Toniolo and Linder, Equation (7)

Percentage Accurate: 33.9% → 85.0%

Time: 19.4s

Alternatives: 10

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: 33.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot \left(t\_m \cdot t\_m\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.5 \cdot 10^{-173}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{l\_m \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{elif}\;t\_m \leq 4.5 \cdot 10^{+59}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{t\_2 - \frac{\left(t\_2 + l\_m \cdot l\_m\right) \cdot \left(\frac{-1}{x} + \left(-2 + \frac{-1}{x}\right)\right)}{x}}{2}}}{t\_m}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \end{array} \]

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_s
    (if (<= t_m 4.5e-173)
      (/ (* t_m (sqrt 2.0)) (* l_m (sqrt (+ (/ 2.0 x) (/ 2.0 (* x x))))))
      (if (<= t_m 4.5e+59)
        (/
         1.0
         (/
          (sqrt
           (/
            (-
             t_2
             (/ (* (+ t_2 (* l_m l_m)) (+ (/ -1.0 x) (+ -2.0 (/ -1.0 x)))) x))
            2.0))
          t_m))
        (+ 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) {
	double t_2 = 2.0 * (t_m * t_m);
	double tmp;
	if (t_m <= 4.5e-173) {
		tmp = (t_m * sqrt(2.0)) / (l_m * sqrt(((2.0 / x) + (2.0 / (x * x)))));
	} else if (t_m <= 4.5e+59) {
		tmp = 1.0 / (sqrt(((t_2 - (((t_2 + (l_m * l_m)) * ((-1.0 / x) + (-2.0 + (-1.0 / x)))) / x)) / 2.0)) / t_m);
	} else {
		tmp = 1.0 + (-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) :: tmp
    t_2 = 2.0d0 * (t_m * t_m)
    if (t_m <= 4.5d-173) then
        tmp = (t_m * sqrt(2.0d0)) / (l_m * sqrt(((2.0d0 / x) + (2.0d0 / (x * x)))))
    else if (t_m <= 4.5d+59) then
        tmp = 1.0d0 / (sqrt(((t_2 - (((t_2 + (l_m * l_m)) * (((-1.0d0) / x) + ((-2.0d0) + ((-1.0d0) / x)))) / x)) / 2.0d0)) / t_m)
    else
        tmp = 1.0d0 + ((-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 tmp;
	if (t_m <= 4.5e-173) {
		tmp = (t_m * Math.sqrt(2.0)) / (l_m * Math.sqrt(((2.0 / x) + (2.0 / (x * x)))));
	} else if (t_m <= 4.5e+59) {
		tmp = 1.0 / (Math.sqrt(((t_2 - (((t_2 + (l_m * l_m)) * ((-1.0 / x) + (-2.0 + (-1.0 / x)))) / x)) / 2.0)) / t_m);
	} else {
		tmp = 1.0 + (-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)
	tmp = 0
	if t_m <= 4.5e-173:
		tmp = (t_m * math.sqrt(2.0)) / (l_m * math.sqrt(((2.0 / x) + (2.0 / (x * x)))))
	elif t_m <= 4.5e+59:
		tmp = 1.0 / (math.sqrt(((t_2 - (((t_2 + (l_m * l_m)) * ((-1.0 / x) + (-2.0 + (-1.0 / x)))) / x)) / 2.0)) / t_m)
	else:
		tmp = 1.0 + (-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))
	tmp = 0.0
	if (t_m <= 4.5e-173)
		tmp = Float64(Float64(t_m * sqrt(2.0)) / Float64(l_m * sqrt(Float64(Float64(2.0 / x) + Float64(2.0 / Float64(x * x))))));
	elseif (t_m <= 4.5e+59)
		tmp = Float64(1.0 / Float64(sqrt(Float64(Float64(t_2 - Float64(Float64(Float64(t_2 + Float64(l_m * l_m)) * Float64(Float64(-1.0 / x) + Float64(-2.0 + Float64(-1.0 / x)))) / x)) / 2.0)) / t_m));
	else
		tmp = Float64(1.0 + Float64(-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);
	tmp = 0.0;
	if (t_m <= 4.5e-173)
		tmp = (t_m * sqrt(2.0)) / (l_m * sqrt(((2.0 / x) + (2.0 / (x * x)))));
	elseif (t_m <= 4.5e+59)
		tmp = 1.0 / (sqrt(((t_2 - (((t_2 + (l_m * l_m)) * ((-1.0 / x) + (-2.0 + (-1.0 / x)))) / x)) / 2.0)) / t_m);
	else
		tmp = 1.0 + (-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]}, N[(t$95$s * If[LessEqual[t$95$m, 4.5e-173], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(l$95$m * N[Sqrt[N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.5e+59], N[(1.0 / N[(N[Sqrt[N[(N[(t$95$2 - N[(N[(N[(t$95$2 + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(-1.0 / x), $MachinePrecision] + N[(-2.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 4.50000000000000018e-173

   1. Initial program 34.1%

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

   3. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

   5. Simplified 65.2%

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

   6. Taylor expanded in l around inf

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 19.3%

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

   8. Simplified 19.3%

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

   if 4.50000000000000018e-173 < t < 4.49999999999999959e59

   1. Initial program 58.0%

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

   3. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

   5. Simplified 88.2%

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

   7. Applied egg-rr 88.6%

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

   if 4.49999999999999959e59 < t

   1. Initial program 19.8%

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

   3. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

   5. Simplified 23.0%

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

   7. Applied egg-rr 23.1%

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

   8. Taylor expanded in x around inf

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

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

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

      2. associate-*r/ N/A

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

      3. times-frac N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. rem-square-sqrt N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right)\right), \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \]

      18. *-lowering-*.f64 21.7%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   10. Simplified 21.7%

       \[\leadsto \color{blue}{1 + \frac{-0.25}{t \cdot t} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x \cdot 0.5}} \]

   11. Taylor expanded in t around inf

       \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{x}\right)}\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 91.4%

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

   13. Simplified 91.4%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{-173}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+59}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{2 \cdot \left(t \cdot t\right) - \frac{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right) \cdot \left(\frac{-1}{x} + \left(-2 + \frac{-1}{x}\right)\right)}{x}}{2}}}{t}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot \left(t\_m \cdot t\_m\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-172}:\\ \;\;\;\;\frac{t\_m}{l\_m \cdot \sqrt{\frac{\frac{1}{x} \cdot \left(2 + \frac{2}{x}\right)}{2}}}\\ \mathbf{elif}\;t\_m \leq 3.9 \cdot 10^{+59}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{t\_2 - \frac{\left(t\_2 + l\_m \cdot l\_m\right) \cdot \left(\frac{-1}{x} + \left(-2 + \frac{-1}{x}\right)\right)}{x}}{2}}}{t\_m}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \end{array} \]

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_s
    (if (<= t_m 1e-172)
      (/ t_m (* l_m (sqrt (/ (* (/ 1.0 x) (+ 2.0 (/ 2.0 x))) 2.0))))
      (if (<= t_m 3.9e+59)
        (/
         1.0
         (/
          (sqrt
           (/
            (-
             t_2
             (/ (* (+ t_2 (* l_m l_m)) (+ (/ -1.0 x) (+ -2.0 (/ -1.0 x)))) x))
            2.0))
          t_m))
        (+ 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) {
	double t_2 = 2.0 * (t_m * t_m);
	double tmp;
	if (t_m <= 1e-172) {
		tmp = t_m / (l_m * sqrt((((1.0 / x) * (2.0 + (2.0 / x))) / 2.0)));
	} else if (t_m <= 3.9e+59) {
		tmp = 1.0 / (sqrt(((t_2 - (((t_2 + (l_m * l_m)) * ((-1.0 / x) + (-2.0 + (-1.0 / x)))) / x)) / 2.0)) / t_m);
	} else {
		tmp = 1.0 + (-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) :: tmp
    t_2 = 2.0d0 * (t_m * t_m)
    if (t_m <= 1d-172) then
        tmp = t_m / (l_m * sqrt((((1.0d0 / x) * (2.0d0 + (2.0d0 / x))) / 2.0d0)))
    else if (t_m <= 3.9d+59) then
        tmp = 1.0d0 / (sqrt(((t_2 - (((t_2 + (l_m * l_m)) * (((-1.0d0) / x) + ((-2.0d0) + ((-1.0d0) / x)))) / x)) / 2.0d0)) / t_m)
    else
        tmp = 1.0d0 + ((-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 tmp;
	if (t_m <= 1e-172) {
		tmp = t_m / (l_m * Math.sqrt((((1.0 / x) * (2.0 + (2.0 / x))) / 2.0)));
	} else if (t_m <= 3.9e+59) {
		tmp = 1.0 / (Math.sqrt(((t_2 - (((t_2 + (l_m * l_m)) * ((-1.0 / x) + (-2.0 + (-1.0 / x)))) / x)) / 2.0)) / t_m);
	} else {
		tmp = 1.0 + (-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)
	tmp = 0
	if t_m <= 1e-172:
		tmp = t_m / (l_m * math.sqrt((((1.0 / x) * (2.0 + (2.0 / x))) / 2.0)))
	elif t_m <= 3.9e+59:
		tmp = 1.0 / (math.sqrt(((t_2 - (((t_2 + (l_m * l_m)) * ((-1.0 / x) + (-2.0 + (-1.0 / x)))) / x)) / 2.0)) / t_m)
	else:
		tmp = 1.0 + (-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))
	tmp = 0.0
	if (t_m <= 1e-172)
		tmp = Float64(t_m / Float64(l_m * sqrt(Float64(Float64(Float64(1.0 / x) * Float64(2.0 + Float64(2.0 / x))) / 2.0))));
	elseif (t_m <= 3.9e+59)
		tmp = Float64(1.0 / Float64(sqrt(Float64(Float64(t_2 - Float64(Float64(Float64(t_2 + Float64(l_m * l_m)) * Float64(Float64(-1.0 / x) + Float64(-2.0 + Float64(-1.0 / x)))) / x)) / 2.0)) / t_m));
	else
		tmp = Float64(1.0 + Float64(-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);
	tmp = 0.0;
	if (t_m <= 1e-172)
		tmp = t_m / (l_m * sqrt((((1.0 / x) * (2.0 + (2.0 / x))) / 2.0)));
	elseif (t_m <= 3.9e+59)
		tmp = 1.0 / (sqrt(((t_2 - (((t_2 + (l_m * l_m)) * ((-1.0 / x) + (-2.0 + (-1.0 / x)))) / x)) / 2.0)) / t_m);
	else
		tmp = 1.0 + (-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]}, N[(t$95$s * If[LessEqual[t$95$m, 1e-172], N[(t$95$m / N[(l$95$m * N[Sqrt[N[(N[(N[(1.0 / x), $MachinePrecision] * N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.9e+59], N[(1.0 / N[(N[Sqrt[N[(N[(t$95$2 - N[(N[(N[(t$95$2 + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(-1.0 / x), $MachinePrecision] + N[(-2.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1e-172

   1. Initial program 34.1%

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

   3. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

   5. Simplified 65.2%

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

   6. Taylor expanded in l around inf

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 19.3%

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

   8. Simplified 19.3%

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

   10. Applied egg-rr 19.3%

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

   if 1e-172 < t < 3.90000000000000021e59

   1. Initial program 58.0%

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

   3. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

   5. Simplified 88.2%

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

   7. Applied egg-rr 88.6%

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

   if 3.90000000000000021e59 < t

   1. Initial program 19.8%

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

   3. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

   5. Simplified 23.0%

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

   7. Applied egg-rr 23.1%

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

   8. Taylor expanded in x around inf

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

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

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

      2. associate-*r/ N/A

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

      3. times-frac N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. rem-square-sqrt N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right)\right), \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \]

      18. *-lowering-*.f64 21.7%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   10. Simplified 21.7%

       \[\leadsto \color{blue}{1 + \frac{-0.25}{t \cdot t} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x \cdot 0.5}} \]

   11. Taylor expanded in t around inf

       \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{x}\right)}\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 91.4%

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

   13. Simplified 91.4%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-172}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{\frac{1}{x} \cdot \left(2 + \frac{2}{x}\right)}{2}}}\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{+59}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{2 \cdot \left(t \cdot t\right) - \frac{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right) \cdot \left(\frac{-1}{x} + \left(-2 + \frac{-1}{x}\right)\right)}{x}}{2}}}{t}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.2 \cdot 10^{-164}:\\ \;\;\;\;\frac{t\_m}{l\_m \cdot \sqrt{\frac{\frac{1}{x} \cdot \left(2 + \frac{2}{x}\right)}{2}}}\\ \mathbf{elif}\;t\_m \leq 1.65 \cdot 10^{+59}:\\ \;\;\;\;t\_m \cdot {\left(t\_m \cdot t\_m - \frac{\left(-2 + \frac{-2}{x}\right) \cdot \left(l\_m \cdot l\_m + t\_m \cdot \left(t\_m \cdot 2\right)\right)}{2 \cdot x}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

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 1.2e-164)
    (/ t_m (* l_m (sqrt (/ (* (/ 1.0 x) (+ 2.0 (/ 2.0 x))) 2.0))))
    (if (<= t_m 1.65e+59)
      (*
       t_m
       (pow
        (-
         (* t_m t_m)
         (/
          (* (+ -2.0 (/ -2.0 x)) (+ (* l_m l_m) (* t_m (* t_m 2.0))))
          (* 2.0 x)))
        -0.5))
      (+ 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) {
	double tmp;
	if (t_m <= 1.2e-164) {
		tmp = t_m / (l_m * sqrt((((1.0 / x) * (2.0 + (2.0 / x))) / 2.0)));
	} else if (t_m <= 1.65e+59) {
		tmp = t_m * pow(((t_m * t_m) - (((-2.0 + (-2.0 / x)) * ((l_m * l_m) + (t_m * (t_m * 2.0)))) / (2.0 * x))), -0.5);
	} else {
		tmp = 1.0 + (-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 <= 1.2d-164) then
        tmp = t_m / (l_m * sqrt((((1.0d0 / x) * (2.0d0 + (2.0d0 / x))) / 2.0d0)))
    else if (t_m <= 1.65d+59) then
        tmp = t_m * (((t_m * t_m) - ((((-2.0d0) + ((-2.0d0) / x)) * ((l_m * l_m) + (t_m * (t_m * 2.0d0)))) / (2.0d0 * x))) ** (-0.5d0))
    else
        tmp = 1.0d0 + ((-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 <= 1.2e-164) {
		tmp = t_m / (l_m * Math.sqrt((((1.0 / x) * (2.0 + (2.0 / x))) / 2.0)));
	} else if (t_m <= 1.65e+59) {
		tmp = t_m * Math.pow(((t_m * t_m) - (((-2.0 + (-2.0 / x)) * ((l_m * l_m) + (t_m * (t_m * 2.0)))) / (2.0 * x))), -0.5);
	} else {
		tmp = 1.0 + (-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 <= 1.2e-164:
		tmp = t_m / (l_m * math.sqrt((((1.0 / x) * (2.0 + (2.0 / x))) / 2.0)))
	elif t_m <= 1.65e+59:
		tmp = t_m * math.pow(((t_m * t_m) - (((-2.0 + (-2.0 / x)) * ((l_m * l_m) + (t_m * (t_m * 2.0)))) / (2.0 * x))), -0.5)
	else:
		tmp = 1.0 + (-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 <= 1.2e-164)
		tmp = Float64(t_m / Float64(l_m * sqrt(Float64(Float64(Float64(1.0 / x) * Float64(2.0 + Float64(2.0 / x))) / 2.0))));
	elseif (t_m <= 1.65e+59)
		tmp = Float64(t_m * (Float64(Float64(t_m * t_m) - Float64(Float64(Float64(-2.0 + Float64(-2.0 / x)) * Float64(Float64(l_m * l_m) + Float64(t_m * Float64(t_m * 2.0)))) / Float64(2.0 * x))) ^ -0.5));
	else
		tmp = Float64(1.0 + Float64(-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 <= 1.2e-164)
		tmp = t_m / (l_m * sqrt((((1.0 / x) * (2.0 + (2.0 / x))) / 2.0)));
	elseif (t_m <= 1.65e+59)
		tmp = t_m * (((t_m * t_m) - (((-2.0 + (-2.0 / x)) * ((l_m * l_m) + (t_m * (t_m * 2.0)))) / (2.0 * x))) ^ -0.5);
	else
		tmp = 1.0 + (-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, 1.2e-164], N[(t$95$m / N[(l$95$m * N[Sqrt[N[(N[(N[(1.0 / x), $MachinePrecision] * N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.65e+59], N[(t$95$m * N[Power[N[(N[(t$95$m * t$95$m), $MachinePrecision] - N[(N[(N[(-2.0 + N[(-2.0 / x), $MachinePrecision]), $MachinePrecision] * N[(N[(l$95$m * l$95$m), $MachinePrecision] + N[(t$95$m * N[(t$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.19999999999999992e-164

   1. Initial program 34.1%

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

   3. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

   5. Simplified 65.2%

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

   6. Taylor expanded in l around inf

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 19.3%

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

   8. Simplified 19.3%

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

   10. Applied egg-rr 19.3%

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

   if 1.19999999999999992e-164 < t < 1.65e59

   1. Initial program 58.0%

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

   3. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

   5. Simplified 88.2%

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

   7. Applied egg-rr 88.6%

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

      1. associate-/r/ N/A

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

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

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

   9. Applied egg-rr 88.4%

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

   if 1.65e59 < t

   1. Initial program 19.8%

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

   3. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

   5. Simplified 23.0%

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

   7. Applied egg-rr 23.1%

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

   8. Taylor expanded in x around inf

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

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

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

      2. associate-*r/ N/A

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

      3. times-frac N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. rem-square-sqrt N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right)\right), \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \]

      18. *-lowering-*.f64 21.7%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   10. Simplified 21.7%

       \[\leadsto \color{blue}{1 + \frac{-0.25}{t \cdot t} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x \cdot 0.5}} \]

   11. Taylor expanded in t around inf

       \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{x}\right)}\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 91.4%

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

   13. Simplified 91.4%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{-164}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{\frac{1}{x} \cdot \left(2 + \frac{2}{x}\right)}{2}}}\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{+59}:\\ \;\;\;\;t \cdot {\left(t \cdot t - \frac{\left(-2 + \frac{-2}{x}\right) \cdot \left(\ell \cdot \ell + t \cdot \left(t \cdot 2\right)\right)}{2 \cdot x}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-172}:\\ \;\;\;\;\frac{t\_m}{l\_m \cdot \sqrt{\frac{\frac{1}{x} \cdot \left(2 + \frac{2}{x}\right)}{2}}}\\ \mathbf{elif}\;t\_m \leq 4.7 \cdot 10^{+59}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{2 \cdot \left(t\_m \cdot t\_m\right)}{x} + \left(t\_m \cdot t\_m + \frac{l\_m \cdot l\_m}{x}\right)}}{t\_m}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

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 1e-172)
    (/ t_m (* l_m (sqrt (/ (* (/ 1.0 x) (+ 2.0 (/ 2.0 x))) 2.0))))
    (if (<= t_m 4.7e+59)
      (/
       1.0
       (/
        (sqrt (+ (/ (* 2.0 (* t_m t_m)) x) (+ (* t_m t_m) (/ (* l_m l_m) x))))
        t_m))
      (+ 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) {
	double tmp;
	if (t_m <= 1e-172) {
		tmp = t_m / (l_m * sqrt((((1.0 / x) * (2.0 + (2.0 / x))) / 2.0)));
	} else if (t_m <= 4.7e+59) {
		tmp = 1.0 / (sqrt((((2.0 * (t_m * t_m)) / x) + ((t_m * t_m) + ((l_m * l_m) / x)))) / t_m);
	} else {
		tmp = 1.0 + (-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 <= 1d-172) then
        tmp = t_m / (l_m * sqrt((((1.0d0 / x) * (2.0d0 + (2.0d0 / x))) / 2.0d0)))
    else if (t_m <= 4.7d+59) then
        tmp = 1.0d0 / (sqrt((((2.0d0 * (t_m * t_m)) / x) + ((t_m * t_m) + ((l_m * l_m) / x)))) / t_m)
    else
        tmp = 1.0d0 + ((-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 <= 1e-172) {
		tmp = t_m / (l_m * Math.sqrt((((1.0 / x) * (2.0 + (2.0 / x))) / 2.0)));
	} else if (t_m <= 4.7e+59) {
		tmp = 1.0 / (Math.sqrt((((2.0 * (t_m * t_m)) / x) + ((t_m * t_m) + ((l_m * l_m) / x)))) / t_m);
	} else {
		tmp = 1.0 + (-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 <= 1e-172:
		tmp = t_m / (l_m * math.sqrt((((1.0 / x) * (2.0 + (2.0 / x))) / 2.0)))
	elif t_m <= 4.7e+59:
		tmp = 1.0 / (math.sqrt((((2.0 * (t_m * t_m)) / x) + ((t_m * t_m) + ((l_m * l_m) / x)))) / t_m)
	else:
		tmp = 1.0 + (-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 <= 1e-172)
		tmp = Float64(t_m / Float64(l_m * sqrt(Float64(Float64(Float64(1.0 / x) * Float64(2.0 + Float64(2.0 / x))) / 2.0))));
	elseif (t_m <= 4.7e+59)
		tmp = Float64(1.0 / Float64(sqrt(Float64(Float64(Float64(2.0 * Float64(t_m * t_m)) / x) + Float64(Float64(t_m * t_m) + Float64(Float64(l_m * l_m) / x)))) / t_m));
	else
		tmp = Float64(1.0 + Float64(-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 <= 1e-172)
		tmp = t_m / (l_m * sqrt((((1.0 / x) * (2.0 + (2.0 / x))) / 2.0)));
	elseif (t_m <= 4.7e+59)
		tmp = 1.0 / (sqrt((((2.0 * (t_m * t_m)) / x) + ((t_m * t_m) + ((l_m * l_m) / x)))) / t_m);
	else
		tmp = 1.0 + (-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, 1e-172], N[(t$95$m / N[(l$95$m * N[Sqrt[N[(N[(N[(1.0 / x), $MachinePrecision] * N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.7e+59], N[(1.0 / N[(N[Sqrt[N[(N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] + N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1e-172

   1. Initial program 34.1%

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

   3. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

   5. Simplified 65.2%

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

   6. Taylor expanded in l around inf

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 19.3%

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

   8. Simplified 19.3%

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

   10. Applied egg-rr 19.3%

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

   if 1e-172 < t < 4.7e59

   1. Initial program 58.0%

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

   3. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

   5. Simplified 88.2%

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

   7. Applied egg-rr 88.6%

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

   8. Taylor expanded in x around inf

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

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

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

      2. associate-*r/ N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. +-commutative N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 88.0%

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

   10. Simplified 88.0%

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

   if 4.7e59 < t

   1. Initial program 19.8%

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

   3. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

   5. Simplified 23.0%

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

   7. Applied egg-rr 23.1%

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

   8. Taylor expanded in x around inf

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

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

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

      2. associate-*r/ N/A

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

      3. times-frac N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. rem-square-sqrt N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right)\right), \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \]

      18. *-lowering-*.f64 21.7%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   10. Simplified 21.7%

       \[\leadsto \color{blue}{1 + \frac{-0.25}{t \cdot t} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x \cdot 0.5}} \]

   11. Taylor expanded in t around inf

       \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{x}\right)}\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 91.4%

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

   13. Simplified 91.4%

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

Alternative 5: 79.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6 \cdot 10^{-156}:\\ \;\;\;\;\frac{t\_m}{l\_m \cdot \sqrt{\frac{\frac{1}{x} \cdot \left(2 + \frac{2}{x}\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

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 6e-156)
    (/ t_m (* l_m (sqrt (/ (* (/ 1.0 x) (+ 2.0 (/ 2.0 x))) 2.0))))
    (+ 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) {
	double tmp;
	if (t_m <= 6e-156) {
		tmp = t_m / (l_m * sqrt((((1.0 / x) * (2.0 + (2.0 / x))) / 2.0)));
	} else {
		tmp = 1.0 + (-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 <= 6d-156) then
        tmp = t_m / (l_m * sqrt((((1.0d0 / x) * (2.0d0 + (2.0d0 / x))) / 2.0d0)))
    else
        tmp = 1.0d0 + ((-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 <= 6e-156) {
		tmp = t_m / (l_m * Math.sqrt((((1.0 / x) * (2.0 + (2.0 / x))) / 2.0)));
	} else {
		tmp = 1.0 + (-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 <= 6e-156:
		tmp = t_m / (l_m * math.sqrt((((1.0 / x) * (2.0 + (2.0 / x))) / 2.0)))
	else:
		tmp = 1.0 + (-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 <= 6e-156)
		tmp = Float64(t_m / Float64(l_m * sqrt(Float64(Float64(Float64(1.0 / x) * Float64(2.0 + Float64(2.0 / x))) / 2.0))));
	else
		tmp = Float64(1.0 + Float64(-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 <= 6e-156)
		tmp = t_m / (l_m * sqrt((((1.0 / x) * (2.0 + (2.0 / x))) / 2.0)));
	else
		tmp = 1.0 + (-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, 6e-156], N[(t$95$m / N[(l$95$m * N[Sqrt[N[(N[(N[(1.0 / x), $MachinePrecision] * N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6e-156

   1. Initial program 33.6%

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

   3. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

   5. Simplified 65.7%

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

   6. Taylor expanded in l around inf

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 20.4%

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

   8. Simplified 20.4%

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

   10. Applied egg-rr 20.4%

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

   if 6e-156 < t

   1. Initial program 37.2%

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

   3. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

   5. Simplified 50.8%

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

   7. Applied egg-rr 51.1%

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

   8. Taylor expanded in x around inf

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

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

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

      2. associate-*r/ N/A

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

      3. times-frac N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. rem-square-sqrt N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right)\right), \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \]

      18. *-lowering-*.f64 42.2%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   10. Simplified 42.2%

       \[\leadsto \color{blue}{1 + \frac{-0.25}{t \cdot t} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x \cdot 0.5}} \]

   11. Taylor expanded in t around inf

       \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{x}\right)}\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 82.6%

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

   13. Simplified 82.6%

       \[\leadsto 1 + \color{blue}{\frac{-1}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 79.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.6 \cdot 10^{-154}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{\frac{2}{\frac{1}{x} \cdot \left(2 + \frac{2}{x}\right)}}}{l\_m}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

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 1.6e-154)
    (* t_m (/ (sqrt (/ 2.0 (* (/ 1.0 x) (+ 2.0 (/ 2.0 x))))) l_m))
    (+ 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) {
	double tmp;
	if (t_m <= 1.6e-154) {
		tmp = t_m * (sqrt((2.0 / ((1.0 / x) * (2.0 + (2.0 / x))))) / l_m);
	} else {
		tmp = 1.0 + (-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 <= 1.6d-154) then
        tmp = t_m * (sqrt((2.0d0 / ((1.0d0 / x) * (2.0d0 + (2.0d0 / x))))) / l_m)
    else
        tmp = 1.0d0 + ((-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 <= 1.6e-154) {
		tmp = t_m * (Math.sqrt((2.0 / ((1.0 / x) * (2.0 + (2.0 / x))))) / l_m);
	} else {
		tmp = 1.0 + (-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 <= 1.6e-154:
		tmp = t_m * (math.sqrt((2.0 / ((1.0 / x) * (2.0 + (2.0 / x))))) / l_m)
	else:
		tmp = 1.0 + (-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 <= 1.6e-154)
		tmp = Float64(t_m * Float64(sqrt(Float64(2.0 / Float64(Float64(1.0 / x) * Float64(2.0 + Float64(2.0 / x))))) / l_m));
	else
		tmp = Float64(1.0 + Float64(-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 <= 1.6e-154)
		tmp = t_m * (sqrt((2.0 / ((1.0 / x) * (2.0 + (2.0 / x))))) / l_m);
	else
		tmp = 1.0 + (-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, 1.6e-154], N[(t$95$m * N[(N[Sqrt[N[(2.0 / N[(N[(1.0 / x), $MachinePrecision] * N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.60000000000000002e-154

   1. Initial program 33.6%

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

   3. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

   5. Simplified 65.7%

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

   6. Taylor expanded in l around inf

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 20.4%

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

   8. Simplified 20.4%

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

   10. Applied egg-rr 20.4%

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

   if 1.60000000000000002e-154 < t

   1. Initial program 37.2%

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

   3. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

   5. Simplified 50.8%

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

   7. Applied egg-rr 51.1%

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

   8. Taylor expanded in x around inf

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

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

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

      2. associate-*r/ N/A

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

      3. times-frac N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. rem-square-sqrt N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right)\right), \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \]

      18. *-lowering-*.f64 42.2%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   10. Simplified 42.2%

       \[\leadsto \color{blue}{1 + \frac{-0.25}{t \cdot t} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x \cdot 0.5}} \]

   11. Taylor expanded in t around inf

       \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{x}\right)}\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 82.6%

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

   13. Simplified 82.6%

       \[\leadsto 1 + \color{blue}{\frac{-1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{-154}:\\ \;\;\;\;t \cdot \frac{\sqrt{\frac{2}{\frac{1}{x} \cdot \left(2 + \frac{2}{x}\right)}}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-155}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{l\_m}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

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 9e-155) (/ (* t_m (sqrt x)) l_m) (+ 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) {
	double tmp;
	if (t_m <= 9e-155) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else {
		tmp = 1.0 + (-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 <= 9d-155) then
        tmp = (t_m * sqrt(x)) / l_m
    else
        tmp = 1.0d0 + ((-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 <= 9e-155) {
		tmp = (t_m * Math.sqrt(x)) / l_m;
	} else {
		tmp = 1.0 + (-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 <= 9e-155:
		tmp = (t_m * math.sqrt(x)) / l_m
	else:
		tmp = 1.0 + (-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 <= 9e-155)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	else
		tmp = Float64(1.0 + Float64(-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 <= 9e-155)
		tmp = (t_m * sqrt(x)) / l_m;
	else
		tmp = 1.0 + (-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, 9e-155], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.0000000000000007e-155

   1. Initial program 33.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing
   3. 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. *-commutative N/A

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

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

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

   4. Applied egg-rr 33.5%

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

   5. Taylor expanded in l around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. associate--l+ N/A

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

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

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

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

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

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

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. +-lowering-+.f64 14.3%

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

   7. Simplified 14.3%

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

   8. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 28.1%

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

   10. Simplified 28.1%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{t}{\ell} \cdot \sqrt{x}} \]
   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 20.3%

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

   13. Simplified 20.3%

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

   if 9.0000000000000007e-155 < t

   1. Initial program 37.2%

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

   3. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

   5. Simplified 50.8%

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

   7. Applied egg-rr 51.1%

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

   8. Taylor expanded in x around inf

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

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

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

      2. associate-*r/ N/A

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

      3. times-frac N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. rem-square-sqrt N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right)\right), \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \]

      18. *-lowering-*.f64 42.2%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   10. Simplified 42.2%

       \[\leadsto \color{blue}{1 + \frac{-0.25}{t \cdot t} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x \cdot 0.5}} \]

   11. Taylor expanded in t around inf

       \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{x}\right)}\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 82.6%

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

   13. Simplified 82.6%

       \[\leadsto 1 + \color{blue}{\frac{-1}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 79.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.35 \cdot 10^{-154}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{x}}{l\_m}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

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 1.35e-154) (* t_m (/ (sqrt x) l_m)) (+ 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) {
	double tmp;
	if (t_m <= 1.35e-154) {
		tmp = t_m * (sqrt(x) / l_m);
	} else {
		tmp = 1.0 + (-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 <= 1.35d-154) then
        tmp = t_m * (sqrt(x) / l_m)
    else
        tmp = 1.0d0 + ((-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 <= 1.35e-154) {
		tmp = t_m * (Math.sqrt(x) / l_m);
	} else {
		tmp = 1.0 + (-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 <= 1.35e-154:
		tmp = t_m * (math.sqrt(x) / l_m)
	else:
		tmp = 1.0 + (-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 <= 1.35e-154)
		tmp = Float64(t_m * Float64(sqrt(x) / l_m));
	else
		tmp = Float64(1.0 + Float64(-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 <= 1.35e-154)
		tmp = t_m * (sqrt(x) / l_m);
	else
		tmp = 1.0 + (-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, 1.35e-154], N[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.34999999999999995e-154

   1. Initial program 33.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing
   3. 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. *-commutative N/A

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

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

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

   4. Applied egg-rr 33.5%

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

   5. Taylor expanded in l around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. associate--l+ N/A

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

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

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

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

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

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

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. +-lowering-+.f64 14.3%

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

   7. Simplified 14.3%

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

   8. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 28.1%

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

   10. Simplified 28.1%

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

   11. Taylor expanded in x around inf

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

       1. associate-*l/ N/A

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

       2. *-lft-identity N/A

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

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

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

       4. sqrt-lowering-sqrt.f64 20.3%

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

   13. Simplified 20.3%

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

   if 1.34999999999999995e-154 < t

   1. Initial program 37.2%

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

   3. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

   5. Simplified 50.8%

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

   7. Applied egg-rr 51.1%

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

   8. Taylor expanded in x around inf

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

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

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

      2. associate-*r/ N/A

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

      3. times-frac N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. rem-square-sqrt N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right)\right), \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \]

      18. *-lowering-*.f64 42.2%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   10. Simplified 42.2%

       \[\leadsto \color{blue}{1 + \frac{-0.25}{t \cdot t} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x \cdot 0.5}} \]

   11. Taylor expanded in t around inf

       \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{x}\right)}\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 82.6%

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

   13. Simplified 82.6%

       \[\leadsto 1 + \color{blue}{\frac{-1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.35 \cdot 10^{-154}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.1% 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 35.2%

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

3. Taylor expanded in x around -inf

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

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

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

5. Simplified 59.0%

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

7. Applied egg-rr 59.0%

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

8. Taylor expanded in x around inf

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

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

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

   2. associate-*r/ N/A

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

   3. times-frac N/A

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

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

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

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

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

   6. unpow2 N/A

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

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

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

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

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

   9. +-commutative N/A

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

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

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

   11. unpow2 N/A

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

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

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

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

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

   14. unpow2 N/A

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

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

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

   16. unpow2 N/A

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

   17. rem-square-sqrt N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right)\right), \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \]

   18. *-lowering-*.f64 19.5%

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

10. Simplified 19.5%

    \[\leadsto \color{blue}{1 + \frac{-0.25}{t \cdot t} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x \cdot 0.5}} \]

11. Taylor expanded in t around inf

    \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{x}\right)}\right) \]
12. Step-by-step derivation

    1. /-lowering-/.f64 39.5%

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

13. Simplified 39.5%

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

Alternative 10: 75.4% 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 35.2%

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

3. Taylor expanded in x around inf

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

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

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

   2. sqrt-lowering-sqrt.f64 39.3%

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

5. Simplified 39.3%

   \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \sqrt{2}}} \]
6. Step-by-step derivation

   1. *-commutative N/A

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

   2. *-inverses 39.3%

      \[\leadsto 1 \]

7. Applied egg-rr 39.3%

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

Reproduce

herbie shell --seed 2024164 
(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)))))