Toniolo and Linder, Equation (7)

Percentage Accurate: 33.5% → 80.7%

Time: 18.2s

Alternatives: 7

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: 33.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot \left(t\_m \cdot t\_m\right)\\ t_3 := l\_m \cdot l\_m + t\_2\\ t_4 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.5 \cdot 10^{-247}:\\ \;\;\;\;\frac{t\_4}{t\_4 + \frac{0.5 \cdot \left(2 \cdot t\_3\right)}{t\_m \cdot \left(\sqrt{2} \cdot x\right)}}\\ \mathbf{elif}\;t\_m \leq 1.22 \cdot 10^{-116}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 7.2 \cdot 10^{+92}:\\ \;\;\;\;t\_m \cdot {\left(\frac{\left(t\_2 + \frac{t\_2}{x}\right) + \frac{1}{x} \cdot \left(l\_m \cdot l\_m + t\_3\right)}{2}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \end{array} \]

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 (* t_m t_m)))
        (t_3 (+ (* l_m l_m) t_2))
        (t_4 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 8.5e-247)
      (/ t_4 (+ t_4 (/ (* 0.5 (* 2.0 t_3)) (* t_m (* (sqrt 2.0) x)))))
      (if (<= t_m 1.22e-116)
        (/ (* t_m (sqrt x)) l_m)
        (if (<= t_m 7.2e+92)
          (*
           t_m
           (pow
            (/ (+ (+ t_2 (/ t_2 x)) (* (/ 1.0 x) (+ (* l_m l_m) t_3))) 2.0)
            -0.5))
          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 * (t_m * t_m);
	double t_3 = (l_m * l_m) + t_2;
	double t_4 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 8.5e-247) {
		tmp = t_4 / (t_4 + ((0.5 * (2.0 * t_3)) / (t_m * (sqrt(2.0) * x))));
	} else if (t_m <= 1.22e-116) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else if (t_m <= 7.2e+92) {
		tmp = t_m * pow((((t_2 + (t_2 / x)) + ((1.0 / x) * ((l_m * l_m) + t_3))) / 2.0), -0.5);
	} else {
		tmp = 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) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m * t_m)
    t_3 = (l_m * l_m) + t_2
    t_4 = t_m * sqrt(2.0d0)
    if (t_m <= 8.5d-247) then
        tmp = t_4 / (t_4 + ((0.5d0 * (2.0d0 * t_3)) / (t_m * (sqrt(2.0d0) * x))))
    else if (t_m <= 1.22d-116) then
        tmp = (t_m * sqrt(x)) / l_m
    else if (t_m <= 7.2d+92) then
        tmp = t_m * ((((t_2 + (t_2 / x)) + ((1.0d0 / x) * ((l_m * l_m) + t_3))) / 2.0d0) ** (-0.5d0))
    else
        tmp = 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 * (t_m * t_m);
	double t_3 = (l_m * l_m) + t_2;
	double t_4 = t_m * Math.sqrt(2.0);
	double tmp;
	if (t_m <= 8.5e-247) {
		tmp = t_4 / (t_4 + ((0.5 * (2.0 * t_3)) / (t_m * (Math.sqrt(2.0) * x))));
	} else if (t_m <= 1.22e-116) {
		tmp = (t_m * Math.sqrt(x)) / l_m;
	} else if (t_m <= 7.2e+92) {
		tmp = t_m * Math.pow((((t_2 + (t_2 / x)) + ((1.0 / x) * ((l_m * l_m) + t_3))) / 2.0), -0.5);
	} else {
		tmp = 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 * (t_m * t_m)
	t_3 = (l_m * l_m) + t_2
	t_4 = t_m * math.sqrt(2.0)
	tmp = 0
	if t_m <= 8.5e-247:
		tmp = t_4 / (t_4 + ((0.5 * (2.0 * t_3)) / (t_m * (math.sqrt(2.0) * x))))
	elif t_m <= 1.22e-116:
		tmp = (t_m * math.sqrt(x)) / l_m
	elif t_m <= 7.2e+92:
		tmp = t_m * math.pow((((t_2 + (t_2 / x)) + ((1.0 / x) * ((l_m * l_m) + t_3))) / 2.0), -0.5)
	else:
		tmp = 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 * Float64(t_m * t_m))
	t_3 = Float64(Float64(l_m * l_m) + t_2)
	t_4 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 8.5e-247)
		tmp = Float64(t_4 / Float64(t_4 + Float64(Float64(0.5 * Float64(2.0 * t_3)) / Float64(t_m * Float64(sqrt(2.0) * x)))));
	elseif (t_m <= 1.22e-116)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	elseif (t_m <= 7.2e+92)
		tmp = Float64(t_m * (Float64(Float64(Float64(t_2 + Float64(t_2 / x)) + Float64(Float64(1.0 / x) * Float64(Float64(l_m * l_m) + t_3))) / 2.0) ^ -0.5));
	else
		tmp = 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 * t_m);
	t_3 = (l_m * l_m) + t_2;
	t_4 = t_m * sqrt(2.0);
	tmp = 0.0;
	if (t_m <= 8.5e-247)
		tmp = t_4 / (t_4 + ((0.5 * (2.0 * t_3)) / (t_m * (sqrt(2.0) * x))));
	elseif (t_m <= 1.22e-116)
		tmp = (t_m * sqrt(x)) / l_m;
	elseif (t_m <= 7.2e+92)
		tmp = t_m * ((((t_2 + (t_2 / x)) + ((1.0 / x) * ((l_m * l_m) + t_3))) / 2.0) ^ -0.5);
	else
		tmp = 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[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(l$95$m * l$95$m), $MachinePrecision] + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 8.5e-247], N[(t$95$4 / N[(t$95$4 + N[(N[(0.5 * N[(2.0 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.22e-116], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 7.2e+92], N[(t$95$m * N[Power[N[(N[(N[(t$95$2 + N[(t$95$2 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] * N[(N[(l$95$m * l$95$m), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], 1.0]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < 8.5000000000000003e-247

   1. Initial program 29.9%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

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

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

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

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

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

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

      5. associate-*r/ N/A

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

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

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

   5. Simplified 7.5%

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

   if 8.5000000000000003e-247 < t < 1.22e-116

   1. Initial program 6.4%

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

   3. Taylor expanded in l around inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

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

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

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

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

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

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. +-lowering-+.f64 1.8%

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

   5. Simplified 1.8%

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

   6. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 50.0%

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

   8. Simplified 50.0%

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

   9. Taylor expanded in t around 0

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

       1. associate-*l/ N/A

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

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

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

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

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

       4. sqrt-lowering-sqrt.f64 50.4%

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

   11. Simplified 50.4%

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

   if 1.22e-116 < t < 7.2e92

   1. Initial program 67.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. sqrt-undiv N/A

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

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

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

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

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

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

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

   4. Applied egg-rr 68.0%

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

   5. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

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

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

      3. associate-*r/ N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

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

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

   7. Simplified 92.6%

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

      1. pow1/2 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

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

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

   9. Applied egg-rr 92.6%

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

   if 7.2e92 < t

   1. Initial program 18.3%

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

   3. Taylor expanded in x around inf

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

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

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

      2. sqrt-lowering-sqrt.f64 94.9%

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

   5. Simplified 94.9%

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

      1. *-commutative N/A

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

      2. *-inverses 94.9%

         \[\leadsto 1 \]

   7. Applied egg-rr 94.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{-247}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + \frac{0.5 \cdot \left(2 \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}{t \cdot \left(\sqrt{2} \cdot x\right)}}\\ \mathbf{elif}\;t \leq 1.22 \cdot 10^{-116}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+92}:\\ \;\;\;\;t \cdot {\left(\frac{\left(2 \cdot \left(t \cdot t\right) + \frac{2 \cdot \left(t \cdot t\right)}{x}\right) + \frac{1}{x} \cdot \left(\ell \cdot \ell + \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}{2}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot \left(t\_m \cdot t\_m\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.22 \cdot 10^{-116}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{l\_m \cdot \sqrt{\frac{2 + \frac{2}{x}}{x}}}\\ \mathbf{elif}\;t\_m \leq 4.4 \cdot 10^{+92}:\\ \;\;\;\;t\_m \cdot {\left(\frac{\left(t\_2 + \frac{t\_2}{x}\right) + \frac{1}{x} \cdot \left(l\_m \cdot l\_m + \left(l\_m \cdot l\_m + t\_2\right)\right)}{2}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \end{array} \]

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 (* t_m t_m))))
   (*
    t_s
    (if (<= t_m 1.22e-116)
      (/ (* t_m (sqrt 2.0)) (* l_m (sqrt (/ (+ 2.0 (/ 2.0 x)) x))))
      (if (<= t_m 4.4e+92)
        (*
         t_m
         (pow
          (/
           (+
            (+ t_2 (/ t_2 x))
            (* (/ 1.0 x) (+ (* l_m l_m) (+ (* l_m l_m) t_2))))
           2.0)
          -0.5))
        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 * (t_m * t_m);
	double tmp;
	if (t_m <= 1.22e-116) {
		tmp = (t_m * sqrt(2.0)) / (l_m * sqrt(((2.0 + (2.0 / x)) / x)));
	} else if (t_m <= 4.4e+92) {
		tmp = t_m * pow((((t_2 + (t_2 / x)) + ((1.0 / x) * ((l_m * l_m) + ((l_m * l_m) + t_2)))) / 2.0), -0.5);
	} else {
		tmp = 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 * t_m)
    if (t_m <= 1.22d-116) then
        tmp = (t_m * sqrt(2.0d0)) / (l_m * sqrt(((2.0d0 + (2.0d0 / x)) / x)))
    else if (t_m <= 4.4d+92) then
        tmp = t_m * ((((t_2 + (t_2 / x)) + ((1.0d0 / x) * ((l_m * l_m) + ((l_m * l_m) + t_2)))) / 2.0d0) ** (-0.5d0))
    else
        tmp = 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 * (t_m * t_m);
	double tmp;
	if (t_m <= 1.22e-116) {
		tmp = (t_m * Math.sqrt(2.0)) / (l_m * Math.sqrt(((2.0 + (2.0 / x)) / x)));
	} else if (t_m <= 4.4e+92) {
		tmp = t_m * Math.pow((((t_2 + (t_2 / x)) + ((1.0 / x) * ((l_m * l_m) + ((l_m * l_m) + t_2)))) / 2.0), -0.5);
	} else {
		tmp = 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 * (t_m * t_m)
	tmp = 0
	if t_m <= 1.22e-116:
		tmp = (t_m * math.sqrt(2.0)) / (l_m * math.sqrt(((2.0 + (2.0 / x)) / x)))
	elif t_m <= 4.4e+92:
		tmp = t_m * math.pow((((t_2 + (t_2 / x)) + ((1.0 / x) * ((l_m * l_m) + ((l_m * l_m) + t_2)))) / 2.0), -0.5)
	else:
		tmp = 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 * Float64(t_m * t_m))
	tmp = 0.0
	if (t_m <= 1.22e-116)
		tmp = Float64(Float64(t_m * sqrt(2.0)) / Float64(l_m * sqrt(Float64(Float64(2.0 + Float64(2.0 / x)) / x))));
	elseif (t_m <= 4.4e+92)
		tmp = Float64(t_m * (Float64(Float64(Float64(t_2 + Float64(t_2 / x)) + Float64(Float64(1.0 / x) * Float64(Float64(l_m * l_m) + Float64(Float64(l_m * l_m) + t_2)))) / 2.0) ^ -0.5));
	else
		tmp = 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 * t_m);
	tmp = 0.0;
	if (t_m <= 1.22e-116)
		tmp = (t_m * sqrt(2.0)) / (l_m * sqrt(((2.0 + (2.0 / x)) / x)));
	elseif (t_m <= 4.4e+92)
		tmp = t_m * ((((t_2 + (t_2 / x)) + ((1.0 / x) * ((l_m * l_m) + ((l_m * l_m) + t_2)))) / 2.0) ^ -0.5);
	else
		tmp = 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[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.22e-116], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(l$95$m * N[Sqrt[N[(N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.4e+92], N[(t$95$m * N[Power[N[(N[(N[(t$95$2 + N[(t$95$2 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] * N[(N[(l$95$m * l$95$m), $MachinePrecision] + N[(N[(l$95$m * l$95$m), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], 1.0]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.22e-116

   1. Initial program 26.5%

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

   3. Taylor expanded in l around inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

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

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

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

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

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

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. +-lowering-+.f64 1.8%

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

   5. Simplified 1.8%

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

   6. Taylor expanded in x around inf

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

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

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 20.1%

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

   8. Simplified 20.1%

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

   if 1.22e-116 < t < 4.39999999999999984e92

   1. Initial program 67.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. sqrt-undiv N/A

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

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

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

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

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

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

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

   4. Applied egg-rr 68.0%

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

   5. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

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

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

      3. associate-*r/ N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

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

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

   7. Simplified 92.6%

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

      1. pow1/2 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

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

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

   9. Applied egg-rr 92.6%

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

   if 4.39999999999999984e92 < t

   1. Initial program 18.3%

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

   3. Taylor expanded in x around inf

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

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

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

      2. sqrt-lowering-sqrt.f64 94.9%

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

   5. Simplified 94.9%

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

      1. *-commutative N/A

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

      2. *-inverses 94.9%

         \[\leadsto 1 \]

   7. Applied egg-rr 94.9%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.22 \cdot 10^{-116}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2 + \frac{2}{x}}{x}}}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{+92}:\\ \;\;\;\;t \cdot {\left(\frac{\left(2 \cdot \left(t \cdot t\right) + \frac{2 \cdot \left(t \cdot t\right)}{x}\right) + \frac{1}{x} \cdot \left(\ell \cdot \ell + \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}{2}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot \left(t\_m \cdot t\_m\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.22 \cdot 10^{-116}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 7.5 \cdot 10^{+92}:\\ \;\;\;\;t\_m \cdot {\left(\frac{\left(t\_2 + \frac{t\_2}{x}\right) + \frac{1}{x} \cdot \left(l\_m \cdot l\_m + \left(l\_m \cdot l\_m + t\_2\right)\right)}{2}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \end{array} \]

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 (* t_m t_m))))
   (*
    t_s
    (if (<= t_m 1.22e-116)
      (/ (* t_m (sqrt x)) l_m)
      (if (<= t_m 7.5e+92)
        (*
         t_m
         (pow
          (/
           (+
            (+ t_2 (/ t_2 x))
            (* (/ 1.0 x) (+ (* l_m l_m) (+ (* l_m l_m) t_2))))
           2.0)
          -0.5))
        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 * (t_m * t_m);
	double tmp;
	if (t_m <= 1.22e-116) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else if (t_m <= 7.5e+92) {
		tmp = t_m * pow((((t_2 + (t_2 / x)) + ((1.0 / x) * ((l_m * l_m) + ((l_m * l_m) + t_2)))) / 2.0), -0.5);
	} else {
		tmp = 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 * t_m)
    if (t_m <= 1.22d-116) then
        tmp = (t_m * sqrt(x)) / l_m
    else if (t_m <= 7.5d+92) then
        tmp = t_m * ((((t_2 + (t_2 / x)) + ((1.0d0 / x) * ((l_m * l_m) + ((l_m * l_m) + t_2)))) / 2.0d0) ** (-0.5d0))
    else
        tmp = 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 * (t_m * t_m);
	double tmp;
	if (t_m <= 1.22e-116) {
		tmp = (t_m * Math.sqrt(x)) / l_m;
	} else if (t_m <= 7.5e+92) {
		tmp = t_m * Math.pow((((t_2 + (t_2 / x)) + ((1.0 / x) * ((l_m * l_m) + ((l_m * l_m) + t_2)))) / 2.0), -0.5);
	} else {
		tmp = 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 * (t_m * t_m)
	tmp = 0
	if t_m <= 1.22e-116:
		tmp = (t_m * math.sqrt(x)) / l_m
	elif t_m <= 7.5e+92:
		tmp = t_m * math.pow((((t_2 + (t_2 / x)) + ((1.0 / x) * ((l_m * l_m) + ((l_m * l_m) + t_2)))) / 2.0), -0.5)
	else:
		tmp = 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 * Float64(t_m * t_m))
	tmp = 0.0
	if (t_m <= 1.22e-116)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	elseif (t_m <= 7.5e+92)
		tmp = Float64(t_m * (Float64(Float64(Float64(t_2 + Float64(t_2 / x)) + Float64(Float64(1.0 / x) * Float64(Float64(l_m * l_m) + Float64(Float64(l_m * l_m) + t_2)))) / 2.0) ^ -0.5));
	else
		tmp = 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 * t_m);
	tmp = 0.0;
	if (t_m <= 1.22e-116)
		tmp = (t_m * sqrt(x)) / l_m;
	elseif (t_m <= 7.5e+92)
		tmp = t_m * ((((t_2 + (t_2 / x)) + ((1.0 / x) * ((l_m * l_m) + ((l_m * l_m) + t_2)))) / 2.0) ^ -0.5);
	else
		tmp = 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[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.22e-116], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 7.5e+92], N[(t$95$m * N[Power[N[(N[(N[(t$95$2 + N[(t$95$2 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] * N[(N[(l$95$m * l$95$m), $MachinePrecision] + N[(N[(l$95$m * l$95$m), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], 1.0]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.22e-116

   1. Initial program 26.5%

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

   3. Taylor expanded in l around inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

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

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

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

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

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

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. +-lowering-+.f64 1.8%

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

   5. Simplified 1.8%

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

   6. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 20.1%

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

   8. Simplified 20.1%

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

   9. Taylor expanded in t around 0

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

       1. associate-*l/ N/A

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

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

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

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

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

       4. sqrt-lowering-sqrt.f64 20.2%

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

   11. Simplified 20.2%

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

   if 1.22e-116 < t < 7.49999999999999946e92

   1. Initial program 67.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. sqrt-undiv N/A

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

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

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

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

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

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

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

   4. Applied egg-rr 68.0%

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

   5. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

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

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

      3. associate-*r/ N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

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

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

   7. Simplified 92.6%

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

      1. pow1/2 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

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

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

   9. Applied egg-rr 92.6%

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

   if 7.49999999999999946e92 < t

   1. Initial program 18.3%

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

   3. Taylor expanded in x around inf

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

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

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

      2. sqrt-lowering-sqrt.f64 94.9%

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

   5. Simplified 94.9%

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

      1. *-commutative N/A

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

      2. *-inverses 94.9%

         \[\leadsto 1 \]

   7. Applied egg-rr 94.9%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.22 \cdot 10^{-116}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+92}:\\ \;\;\;\;t \cdot {\left(\frac{\left(2 \cdot \left(t \cdot t\right) + \frac{2 \cdot \left(t \cdot t\right)}{x}\right) + \frac{1}{x} \cdot \left(\ell \cdot \ell + \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}{2}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot \left(t\_m \cdot t\_m\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.22 \cdot 10^{-116}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 5.4 \cdot 10^{+92}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2}{\frac{t\_2}{x} + \left(\left(t\_2 + \frac{l\_m \cdot l\_m}{x}\right) + \frac{l\_m \cdot l\_m + t\_2}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \end{array} \]

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 (* t_m t_m))))
   (*
    t_s
    (if (<= t_m 1.22e-116)
      (/ (* t_m (sqrt x)) l_m)
      (if (<= t_m 5.4e+92)
        (*
         t_m
         (sqrt
          (/
           2.0
           (+
            (/ t_2 x)
            (+ (+ t_2 (/ (* l_m l_m) x)) (/ (+ (* l_m l_m) t_2) 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 * (t_m * t_m);
	double tmp;
	if (t_m <= 1.22e-116) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else if (t_m <= 5.4e+92) {
		tmp = t_m * sqrt((2.0 / ((t_2 / x) + ((t_2 + ((l_m * l_m) / x)) + (((l_m * l_m) + t_2) / x)))));
	} else {
		tmp = 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 * t_m)
    if (t_m <= 1.22d-116) then
        tmp = (t_m * sqrt(x)) / l_m
    else if (t_m <= 5.4d+92) then
        tmp = t_m * sqrt((2.0d0 / ((t_2 / x) + ((t_2 + ((l_m * l_m) / x)) + (((l_m * l_m) + t_2) / x)))))
    else
        tmp = 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 * (t_m * t_m);
	double tmp;
	if (t_m <= 1.22e-116) {
		tmp = (t_m * Math.sqrt(x)) / l_m;
	} else if (t_m <= 5.4e+92) {
		tmp = t_m * Math.sqrt((2.0 / ((t_2 / x) + ((t_2 + ((l_m * l_m) / x)) + (((l_m * l_m) + t_2) / x)))));
	} else {
		tmp = 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 * (t_m * t_m)
	tmp = 0
	if t_m <= 1.22e-116:
		tmp = (t_m * math.sqrt(x)) / l_m
	elif t_m <= 5.4e+92:
		tmp = t_m * math.sqrt((2.0 / ((t_2 / x) + ((t_2 + ((l_m * l_m) / x)) + (((l_m * l_m) + t_2) / x)))))
	else:
		tmp = 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 * Float64(t_m * t_m))
	tmp = 0.0
	if (t_m <= 1.22e-116)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	elseif (t_m <= 5.4e+92)
		tmp = Float64(t_m * sqrt(Float64(2.0 / Float64(Float64(t_2 / x) + Float64(Float64(t_2 + Float64(Float64(l_m * l_m) / x)) + Float64(Float64(Float64(l_m * l_m) + t_2) / x))))));
	else
		tmp = 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 * t_m);
	tmp = 0.0;
	if (t_m <= 1.22e-116)
		tmp = (t_m * sqrt(x)) / l_m;
	elseif (t_m <= 5.4e+92)
		tmp = t_m * sqrt((2.0 / ((t_2 / x) + ((t_2 + ((l_m * l_m) / x)) + (((l_m * l_m) + t_2) / x)))));
	else
		tmp = 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[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.22e-116], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 5.4e+92], N[(t$95$m * N[Sqrt[N[(2.0 / N[(N[(t$95$2 / x), $MachinePrecision] + N[(N[(t$95$2 + N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] + t$95$2), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.22e-116

   1. Initial program 26.5%

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

   3. Taylor expanded in l around inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

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

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

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

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

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

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. +-lowering-+.f64 1.8%

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

   5. Simplified 1.8%

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

   6. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 20.1%

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

   8. Simplified 20.1%

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

   9. Taylor expanded in t around 0

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

       1. associate-*l/ N/A

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

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

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

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

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

       4. sqrt-lowering-sqrt.f64 20.2%

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

   11. Simplified 20.2%

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

   if 1.22e-116 < t < 5.3999999999999999e92

   1. Initial program 67.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. sqrt-undiv N/A

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

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

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

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

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

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

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

   4. Applied egg-rr 68.0%

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

   5. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

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

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

      3. associate-*r/ N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

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

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

   7. Simplified 92.6%

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

   if 5.3999999999999999e92 < t

   1. Initial program 18.3%

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

   3. Taylor expanded in x around inf

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

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

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

      2. sqrt-lowering-sqrt.f64 94.9%

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

   5. Simplified 94.9%

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

      1. *-commutative N/A

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

      2. *-inverses 94.9%

         \[\leadsto 1 \]

   7. Applied egg-rr 94.9%

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

Alternative 5: 81.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.22 \cdot 10^{-116}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 3.4 \cdot 10^{+92}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2}{\left(t\_m \cdot t\_m\right) \cdot \left(2 + \frac{4}{x}\right) + \frac{2 \cdot \left(l\_m \cdot l\_m\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.22e-116)
    (/ (* t_m (sqrt x)) l_m)
    (if (<= t_m 3.4e+92)
      (*
       t_m
       (sqrt
        (/
         2.0
         (+ (* (* t_m t_m) (+ 2.0 (/ 4.0 x))) (/ (* 2.0 (* l_m l_m)) x)))))
      1.0))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.22e-116) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else if (t_m <= 3.4e+92) {
		tmp = t_m * sqrt((2.0 / (((t_m * t_m) * (2.0 + (4.0 / x))) + ((2.0 * (l_m * l_m)) / x))));
	} else {
		tmp = 1.0;
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1.22d-116) then
        tmp = (t_m * sqrt(x)) / l_m
    else if (t_m <= 3.4d+92) then
        tmp = t_m * sqrt((2.0d0 / (((t_m * t_m) * (2.0d0 + (4.0d0 / x))) + ((2.0d0 * (l_m * l_m)) / x))))
    else
        tmp = 1.0d0
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.22e-116) {
		tmp = (t_m * Math.sqrt(x)) / l_m;
	} else if (t_m <= 3.4e+92) {
		tmp = t_m * Math.sqrt((2.0 / (((t_m * t_m) * (2.0 + (4.0 / x))) + ((2.0 * (l_m * l_m)) / x))));
	} else {
		tmp = 1.0;
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 1.22e-116:
		tmp = (t_m * math.sqrt(x)) / l_m
	elif t_m <= 3.4e+92:
		tmp = t_m * math.sqrt((2.0 / (((t_m * t_m) * (2.0 + (4.0 / x))) + ((2.0 * (l_m * l_m)) / x))))
	else:
		tmp = 1.0
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 1.22e-116)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	elseif (t_m <= 3.4e+92)
		tmp = Float64(t_m * sqrt(Float64(2.0 / Float64(Float64(Float64(t_m * t_m) * Float64(2.0 + Float64(4.0 / x))) + Float64(Float64(2.0 * Float64(l_m * l_m)) / x)))));
	else
		tmp = 1.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 1.22e-116)
		tmp = (t_m * sqrt(x)) / l_m;
	elseif (t_m <= 3.4e+92)
		tmp = t_m * sqrt((2.0 / (((t_m * t_m) * (2.0 + (4.0 / x))) + ((2.0 * (l_m * l_m)) / x))));
	else
		tmp = 1.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.22e-116], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 3.4e+92], N[(t$95$m * N[Sqrt[N[(2.0 / N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.22e-116

   1. Initial program 26.5%

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

   3. Taylor expanded in l around inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

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

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

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

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

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

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. +-lowering-+.f64 1.8%

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

   5. Simplified 1.8%

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

   6. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 20.1%

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

   8. Simplified 20.1%

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

   9. Taylor expanded in t around 0

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

       1. associate-*l/ N/A

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

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

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

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

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

       4. sqrt-lowering-sqrt.f64 20.2%

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

   11. Simplified 20.2%

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

   if 1.22e-116 < t < 3.3999999999999998e92

   1. Initial program 67.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. sqrt-undiv N/A

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

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

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

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

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

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

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

   4. Applied egg-rr 68.0%

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

   5. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

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

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

      3. associate-*r/ N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

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

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

   7. Simplified 92.6%

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

   8. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

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

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

      10. associate-*r/ N/A

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 92.5%

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

   10. Simplified 92.5%

       \[\leadsto t \cdot \sqrt{\frac{2}{\color{blue}{\left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right) + \frac{2 \cdot \left(\ell \cdot \ell\right)}{x}}}} \]

   if 3.3999999999999998e92 < t

   1. Initial program 18.3%

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

   3. Taylor expanded in x around inf

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

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

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

      2. sqrt-lowering-sqrt.f64 94.9%

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

   5. Simplified 94.9%

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

      1. *-commutative N/A

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

      2. *-inverses 94.9%

         \[\leadsto 1 \]

   7. Applied egg-rr 94.9%

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

Alternative 6: 78.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-109}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{l\_m}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (* t_s (if (<= t_m 4.2e-109) (/ (* t_m (sqrt x)) l_m) 1.0)))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 4.2e-109) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else {
		tmp = 1.0;
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 4.2d-109) then
        tmp = (t_m * sqrt(x)) / l_m
    else
        tmp = 1.0d0
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 4.2e-109) {
		tmp = (t_m * Math.sqrt(x)) / l_m;
	} else {
		tmp = 1.0;
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 4.2e-109:
		tmp = (t_m * math.sqrt(x)) / l_m
	else:
		tmp = 1.0
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 4.2e-109)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	else
		tmp = 1.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 4.2e-109)
		tmp = (t_m * sqrt(x)) / l_m;
	else
		tmp = 1.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 4.2e-109], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], 1.0]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.19999999999999992e-109

   1. Initial program 27.0%

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

   3. Taylor expanded in l around inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

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

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

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

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

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

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. +-lowering-+.f64 1.8%

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

   5. Simplified 1.8%

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

   6. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 20.0%

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

   8. Simplified 20.0%

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

   9. Taylor expanded in t around 0

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

       1. associate-*l/ N/A

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

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

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

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

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

       4. sqrt-lowering-sqrt.f64 20.1%

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

   11. Simplified 20.1%

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

   if 4.19999999999999992e-109 < t

   1. Initial program 38.7%

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

   3. Taylor expanded in x around inf

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

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

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

      2. sqrt-lowering-sqrt.f64 87.7%

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

   5. Simplified 87.7%

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

      1. *-commutative N/A

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

      2. *-inverses 87.7%

         \[\leadsto 1 \]

   7. Applied egg-rr 87.7%

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

Alternative 7: 76.4% accurate, 225.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.2%

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

3. Taylor expanded in x around inf

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

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

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

   2. sqrt-lowering-sqrt.f64 37.1%

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

5. Simplified 37.1%

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

   1. *-commutative N/A

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

   2. *-inverses 37.1%

      \[\leadsto 1 \]

7. Applied egg-rr 37.1%

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

Reproduce

herbie shell --seed 2024158 
(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)))))