_m (sqrt x)) l_m)
      (if (<= t_m 7.5e-155)
        (+ 1.0 (/ (- -1.0 (/ -0.5 x)) x))
        (if (<= t_m 2.15e+43)
          (*
           t_m
           (sqrt
            (/
             2.0
             (+
              t_2
              (/
               (+
                (+ (/ t_3 x) (+ t_3 t_3))
                (+ (/ (* l_m l_m) x) (* 2.0 (/ (* t_m t_m) x))))
               x)))))
          (+ 1.0 (/ (+ (/ (+ 0.5 (/ -0.5 x)) x) -1.0) x))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * (t_m * t_m);
	double t_3 = (l_m * l_m) + t_2;
	double tmp;
	if (t_m <= 4.2e-215) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else if (t_m <= 7.5e-155) {
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	} else if (t_m <= 2.15e+43) {
		tmp = t_m * sqrt((2.0 / (t_2 + ((((t_3 / x) + (t_3 + t_3)) + (((l_m * l_m) / x) + (2.0 * ((t_m * t_m) / x)))) / x))));
	} else {
		tmp = 1.0 + ((((0.5 + (-0.5 / x)) / x) + -1.0) / x);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m * t_m)
    t_3 = (l_m * l_m) + t_2
    if (t_m <= 4.2d-215) then
        tmp = (t_m * sqrt(x)) / l_m
    else if (t_m <= 7.5d-155) then
        tmp = 1.0d0 + (((-1.0d0) - ((-0.5d0) / x)) / x)
    else if (t_m <= 2.15d+43) then
        tmp = t_m * sqrt((2.0d0 / (t_2 + ((((t_3 / x) + (t_3 + t_3)) + (((l_m * l_m) / x) + (2.0d0 * ((t_m * t_m) / x)))) / x))))
    else
        tmp = 1.0d0 + ((((0.5d0 + ((-0.5d0) / x)) / x) + (-1.0d0)) / x)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * (t_m * t_m);
	double t_3 = (l_m * l_m) + t_2;
	double tmp;
	if (t_m <= 4.2e-215) {
		tmp = (t_m * Math.sqrt(x)) / l_m;
	} else if (t_m <= 7.5e-155) {
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	} else if (t_m <= 2.15e+43) {
		tmp = t_m * Math.sqrt((2.0 / (t_2 + ((((t_3 / x) + (t_3 + t_3)) + (((l_m * l_m) / x) + (2.0 * ((t_m * t_m) / x)))) / x))));
	} else {
		tmp = 1.0 + ((((0.5 + (-0.5 / x)) / x) + -1.0) / x);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = 2.0 * (t_m * t_m)
	t_3 = (l_m * l_m) + t_2
	tmp = 0
	if t_m <= 4.2e-215:
		tmp = (t_m * math.sqrt(x)) / l_m
	elif t_m <= 7.5e-155:
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x)
	elif t_m <= 2.15e+43:
		tmp = t_m * math.sqrt((2.0 / (t_2 + ((((t_3 / x) + (t_3 + t_3)) + (((l_m * l_m) / x) + (2.0 * ((t_m * t_m) / x)))) / x))))
	else:
		tmp = 1.0 + ((((0.5 + (-0.5 / x)) / x) + -1.0) / x)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(2.0 * Float64(t_m * t_m))
	t_3 = Float64(Float64(l_m * l_m) + t_2)
	tmp = 0.0
	if (t_m <= 4.2e-215)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	elseif (t_m <= 7.5e-155)
		tmp = Float64(1.0 + Float64(Float64(-1.0 - Float64(-0.5 / x)) / x));
	elseif (t_m <= 2.15e+43)
		tmp = Float64(t_m * sqrt(Float64(2.0 / Float64(t_2 + Float64(Float64(Float64(Float64(t_3 / x) + Float64(t_3 + t_3)) + Float64(Float64(Float64(l_m * l_m) / x) + Float64(2.0 * Float64(Float64(t_m * t_m) / x)))) / x)))));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(0.5 + Float64(-0.5 / x)) / x) + -1.0) / x));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = 2.0 * (t_m * t_m);
	t_3 = (l_m * l_m) + t_2;
	tmp = 0.0;
	if (t_m <= 4.2e-215)
		tmp = (t_m * sqrt(x)) / l_m;
	elseif (t_m <= 7.5e-155)
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	elseif (t_m <= 2.15e+43)
		tmp = t_m * sqrt((2.0 / (t_2 + ((((t_3 / x) + (t_3 + t_3)) + (((l_m * l_m) / x) + (2.0 * ((t_m * t_m) / x)))) / x))));
	else
		tmp = 1.0 + ((((0.5 + (-0.5 / x)) / x) + -1.0) / x);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(l$95$m * l$95$m), $MachinePrecision] + t$95$2), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.2e-215], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 7.5e-155], N[(1.0 + N[(N[(-1.0 - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.15e+43], N[(t$95$m * N[Sqrt[N[(2.0 / N[(t$95$2 + N[(N[(N[(N[(t$95$3 / x), $MachinePrecision] + N[(t$95$3 + t$95$3), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 4.2e-215

   1. Initial program 31.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. sqrt-undiv N/A

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

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

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

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

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

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

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

   4. Applied egg-rr 31.0%

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

   5. Taylor expanded in t around 0

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

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

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. +-commutative N/A

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 3.0%

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

   7. Simplified 3.0%

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

   8. Taylor expanded in x around inf

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

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

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

      2. cancel-sign-sub-inv N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 18.5%

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

   10. Simplified 18.5%

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

   11. Taylor expanded in t around 0

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

       1. associate-*l/ N/A

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

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

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

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

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

       4. sqrt-lowering-sqrt.f64 19.9%

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

   13. Simplified 19.9%

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

   if 4.2e-215 < t < 7.5000000000000006e-155

   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. Add Preprocessing

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 75.9%

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

   5. Simplified 75.9%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right) - \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right), \color{blue}{\left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right), \left(\color{blue}{\frac{1}{x}} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{\color{blue}{x}} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\frac{1}{2} \cdot 1}{\color{blue}{{x}^{3}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\frac{1}{2}}{{\color{blue}{x}}^{3}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{3}\right)}\right)\right)\right) \]

      11. cube-mult N/A

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 76.1%

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

   8. Simplified 76.1%

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

   9. Taylor expanded in x around -inf

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

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

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

       2. mul-1-neg N/A

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

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

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

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

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

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

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

       6. mul-1-neg N/A

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

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

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

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

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

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

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

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

       12. /-lowering-/.f64 76.1%

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

   11. Simplified 76.1%

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

   12. Taylor expanded in x around -inf

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

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

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

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

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

       5. sub-neg N/A

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

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

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

       7. associate-*r/ N/A

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

       8. metadata-eval N/A

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

       9. distribute-neg-frac N/A

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

       10. metadata-eval N/A

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

       11. /-lowering-/.f64 76.1%

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

   14. Simplified 76.1%

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

   if 7.5000000000000006e-155 < t < 2.15e43

   1. Initial program 49.9%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. sqrt-undiv N/A

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

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

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

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

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

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

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

   4. Applied egg-rr 50.1%

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

   5. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. mul-1-neg N/A

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

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

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

   7. Simplified 88.1%

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

   if 2.15e43 < t

   1. Initial program 40.0%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 97.0%

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

   5. Simplified 97.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right) - \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right), \color{blue}{\left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right), \left(\color{blue}{\frac{1}{x}} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{\color{blue}{x}} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\frac{1}{2} \cdot 1}{\color{blue}{{x}^{3}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\frac{1}{2}}{{\color{blue}{x}}^{3}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{3}\right)}\right)\right)\right) \]

      11. cube-mult N/A

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 97.1%

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

   8. Simplified 97.1%

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

   9. Taylor expanded in x around -inf

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

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

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

       2. mul-1-neg N/A

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

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

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

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

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

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

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

       6. mul-1-neg N/A

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

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

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

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

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

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

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

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

       12. /-lowering-/.f64 97.1%

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

   11. Simplified 97.1%

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

       1. unsub-neg N/A

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

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

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

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

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

       4. unsub-neg N/A

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

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

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

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

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

       7. sub-neg N/A

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

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

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

       9. distribute-neg-frac N/A

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

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

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

       11. metadata-eval 97.1%

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

   13. Applied egg-rr 97.1%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-215}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;1 + \frac{-1 - \frac{-0.5}{x}}{x}\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{+43}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{2 \cdot \left(t \cdot t\right) + \frac{\left(\frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x} + \left(\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) + \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \frac{t \cdot t}{x}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{0.5 + \frac{-0.5}{x}}{x} + -1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.3 \cdot 10^{-215}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 1.85 \cdot 10^{-155}:\\ \;\;\;\;1 + \frac{-1 - \frac{-0.5}{x}}{x}\\ \mathbf{elif}\;t\_m \leq 2 \cdot 10^{+43}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2}{2 \cdot \left(t\_m \cdot t\_m\right) - \frac{-2 \cdot \left(l\_m \cdot l\_m + \left(t\_m \cdot t\_m + t\_m \cdot t\_m\right)\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{0.5 + \frac{-0.5}{x}}{x} + -1}{x}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.3e-215)
    (/ (* t_m (sqrt x)) l_m)
    (if (<= t_m 1.85e-155)
      (+ 1.0 (/ (- -1.0 (/ -0.5 x)) x))
      (if (<= t_m 2e+43)
        (*
         t_m
         (sqrt
          (/
           2.0
           (-
            (* 2.0 (* t_m t_m))
            (/ (* -2.0 (+ (* l_m l_m) (+ (* t_m t_m) (* t_m t_m)))) x)))))
        (+ 1.0 (/ (+ (/ (+ 0.5 (/ -0.5 x)) x) -1.0) x)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.3e-215) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else if (t_m <= 1.85e-155) {
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	} else if (t_m <= 2e+43) {
		tmp = t_m * sqrt((2.0 / ((2.0 * (t_m * t_m)) - ((-2.0 * ((l_m * l_m) + ((t_m * t_m) + (t_m * t_m)))) / x))));
	} else {
		tmp = 1.0 + ((((0.5 + (-0.5 / x)) / x) + -1.0) / x);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1.3d-215) then
        tmp = (t_m * sqrt(x)) / l_m
    else if (t_m <= 1.85d-155) then
        tmp = 1.0d0 + (((-1.0d0) - ((-0.5d0) / x)) / x)
    else if (t_m <= 2d+43) then
        tmp = t_m * sqrt((2.0d0 / ((2.0d0 * (t_m * t_m)) - (((-2.0d0) * ((l_m * l_m) + ((t_m * t_m) + (t_m * t_m)))) / x))))
    else
        tmp = 1.0d0 + ((((0.5d0 + ((-0.5d0) / x)) / x) + (-1.0d0)) / x)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.3e-215) {
		tmp = (t_m * Math.sqrt(x)) / l_m;
	} else if (t_m <= 1.85e-155) {
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	} else if (t_m <= 2e+43) {
		tmp = t_m * Math.sqrt((2.0 / ((2.0 * (t_m * t_m)) - ((-2.0 * ((l_m * l_m) + ((t_m * t_m) + (t_m * t_m)))) / x))));
	} else {
		tmp = 1.0 + ((((0.5 + (-0.5 / x)) / x) + -1.0) / x);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 1.3e-215:
		tmp = (t_m * math.sqrt(x)) / l_m
	elif t_m <= 1.85e-155:
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x)
	elif t_m <= 2e+43:
		tmp = t_m * math.sqrt((2.0 / ((2.0 * (t_m * t_m)) - ((-2.0 * ((l_m * l_m) + ((t_m * t_m) + (t_m * t_m)))) / x))))
	else:
		tmp = 1.0 + ((((0.5 + (-0.5 / x)) / x) + -1.0) / x)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 1.3e-215)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	elseif (t_m <= 1.85e-155)
		tmp = Float64(1.0 + Float64(Float64(-1.0 - Float64(-0.5 / x)) / x));
	elseif (t_m <= 2e+43)
		tmp = Float64(t_m * sqrt(Float64(2.0 / Float64(Float64(2.0 * Float64(t_m * t_m)) - Float64(Float64(-2.0 * Float64(Float64(l_m * l_m) + Float64(Float64(t_m * t_m) + Float64(t_m * t_m)))) / x)))));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(0.5 + Float64(-0.5 / x)) / x) + -1.0) / x));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 1.3e-215)
		tmp = (t_m * sqrt(x)) / l_m;
	elseif (t_m <= 1.85e-155)
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	elseif (t_m <= 2e+43)
		tmp = t_m * sqrt((2.0 / ((2.0 * (t_m * t_m)) - ((-2.0 * ((l_m * l_m) + ((t_m * t_m) + (t_m * t_m)))) / x))));
	else
		tmp = 1.0 + ((((0.5 + (-0.5 / x)) / x) + -1.0) / x);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.3e-215], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 1.85e-155], N[(1.0 + N[(N[(-1.0 - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2e+43], N[(t$95$m * N[Sqrt[N[(2.0 / N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(-2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] + N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 1.3e-215

   1. Initial program 31.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. sqrt-undiv N/A

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

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

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

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

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

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

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

   4. Applied egg-rr 31.0%

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

   5. Taylor expanded in t around 0

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

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

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. +-commutative N/A

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 3.0%

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

   7. Simplified 3.0%

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

   8. Taylor expanded in x around inf

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

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

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

      2. cancel-sign-sub-inv N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 18.5%

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

   10. Simplified 18.5%

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

   11. Taylor expanded in t around 0

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

       1. associate-*l/ N/A

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

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

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

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

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

       4. sqrt-lowering-sqrt.f64 19.9%

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

   13. Simplified 19.9%

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

   if 1.3e-215 < t < 1.85e-155

   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. Add Preprocessing

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 75.9%

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

   5. Simplified 75.9%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right) - \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right), \color{blue}{\left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right), \left(\color{blue}{\frac{1}{x}} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{\color{blue}{x}} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\frac{1}{2} \cdot 1}{\color{blue}{{x}^{3}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\frac{1}{2}}{{\color{blue}{x}}^{3}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{3}\right)}\right)\right)\right) \]

      11. cube-mult N/A

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 76.1%

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

   8. Simplified 76.1%

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

   9. Taylor expanded in x around -inf

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

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

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

       2. mul-1-neg N/A

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

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

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

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

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

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

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

       6. mul-1-neg N/A

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

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

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

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

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

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

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

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

       12. /-lowering-/.f64 76.1%

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

   11. Simplified 76.1%

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

   12. Taylor expanded in x around -inf

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

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

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

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

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

       5. sub-neg N/A

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

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

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

       7. associate-*r/ N/A

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

       8. metadata-eval N/A

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

       9. distribute-neg-frac N/A

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

       10. metadata-eval N/A

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

       11. /-lowering-/.f64 76.1%

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

   14. Simplified 76.1%

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

   if 1.85e-155 < t < 2.00000000000000003e43

   1. Initial program 49.9%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. sqrt-undiv N/A

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

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

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

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

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

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

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

   4. Applied egg-rr 50.1%

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

   5. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. associate--l+ N/A

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

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

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

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

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

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

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

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

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

   7. Simplified 63.9%

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

   8. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

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

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

   10. Simplified 88.0%

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

   if 2.00000000000000003e43 < t

   1. Initial program 40.0%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 97.0%

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

   5. Simplified 97.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right) - \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right), \color{blue}{\left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right), \left(\color{blue}{\frac{1}{x}} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{\color{blue}{x}} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\frac{1}{2} \cdot 1}{\color{blue}{{x}^{3}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\frac{1}{2}}{{\color{blue}{x}}^{3}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{3}\right)}\right)\right)\right) \]

      11. cube-mult N/A

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 97.1%

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

   8. Simplified 97.1%

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

   9. Taylor expanded in x around -inf

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

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

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

       2. mul-1-neg N/A

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

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\left(\frac{1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}}{x}\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right), x\right)\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(-1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right), x\right)\right)\right) \]

       6. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right)\right), x\right)\right)\right) \]

       7. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\left(\frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right)\right), x\right)\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}\right), x\right)\right)\right), x\right)\right)\right) \]

       9. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), x\right)\right)\right), x\right)\right)\right) \]

       10. associate-*r/ N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right), x\right)\right)\right), x\right)\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{x}\right)\right), x\right)\right)\right), x\right)\right)\right) \]

       12. /-lowering-/.f64 97.1%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), x\right)\right)\right), x\right)\right)\right) \]

   11. Simplified 97.1%

       \[\leadsto \color{blue}{1 + \left(-\frac{1 + \left(-\frac{0.5 - \frac{0.5}{x}}{x}\right)}{x}\right)} \]
   12. Step-by-step derivation

       1. unsub-neg N/A

          \[\leadsto 1 - \color{blue}{\frac{1 + \left(\mathsf{neg}\left(\frac{\frac{1}{2} - \frac{\frac{1}{2}}{x}}{x}\right)\right)}{x}} \]

       2. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1 + \left(\mathsf{neg}\left(\frac{\frac{1}{2} - \frac{\frac{1}{2}}{x}}{x}\right)\right)}{x}\right)}\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{\frac{1}{2} - \frac{\frac{1}{2}}{x}}{x}\right)\right)\right), \color{blue}{x}\right)\right) \]

       4. unsub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(1 - \frac{\frac{1}{2} - \frac{\frac{1}{2}}{x}}{x}\right), x\right)\right) \]

       5. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{1}{2} - \frac{\frac{1}{2}}{x}}{x}\right)\right), x\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} - \frac{\frac{1}{2}}{x}\right), x\right)\right), x\right)\right) \]

       7. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{x}\right)\right)\right), x\right)\right), x\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{x}\right)\right)\right), x\right)\right), x\right)\right) \]

       9. distribute-neg-frac N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}\right)\right), x\right)\right), x\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right), x\right)\right), x\right)\right), x\right)\right) \]

       11. metadata-eval 97.1%

           \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), x\right)\right), x\right)\right) \]

   13. Applied egg-rr 97.1%

       \[\leadsto \color{blue}{1 - \frac{1 - \frac{0.5 + \frac{-0.5}{x}}{x}}{x}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{-215}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{-155}:\\ \;\;\;\;1 + \frac{-1 - \frac{-0.5}{x}}{x}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+43}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{2 \cdot \left(t \cdot t\right) - \frac{-2 \cdot \left(\ell \cdot \ell + \left(t \cdot t + t \cdot t\right)\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{0.5 + \frac{-0.5}{x}}{x} + -1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 7.5 \cdot 10^{+194}:\\ \;\;\;\;1 + \frac{\frac{0.5 + \frac{-0.5}{x}}{x} + -1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{x}}{l\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 7.5e+194)
    (+ 1.0 (/ (+ (/ (+ 0.5 (/ -0.5 x)) x) -1.0) x))
    (* t_m (/ (sqrt x) l_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 7.5e+194) {
		tmp = 1.0 + ((((0.5 + (-0.5 / x)) / x) + -1.0) / x);
	} else {
		tmp = t_m * (sqrt(x) / l_m);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l_m <= 7.5d+194) then
        tmp = 1.0d0 + ((((0.5d0 + ((-0.5d0) / x)) / x) + (-1.0d0)) / x)
    else
        tmp = t_m * (sqrt(x) / l_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 7.5e+194) {
		tmp = 1.0 + ((((0.5 + (-0.5 / x)) / x) + -1.0) / x);
	} else {
		tmp = t_m * (Math.sqrt(x) / l_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if l_m <= 7.5e+194:
		tmp = 1.0 + ((((0.5 + (-0.5 / x)) / x) + -1.0) / x)
	else:
		tmp = t_m * (math.sqrt(x) / l_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (l_m <= 7.5e+194)
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(0.5 + Float64(-0.5 / x)) / x) + -1.0) / x));
	else
		tmp = Float64(t_m * Float64(sqrt(x) / l_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (l_m <= 7.5e+194)
		tmp = 1.0 + ((((0.5 + (-0.5 / x)) / x) + -1.0) / x);
	else
		tmp = t_m * (sqrt(x) / l_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 7.5e+194], N[(1.0 + N[(N[(N[(N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 7.5000000000000002e194

   1. Initial program 37.7%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 43.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

   5. Simplified 43.3%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x + -1}}}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right) - \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right), \color{blue}{\left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right), \left(\color{blue}{\frac{1}{x}} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{\color{blue}{x}} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\frac{1}{2} \cdot 1}{\color{blue}{{x}^{3}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\frac{1}{2}}{{\color{blue}{x}}^{3}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{3}\right)}\right)\right)\right) \]

      11. cube-mult N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{1}{2}, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 43.4%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

   8. Simplified 43.4%

      \[\leadsto \color{blue}{\left(1 + \frac{0.5}{x \cdot x}\right) - \left(\frac{1}{x} + \frac{0.5}{x \cdot \left(x \cdot x\right)}\right)} \]

   9. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}}{x}} \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(-1 \cdot \frac{1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}}{x}\right)}\right) \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}}{x}\right)\right)\right) \]

       3. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\left(\frac{1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}}{x}\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right), x\right)\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(-1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right), x\right)\right)\right) \]

       6. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right)\right), x\right)\right)\right) \]

       7. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\left(\frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right)\right), x\right)\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}\right), x\right)\right)\right), x\right)\right)\right) \]

       9. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), x\right)\right)\right), x\right)\right)\right) \]

       10. associate-*r/ N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right), x\right)\right)\right), x\right)\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{x}\right)\right), x\right)\right)\right), x\right)\right)\right) \]

       12. /-lowering-/.f64 43.4%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), x\right)\right)\right), x\right)\right)\right) \]

   11. Simplified 43.4%

       \[\leadsto \color{blue}{1 + \left(-\frac{1 + \left(-\frac{0.5 - \frac{0.5}{x}}{x}\right)}{x}\right)} \]
   12. Step-by-step derivation

       1. unsub-neg N/A

          \[\leadsto 1 - \color{blue}{\frac{1 + \left(\mathsf{neg}\left(\frac{\frac{1}{2} - \frac{\frac{1}{2}}{x}}{x}\right)\right)}{x}} \]

       2. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1 + \left(\mathsf{neg}\left(\frac{\frac{1}{2} - \frac{\frac{1}{2}}{x}}{x}\right)\right)}{x}\right)}\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{\frac{1}{2} - \frac{\frac{1}{2}}{x}}{x}\right)\right)\right), \color{blue}{x}\right)\right) \]

       4. unsub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(1 - \frac{\frac{1}{2} - \frac{\frac{1}{2}}{x}}{x}\right), x\right)\right) \]

       5. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{1}{2} - \frac{\frac{1}{2}}{x}}{x}\right)\right), x\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} - \frac{\frac{1}{2}}{x}\right), x\right)\right), x\right)\right) \]

       7. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{x}\right)\right)\right), x\right)\right), x\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{x}\right)\right)\right), x\right)\right), x\right)\right) \]

       9. distribute-neg-frac N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}\right)\right), x\right)\right), x\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right), x\right)\right), x\right)\right), x\right)\right) \]

       11. metadata-eval 43.4%

           \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), x\right)\right), x\right)\right) \]

   13. Applied egg-rr 43.4%

       \[\leadsto \color{blue}{1 - \frac{1 - \frac{0.5 + \frac{-0.5}{x}}{x}}{x}} \]

   if 7.5000000000000002e194 < l

   1. Initial program 0.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]

      2. associate-/l* N/A

         \[\leadsto t \cdot \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)}\right) \]

      4. sqrt-undiv N/A

         \[\leadsto \mathsf{*.f64}\left(t, \left(\sqrt{\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{t \cdot \sqrt{\frac{2}{\frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{\frac{x + -1}{x + 1}} - \ell \cdot \ell}}} \]

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{2}{\frac{{\ell}^{2} \cdot \left(1 + x\right)}{x - 1} - {\ell}^{2}}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{{\ell}^{2} \cdot \left(1 + x\right)}{x - 1} - {\ell}^{2}\right)\right)\right)\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\left(\frac{{\ell}^{2} \cdot \left(1 + x\right)}{x - 1}\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2} \cdot \left(1 + x\right)\right), \left(x - 1\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(1 + x\right)\right), \left(x - 1\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(1 + x\right)\right), \left(x - 1\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(1 + x\right)\right), \left(x - 1\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(x + 1\right)\right), \left(x - 1\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(x, 1\right)\right), \left(x - 1\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(x, 1\right)\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(x, 1\right)\right), \left(x + -1\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{+.f64}\left(x, -1\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{+.f64}\left(x, -1\right)\right), \left(\ell \cdot \ell\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 0.0%

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right) \]

   7. Simplified 0.0%

      \[\leadsto t \cdot \sqrt{\color{blue}{\frac{2}{\frac{\left(\ell \cdot \ell\right) \cdot \left(x + 1\right)}{x + -1} - \ell \cdot \ell}}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{\ell}^{2} - -1 \cdot {\ell}^{2}}{x}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left({\ell}^{2} - -1 \cdot {\ell}^{2}\right), x\right)\right)\right)\right) \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left({\ell}^{2} + \left(\mathsf{neg}\left(-1\right)\right) \cdot {\ell}^{2}\right), x\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({\ell}^{2}\right), \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot {\ell}^{2}\right)\right), x\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\ell \cdot \ell\right), \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot {\ell}^{2}\right)\right), x\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot {\ell}^{2}\right)\right), x\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(1 \cdot {\ell}^{2}\right)\right), x\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(1, \left({\ell}^{2}\right)\right)\right), x\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(1, \left(\ell \cdot \ell\right)\right)\right), x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 21.8%

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(1, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), x\right)\right)\right)\right) \]

   10. Simplified 21.8%

       \[\leadsto t \cdot \sqrt{\frac{2}{\color{blue}{\frac{\ell \cdot \ell + 1 \cdot \left(\ell \cdot \ell\right)}{x}}}} \]

   11. Taylor expanded in l around 0

       \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{1}{\ell} \cdot \sqrt{x}\right)}\right) \]
   12. Step-by-step derivation

       1. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(t, \left(\frac{1 \cdot \sqrt{x}}{\color{blue}{\ell}}\right)\right) \]

       2. *-lft-identity N/A

          \[\leadsto \mathsf{*.f64}\left(t, \left(\frac{\sqrt{x}}{\ell}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\left(\sqrt{x}\right), \color{blue}{\ell}\right)\right) \]

       4. sqrt-lowering-sqrt.f64 80.9%

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), \ell\right)\right) \]

   13. Simplified 80.9%

       \[\leadsto t \cdot \color{blue}{\frac{\sqrt{x}}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.5 \cdot 10^{+194}:\\ \;\;\;\;1 + \frac{\frac{0.5 + \frac{-0.5}{x}}{x} + -1}{x}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.2% accurate, 17.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(1 + \frac{\frac{0.5 + \frac{-0.5}{x}}{x} + -1}{x}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (* t_s (+ 1.0 (/ (+ (/ (+ 0.5 (/ -0.5 x)) x) -1.0) x))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 + ((((0.5 + (-0.5 / x)) / x) + -1.0) / x));
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * (1.0d0 + ((((0.5d0 + ((-0.5d0) / x)) / x) + (-1.0d0)) / x))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 + ((((0.5 + (-0.5 / x)) / x) + -1.0) / x));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * (1.0 + ((((0.5 + (-0.5 / x)) / x) + -1.0) / x))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * Float64(1.0 + Float64(Float64(Float64(Float64(0.5 + Float64(-0.5 / x)) / x) + -1.0) / x)))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * (1.0 + ((((0.5 + (-0.5 / x)) / x) + -1.0) / x));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * N[(1.0 + N[(N[(N[(N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.4%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Add Preprocessing

3. Taylor expanded in l around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

   3. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f64 41.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

5. Simplified 41.6%

   \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x + -1}}}} \]

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right) - \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)} \]
7. Step-by-step derivation

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right), \color{blue}{\left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right), \left(\color{blue}{\frac{1}{x}} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{\color{blue}{x}} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]

   8. associate-*r/ N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\frac{1}{2} \cdot 1}{\color{blue}{{x}^{3}}}\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\frac{1}{2}}{{\color{blue}{x}}^{3}}\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{3}\right)}\right)\right)\right) \]

   11. cube-mult N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{1}{2}, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f64 41.6%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

8. Simplified 41.6%

   \[\leadsto \color{blue}{\left(1 + \frac{0.5}{x \cdot x}\right) - \left(\frac{1}{x} + \frac{0.5}{x \cdot \left(x \cdot x\right)}\right)} \]

9. Taylor expanded in x around -inf

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}}{x}} \]
10. Step-by-step derivation

    1. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(-1 \cdot \frac{1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}}{x}\right)}\right) \]

    2. mul-1-neg N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}}{x}\right)\right)\right) \]

    3. neg-lowering-neg.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\left(\frac{1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}}{x}\right)\right)\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right), x\right)\right)\right) \]

    5. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(-1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right), x\right)\right)\right) \]

    6. mul-1-neg N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right)\right), x\right)\right)\right) \]

    7. neg-lowering-neg.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\left(\frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right)\right), x\right)\right)\right) \]

    8. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}\right), x\right)\right)\right), x\right)\right)\right) \]

    9. --lowering--.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), x\right)\right)\right), x\right)\right)\right) \]

    10. associate-*r/ N/A

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right), x\right)\right)\right), x\right)\right)\right) \]

    11. metadata-eval N/A

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{x}\right)\right), x\right)\right)\right), x\right)\right)\right) \]

    12. /-lowering-/.f64 41.6%

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), x\right)\right)\right), x\right)\right)\right) \]

11. Simplified 41.6%

    \[\leadsto \color{blue}{1 + \left(-\frac{1 + \left(-\frac{0.5 - \frac{0.5}{x}}{x}\right)}{x}\right)} \]
12. Step-by-step derivation

    1. unsub-neg N/A

       \[\leadsto 1 - \color{blue}{\frac{1 + \left(\mathsf{neg}\left(\frac{\frac{1}{2} - \frac{\frac{1}{2}}{x}}{x}\right)\right)}{x}} \]

    2. --lowering--.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1 + \left(\mathsf{neg}\left(\frac{\frac{1}{2} - \frac{\frac{1}{2}}{x}}{x}\right)\right)}{x}\right)}\right) \]

    3. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{\frac{1}{2} - \frac{\frac{1}{2}}{x}}{x}\right)\right)\right), \color{blue}{x}\right)\right) \]

    4. unsub-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(1 - \frac{\frac{1}{2} - \frac{\frac{1}{2}}{x}}{x}\right), x\right)\right) \]

    5. --lowering--.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{1}{2} - \frac{\frac{1}{2}}{x}}{x}\right)\right), x\right)\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} - \frac{\frac{1}{2}}{x}\right), x\right)\right), x\right)\right) \]

    7. sub-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{x}\right)\right)\right), x\right)\right), x\right)\right) \]

    8. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{x}\right)\right)\right), x\right)\right), x\right)\right) \]

    9. distribute-neg-frac N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}\right)\right), x\right)\right), x\right)\right) \]

    10. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right), x\right)\right), x\right)\right), x\right)\right) \]

    11. metadata-eval 41.6%

        \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), x\right)\right), x\right)\right) \]

13. Applied egg-rr 41.6%

    \[\leadsto \color{blue}{1 - \frac{1 - \frac{0.5 + \frac{-0.5}{x}}{x}}{x}} \]

14. Final simplification 41.6%

    \[\leadsto 1 + \frac{\frac{0.5 + \frac{-0.5}{x}}{x} + -1}{x} \]
15. Add Preprocessing

Alternative 7: 76.1% accurate, 25.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(1 + \frac{-1 - \frac{-0.5}{x}}{x}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (* t_s (+ 1.0 (/ (- -1.0 (/ -0.5 x)) x))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 + ((-1.0 - (-0.5 / x)) / x));
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * (1.0d0 + (((-1.0d0) - ((-0.5d0) / x)) / x))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 + ((-1.0 - (-0.5 / x)) / x));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * (1.0 + ((-1.0 - (-0.5 / x)) / x))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * Float64(1.0 + Float64(Float64(-1.0 - Float64(-0.5 / x)) / x)))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * (1.0 + ((-1.0 - (-0.5 / x)) / x));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * N[(1.0 + N[(N[(-1.0 - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.4%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Add Preprocessing

3. Taylor expanded in l around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

   3. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f64 41.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

5. Simplified 41.6%

   \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x + -1}}}} \]

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right) - \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)} \]
7. Step-by-step derivation

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right), \color{blue}{\left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right), \left(\color{blue}{\frac{1}{x}} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{\color{blue}{x}} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]

   8. associate-*r/ N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\frac{1}{2} \cdot 1}{\color{blue}{{x}^{3}}}\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\frac{1}{2}}{{\color{blue}{x}}^{3}}\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{3}\right)}\right)\right)\right) \]

   11. cube-mult N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{1}{2}, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f64 41.6%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

8. Simplified 41.6%

   \[\leadsto \color{blue}{\left(1 + \frac{0.5}{x \cdot x}\right) - \left(\frac{1}{x} + \frac{0.5}{x \cdot \left(x \cdot x\right)}\right)} \]

9. Taylor expanded in x around -inf

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}}{x}} \]
10. Step-by-step derivation

    1. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(-1 \cdot \frac{1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}}{x}\right)}\right) \]

    2. mul-1-neg N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}}{x}\right)\right)\right) \]

    3. neg-lowering-neg.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\left(\frac{1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}}{x}\right)\right)\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right), x\right)\right)\right) \]

    5. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(-1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right), x\right)\right)\right) \]

    6. mul-1-neg N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right)\right), x\right)\right)\right) \]

    7. neg-lowering-neg.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\left(\frac{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right)\right), x\right)\right)\right) \]

    8. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}\right), x\right)\right)\right), x\right)\right)\right) \]

    9. --lowering--.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), x\right)\right)\right), x\right)\right)\right) \]

    10. associate-*r/ N/A

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right), x\right)\right)\right), x\right)\right)\right) \]

    11. metadata-eval N/A

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{x}\right)\right), x\right)\right)\right), x\right)\right)\right) \]

    12. /-lowering-/.f64 41.6%

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), x\right)\right)\right), x\right)\right)\right) \]

11. Simplified 41.6%

    \[\leadsto \color{blue}{1 + \left(-\frac{1 + \left(-\frac{0.5 - \frac{0.5}{x}}{x}\right)}{x}\right)} \]

12. Taylor expanded in x around -inf

    \[\leadsto \color{blue}{1 + -1 \cdot \frac{1 - \frac{1}{2} \cdot \frac{1}{x}}{x}} \]
13. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto 1 + \left(\mathsf{neg}\left(\frac{1 - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right) \]

    2. unsub-neg N/A

       \[\leadsto 1 - \color{blue}{\frac{1 - \frac{1}{2} \cdot \frac{1}{x}}{x}} \]

    3. --lowering--.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1 - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)}\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(1 - \frac{1}{2} \cdot \frac{1}{x}\right), \color{blue}{x}\right)\right) \]

    5. sub-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right)\right) \]

    6. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right)\right) \]

    7. associate-*r/ N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right)\right), x\right)\right) \]

    8. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{x}\right)\right)\right), x\right)\right) \]

    9. distribute-neg-frac N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}\right)\right), x\right)\right) \]

    10. metadata-eval N/A

        \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{2}}{x}\right)\right), x\right)\right) \]

    11. /-lowering-/.f64 41.6%

        \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), x\right)\right) \]

14. Simplified 41.6%

    \[\leadsto \color{blue}{1 - \frac{1 + \frac{-0.5}{x}}{x}} \]

15. Final simplification 41.6%

    \[\leadsto 1 + \frac{-1 - \frac{-0.5}{x}}{x} \]
16. Add Preprocessing

Alternative 8: 75.8% accurate, 45.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(1 + \frac{-1}{x}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m) :precision binary64 (* t_s (+ 1.0 (/ -1.0 x))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 + (-1.0 / x));
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * (1.0d0 + ((-1.0d0) / x))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 + (-1.0 / x));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * (1.0 + (-1.0 / x))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * Float64(1.0 + Float64(-1.0 / x)))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * (1.0 + (-1.0 / x));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.4%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Add Preprocessing

3. Taylor expanded in l around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

   3. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f64 41.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

5. Simplified 41.6%

   \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x + -1}}}} \]

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
7. Step-by-step derivation

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1}{x}\right)}\right) \]

   2. /-lowering-/.f64 41.5%

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{x}\right)\right) \]

8. Simplified 41.5%

   \[\leadsto \color{blue}{1 - \frac{1}{x}} \]

9. Final simplification 41.5%

   \[\leadsto 1 + \frac{-1}{x} \]
10. Add Preprocessing

Alternative 9: 75.1% accurate, 225.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot 1 \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m) :precision binary64 (* t_s 1.0))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * 1.0;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * 1.0d0
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * 1.0;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * 1.0

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * 1.0)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * 1.0;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * 1.0), $MachinePrecision]

Derivation

1. Initial program 35.4%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Add Preprocessing

3. Taylor expanded in l around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

   3. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f64 41.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

5. Simplified 41.6%

   \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x + -1}}}} \]

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 40.8%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(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)))))