Toniolo and Linder, Equation (7)

Percentage Accurate: 33.2% → 84.6%

Time: 27.1s

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: 33.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.4 \cdot 10^{-189}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 1.2 \cdot 10^{-174}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 3.3 \cdot 10^{-161}:\\ \;\;\;\;\frac{\left(t\_m \cdot \sqrt{x \cdot 0.5}\right) \cdot \sqrt{2}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 1.32 \cdot 10^{+36}:\\ \;\;\;\;\frac{\sqrt{t\_2}}{\sqrt{\left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_2 + \frac{{l\_m}^{2}}{x}\right)\right) + \frac{t\_2 + {l\_m}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \end{array} \end{array} \]

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 (pow t_m 2.0))))
   (*
    t_s
    (if (<= t_m 6.4e-189)
      (/ (* t_m (sqrt x)) l_m)
      (if (<= t_m 1.2e-174)
        1.0
        (if (<= t_m 3.3e-161)
          (/ (* (* t_m (sqrt (* x 0.5))) (sqrt 2.0)) l_m)
          (if (<= t_m 1.32e+36)
            (/
             (sqrt t_2)
             (sqrt
              (+
               (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l_m 2.0) x)))
               (/ (+ t_2 (pow l_m 2.0)) x))))
            (/ 1.0 (sqrt (/ (+ x 1.0) (+ x -1.0)))))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double tmp;
	if (t_m <= 6.4e-189) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else if (t_m <= 1.2e-174) {
		tmp = 1.0;
	} else if (t_m <= 3.3e-161) {
		tmp = ((t_m * sqrt((x * 0.5))) * sqrt(2.0)) / l_m;
	} else if (t_m <= 1.32e+36) {
		tmp = sqrt(t_2) / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l_m, 2.0) / x))) + ((t_2 + pow(l_m, 2.0)) / x)));
	} else {
		tmp = 1.0 / sqrt(((x + 1.0) / (x + -1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    if (t_m <= 6.4d-189) then
        tmp = (t_m * sqrt(x)) / l_m
    else if (t_m <= 1.2d-174) then
        tmp = 1.0d0
    else if (t_m <= 3.3d-161) then
        tmp = ((t_m * sqrt((x * 0.5d0))) * sqrt(2.0d0)) / l_m
    else if (t_m <= 1.32d+36) then
        tmp = sqrt(t_2) / sqrt((((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_2 + ((l_m ** 2.0d0) / x))) + ((t_2 + (l_m ** 2.0d0)) / x)))
    else
        tmp = 1.0d0 / sqrt(((x + 1.0d0) / (x + (-1.0d0))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * Math.pow(t_m, 2.0);
	double tmp;
	if (t_m <= 6.4e-189) {
		tmp = (t_m * Math.sqrt(x)) / l_m;
	} else if (t_m <= 1.2e-174) {
		tmp = 1.0;
	} else if (t_m <= 3.3e-161) {
		tmp = ((t_m * Math.sqrt((x * 0.5))) * Math.sqrt(2.0)) / l_m;
	} else if (t_m <= 1.32e+36) {
		tmp = Math.sqrt(t_2) / Math.sqrt((((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_2 + (Math.pow(l_m, 2.0) / x))) + ((t_2 + Math.pow(l_m, 2.0)) / x)));
	} else {
		tmp = 1.0 / Math.sqrt(((x + 1.0) / (x + -1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = 2.0 * math.pow(t_m, 2.0)
	tmp = 0
	if t_m <= 6.4e-189:
		tmp = (t_m * math.sqrt(x)) / l_m
	elif t_m <= 1.2e-174:
		tmp = 1.0
	elif t_m <= 3.3e-161:
		tmp = ((t_m * math.sqrt((x * 0.5))) * math.sqrt(2.0)) / l_m
	elif t_m <= 1.32e+36:
		tmp = math.sqrt(t_2) / math.sqrt((((2.0 * (math.pow(t_m, 2.0) / x)) + (t_2 + (math.pow(l_m, 2.0) / x))) + ((t_2 + math.pow(l_m, 2.0)) / x)))
	else:
		tmp = 1.0 / math.sqrt(((x + 1.0) / (x + -1.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(2.0 * (t_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 6.4e-189)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	elseif (t_m <= 1.2e-174)
		tmp = 1.0;
	elseif (t_m <= 3.3e-161)
		tmp = Float64(Float64(Float64(t_m * sqrt(Float64(x * 0.5))) * sqrt(2.0)) / l_m);
	elseif (t_m <= 1.32e+36)
		tmp = Float64(sqrt(t_2) / sqrt(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l_m ^ 2.0) / x))) + Float64(Float64(t_2 + (l_m ^ 2.0)) / x))));
	else
		tmp = Float64(1.0 / sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = 2.0 * (t_m ^ 2.0);
	tmp = 0.0;
	if (t_m <= 6.4e-189)
		tmp = (t_m * sqrt(x)) / l_m;
	elseif (t_m <= 1.2e-174)
		tmp = 1.0;
	elseif (t_m <= 3.3e-161)
		tmp = ((t_m * sqrt((x * 0.5))) * sqrt(2.0)) / l_m;
	elseif (t_m <= 1.32e+36)
		tmp = sqrt(t_2) / sqrt((((2.0 * ((t_m ^ 2.0) / x)) + (t_2 + ((l_m ^ 2.0) / x))) + ((t_2 + (l_m ^ 2.0)) / x)));
	else
		tmp = 1.0 / sqrt(((x + 1.0) / (x + -1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 6.4e-189], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 1.2e-174], 1.0, If[LessEqual[t$95$m, 3.3e-161], N[(N[(N[(t$95$m * N[Sqrt[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 1.32e+36], N[(N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 5 regimes

2. if t < 6.4000000000000001e-189

   1. Initial program 29.9%

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

   3. Simplified 29.8%

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

   5. Taylor expanded in l around inf 4.7%

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

      1. *-commutative 4.7%

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

      2. associate--l+ 10.4%

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

      3. sub-neg 10.4%

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

      4. metadata-eval 10.4%

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

      5. +-commutative 10.4%

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

      6. sub-neg 10.4%

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

      7. metadata-eval 10.4%

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

      8. +-commutative 10.4%

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

      9. associate-/l* 10.4%

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

   7. Simplified 10.4%

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

   8. Taylor expanded in x around inf 17.6%

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

      1. *-commutative 17.6%

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

   10. Simplified 17.6%

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

       1. pow1 17.6%

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

   12. Applied egg-rr 17.6%

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

       1. unpow1 17.6%

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

       2. *-commutative 17.6%

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

       3. associate-*l* 20.6%

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

   14. Simplified 20.6%

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

       1. associate-*r* 17.6%

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

       2. *-commutative 17.6%

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

       3. sqrt-prod 17.5%

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

       4. associate-*l* 17.6%

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

       5. associate-*l* 17.5%

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

       6. *-commutative 17.5%

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

       7. associate-/l* 17.5%

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

       8. *-commutative 17.5%

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

       9. associate-*l/ 20.6%

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

       10. associate-*l* 20.5%

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

       11. sqrt-unprod 20.7%

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

       12. metadata-eval 20.7%

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

       13. metadata-eval 20.7%

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

       14. *-commutative 20.7%

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

       15. *-un-lft-identity 20.7%

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

   16. Applied egg-rr 20.7%

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

   if 6.4000000000000001e-189 < t < 1.2e-174

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in l around 0 99.6%

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

   6. Taylor expanded in x around inf 99.6%

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

   if 1.2e-174 < t < 3.2999999999999998e-161

   1. Initial program 1.2%

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

   3. Simplified 1.2%

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

   5. Taylor expanded in l around inf 0.7%

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

      1. *-commutative 0.7%

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

      2. associate--l+ 11.4%

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

      3. sub-neg 11.4%

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

      4. metadata-eval 11.4%

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

      5. +-commutative 11.4%

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

      6. sub-neg 11.4%

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

      7. metadata-eval 11.4%

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

      8. +-commutative 11.4%

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

      9. associate-/l* 11.4%

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

   7. Simplified 11.4%

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

   8. Taylor expanded in x around inf 27.3%

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

      1. *-commutative 27.3%

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

   10. Simplified 27.3%

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

       1. pow1 27.3%

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

   12. Applied egg-rr 27.3%

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

       1. unpow1 27.3%

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

       2. *-commutative 27.3%

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

       3. associate-*l* 26.4%

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

   14. Simplified 26.4%

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

       1. associate-*r* 27.3%

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

       2. *-commutative 27.3%

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

       3. associate-*r* 26.4%

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

       4. associate-*r/ 26.4%

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

       5. *-commutative 26.4%

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

   16. Applied egg-rr 26.4%

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

   if 3.2999999999999998e-161 < t < 1.3200000000000001e36

   1. Initial program 56.8%

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

      1. add-sqr-sqrt 56.7%

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

      2. sqrt-prod 56.8%

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

      3. sqrt-prod 57.3%

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

      4. pow1/2 57.3%

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

      5. pow2 57.3%

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

   4. Applied egg-rr 57.3%

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

      1. unpow1/2 57.3%

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

   6. Simplified 57.3%

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

   7. Taylor expanded in x around inf 90.7%

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

   if 1.3200000000000001e36 < t

   1. Initial program 32.6%

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

   3. Simplified 32.5%

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

   5. Taylor expanded in l around 0 96.2%

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

   6. Taylor expanded in t around 0 96.4%

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

      1. expm1-log1p-u 96.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{x - 1}{1 + x}}\right)\right)} \]

      2. expm1-undefine 96.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{x - 1}{1 + x}}\right)} - 1} \]

      3. sub-neg 96.4%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}}\right)} - 1 \]

      4. metadata-eval 96.4%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\frac{x + \color{blue}{-1}}{1 + x}}\right)} - 1 \]

      5. +-commutative 96.4%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\frac{x + -1}{\color{blue}{x + 1}}}\right)} - 1 \]

   8. Applied egg-rr 96.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{x + -1}{x + 1}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-define 96.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{x + -1}{x + 1}}\right)\right)} \]

      2. expm1-log1p-u 96.4%

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

      3. clear-num 96.4%

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

      4. sqrt-div 96.4%

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

      5. metadata-eval 96.4%

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

   10. Applied egg-rr 96.4%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{x + 1}{x + -1}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.4 \cdot 10^{-189}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-174}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-161}:\\ \;\;\;\;\frac{\left(t \cdot \sqrt{x \cdot 0.5}\right) \cdot \sqrt{2}}{\ell}\\ \mathbf{elif}\;t \leq 1.32 \cdot 10^{+36}:\\ \;\;\;\;\frac{\sqrt{2 \cdot {t}^{2}}}{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.1 \cdot 10^{-188}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 1.45 \cdot 10^{-174}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 1.65 \cdot 10^{-161}:\\ \;\;\;\;\frac{\left(t\_m \cdot \sqrt{x \cdot 0.5}\right) \cdot \sqrt{2}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 5.4 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{\left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_2 + \frac{{l\_m}^{2}}{x}\right)\right) + \frac{t\_2 + {l\_m}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \end{array} \end{array} \]

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 (pow t_m 2.0))))
   (*
    t_s
    (if (<= t_m 1.1e-188)
      (/ (* t_m (sqrt x)) l_m)
      (if (<= t_m 1.45e-174)
        1.0
        (if (<= t_m 1.65e-161)
          (/ (* (* t_m (sqrt (* x 0.5))) (sqrt 2.0)) l_m)
          (if (<= t_m 5.4e+33)
            (*
             (sqrt 2.0)
             (/
              t_m
              (sqrt
               (+
                (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l_m 2.0) x)))
                (/ (+ t_2 (pow l_m 2.0)) x)))))
            (/ 1.0 (sqrt (/ (+ x 1.0) (+ x -1.0)))))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double tmp;
	if (t_m <= 1.1e-188) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else if (t_m <= 1.45e-174) {
		tmp = 1.0;
	} else if (t_m <= 1.65e-161) {
		tmp = ((t_m * sqrt((x * 0.5))) * sqrt(2.0)) / l_m;
	} else if (t_m <= 5.4e+33) {
		tmp = sqrt(2.0) * (t_m / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l_m, 2.0) / x))) + ((t_2 + pow(l_m, 2.0)) / x))));
	} else {
		tmp = 1.0 / sqrt(((x + 1.0) / (x + -1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    if (t_m <= 1.1d-188) then
        tmp = (t_m * sqrt(x)) / l_m
    else if (t_m <= 1.45d-174) then
        tmp = 1.0d0
    else if (t_m <= 1.65d-161) then
        tmp = ((t_m * sqrt((x * 0.5d0))) * sqrt(2.0d0)) / l_m
    else if (t_m <= 5.4d+33) then
        tmp = sqrt(2.0d0) * (t_m / sqrt((((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_2 + ((l_m ** 2.0d0) / x))) + ((t_2 + (l_m ** 2.0d0)) / x))))
    else
        tmp = 1.0d0 / sqrt(((x + 1.0d0) / (x + (-1.0d0))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * Math.pow(t_m, 2.0);
	double tmp;
	if (t_m <= 1.1e-188) {
		tmp = (t_m * Math.sqrt(x)) / l_m;
	} else if (t_m <= 1.45e-174) {
		tmp = 1.0;
	} else if (t_m <= 1.65e-161) {
		tmp = ((t_m * Math.sqrt((x * 0.5))) * Math.sqrt(2.0)) / l_m;
	} else if (t_m <= 5.4e+33) {
		tmp = Math.sqrt(2.0) * (t_m / Math.sqrt((((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_2 + (Math.pow(l_m, 2.0) / x))) + ((t_2 + Math.pow(l_m, 2.0)) / x))));
	} else {
		tmp = 1.0 / Math.sqrt(((x + 1.0) / (x + -1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = 2.0 * math.pow(t_m, 2.0)
	tmp = 0
	if t_m <= 1.1e-188:
		tmp = (t_m * math.sqrt(x)) / l_m
	elif t_m <= 1.45e-174:
		tmp = 1.0
	elif t_m <= 1.65e-161:
		tmp = ((t_m * math.sqrt((x * 0.5))) * math.sqrt(2.0)) / l_m
	elif t_m <= 5.4e+33:
		tmp = math.sqrt(2.0) * (t_m / math.sqrt((((2.0 * (math.pow(t_m, 2.0) / x)) + (t_2 + (math.pow(l_m, 2.0) / x))) + ((t_2 + math.pow(l_m, 2.0)) / x))))
	else:
		tmp = 1.0 / math.sqrt(((x + 1.0) / (x + -1.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(2.0 * (t_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 1.1e-188)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	elseif (t_m <= 1.45e-174)
		tmp = 1.0;
	elseif (t_m <= 1.65e-161)
		tmp = Float64(Float64(Float64(t_m * sqrt(Float64(x * 0.5))) * sqrt(2.0)) / l_m);
	elseif (t_m <= 5.4e+33)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l_m ^ 2.0) / x))) + Float64(Float64(t_2 + (l_m ^ 2.0)) / x)))));
	else
		tmp = Float64(1.0 / sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = 2.0 * (t_m ^ 2.0);
	tmp = 0.0;
	if (t_m <= 1.1e-188)
		tmp = (t_m * sqrt(x)) / l_m;
	elseif (t_m <= 1.45e-174)
		tmp = 1.0;
	elseif (t_m <= 1.65e-161)
		tmp = ((t_m * sqrt((x * 0.5))) * sqrt(2.0)) / l_m;
	elseif (t_m <= 5.4e+33)
		tmp = sqrt(2.0) * (t_m / sqrt((((2.0 * ((t_m ^ 2.0) / x)) + (t_2 + ((l_m ^ 2.0) / x))) + ((t_2 + (l_m ^ 2.0)) / x))));
	else
		tmp = 1.0 / sqrt(((x + 1.0) / (x + -1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.1e-188], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 1.45e-174], 1.0, If[LessEqual[t$95$m, 1.65e-161], N[(N[(N[(t$95$m * N[Sqrt[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 5.4e+33], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 5 regimes

2. if t < 1.1e-188

   1. Initial program 29.9%

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

   3. Simplified 29.8%

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

   5. Taylor expanded in l around inf 4.7%

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

      1. *-commutative 4.7%

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

      2. associate--l+ 10.4%

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

      3. sub-neg 10.4%

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

      4. metadata-eval 10.4%

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

      5. +-commutative 10.4%

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

      6. sub-neg 10.4%

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

      7. metadata-eval 10.4%

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

      8. +-commutative 10.4%

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

      9. associate-/l* 10.4%

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

   7. Simplified 10.4%

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

   8. Taylor expanded in x around inf 17.6%

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

      1. *-commutative 17.6%

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

   10. Simplified 17.6%

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

       1. pow1 17.6%

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

   12. Applied egg-rr 17.6%

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

       1. unpow1 17.6%

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

       2. *-commutative 17.6%

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

       3. associate-*l* 20.6%

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

   14. Simplified 20.6%

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

       1. associate-*r* 17.6%

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

       2. *-commutative 17.6%

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

       3. sqrt-prod 17.5%

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

       4. associate-*l* 17.6%

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

       5. associate-*l* 17.5%

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

       6. *-commutative 17.5%

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

       7. associate-/l* 17.5%

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

       8. *-commutative 17.5%

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

       9. associate-*l/ 20.6%

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

       10. associate-*l* 20.5%

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

       11. sqrt-unprod 20.7%

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

       12. metadata-eval 20.7%

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

       13. metadata-eval 20.7%

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

       14. *-commutative 20.7%

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

       15. *-un-lft-identity 20.7%

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

   16. Applied egg-rr 20.7%

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

   if 1.1e-188 < t < 1.45000000000000005e-174

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in l around 0 99.6%

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

   6. Taylor expanded in x around inf 99.6%

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

   if 1.45000000000000005e-174 < t < 1.6499999999999999e-161

   1. Initial program 1.2%

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

   3. Simplified 1.2%

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

   5. Taylor expanded in l around inf 0.7%

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

      1. *-commutative 0.7%

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

      2. associate--l+ 11.4%

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

      3. sub-neg 11.4%

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

      4. metadata-eval 11.4%

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

      5. +-commutative 11.4%

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

      6. sub-neg 11.4%

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

      7. metadata-eval 11.4%

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

      8. +-commutative 11.4%

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

      9. associate-/l* 11.4%

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

   7. Simplified 11.4%

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

   8. Taylor expanded in x around inf 27.3%

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

      1. *-commutative 27.3%

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

   10. Simplified 27.3%

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

       1. pow1 27.3%

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

   12. Applied egg-rr 27.3%

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

       1. unpow1 27.3%

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

       2. *-commutative 27.3%

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

       3. associate-*l* 26.4%

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

   14. Simplified 26.4%

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

       1. associate-*r* 27.3%

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

       2. *-commutative 27.3%

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

       3. associate-*r* 26.4%

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

       4. associate-*r/ 26.4%

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

       5. *-commutative 26.4%

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

   16. Applied egg-rr 26.4%

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

   if 1.6499999999999999e-161 < t < 5.39999999999999982e33

   1. Initial program 55.6%

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

   3. Simplified 55.5%

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

   5. Taylor expanded in x around inf 90.1%

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

   if 5.39999999999999982e33 < t

   1. Initial program 33.6%

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

   3. Simplified 33.5%

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

   5. Taylor expanded in l around 0 96.3%

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

   6. Taylor expanded in t around 0 96.4%

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

      1. expm1-log1p-u 96.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{x - 1}{1 + x}}\right)\right)} \]

      2. expm1-undefine 96.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{x - 1}{1 + x}}\right)} - 1} \]

      3. sub-neg 96.4%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}}\right)} - 1 \]

      4. metadata-eval 96.4%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\frac{x + \color{blue}{-1}}{1 + x}}\right)} - 1 \]

      5. +-commutative 96.4%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\frac{x + -1}{\color{blue}{x + 1}}}\right)} - 1 \]

   8. Applied egg-rr 96.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{x + -1}{x + 1}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-define 96.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{x + -1}{x + 1}}\right)\right)} \]

      2. expm1-log1p-u 96.4%

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

      3. clear-num 96.4%

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

      4. sqrt-div 96.4%

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

      5. metadata-eval 96.4%

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

   10. Applied egg-rr 96.4%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{x + 1}{x + -1}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-188}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-174}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-161}:\\ \;\;\;\;\frac{\left(t \cdot \sqrt{x \cdot 0.5}\right) \cdot \sqrt{2}}{\ell}\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{l\_m}^{2}}{x}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.7 \cdot 10^{-188}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 7.8 \cdot 10^{-175}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 1.02 \cdot 10^{-161}:\\ \;\;\;\;\frac{\left(t\_m \cdot \sqrt{x \cdot 0.5}\right) \cdot \sqrt{2}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 4.6 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{t\_2 + \left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(2 \cdot {t\_m}^{2} + t\_2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \end{array} \end{array} \]

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 (/ (pow l_m 2.0) x)))
   (*
    t_s
    (if (<= t_m 2.7e-188)
      (/ (* t_m (sqrt x)) l_m)
      (if (<= t_m 7.8e-175)
        1.0
        (if (<= t_m 1.02e-161)
          (/ (* (* t_m (sqrt (* x 0.5))) (sqrt 2.0)) l_m)
          (if (<= t_m 4.6e+33)
            (*
             (sqrt 2.0)
             (/
              t_m
              (sqrt
               (+
                t_2
                (+
                 (* 2.0 (/ (pow t_m 2.0) x))
                 (+ (* 2.0 (pow t_m 2.0)) t_2))))))
            (/ 1.0 (sqrt (/ (+ x 1.0) (+ x -1.0)))))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = pow(l_m, 2.0) / x;
	double tmp;
	if (t_m <= 2.7e-188) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else if (t_m <= 7.8e-175) {
		tmp = 1.0;
	} else if (t_m <= 1.02e-161) {
		tmp = ((t_m * sqrt((x * 0.5))) * sqrt(2.0)) / l_m;
	} else if (t_m <= 4.6e+33) {
		tmp = sqrt(2.0) * (t_m / sqrt((t_2 + ((2.0 * (pow(t_m, 2.0) / x)) + ((2.0 * pow(t_m, 2.0)) + t_2)))));
	} else {
		tmp = 1.0 / sqrt(((x + 1.0) / (x + -1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (l_m ** 2.0d0) / x
    if (t_m <= 2.7d-188) then
        tmp = (t_m * sqrt(x)) / l_m
    else if (t_m <= 7.8d-175) then
        tmp = 1.0d0
    else if (t_m <= 1.02d-161) then
        tmp = ((t_m * sqrt((x * 0.5d0))) * sqrt(2.0d0)) / l_m
    else if (t_m <= 4.6d+33) then
        tmp = sqrt(2.0d0) * (t_m / sqrt((t_2 + ((2.0d0 * ((t_m ** 2.0d0) / x)) + ((2.0d0 * (t_m ** 2.0d0)) + t_2)))))
    else
        tmp = 1.0d0 / sqrt(((x + 1.0d0) / (x + (-1.0d0))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = Math.pow(l_m, 2.0) / x;
	double tmp;
	if (t_m <= 2.7e-188) {
		tmp = (t_m * Math.sqrt(x)) / l_m;
	} else if (t_m <= 7.8e-175) {
		tmp = 1.0;
	} else if (t_m <= 1.02e-161) {
		tmp = ((t_m * Math.sqrt((x * 0.5))) * Math.sqrt(2.0)) / l_m;
	} else if (t_m <= 4.6e+33) {
		tmp = Math.sqrt(2.0) * (t_m / Math.sqrt((t_2 + ((2.0 * (Math.pow(t_m, 2.0) / x)) + ((2.0 * Math.pow(t_m, 2.0)) + t_2)))));
	} else {
		tmp = 1.0 / Math.sqrt(((x + 1.0) / (x + -1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = math.pow(l_m, 2.0) / x
	tmp = 0
	if t_m <= 2.7e-188:
		tmp = (t_m * math.sqrt(x)) / l_m
	elif t_m <= 7.8e-175:
		tmp = 1.0
	elif t_m <= 1.02e-161:
		tmp = ((t_m * math.sqrt((x * 0.5))) * math.sqrt(2.0)) / l_m
	elif t_m <= 4.6e+33:
		tmp = math.sqrt(2.0) * (t_m / math.sqrt((t_2 + ((2.0 * (math.pow(t_m, 2.0) / x)) + ((2.0 * math.pow(t_m, 2.0)) + t_2)))))
	else:
		tmp = 1.0 / math.sqrt(((x + 1.0) / (x + -1.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64((l_m ^ 2.0) / x)
	tmp = 0.0
	if (t_m <= 2.7e-188)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	elseif (t_m <= 7.8e-175)
		tmp = 1.0;
	elseif (t_m <= 1.02e-161)
		tmp = Float64(Float64(Float64(t_m * sqrt(Float64(x * 0.5))) * sqrt(2.0)) / l_m);
	elseif (t_m <= 4.6e+33)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(t_2 + Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(Float64(2.0 * (t_m ^ 2.0)) + t_2))))));
	else
		tmp = Float64(1.0 / sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = (l_m ^ 2.0) / x;
	tmp = 0.0;
	if (t_m <= 2.7e-188)
		tmp = (t_m * sqrt(x)) / l_m;
	elseif (t_m <= 7.8e-175)
		tmp = 1.0;
	elseif (t_m <= 1.02e-161)
		tmp = ((t_m * sqrt((x * 0.5))) * sqrt(2.0)) / l_m;
	elseif (t_m <= 4.6e+33)
		tmp = sqrt(2.0) * (t_m / sqrt((t_2 + ((2.0 * ((t_m ^ 2.0) / x)) + ((2.0 * (t_m ^ 2.0)) + t_2)))));
	else
		tmp = 1.0 / sqrt(((x + 1.0) / (x + -1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.7e-188], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 7.8e-175], 1.0, If[LessEqual[t$95$m, 1.02e-161], N[(N[(N[(t$95$m * N[Sqrt[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 4.6e+33], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(t$95$2 + N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 5 regimes

2. if t < 2.7000000000000001e-188

   1. Initial program 29.9%

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

   3. Simplified 29.8%

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

   5. Taylor expanded in l around inf 4.7%

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

      1. *-commutative 4.7%

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

      2. associate--l+ 10.4%

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

      3. sub-neg 10.4%

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

      4. metadata-eval 10.4%

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

      5. +-commutative 10.4%

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

      6. sub-neg 10.4%

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

      7. metadata-eval 10.4%

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

      8. +-commutative 10.4%

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

      9. associate-/l* 10.4%

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

   7. Simplified 10.4%

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

   8. Taylor expanded in x around inf 17.6%

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

      1. *-commutative 17.6%

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

   10. Simplified 17.6%

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

       1. pow1 17.6%

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

   12. Applied egg-rr 17.6%

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

       1. unpow1 17.6%

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

       2. *-commutative 17.6%

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

       3. associate-*l* 20.6%

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

   14. Simplified 20.6%

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

       1. associate-*r* 17.6%

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

       2. *-commutative 17.6%

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

       3. sqrt-prod 17.5%

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

       4. associate-*l* 17.6%

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

       5. associate-*l* 17.5%

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

       6. *-commutative 17.5%

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

       7. associate-/l* 17.5%

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

       8. *-commutative 17.5%

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

       9. associate-*l/ 20.6%

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

       10. associate-*l* 20.5%

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

       11. sqrt-unprod 20.7%

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

       12. metadata-eval 20.7%

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

       13. metadata-eval 20.7%

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

       14. *-commutative 20.7%

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

       15. *-un-lft-identity 20.7%

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

   16. Applied egg-rr 20.7%

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

   if 2.7000000000000001e-188 < t < 7.79999999999999997e-175

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in l around 0 99.6%

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

   6. Taylor expanded in x around inf 99.6%

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

   if 7.79999999999999997e-175 < t < 1.0199999999999999e-161

   1. Initial program 1.2%

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

   3. Simplified 1.2%

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

   5. Taylor expanded in l around inf 0.7%

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

      1. *-commutative 0.7%

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

      2. associate--l+ 11.4%

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

      3. sub-neg 11.4%

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

      4. metadata-eval 11.4%

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

      5. +-commutative 11.4%

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

      6. sub-neg 11.4%

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

      7. metadata-eval 11.4%

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

      8. +-commutative 11.4%

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

      9. associate-/l* 11.4%

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

   7. Simplified 11.4%

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

   8. Taylor expanded in x around inf 27.3%

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

      1. *-commutative 27.3%

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

   10. Simplified 27.3%

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

       1. pow1 27.3%

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

   12. Applied egg-rr 27.3%

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

       1. unpow1 27.3%

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

       2. *-commutative 27.3%

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

       3. associate-*l* 26.4%

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

   14. Simplified 26.4%

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

       1. associate-*r* 27.3%

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

       2. *-commutative 27.3%

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

       3. associate-*r* 26.4%

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

       4. associate-*r/ 26.4%

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

       5. *-commutative 26.4%

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

   16. Applied egg-rr 26.4%

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

   if 1.0199999999999999e-161 < t < 4.60000000000000021e33

   1. Initial program 55.6%

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

   3. Simplified 55.5%

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

   5. Taylor expanded in x around inf 90.1%

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

   6. Taylor expanded in t around 0 89.6%

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

   if 4.60000000000000021e33 < t

   1. Initial program 33.6%

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

   3. Simplified 33.5%

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

   5. Taylor expanded in l around 0 96.3%

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

   6. Taylor expanded in t around 0 96.4%

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

      1. expm1-log1p-u 96.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{x - 1}{1 + x}}\right)\right)} \]

      2. expm1-undefine 96.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{x - 1}{1 + x}}\right)} - 1} \]

      3. sub-neg 96.4%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}}\right)} - 1 \]

      4. metadata-eval 96.4%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\frac{x + \color{blue}{-1}}{1 + x}}\right)} - 1 \]

      5. +-commutative 96.4%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\frac{x + -1}{\color{blue}{x + 1}}}\right)} - 1 \]

   8. Applied egg-rr 96.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{x + -1}{x + 1}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-define 96.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{x + -1}{x + 1}}\right)\right)} \]

      2. expm1-log1p-u 96.4%

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

      3. clear-num 96.4%

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

      4. sqrt-div 96.4%

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

      5. metadata-eval 96.4%

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

   10. Applied egg-rr 96.4%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{x + 1}{x + -1}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.7 \cdot 10^{-188}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{-175}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-161}:\\ \;\;\;\;\frac{\left(t \cdot \sqrt{x \cdot 0.5}\right) \cdot \sqrt{2}}{\ell}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m \cdot \sqrt{x}}{l\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.3 \cdot 10^{-188}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 1.16 \cdot 10^{-174}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 2.05 \cdot 10^{-144}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (/ (* t_m (sqrt x)) l_m)))
   (*
    t_s
    (if (<= t_m 5.3e-188)
      t_2
      (if (<= t_m 1.16e-174)
        1.0
        (if (<= t_m 2.05e-144)
          t_2
          (/ 1.0 (sqrt (/ (+ x 1.0) (+ x -1.0))))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = (t_m * sqrt(x)) / l_m;
	double tmp;
	if (t_m <= 5.3e-188) {
		tmp = t_2;
	} else if (t_m <= 1.16e-174) {
		tmp = 1.0;
	} else if (t_m <= 2.05e-144) {
		tmp = t_2;
	} else {
		tmp = 1.0 / sqrt(((x + 1.0) / (x + -1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (t_m * sqrt(x)) / l_m
    if (t_m <= 5.3d-188) then
        tmp = t_2
    else if (t_m <= 1.16d-174) then
        tmp = 1.0d0
    else if (t_m <= 2.05d-144) then
        tmp = t_2
    else
        tmp = 1.0d0 / sqrt(((x + 1.0d0) / (x + (-1.0d0))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = (t_m * Math.sqrt(x)) / l_m;
	double tmp;
	if (t_m <= 5.3e-188) {
		tmp = t_2;
	} else if (t_m <= 1.16e-174) {
		tmp = 1.0;
	} else if (t_m <= 2.05e-144) {
		tmp = t_2;
	} else {
		tmp = 1.0 / Math.sqrt(((x + 1.0) / (x + -1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = (t_m * math.sqrt(x)) / l_m
	tmp = 0
	if t_m <= 5.3e-188:
		tmp = t_2
	elif t_m <= 1.16e-174:
		tmp = 1.0
	elif t_m <= 2.05e-144:
		tmp = t_2
	else:
		tmp = 1.0 / math.sqrt(((x + 1.0) / (x + -1.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(Float64(t_m * sqrt(x)) / l_m)
	tmp = 0.0
	if (t_m <= 5.3e-188)
		tmp = t_2;
	elseif (t_m <= 1.16e-174)
		tmp = 1.0;
	elseif (t_m <= 2.05e-144)
		tmp = t_2;
	else
		tmp = Float64(1.0 / sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = (t_m * sqrt(x)) / l_m;
	tmp = 0.0;
	if (t_m <= 5.3e-188)
		tmp = t_2;
	elseif (t_m <= 1.16e-174)
		tmp = 1.0;
	elseif (t_m <= 2.05e-144)
		tmp = t_2;
	else
		tmp = 1.0 / sqrt(((x + 1.0) / (x + -1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 5.3e-188], t$95$2, If[LessEqual[t$95$m, 1.16e-174], 1.0, If[LessEqual[t$95$m, 2.05e-144], t$95$2, N[(1.0 / N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 5.30000000000000014e-188 or 1.16e-174 < t < 2.05e-144

   1. Initial program 28.4%

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

   3. Simplified 28.4%

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

   5. Taylor expanded in l around inf 4.5%

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

      1. *-commutative 4.5%

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

      2. associate--l+ 11.0%

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

      3. sub-neg 11.0%

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

      4. metadata-eval 11.0%

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

      5. +-commutative 11.0%

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

      6. sub-neg 11.0%

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

      7. metadata-eval 11.0%

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

      8. +-commutative 11.0%

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

      9. associate-/l* 11.0%

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

   7. Simplified 11.0%

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

   8. Taylor expanded in x around inf 18.9%

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

      1. *-commutative 18.9%

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

   10. Simplified 18.9%

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

       1. pow1 18.9%

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

   12. Applied egg-rr 18.9%

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

       1. unpow1 18.9%

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

       2. *-commutative 18.9%

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

       3. associate-*l* 21.7%

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

   14. Simplified 21.7%

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

       1. associate-*r* 18.9%

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

       2. *-commutative 18.9%

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

       3. sqrt-prod 18.8%

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

       4. associate-*l* 18.8%

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

       5. associate-*l* 18.8%

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

       6. *-commutative 18.8%

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

       7. associate-/l* 18.8%

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

       8. *-commutative 18.8%

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

       9. associate-*l/ 21.7%

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

       10. associate-*l* 21.6%

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

       11. sqrt-unprod 21.8%

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

       12. metadata-eval 21.8%

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

       13. metadata-eval 21.8%

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

       14. *-commutative 21.8%

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

       15. *-un-lft-identity 21.8%

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

   16. Applied egg-rr 21.8%

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

   if 5.30000000000000014e-188 < t < 1.16e-174

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in l around 0 99.6%

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

   6. Taylor expanded in x around inf 99.6%

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

   if 2.05e-144 < t

   1. Initial program 42.4%

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

   3. Simplified 42.3%

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

   5. Taylor expanded in l around 0 87.7%

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

   6. Taylor expanded in t around 0 87.8%

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

      1. expm1-log1p-u 87.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{x - 1}{1 + x}}\right)\right)} \]

      2. expm1-undefine 87.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{x - 1}{1 + x}}\right)} - 1} \]

      3. sub-neg 87.8%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}}\right)} - 1 \]

      4. metadata-eval 87.8%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\frac{x + \color{blue}{-1}}{1 + x}}\right)} - 1 \]

      5. +-commutative 87.8%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\frac{x + -1}{\color{blue}{x + 1}}}\right)} - 1 \]

   8. Applied egg-rr 87.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{x + -1}{x + 1}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-define 87.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{x + -1}{x + 1}}\right)\right)} \]

      2. expm1-log1p-u 87.8%

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

      3. clear-num 87.8%

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

      4. sqrt-div 87.8%

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

      5. metadata-eval 87.8%

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

   10. Applied egg-rr 87.8%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.3 \cdot 10^{-188}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.16 \cdot 10^{-174}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-144}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m \cdot \sqrt{x}}{l\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3 \cdot 10^{-188}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 9.2 \cdot 10^{-175}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 1.25 \cdot 10^{-143}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (/ (* t_m (sqrt x)) l_m)))
   (*
    t_s
    (if (<= t_m 3e-188)
      t_2
      (if (<= t_m 9.2e-175)
        1.0
        (if (<= t_m 1.25e-143) t_2 (sqrt (/ (+ x -1.0) (+ x 1.0)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = (t_m * sqrt(x)) / l_m;
	double tmp;
	if (t_m <= 3e-188) {
		tmp = t_2;
	} else if (t_m <= 9.2e-175) {
		tmp = 1.0;
	} else if (t_m <= 1.25e-143) {
		tmp = t_2;
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (t_m * sqrt(x)) / l_m
    if (t_m <= 3d-188) then
        tmp = t_2
    else if (t_m <= 9.2d-175) then
        tmp = 1.0d0
    else if (t_m <= 1.25d-143) then
        tmp = t_2
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = (t_m * Math.sqrt(x)) / l_m;
	double tmp;
	if (t_m <= 3e-188) {
		tmp = t_2;
	} else if (t_m <= 9.2e-175) {
		tmp = 1.0;
	} else if (t_m <= 1.25e-143) {
		tmp = t_2;
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = (t_m * math.sqrt(x)) / l_m
	tmp = 0
	if t_m <= 3e-188:
		tmp = t_2
	elif t_m <= 9.2e-175:
		tmp = 1.0
	elif t_m <= 1.25e-143:
		tmp = t_2
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(Float64(t_m * sqrt(x)) / l_m)
	tmp = 0.0
	if (t_m <= 3e-188)
		tmp = t_2;
	elseif (t_m <= 9.2e-175)
		tmp = 1.0;
	elseif (t_m <= 1.25e-143)
		tmp = t_2;
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = (t_m * sqrt(x)) / l_m;
	tmp = 0.0;
	if (t_m <= 3e-188)
		tmp = t_2;
	elseif (t_m <= 9.2e-175)
		tmp = 1.0;
	elseif (t_m <= 1.25e-143)
		tmp = t_2;
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3e-188], t$95$2, If[LessEqual[t$95$m, 9.2e-175], 1.0, If[LessEqual[t$95$m, 1.25e-143], t$95$2, N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 3.00000000000000017e-188 or 9.2e-175 < t < 1.2500000000000001e-143

   1. Initial program 28.4%

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

   3. Simplified 28.4%

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

   5. Taylor expanded in l around inf 4.5%

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

      1. *-commutative 4.5%

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

      2. associate--l+ 11.0%

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

      3. sub-neg 11.0%

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

      4. metadata-eval 11.0%

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

      5. +-commutative 11.0%

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

      6. sub-neg 11.0%

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

      7. metadata-eval 11.0%

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

      8. +-commutative 11.0%

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

      9. associate-/l* 11.0%

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

   7. Simplified 11.0%

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

   8. Taylor expanded in x around inf 18.9%

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

      1. *-commutative 18.9%

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

   10. Simplified 18.9%

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

       1. pow1 18.9%

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

   12. Applied egg-rr 18.9%

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

       1. unpow1 18.9%

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

       2. *-commutative 18.9%

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

       3. associate-*l* 21.7%

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

   14. Simplified 21.7%

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

       1. associate-*r* 18.9%

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

       2. *-commutative 18.9%

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

       3. sqrt-prod 18.8%

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

       4. associate-*l* 18.8%

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

       5. associate-*l* 18.8%

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

       6. *-commutative 18.8%

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

       7. associate-/l* 18.8%

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

       8. *-commutative 18.8%

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

       9. associate-*l/ 21.7%

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

       10. associate-*l* 21.6%

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

       11. sqrt-unprod 21.8%

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

       12. metadata-eval 21.8%

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

       13. metadata-eval 21.8%

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

       14. *-commutative 21.8%

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

       15. *-un-lft-identity 21.8%

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

   16. Applied egg-rr 21.8%

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

   if 3.00000000000000017e-188 < t < 9.2e-175

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in l around 0 99.6%

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

   6. Taylor expanded in x around inf 99.6%

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

   if 1.2500000000000001e-143 < t

   1. Initial program 42.4%

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

   3. Simplified 42.3%

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

   5. Taylor expanded in l around 0 87.7%

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

   6. Taylor expanded in t around 0 87.8%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-188}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-175}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-143}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \frac{\sqrt{x}}{l\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.1 \cdot 10^{-188}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 6 \cdot 10^{-175}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 9.8 \cdot 10^{-144}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1 + \frac{0.5}{x}}{x}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* t_m (/ (sqrt x) l_m))))
   (*
    t_s
    (if (<= t_m 1.1e-188)
      t_2
      (if (<= t_m 6e-175)
        1.0
        (if (<= t_m 9.8e-144) t_2 (+ 1.0 (/ (+ -1.0 (/ 0.5 x)) x))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * (sqrt(x) / l_m);
	double tmp;
	if (t_m <= 1.1e-188) {
		tmp = t_2;
	} else if (t_m <= 6e-175) {
		tmp = 1.0;
	} else if (t_m <= 9.8e-144) {
		tmp = t_2;
	} else {
		tmp = 1.0 + ((-1.0 + (0.5 / x)) / 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 = t_m * (sqrt(x) / l_m)
    if (t_m <= 1.1d-188) then
        tmp = t_2
    else if (t_m <= 6d-175) then
        tmp = 1.0d0
    else if (t_m <= 9.8d-144) then
        tmp = t_2
    else
        tmp = 1.0d0 + (((-1.0d0) + (0.5d0 / x)) / x)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * (Math.sqrt(x) / l_m);
	double tmp;
	if (t_m <= 1.1e-188) {
		tmp = t_2;
	} else if (t_m <= 6e-175) {
		tmp = 1.0;
	} else if (t_m <= 9.8e-144) {
		tmp = t_2;
	} else {
		tmp = 1.0 + ((-1.0 + (0.5 / x)) / x);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = t_m * (math.sqrt(x) / l_m)
	tmp = 0
	if t_m <= 1.1e-188:
		tmp = t_2
	elif t_m <= 6e-175:
		tmp = 1.0
	elif t_m <= 9.8e-144:
		tmp = t_2
	else:
		tmp = 1.0 + ((-1.0 + (0.5 / x)) / x)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(t_m * Float64(sqrt(x) / l_m))
	tmp = 0.0
	if (t_m <= 1.1e-188)
		tmp = t_2;
	elseif (t_m <= 6e-175)
		tmp = 1.0;
	elseif (t_m <= 9.8e-144)
		tmp = t_2;
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 + Float64(0.5 / x)) / x));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = t_m * (sqrt(x) / l_m);
	tmp = 0.0;
	if (t_m <= 1.1e-188)
		tmp = t_2;
	elseif (t_m <= 6e-175)
		tmp = 1.0;
	elseif (t_m <= 9.8e-144)
		tmp = t_2;
	else
		tmp = 1.0 + ((-1.0 + (0.5 / x)) / x);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.1e-188], t$95$2, If[LessEqual[t$95$m, 6e-175], 1.0, If[LessEqual[t$95$m, 9.8e-144], t$95$2, N[(1.0 + N[(N[(-1.0 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.1e-188 or 6e-175 < t < 9.8000000000000002e-144

   1. Initial program 28.4%

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

   3. Simplified 28.4%

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

   5. Taylor expanded in l around inf 4.5%

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

      1. *-commutative 4.5%

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

      2. associate--l+ 11.0%

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

      3. sub-neg 11.0%

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

      4. metadata-eval 11.0%

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

      5. +-commutative 11.0%

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

      6. sub-neg 11.0%

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

      7. metadata-eval 11.0%

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

      8. +-commutative 11.0%

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

      9. associate-/l* 11.0%

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

   7. Simplified 11.0%

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

   8. Taylor expanded in x around inf 18.9%

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

      1. *-commutative 18.9%

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

   10. Simplified 18.9%

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

       1. pow1 18.9%

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

   12. Applied egg-rr 18.9%

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

       1. unpow1 18.9%

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

       2. *-commutative 18.9%

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

       3. associate-*l* 21.7%

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

   14. Simplified 21.7%

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

       1. associate-*l/ 21.7%

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

       2. clear-num 21.7%

          \[\leadsto t \cdot \color{blue}{\frac{1}{\frac{\ell}{\sqrt{2} \cdot \sqrt{x \cdot 0.5}}}} \]

       3. sqrt-unprod 21.7%

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

   16. Applied egg-rr 21.7%

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

       1. associate-/r/ 21.7%

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

       2. associate-*l/ 21.7%

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

       3. *-commutative 21.7%

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

       4. associate-*r* 21.7%

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

       5. metadata-eval 21.7%

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

       6. *-lft-identity 21.7%

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

       7. *-lft-identity 21.7%

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

   18. Simplified 21.7%

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

   if 1.1e-188 < t < 6e-175

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in l around 0 99.6%

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

   6. Taylor expanded in x around inf 99.6%

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

   if 9.8000000000000002e-144 < t

   1. Initial program 42.4%

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

   3. Simplified 42.3%

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

   5. Taylor expanded in l around 0 87.7%

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

   6. Taylor expanded in t around 0 87.8%

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

   7. Taylor expanded in x around inf 87.5%

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

      1. associate--l+ 87.5%

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

      2. unpow2 87.5%

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

      3. associate-/r* 87.5%

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

      4. metadata-eval 87.5%

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

      5. metadata-eval 87.5%

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

      6. metadata-eval 87.5%

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

      7. rem-square-sqrt 0.0%

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

      8. unpow2 0.0%

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

      9. associate-*l/ 0.0%

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

      10. *-commutative 0.0%

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

      11. div-sub 0.0%

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

   9. Simplified 87.5%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-188}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-175}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{-144}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1 + \frac{0.5}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m \cdot \sqrt{x}}{l\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.4 \cdot 10^{-189}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 6.5 \cdot 10^{-175}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 6.6 \cdot 10^{-145}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1 + \frac{0.5}{x}}{x}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (/ (* t_m (sqrt x)) l_m)))
   (*
    t_s
    (if (<= t_m 2.4e-189)
      t_2
      (if (<= t_m 6.5e-175)
        1.0
        (if (<= t_m 6.6e-145) t_2 (+ 1.0 (/ (+ -1.0 (/ 0.5 x)) x))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = (t_m * sqrt(x)) / l_m;
	double tmp;
	if (t_m <= 2.4e-189) {
		tmp = t_2;
	} else if (t_m <= 6.5e-175) {
		tmp = 1.0;
	} else if (t_m <= 6.6e-145) {
		tmp = t_2;
	} else {
		tmp = 1.0 + ((-1.0 + (0.5 / x)) / 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 = (t_m * sqrt(x)) / l_m
    if (t_m <= 2.4d-189) then
        tmp = t_2
    else if (t_m <= 6.5d-175) then
        tmp = 1.0d0
    else if (t_m <= 6.6d-145) then
        tmp = t_2
    else
        tmp = 1.0d0 + (((-1.0d0) + (0.5d0 / x)) / x)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = (t_m * Math.sqrt(x)) / l_m;
	double tmp;
	if (t_m <= 2.4e-189) {
		tmp = t_2;
	} else if (t_m <= 6.5e-175) {
		tmp = 1.0;
	} else if (t_m <= 6.6e-145) {
		tmp = t_2;
	} else {
		tmp = 1.0 + ((-1.0 + (0.5 / x)) / x);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = (t_m * math.sqrt(x)) / l_m
	tmp = 0
	if t_m <= 2.4e-189:
		tmp = t_2
	elif t_m <= 6.5e-175:
		tmp = 1.0
	elif t_m <= 6.6e-145:
		tmp = t_2
	else:
		tmp = 1.0 + ((-1.0 + (0.5 / x)) / 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(Float64(t_m * sqrt(x)) / l_m)
	tmp = 0.0
	if (t_m <= 2.4e-189)
		tmp = t_2;
	elseif (t_m <= 6.5e-175)
		tmp = 1.0;
	elseif (t_m <= 6.6e-145)
		tmp = t_2;
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 + Float64(0.5 / x)) / x));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = (t_m * sqrt(x)) / l_m;
	tmp = 0.0;
	if (t_m <= 2.4e-189)
		tmp = t_2;
	elseif (t_m <= 6.5e-175)
		tmp = 1.0;
	elseif (t_m <= 6.6e-145)
		tmp = t_2;
	else
		tmp = 1.0 + ((-1.0 + (0.5 / x)) / 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[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.4e-189], t$95$2, If[LessEqual[t$95$m, 6.5e-175], 1.0, If[LessEqual[t$95$m, 6.6e-145], t$95$2, N[(1.0 + N[(N[(-1.0 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.3999999999999998e-189 or 6.5000000000000005e-175 < t < 6.59999999999999962e-145

   1. Initial program 28.4%

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

   3. Simplified 28.4%

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

   5. Taylor expanded in l around inf 4.5%

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

      1. *-commutative 4.5%

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

      2. associate--l+ 11.0%

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

      3. sub-neg 11.0%

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

      4. metadata-eval 11.0%

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

      5. +-commutative 11.0%

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

      6. sub-neg 11.0%

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

      7. metadata-eval 11.0%

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

      8. +-commutative 11.0%

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

      9. associate-/l* 11.0%

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

   7. Simplified 11.0%

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

   8. Taylor expanded in x around inf 18.9%

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

      1. *-commutative 18.9%

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

   10. Simplified 18.9%

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

       1. pow1 18.9%

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

   12. Applied egg-rr 18.9%

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

       1. unpow1 18.9%

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

       2. *-commutative 18.9%

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

       3. associate-*l* 21.7%

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

   14. Simplified 21.7%

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

       1. associate-*r* 18.9%

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

       2. *-commutative 18.9%

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

       3. sqrt-prod 18.8%

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

       4. associate-*l* 18.8%

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

       5. associate-*l* 18.8%

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

       6. *-commutative 18.8%

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

       7. associate-/l* 18.8%

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

       8. *-commutative 18.8%

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

       9. associate-*l/ 21.7%

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

       10. associate-*l* 21.6%

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

       11. sqrt-unprod 21.8%

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

       12. metadata-eval 21.8%

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

       13. metadata-eval 21.8%

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

       14. *-commutative 21.8%

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

       15. *-un-lft-identity 21.8%

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

   16. Applied egg-rr 21.8%

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

   if 2.3999999999999998e-189 < t < 6.5000000000000005e-175

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in l around 0 99.6%

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

   6. Taylor expanded in x around inf 99.6%

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

   if 6.59999999999999962e-145 < t

   1. Initial program 42.4%

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

   3. Simplified 42.3%

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

   5. Taylor expanded in l around 0 87.7%

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

   6. Taylor expanded in t around 0 87.8%

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

   7. Taylor expanded in x around inf 87.5%

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

      1. associate--l+ 87.5%

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

      2. unpow2 87.5%

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

      3. associate-/r* 87.5%

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

      4. metadata-eval 87.5%

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

      5. metadata-eval 87.5%

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

      6. metadata-eval 87.5%

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

      7. rem-square-sqrt 0.0%

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

      8. unpow2 0.0%

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

      9. associate-*l/ 0.0%

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

      10. *-commutative 0.0%

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

      11. div-sub 0.0%

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

   9. Simplified 87.5%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-189}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-175}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{-145}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1 + \frac{0.5}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.6% accurate, 25.0× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (* t_s (+ 1.0 (/ (+ -1.0 (/ 0.5 x)) x))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 + ((-1.0 + (0.5 / x)) / x));
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * (1.0d0 + (((-1.0d0) + (0.5d0 / x)) / x))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 + ((-1.0 + (0.5 / x)) / x));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * (1.0 + ((-1.0 + (0.5 / x)) / x))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * Float64(1.0 + Float64(Float64(-1.0 + Float64(0.5 / x)) / x)))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * (1.0 + ((-1.0 + (0.5 / x)) / x));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * N[(1.0 + N[(N[(-1.0 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.3%

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

3. Simplified 33.2%

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

5. Taylor expanded in l around 0 36.9%

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

6. Taylor expanded in t around 0 36.9%

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

7. Taylor expanded in x around inf 36.8%

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

   1. associate--l+ 36.8%

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

   2. unpow2 36.8%

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

   3. associate-/r* 36.8%

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

   4. metadata-eval 36.8%

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

   5. metadata-eval 36.8%

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

   6. metadata-eval 36.8%

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

   7. rem-square-sqrt 0.0%

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

   8. unpow2 0.0%

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

   9. associate-*l/ 0.0%

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

   10. *-commutative 0.0%

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

   11. div-sub 0.0%

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

9. Simplified 36.8%

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

10. Final simplification 36.8%

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

Alternative 9: 76.4% 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 33.3%

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

3. Simplified 33.2%

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

5. Taylor expanded in l around 0 36.9%

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

6. Taylor expanded in x around inf 36.7%

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

7. Final simplification 36.7%

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

Alternative 10: 75.8% 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 33.3%

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

3. Simplified 33.2%

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

5. Taylor expanded in l around 0 36.9%

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

6. Taylor expanded in x around inf 36.5%

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

7. Final simplification 36.5%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024060 
(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)))))