Toniolo and Linder, Equation (7)

Percentage Accurate: 34.2% → 84.2%

Time: 18.3s

Alternatives: 12

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: 34.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t_3 := 2 \cdot {t\_m}^{2}\\ t_4 := t\_3 + {l\_m}^{2}\\ t_5 := t\_4 + t\_4\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.4 \cdot 10^{-274}:\\ \;\;\;\;\left(t\_m \cdot \frac{\sqrt{2}}{l\_m}\right) \cdot \sqrt{0.5 \cdot x - 0.5}\\ \mathbf{elif}\;t\_m \leq 3.3 \cdot 10^{-124}:\\ \;\;\;\;\frac{t\_2}{t\_2 + 0.5 \cdot \frac{t\_5}{t\_m \cdot \left(\sqrt{2} \cdot x\right)}}\\ \mathbf{elif}\;t\_m \leq 1.45 \cdot 10^{+35}:\\ \;\;\;\;\frac{t\_2}{\sqrt{t\_3 + \frac{t\_4 + \left(t\_4 + \frac{t\_4 + \left(t\_4 + \frac{t\_5}{x}\right)}{x}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* t_m (sqrt 2.0)))
        (t_3 (* 2.0 (pow t_m 2.0)))
        (t_4 (+ t_3 (pow l_m 2.0)))
        (t_5 (+ t_4 t_4)))
   (*
    t_s
    (if (<= t_m 4.4e-274)
      (* (* t_m (/ (sqrt 2.0) l_m)) (sqrt (- (* 0.5 x) 0.5)))
      (if (<= t_m 3.3e-124)
        (/ t_2 (+ t_2 (* 0.5 (/ t_5 (* t_m (* (sqrt 2.0) x))))))
        (if (<= t_m 1.45e+35)
          (/
           t_2
           (sqrt
            (+ t_3 (/ (+ t_4 (+ t_4 (/ (+ t_4 (+ t_4 (/ t_5 x))) x))) x))))
          (sqrt (/ (+ x -1.0) (+ x 1.0)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * sqrt(2.0);
	double t_3 = 2.0 * pow(t_m, 2.0);
	double t_4 = t_3 + pow(l_m, 2.0);
	double t_5 = t_4 + t_4;
	double tmp;
	if (t_m <= 4.4e-274) {
		tmp = (t_m * (sqrt(2.0) / l_m)) * sqrt(((0.5 * x) - 0.5));
	} else if (t_m <= 3.3e-124) {
		tmp = t_2 / (t_2 + (0.5 * (t_5 / (t_m * (sqrt(2.0) * x)))));
	} else if (t_m <= 1.45e+35) {
		tmp = t_2 / sqrt((t_3 + ((t_4 + (t_4 + ((t_4 + (t_4 + (t_5 / x))) / x))) / x)));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_2 = t_m * sqrt(2.0d0)
    t_3 = 2.0d0 * (t_m ** 2.0d0)
    t_4 = t_3 + (l_m ** 2.0d0)
    t_5 = t_4 + t_4
    if (t_m <= 4.4d-274) then
        tmp = (t_m * (sqrt(2.0d0) / l_m)) * sqrt(((0.5d0 * x) - 0.5d0))
    else if (t_m <= 3.3d-124) then
        tmp = t_2 / (t_2 + (0.5d0 * (t_5 / (t_m * (sqrt(2.0d0) * x)))))
    else if (t_m <= 1.45d+35) then
        tmp = t_2 / sqrt((t_3 + ((t_4 + (t_4 + ((t_4 + (t_4 + (t_5 / x))) / x))) / x)))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * Math.sqrt(2.0);
	double t_3 = 2.0 * Math.pow(t_m, 2.0);
	double t_4 = t_3 + Math.pow(l_m, 2.0);
	double t_5 = t_4 + t_4;
	double tmp;
	if (t_m <= 4.4e-274) {
		tmp = (t_m * (Math.sqrt(2.0) / l_m)) * Math.sqrt(((0.5 * x) - 0.5));
	} else if (t_m <= 3.3e-124) {
		tmp = t_2 / (t_2 + (0.5 * (t_5 / (t_m * (Math.sqrt(2.0) * x)))));
	} else if (t_m <= 1.45e+35) {
		tmp = t_2 / Math.sqrt((t_3 + ((t_4 + (t_4 + ((t_4 + (t_4 + (t_5 / x))) / x))) / x)));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = t_m * math.sqrt(2.0)
	t_3 = 2.0 * math.pow(t_m, 2.0)
	t_4 = t_3 + math.pow(l_m, 2.0)
	t_5 = t_4 + t_4
	tmp = 0
	if t_m <= 4.4e-274:
		tmp = (t_m * (math.sqrt(2.0) / l_m)) * math.sqrt(((0.5 * x) - 0.5))
	elif t_m <= 3.3e-124:
		tmp = t_2 / (t_2 + (0.5 * (t_5 / (t_m * (math.sqrt(2.0) * x)))))
	elif t_m <= 1.45e+35:
		tmp = t_2 / math.sqrt((t_3 + ((t_4 + (t_4 + ((t_4 + (t_4 + (t_5 / x))) / x))) / x)))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(t_m * sqrt(2.0))
	t_3 = Float64(2.0 * (t_m ^ 2.0))
	t_4 = Float64(t_3 + (l_m ^ 2.0))
	t_5 = Float64(t_4 + t_4)
	tmp = 0.0
	if (t_m <= 4.4e-274)
		tmp = Float64(Float64(t_m * Float64(sqrt(2.0) / l_m)) * sqrt(Float64(Float64(0.5 * x) - 0.5)));
	elseif (t_m <= 3.3e-124)
		tmp = Float64(t_2 / Float64(t_2 + Float64(0.5 * Float64(t_5 / Float64(t_m * Float64(sqrt(2.0) * x))))));
	elseif (t_m <= 1.45e+35)
		tmp = Float64(t_2 / sqrt(Float64(t_3 + Float64(Float64(t_4 + Float64(t_4 + Float64(Float64(t_4 + Float64(t_4 + Float64(t_5 / x))) / x))) / x))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = t_m * sqrt(2.0);
	t_3 = 2.0 * (t_m ^ 2.0);
	t_4 = t_3 + (l_m ^ 2.0);
	t_5 = t_4 + t_4;
	tmp = 0.0;
	if (t_m <= 4.4e-274)
		tmp = (t_m * (sqrt(2.0) / l_m)) * sqrt(((0.5 * x) - 0.5));
	elseif (t_m <= 3.3e-124)
		tmp = t_2 / (t_2 + (0.5 * (t_5 / (t_m * (sqrt(2.0) * x)))));
	elseif (t_m <= 1.45e+35)
		tmp = t_2 / sqrt((t_3 + ((t_4 + (t_4 + ((t_4 + (t_4 + (t_5 / x))) / x))) / x)));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 + t$95$4), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.4e-274], N[(N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(0.5 * x), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.3e-124], N[(t$95$2 / N[(t$95$2 + N[(0.5 * N[(t$95$5 / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.45e+35], N[(t$95$2 / N[Sqrt[N[(t$95$3 + N[(N[(t$95$4 + N[(t$95$4 + N[(N[(t$95$4 + N[(t$95$4 + N[(t$95$5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 4.3999999999999999e-274

   1. Initial program 30.5%

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

   3. Simplified 29.1%

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

   5. Taylor expanded in l around inf 3.4%

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

      1. associate-/l* 3.4%

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

      2. associate--l+ 12.4%

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

      3. sub-neg 12.4%

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

      4. metadata-eval 12.4%

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

      5. +-commutative 12.4%

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

      6. sub-neg 12.4%

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

      7. metadata-eval 12.4%

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

      8. +-commutative 12.4%

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

   7. Simplified 12.4%

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

   8. Taylor expanded in x around 0 20.0%

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

   if 4.3999999999999999e-274 < t < 3.29999999999999984e-124

   1. Initial program 21.1%

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

   3. Taylor expanded in x around inf 82.6%

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

   if 3.29999999999999984e-124 < t < 1.44999999999999997e35

   1. Initial program 53.6%

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

   3. Taylor expanded in x around inf 53.5%

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

   4. Taylor expanded in x around -inf 84.3%

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

   if 1.44999999999999997e35 < t

   1. Initial program 31.7%

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

   3. Simplified 31.8%

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

   5. Taylor expanded in t around inf 93.1%

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

   6. Taylor expanded in t around 0 93.3%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.4 \cdot 10^{-274}:\\ \;\;\;\;\left(t \cdot \frac{\sqrt{2}}{\ell}\right) \cdot \sqrt{0.5 \cdot x - 0.5}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-124}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + 0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(\sqrt{2} \cdot x\right)}}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+35}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot {t}^{2} + \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{x}\right)}{x}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t_3 := 2 \cdot {t\_m}^{2}\\ t_4 := t\_3 + {l\_m}^{2}\\ t_5 := t\_4 + t\_4\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.5 \cdot 10^{-273}:\\ \;\;\;\;\left(t\_m \cdot \frac{\sqrt{2}}{l\_m}\right) \cdot \sqrt{0.5 \cdot x - 0.5}\\ \mathbf{elif}\;t\_m \leq 3.3 \cdot 10^{-124}:\\ \;\;\;\;\frac{t\_2}{t\_2 + 0.5 \cdot \frac{t\_5}{t\_m \cdot \left(\sqrt{2} \cdot x\right)}}\\ \mathbf{elif}\;t\_m \leq 7 \cdot 10^{+34}:\\ \;\;\;\;\frac{t\_2}{\sqrt{t\_3 + \frac{\left(2 \cdot \frac{{t\_m}^{2}}{x} + \frac{{l\_m}^{2}}{x}\right) + \left(t\_5 + \frac{t\_4}{x}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* t_m (sqrt 2.0)))
        (t_3 (* 2.0 (pow t_m 2.0)))
        (t_4 (+ t_3 (pow l_m 2.0)))
        (t_5 (+ t_4 t_4)))
   (*
    t_s
    (if (<= t_m 4.5e-273)
      (* (* t_m (/ (sqrt 2.0) l_m)) (sqrt (- (* 0.5 x) 0.5)))
      (if (<= t_m 3.3e-124)
        (/ t_2 (+ t_2 (* 0.5 (/ t_5 (* t_m (* (sqrt 2.0) x))))))
        (if (<= t_m 7e+34)
          (/
           t_2
           (sqrt
            (+
             t_3
             (/
              (+
               (+ (* 2.0 (/ (pow t_m 2.0) x)) (/ (pow l_m 2.0) x))
               (+ t_5 (/ t_4 x)))
              x))))
          (sqrt (/ (+ x -1.0) (+ x 1.0)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * sqrt(2.0);
	double t_3 = 2.0 * pow(t_m, 2.0);
	double t_4 = t_3 + pow(l_m, 2.0);
	double t_5 = t_4 + t_4;
	double tmp;
	if (t_m <= 4.5e-273) {
		tmp = (t_m * (sqrt(2.0) / l_m)) * sqrt(((0.5 * x) - 0.5));
	} else if (t_m <= 3.3e-124) {
		tmp = t_2 / (t_2 + (0.5 * (t_5 / (t_m * (sqrt(2.0) * x)))));
	} else if (t_m <= 7e+34) {
		tmp = t_2 / sqrt((t_3 + ((((2.0 * (pow(t_m, 2.0) / x)) + (pow(l_m, 2.0) / x)) + (t_5 + (t_4 / x))) / x)));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_2 = t_m * sqrt(2.0d0)
    t_3 = 2.0d0 * (t_m ** 2.0d0)
    t_4 = t_3 + (l_m ** 2.0d0)
    t_5 = t_4 + t_4
    if (t_m <= 4.5d-273) then
        tmp = (t_m * (sqrt(2.0d0) / l_m)) * sqrt(((0.5d0 * x) - 0.5d0))
    else if (t_m <= 3.3d-124) then
        tmp = t_2 / (t_2 + (0.5d0 * (t_5 / (t_m * (sqrt(2.0d0) * x)))))
    else if (t_m <= 7d+34) then
        tmp = t_2 / sqrt((t_3 + ((((2.0d0 * ((t_m ** 2.0d0) / x)) + ((l_m ** 2.0d0) / x)) + (t_5 + (t_4 / x))) / x)))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * Math.sqrt(2.0);
	double t_3 = 2.0 * Math.pow(t_m, 2.0);
	double t_4 = t_3 + Math.pow(l_m, 2.0);
	double t_5 = t_4 + t_4;
	double tmp;
	if (t_m <= 4.5e-273) {
		tmp = (t_m * (Math.sqrt(2.0) / l_m)) * Math.sqrt(((0.5 * x) - 0.5));
	} else if (t_m <= 3.3e-124) {
		tmp = t_2 / (t_2 + (0.5 * (t_5 / (t_m * (Math.sqrt(2.0) * x)))));
	} else if (t_m <= 7e+34) {
		tmp = t_2 / Math.sqrt((t_3 + ((((2.0 * (Math.pow(t_m, 2.0) / x)) + (Math.pow(l_m, 2.0) / x)) + (t_5 + (t_4 / x))) / x)));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = t_m * math.sqrt(2.0)
	t_3 = 2.0 * math.pow(t_m, 2.0)
	t_4 = t_3 + math.pow(l_m, 2.0)
	t_5 = t_4 + t_4
	tmp = 0
	if t_m <= 4.5e-273:
		tmp = (t_m * (math.sqrt(2.0) / l_m)) * math.sqrt(((0.5 * x) - 0.5))
	elif t_m <= 3.3e-124:
		tmp = t_2 / (t_2 + (0.5 * (t_5 / (t_m * (math.sqrt(2.0) * x)))))
	elif t_m <= 7e+34:
		tmp = t_2 / math.sqrt((t_3 + ((((2.0 * (math.pow(t_m, 2.0) / x)) + (math.pow(l_m, 2.0) / x)) + (t_5 + (t_4 / x))) / x)))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(t_m * sqrt(2.0))
	t_3 = Float64(2.0 * (t_m ^ 2.0))
	t_4 = Float64(t_3 + (l_m ^ 2.0))
	t_5 = Float64(t_4 + t_4)
	tmp = 0.0
	if (t_m <= 4.5e-273)
		tmp = Float64(Float64(t_m * Float64(sqrt(2.0) / l_m)) * sqrt(Float64(Float64(0.5 * x) - 0.5)));
	elseif (t_m <= 3.3e-124)
		tmp = Float64(t_2 / Float64(t_2 + Float64(0.5 * Float64(t_5 / Float64(t_m * Float64(sqrt(2.0) * x))))));
	elseif (t_m <= 7e+34)
		tmp = Float64(t_2 / sqrt(Float64(t_3 + Float64(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64((l_m ^ 2.0) / x)) + Float64(t_5 + Float64(t_4 / x))) / x))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = t_m * sqrt(2.0);
	t_3 = 2.0 * (t_m ^ 2.0);
	t_4 = t_3 + (l_m ^ 2.0);
	t_5 = t_4 + t_4;
	tmp = 0.0;
	if (t_m <= 4.5e-273)
		tmp = (t_m * (sqrt(2.0) / l_m)) * sqrt(((0.5 * x) - 0.5));
	elseif (t_m <= 3.3e-124)
		tmp = t_2 / (t_2 + (0.5 * (t_5 / (t_m * (sqrt(2.0) * x)))));
	elseif (t_m <= 7e+34)
		tmp = t_2 / sqrt((t_3 + ((((2.0 * ((t_m ^ 2.0) / x)) + ((l_m ^ 2.0) / x)) + (t_5 + (t_4 / x))) / x)));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 + t$95$4), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.5e-273], N[(N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(0.5 * x), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.3e-124], N[(t$95$2 / N[(t$95$2 + N[(0.5 * N[(t$95$5 / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7e+34], N[(t$95$2 / N[Sqrt[N[(t$95$3 + N[(N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$5 + N[(t$95$4 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 4.4999999999999996e-273

   1. Initial program 30.5%

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

   3. Simplified 29.1%

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

   5. Taylor expanded in l around inf 3.4%

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

      1. associate-/l* 3.4%

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

      2. associate--l+ 12.4%

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

      3. sub-neg 12.4%

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

      4. metadata-eval 12.4%

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

      5. +-commutative 12.4%

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

      6. sub-neg 12.4%

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

      7. metadata-eval 12.4%

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

      8. +-commutative 12.4%

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

   7. Simplified 12.4%

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

   8. Taylor expanded in x around 0 20.0%

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

   if 4.4999999999999996e-273 < t < 3.29999999999999984e-124

   1. Initial program 21.1%

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

   3. Taylor expanded in x around inf 82.6%

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

   if 3.29999999999999984e-124 < t < 6.99999999999999996e34

   1. Initial program 53.6%

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

   3. Taylor expanded in x around -inf 83.7%

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

   if 6.99999999999999996e34 < t

   1. Initial program 31.7%

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

   3. Simplified 31.8%

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

   5. Taylor expanded in t around inf 93.1%

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

   6. Taylor expanded in t around 0 93.3%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{-273}:\\ \;\;\;\;\left(t \cdot \frac{\sqrt{2}}{\ell}\right) \cdot \sqrt{0.5 \cdot x - 0.5}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-124}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + 0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(\sqrt{2} \cdot x\right)}}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+34}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot {t}^{2} + \frac{\left(2 \cdot \frac{{t}^{2}}{x} + \frac{{\ell}^{2}}{x}\right) + \left(\left(\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t_3 := 2 \cdot {t\_m}^{2}\\ t_4 := t\_3 + {l\_m}^{2}\\ t_5 := t\_4 + t\_4\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.6 \cdot 10^{-273}:\\ \;\;\;\;\left(t\_m \cdot \frac{\sqrt{2}}{l\_m}\right) \cdot \sqrt{0.5 \cdot x - 0.5}\\ \mathbf{elif}\;t\_m \leq 3.3 \cdot 10^{-124}:\\ \;\;\;\;\frac{t\_2}{t\_2 + 0.5 \cdot \frac{t\_5}{t\_m \cdot \left(\sqrt{2} \cdot x\right)}}\\ \mathbf{elif}\;t\_m \leq 6 \cdot 10^{+33}:\\ \;\;\;\;\frac{t\_2}{\sqrt{t\_3 + \frac{t\_4 + \left(t\_4 + \frac{t\_5}{x}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* t_m (sqrt 2.0)))
        (t_3 (* 2.0 (pow t_m 2.0)))
        (t_4 (+ t_3 (pow l_m 2.0)))
        (t_5 (+ t_4 t_4)))
   (*
    t_s
    (if (<= t_m 1.6e-273)
      (* (* t_m (/ (sqrt 2.0) l_m)) (sqrt (- (* 0.5 x) 0.5)))
      (if (<= t_m 3.3e-124)
        (/ t_2 (+ t_2 (* 0.5 (/ t_5 (* t_m (* (sqrt 2.0) x))))))
        (if (<= t_m 6e+33)
          (/ t_2 (sqrt (+ t_3 (/ (+ t_4 (+ t_4 (/ t_5 x))) x))))
          (sqrt (/ (+ x -1.0) (+ x 1.0)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * sqrt(2.0);
	double t_3 = 2.0 * pow(t_m, 2.0);
	double t_4 = t_3 + pow(l_m, 2.0);
	double t_5 = t_4 + t_4;
	double tmp;
	if (t_m <= 1.6e-273) {
		tmp = (t_m * (sqrt(2.0) / l_m)) * sqrt(((0.5 * x) - 0.5));
	} else if (t_m <= 3.3e-124) {
		tmp = t_2 / (t_2 + (0.5 * (t_5 / (t_m * (sqrt(2.0) * x)))));
	} else if (t_m <= 6e+33) {
		tmp = t_2 / sqrt((t_3 + ((t_4 + (t_4 + (t_5 / x))) / x)));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_2 = t_m * sqrt(2.0d0)
    t_3 = 2.0d0 * (t_m ** 2.0d0)
    t_4 = t_3 + (l_m ** 2.0d0)
    t_5 = t_4 + t_4
    if (t_m <= 1.6d-273) then
        tmp = (t_m * (sqrt(2.0d0) / l_m)) * sqrt(((0.5d0 * x) - 0.5d0))
    else if (t_m <= 3.3d-124) then
        tmp = t_2 / (t_2 + (0.5d0 * (t_5 / (t_m * (sqrt(2.0d0) * x)))))
    else if (t_m <= 6d+33) then
        tmp = t_2 / sqrt((t_3 + ((t_4 + (t_4 + (t_5 / x))) / x)))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * Math.sqrt(2.0);
	double t_3 = 2.0 * Math.pow(t_m, 2.0);
	double t_4 = t_3 + Math.pow(l_m, 2.0);
	double t_5 = t_4 + t_4;
	double tmp;
	if (t_m <= 1.6e-273) {
		tmp = (t_m * (Math.sqrt(2.0) / l_m)) * Math.sqrt(((0.5 * x) - 0.5));
	} else if (t_m <= 3.3e-124) {
		tmp = t_2 / (t_2 + (0.5 * (t_5 / (t_m * (Math.sqrt(2.0) * x)))));
	} else if (t_m <= 6e+33) {
		tmp = t_2 / Math.sqrt((t_3 + ((t_4 + (t_4 + (t_5 / x))) / x)));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = t_m * math.sqrt(2.0)
	t_3 = 2.0 * math.pow(t_m, 2.0)
	t_4 = t_3 + math.pow(l_m, 2.0)
	t_5 = t_4 + t_4
	tmp = 0
	if t_m <= 1.6e-273:
		tmp = (t_m * (math.sqrt(2.0) / l_m)) * math.sqrt(((0.5 * x) - 0.5))
	elif t_m <= 3.3e-124:
		tmp = t_2 / (t_2 + (0.5 * (t_5 / (t_m * (math.sqrt(2.0) * x)))))
	elif t_m <= 6e+33:
		tmp = t_2 / math.sqrt((t_3 + ((t_4 + (t_4 + (t_5 / x))) / x)))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(t_m * sqrt(2.0))
	t_3 = Float64(2.0 * (t_m ^ 2.0))
	t_4 = Float64(t_3 + (l_m ^ 2.0))
	t_5 = Float64(t_4 + t_4)
	tmp = 0.0
	if (t_m <= 1.6e-273)
		tmp = Float64(Float64(t_m * Float64(sqrt(2.0) / l_m)) * sqrt(Float64(Float64(0.5 * x) - 0.5)));
	elseif (t_m <= 3.3e-124)
		tmp = Float64(t_2 / Float64(t_2 + Float64(0.5 * Float64(t_5 / Float64(t_m * Float64(sqrt(2.0) * x))))));
	elseif (t_m <= 6e+33)
		tmp = Float64(t_2 / sqrt(Float64(t_3 + Float64(Float64(t_4 + Float64(t_4 + Float64(t_5 / x))) / x))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = t_m * sqrt(2.0);
	t_3 = 2.0 * (t_m ^ 2.0);
	t_4 = t_3 + (l_m ^ 2.0);
	t_5 = t_4 + t_4;
	tmp = 0.0;
	if (t_m <= 1.6e-273)
		tmp = (t_m * (sqrt(2.0) / l_m)) * sqrt(((0.5 * x) - 0.5));
	elseif (t_m <= 3.3e-124)
		tmp = t_2 / (t_2 + (0.5 * (t_5 / (t_m * (sqrt(2.0) * x)))));
	elseif (t_m <= 6e+33)
		tmp = t_2 / sqrt((t_3 + ((t_4 + (t_4 + (t_5 / x))) / x)));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 + t$95$4), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.6e-273], N[(N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(0.5 * x), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.3e-124], N[(t$95$2 / N[(t$95$2 + N[(0.5 * N[(t$95$5 / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6e+33], N[(t$95$2 / N[Sqrt[N[(t$95$3 + N[(N[(t$95$4 + N[(t$95$4 + N[(t$95$5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.59999999999999995e-273

   1. Initial program 30.5%

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

   3. Simplified 29.1%

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

   5. Taylor expanded in l around inf 3.4%

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

      1. associate-/l* 3.4%

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

      2. associate--l+ 12.4%

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

      3. sub-neg 12.4%

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

      4. metadata-eval 12.4%

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

      5. +-commutative 12.4%

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

      6. sub-neg 12.4%

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

      7. metadata-eval 12.4%

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

      8. +-commutative 12.4%

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

   7. Simplified 12.4%

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

   8. Taylor expanded in x around 0 20.0%

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

   if 1.59999999999999995e-273 < t < 3.29999999999999984e-124

   1. Initial program 21.1%

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

   3. Taylor expanded in x around inf 82.6%

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

   if 3.29999999999999984e-124 < t < 5.99999999999999967e33

   1. Initial program 53.6%

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

   3. Taylor expanded in x around inf 53.5%

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

   4. Taylor expanded in x around -inf 83.7%

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

   if 5.99999999999999967e33 < t

   1. Initial program 31.7%

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

   3. Simplified 31.8%

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

   5. Taylor expanded in t around inf 93.1%

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

   6. Taylor expanded in t around 0 93.3%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{-273}:\\ \;\;\;\;\left(t \cdot \frac{\sqrt{2}}{\ell}\right) \cdot \sqrt{0.5 \cdot x - 0.5}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-124}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + 0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(\sqrt{2} \cdot x\right)}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+33}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot {t}^{2} + \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{x}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t_3 := 2 \cdot {t\_m}^{2}\\ t_4 := t\_3 + {l\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.15 \cdot 10^{-273}:\\ \;\;\;\;\left(t\_m \cdot \frac{\sqrt{2}}{l\_m}\right) \cdot \sqrt{0.5 \cdot x - 0.5}\\ \mathbf{elif}\;t\_m \leq 3.3 \cdot 10^{-124}:\\ \;\;\;\;\frac{t\_2}{t\_2 + 0.5 \cdot \frac{t\_4 + t\_4}{t\_m \cdot \left(\sqrt{2} \cdot x\right)}}\\ \mathbf{elif}\;t\_m \leq 3 \cdot 10^{+35}:\\ \;\;\;\;\frac{\sqrt{t\_3}}{\sqrt{\frac{t\_4}{x} + \left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_3 + \frac{{l\_m}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* t_m (sqrt 2.0)))
        (t_3 (* 2.0 (pow t_m 2.0)))
        (t_4 (+ t_3 (pow l_m 2.0))))
   (*
    t_s
    (if (<= t_m 1.15e-273)
      (* (* t_m (/ (sqrt 2.0) l_m)) (sqrt (- (* 0.5 x) 0.5)))
      (if (<= t_m 3.3e-124)
        (/ t_2 (+ t_2 (* 0.5 (/ (+ t_4 t_4) (* t_m (* (sqrt 2.0) x))))))
        (if (<= t_m 3e+35)
          (/
           (sqrt t_3)
           (sqrt
            (+
             (/ t_4 x)
             (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_3 (/ (pow l_m 2.0) x))))))
          (sqrt (/ (+ x -1.0) (+ x 1.0)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * sqrt(2.0);
	double t_3 = 2.0 * pow(t_m, 2.0);
	double t_4 = t_3 + pow(l_m, 2.0);
	double tmp;
	if (t_m <= 1.15e-273) {
		tmp = (t_m * (sqrt(2.0) / l_m)) * sqrt(((0.5 * x) - 0.5));
	} else if (t_m <= 3.3e-124) {
		tmp = t_2 / (t_2 + (0.5 * ((t_4 + t_4) / (t_m * (sqrt(2.0) * x)))));
	} else if (t_m <= 3e+35) {
		tmp = sqrt(t_3) / sqrt(((t_4 / x) + ((2.0 * (pow(t_m, 2.0) / x)) + (t_3 + (pow(l_m, 2.0) / x)))));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_2 = t_m * sqrt(2.0d0)
    t_3 = 2.0d0 * (t_m ** 2.0d0)
    t_4 = t_3 + (l_m ** 2.0d0)
    if (t_m <= 1.15d-273) then
        tmp = (t_m * (sqrt(2.0d0) / l_m)) * sqrt(((0.5d0 * x) - 0.5d0))
    else if (t_m <= 3.3d-124) then
        tmp = t_2 / (t_2 + (0.5d0 * ((t_4 + t_4) / (t_m * (sqrt(2.0d0) * x)))))
    else if (t_m <= 3d+35) then
        tmp = sqrt(t_3) / sqrt(((t_4 / x) + ((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_3 + ((l_m ** 2.0d0) / x)))))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * Math.sqrt(2.0);
	double t_3 = 2.0 * Math.pow(t_m, 2.0);
	double t_4 = t_3 + Math.pow(l_m, 2.0);
	double tmp;
	if (t_m <= 1.15e-273) {
		tmp = (t_m * (Math.sqrt(2.0) / l_m)) * Math.sqrt(((0.5 * x) - 0.5));
	} else if (t_m <= 3.3e-124) {
		tmp = t_2 / (t_2 + (0.5 * ((t_4 + t_4) / (t_m * (Math.sqrt(2.0) * x)))));
	} else if (t_m <= 3e+35) {
		tmp = Math.sqrt(t_3) / Math.sqrt(((t_4 / x) + ((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_3 + (Math.pow(l_m, 2.0) / x)))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = t_m * math.sqrt(2.0)
	t_3 = 2.0 * math.pow(t_m, 2.0)
	t_4 = t_3 + math.pow(l_m, 2.0)
	tmp = 0
	if t_m <= 1.15e-273:
		tmp = (t_m * (math.sqrt(2.0) / l_m)) * math.sqrt(((0.5 * x) - 0.5))
	elif t_m <= 3.3e-124:
		tmp = t_2 / (t_2 + (0.5 * ((t_4 + t_4) / (t_m * (math.sqrt(2.0) * x)))))
	elif t_m <= 3e+35:
		tmp = math.sqrt(t_3) / math.sqrt(((t_4 / x) + ((2.0 * (math.pow(t_m, 2.0) / x)) + (t_3 + (math.pow(l_m, 2.0) / x)))))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(t_m * sqrt(2.0))
	t_3 = Float64(2.0 * (t_m ^ 2.0))
	t_4 = Float64(t_3 + (l_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 1.15e-273)
		tmp = Float64(Float64(t_m * Float64(sqrt(2.0) / l_m)) * sqrt(Float64(Float64(0.5 * x) - 0.5)));
	elseif (t_m <= 3.3e-124)
		tmp = Float64(t_2 / Float64(t_2 + Float64(0.5 * Float64(Float64(t_4 + t_4) / Float64(t_m * Float64(sqrt(2.0) * x))))));
	elseif (t_m <= 3e+35)
		tmp = Float64(sqrt(t_3) / sqrt(Float64(Float64(t_4 / x) + Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_3 + Float64((l_m ^ 2.0) / x))))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = t_m * sqrt(2.0);
	t_3 = 2.0 * (t_m ^ 2.0);
	t_4 = t_3 + (l_m ^ 2.0);
	tmp = 0.0;
	if (t_m <= 1.15e-273)
		tmp = (t_m * (sqrt(2.0) / l_m)) * sqrt(((0.5 * x) - 0.5));
	elseif (t_m <= 3.3e-124)
		tmp = t_2 / (t_2 + (0.5 * ((t_4 + t_4) / (t_m * (sqrt(2.0) * x)))));
	elseif (t_m <= 3e+35)
		tmp = sqrt(t_3) / sqrt(((t_4 / x) + ((2.0 * ((t_m ^ 2.0) / x)) + (t_3 + ((l_m ^ 2.0) / x)))));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.15e-273], N[(N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(0.5 * x), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.3e-124], N[(t$95$2 / N[(t$95$2 + N[(0.5 * N[(N[(t$95$4 + t$95$4), $MachinePrecision] / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3e+35], N[(N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(N[(t$95$4 / x), $MachinePrecision] + N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.1499999999999999e-273

   1. Initial program 30.5%

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

   3. Simplified 29.1%

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

   5. Taylor expanded in l around inf 3.4%

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

      1. associate-/l* 3.4%

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

      2. associate--l+ 12.4%

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

      3. sub-neg 12.4%

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

      4. metadata-eval 12.4%

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

      5. +-commutative 12.4%

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

      6. sub-neg 12.4%

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

      7. metadata-eval 12.4%

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

      8. +-commutative 12.4%

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

   7. Simplified 12.4%

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

   8. Taylor expanded in x around 0 20.0%

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

   if 1.1499999999999999e-273 < t < 3.29999999999999984e-124

   1. Initial program 21.1%

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

   3. Taylor expanded in x around inf 82.6%

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

   if 3.29999999999999984e-124 < t < 2.99999999999999991e35

   1. Initial program 53.6%

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

   3. Taylor expanded in x around inf 82.6%

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

      1. add-sqr-sqrt 82.4%

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

      2. sqrt-prod 82.6%

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

      3. sqrt-prod 83.0%

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

      4. pow1/2 83.0%

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

      5. pow2 83.0%

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

   5. Applied egg-rr 83.0%

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

      1. unpow1/2 83.0%

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

   7. Simplified 83.0%

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

   if 2.99999999999999991e35 < t

   1. Initial program 31.7%

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

   3. Simplified 31.8%

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

   5. Taylor expanded in t around inf 93.1%

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

   6. Taylor expanded in t around 0 93.3%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.15 \cdot 10^{-273}:\\ \;\;\;\;\left(t \cdot \frac{\sqrt{2}}{\ell}\right) \cdot \sqrt{0.5 \cdot x - 0.5}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-124}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + 0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(\sqrt{2} \cdot x\right)}}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+35}:\\ \;\;\;\;\frac{\sqrt{2 \cdot {t}^{2}}}{\sqrt{\frac{2 \cdot {t}^{2} + {\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5 \cdot 10^{-274}:\\ \;\;\;\;\left(t\_m \cdot \frac{\sqrt{2}}{l\_m}\right) \cdot \sqrt{0.5 \cdot x - 0.5}\\ \mathbf{elif}\;t\_m \leq 2.55 \cdot 10^{-129}:\\ \;\;\;\;\frac{t\_2}{t\_2 + \frac{{l\_m}^{2}}{t\_m \cdot \left(\sqrt{2} \cdot x\right)}}\\ \mathbf{elif}\;t\_m \leq 3.1 \cdot 10^{+34}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{2 \cdot {t\_m}^{2} + {l\_m}^{2}}{x} + \left(\frac{{l\_m}^{2}}{x} + {t\_m}^{2} \cdot \left(2 + 2 \cdot \frac{1}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 5e-274)
      (* (* t_m (/ (sqrt 2.0) l_m)) (sqrt (- (* 0.5 x) 0.5)))
      (if (<= t_m 2.55e-129)
        (/ t_2 (+ t_2 (/ (pow l_m 2.0) (* t_m (* (sqrt 2.0) x)))))
        (if (<= t_m 3.1e+34)
          (/
           t_2
           (sqrt
            (+
             (/ (+ (* 2.0 (pow t_m 2.0)) (pow l_m 2.0)) x)
             (+
              (/ (pow l_m 2.0) x)
              (* (pow t_m 2.0) (+ 2.0 (* 2.0 (/ 1.0 x))))))))
          (sqrt (/ (+ x -1.0) (+ x 1.0)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 5e-274) {
		tmp = (t_m * (sqrt(2.0) / l_m)) * sqrt(((0.5 * x) - 0.5));
	} else if (t_m <= 2.55e-129) {
		tmp = t_2 / (t_2 + (pow(l_m, 2.0) / (t_m * (sqrt(2.0) * x))));
	} else if (t_m <= 3.1e+34) {
		tmp = t_2 / sqrt(((((2.0 * pow(t_m, 2.0)) + pow(l_m, 2.0)) / x) + ((pow(l_m, 2.0) / x) + (pow(t_m, 2.0) * (2.0 + (2.0 * (1.0 / x)))))));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = t_m * sqrt(2.0d0)
    if (t_m <= 5d-274) then
        tmp = (t_m * (sqrt(2.0d0) / l_m)) * sqrt(((0.5d0 * x) - 0.5d0))
    else if (t_m <= 2.55d-129) then
        tmp = t_2 / (t_2 + ((l_m ** 2.0d0) / (t_m * (sqrt(2.0d0) * x))))
    else if (t_m <= 3.1d+34) then
        tmp = t_2 / sqrt(((((2.0d0 * (t_m ** 2.0d0)) + (l_m ** 2.0d0)) / x) + (((l_m ** 2.0d0) / x) + ((t_m ** 2.0d0) * (2.0d0 + (2.0d0 * (1.0d0 / x)))))))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * Math.sqrt(2.0);
	double tmp;
	if (t_m <= 5e-274) {
		tmp = (t_m * (Math.sqrt(2.0) / l_m)) * Math.sqrt(((0.5 * x) - 0.5));
	} else if (t_m <= 2.55e-129) {
		tmp = t_2 / (t_2 + (Math.pow(l_m, 2.0) / (t_m * (Math.sqrt(2.0) * x))));
	} else if (t_m <= 3.1e+34) {
		tmp = t_2 / Math.sqrt(((((2.0 * Math.pow(t_m, 2.0)) + Math.pow(l_m, 2.0)) / x) + ((Math.pow(l_m, 2.0) / x) + (Math.pow(t_m, 2.0) * (2.0 + (2.0 * (1.0 / x)))))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = t_m * math.sqrt(2.0)
	tmp = 0
	if t_m <= 5e-274:
		tmp = (t_m * (math.sqrt(2.0) / l_m)) * math.sqrt(((0.5 * x) - 0.5))
	elif t_m <= 2.55e-129:
		tmp = t_2 / (t_2 + (math.pow(l_m, 2.0) / (t_m * (math.sqrt(2.0) * x))))
	elif t_m <= 3.1e+34:
		tmp = t_2 / math.sqrt(((((2.0 * math.pow(t_m, 2.0)) + math.pow(l_m, 2.0)) / x) + ((math.pow(l_m, 2.0) / x) + (math.pow(t_m, 2.0) * (2.0 + (2.0 * (1.0 / x)))))))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 5e-274)
		tmp = Float64(Float64(t_m * Float64(sqrt(2.0) / l_m)) * sqrt(Float64(Float64(0.5 * x) - 0.5)));
	elseif (t_m <= 2.55e-129)
		tmp = Float64(t_2 / Float64(t_2 + Float64((l_m ^ 2.0) / Float64(t_m * Float64(sqrt(2.0) * x)))));
	elseif (t_m <= 3.1e+34)
		tmp = Float64(t_2 / sqrt(Float64(Float64(Float64(Float64(2.0 * (t_m ^ 2.0)) + (l_m ^ 2.0)) / x) + Float64(Float64((l_m ^ 2.0) / x) + Float64((t_m ^ 2.0) * Float64(2.0 + Float64(2.0 * Float64(1.0 / x))))))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = t_m * sqrt(2.0);
	tmp = 0.0;
	if (t_m <= 5e-274)
		tmp = (t_m * (sqrt(2.0) / l_m)) * sqrt(((0.5 * x) - 0.5));
	elseif (t_m <= 2.55e-129)
		tmp = t_2 / (t_2 + ((l_m ^ 2.0) / (t_m * (sqrt(2.0) * x))));
	elseif (t_m <= 3.1e+34)
		tmp = t_2 / sqrt(((((2.0 * (t_m ^ 2.0)) + (l_m ^ 2.0)) / x) + (((l_m ^ 2.0) / x) + ((t_m ^ 2.0) * (2.0 + (2.0 * (1.0 / x)))))));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 5e-274], N[(N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(0.5 * x), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.55e-129], N[(t$95$2 / N[(t$95$2 + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.1e+34], N[(t$95$2 / N[Sqrt[N[(N[(N[(N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision] + N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(2.0 + N[(2.0 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 5e-274

   1. Initial program 30.5%

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

   3. Simplified 29.1%

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

   5. Taylor expanded in l around inf 3.4%

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

      1. associate-/l* 3.4%

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

      2. associate--l+ 12.4%

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

      3. sub-neg 12.4%

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

      4. metadata-eval 12.4%

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

      5. +-commutative 12.4%

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

      6. sub-neg 12.4%

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

      7. metadata-eval 12.4%

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

      8. +-commutative 12.4%

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

   7. Simplified 12.4%

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

   8. Taylor expanded in x around 0 20.0%

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

   if 5e-274 < t < 2.5499999999999999e-129

   1. Initial program 17.2%

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

   3. Taylor expanded in x around inf 81.7%

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

   4. Taylor expanded in l around inf 81.7%

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

   if 2.5499999999999999e-129 < t < 3.09999999999999977e34

   1. Initial program 54.7%

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

   3. Taylor expanded in x around inf 83.0%

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

   4. Taylor expanded in t around 0 83.1%

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

   if 3.09999999999999977e34 < t

   1. Initial program 31.7%

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

   3. Simplified 31.8%

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

   5. Taylor expanded in t around inf 93.1%

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

   6. Taylor expanded in t around 0 93.3%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-274}:\\ \;\;\;\;\left(t \cdot \frac{\sqrt{2}}{\ell}\right) \cdot \sqrt{0.5 \cdot x - 0.5}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-129}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + \frac{{\ell}^{2}}{t \cdot \left(\sqrt{2} \cdot x\right)}}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+34}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{2 \cdot {t}^{2} + {\ell}^{2}}{x} + \left(\frac{{\ell}^{2}}{x} + {t}^{2} \cdot \left(2 + 2 \cdot \frac{1}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.5 \cdot 10^{-274}:\\ \;\;\;\;\left(t\_m \cdot \frac{\sqrt{2}}{l\_m}\right) \cdot \sqrt{0.5 \cdot x - 0.5}\\ \mathbf{elif}\;t\_m \leq 6.1 \cdot 10^{-44}:\\ \;\;\;\;\frac{t\_2}{t\_2 + \frac{{l\_m}^{2}}{t\_m \cdot \left(\sqrt{2} \cdot x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 6.5e-274)
      (* (* t_m (/ (sqrt 2.0) l_m)) (sqrt (- (* 0.5 x) 0.5)))
      (if (<= t_m 6.1e-44)
        (/ t_2 (+ t_2 (/ (pow l_m 2.0) (* t_m (* (sqrt 2.0) x)))))
        (sqrt (/ (+ x -1.0) (+ x 1.0))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 6.5e-274) {
		tmp = (t_m * (sqrt(2.0) / l_m)) * sqrt(((0.5 * x) - 0.5));
	} else if (t_m <= 6.1e-44) {
		tmp = t_2 / (t_2 + (pow(l_m, 2.0) / (t_m * (sqrt(2.0) * x))));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = t_m * sqrt(2.0d0)
    if (t_m <= 6.5d-274) then
        tmp = (t_m * (sqrt(2.0d0) / l_m)) * sqrt(((0.5d0 * x) - 0.5d0))
    else if (t_m <= 6.1d-44) then
        tmp = t_2 / (t_2 + ((l_m ** 2.0d0) / (t_m * (sqrt(2.0d0) * x))))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * Math.sqrt(2.0);
	double tmp;
	if (t_m <= 6.5e-274) {
		tmp = (t_m * (Math.sqrt(2.0) / l_m)) * Math.sqrt(((0.5 * x) - 0.5));
	} else if (t_m <= 6.1e-44) {
		tmp = t_2 / (t_2 + (Math.pow(l_m, 2.0) / (t_m * (Math.sqrt(2.0) * x))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = t_m * math.sqrt(2.0)
	tmp = 0
	if t_m <= 6.5e-274:
		tmp = (t_m * (math.sqrt(2.0) / l_m)) * math.sqrt(((0.5 * x) - 0.5))
	elif t_m <= 6.1e-44:
		tmp = t_2 / (t_2 + (math.pow(l_m, 2.0) / (t_m * (math.sqrt(2.0) * x))))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 6.5e-274)
		tmp = Float64(Float64(t_m * Float64(sqrt(2.0) / l_m)) * sqrt(Float64(Float64(0.5 * x) - 0.5)));
	elseif (t_m <= 6.1e-44)
		tmp = Float64(t_2 / Float64(t_2 + Float64((l_m ^ 2.0) / Float64(t_m * Float64(sqrt(2.0) * x)))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = t_m * sqrt(2.0);
	tmp = 0.0;
	if (t_m <= 6.5e-274)
		tmp = (t_m * (sqrt(2.0) / l_m)) * sqrt(((0.5 * x) - 0.5));
	elseif (t_m <= 6.1e-44)
		tmp = t_2 / (t_2 + ((l_m ^ 2.0) / (t_m * (sqrt(2.0) * x))));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 6.5e-274], N[(N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(0.5 * x), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.1e-44], N[(t$95$2 / N[(t$95$2 + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 6.49999999999999959e-274

   1. Initial program 30.5%

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

   3. Simplified 29.1%

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

   5. Taylor expanded in l around inf 3.4%

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

      1. associate-/l* 3.4%

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

      2. associate--l+ 12.4%

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

      3. sub-neg 12.4%

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

      4. metadata-eval 12.4%

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

      5. +-commutative 12.4%

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

      6. sub-neg 12.4%

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

      7. metadata-eval 12.4%

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

      8. +-commutative 12.4%

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

   7. Simplified 12.4%

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

   8. Taylor expanded in x around 0 20.0%

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

   if 6.49999999999999959e-274 < t < 6.0999999999999996e-44

   1. Initial program 29.4%

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

   3. Taylor expanded in x around inf 66.0%

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

   4. Taylor expanded in l around inf 65.6%

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

   if 6.0999999999999996e-44 < t

   1. Initial program 41.1%

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

   3. Simplified 38.6%

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

   5. Taylor expanded in t around inf 88.5%

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

   6. Taylor expanded in t around 0 88.7%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.5 \cdot 10^{-274}:\\ \;\;\;\;\left(t \cdot \frac{\sqrt{2}}{\ell}\right) \cdot \sqrt{0.5 \cdot x - 0.5}\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{-44}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + \frac{{\ell}^{2}}{t \cdot \left(\sqrt{2} \cdot x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.06 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{\left(\sqrt{2} \cdot l\_m\right) \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 1.06e+127)
    (sqrt (/ (+ x -1.0) (+ x 1.0)))
    (/ (* t_m (sqrt 2.0)) (* (* (sqrt 2.0) l_m) (sqrt (/ 1.0 x)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 1.06e+127) {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = (t_m * sqrt(2.0)) / ((sqrt(2.0) * l_m) * sqrt((1.0 / x)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l_m <= 1.06d+127) then
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else
        tmp = (t_m * sqrt(2.0d0)) / ((sqrt(2.0d0) * l_m) * sqrt((1.0d0 / x)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 1.06e+127) {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = (t_m * Math.sqrt(2.0)) / ((Math.sqrt(2.0) * l_m) * Math.sqrt((1.0 / x)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if l_m <= 1.06e+127:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	else:
		tmp = (t_m * math.sqrt(2.0)) / ((math.sqrt(2.0) * l_m) * math.sqrt((1.0 / x)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (l_m <= 1.06e+127)
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	else
		tmp = Float64(Float64(t_m * sqrt(2.0)) / Float64(Float64(sqrt(2.0) * l_m) * sqrt(Float64(1.0 / x))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (l_m <= 1.06e+127)
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	else
		tmp = (t_m * sqrt(2.0)) / ((sqrt(2.0) * l_m) * sqrt((1.0 / x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 1.06e+127], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.06000000000000006e127

   1. Initial program 37.7%

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

   3. Simplified 33.5%

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

   5. Taylor expanded in t around inf 38.2%

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

   6. Taylor expanded in t around 0 38.3%

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

   if 1.06000000000000006e127 < l

   1. Initial program 0.4%

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

   3. Taylor expanded in x around inf 30.3%

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

   4. Taylor expanded in l around inf 54.4%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.06 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.22 \cdot 10^{+126}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \left(\frac{\sqrt{2}}{l\_m} \cdot \sqrt{0.5 \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 1.22e+126)
    (sqrt (/ (+ x -1.0) (+ x 1.0)))
    (* t_m (* (/ (sqrt 2.0) l_m) (sqrt (* 0.5 x)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 1.22e+126) {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = t_m * ((sqrt(2.0) / l_m) * sqrt((0.5 * x)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l_m <= 1.22d+126) then
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else
        tmp = t_m * ((sqrt(2.0d0) / l_m) * sqrt((0.5d0 * x)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 1.22e+126) {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = t_m * ((Math.sqrt(2.0) / l_m) * Math.sqrt((0.5 * x)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if l_m <= 1.22e+126:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	else:
		tmp = t_m * ((math.sqrt(2.0) / l_m) * math.sqrt((0.5 * x)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (l_m <= 1.22e+126)
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	else
		tmp = Float64(t_m * Float64(Float64(sqrt(2.0) / l_m) * sqrt(Float64(0.5 * x))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (l_m <= 1.22e+126)
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	else
		tmp = t_m * ((sqrt(2.0) / l_m) * sqrt((0.5 * x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 1.22e+126], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$m * N[(N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * N[Sqrt[N[(0.5 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.21999999999999995e126

   1. Initial program 37.7%

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

   3. Simplified 33.5%

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

   5. Taylor expanded in t around inf 38.2%

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

   6. Taylor expanded in t around 0 38.3%

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

   if 1.21999999999999995e126 < l

   1. Initial program 0.4%

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

   3. Simplified 0.4%

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

   5. Taylor expanded in l around inf 5.4%

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

      1. associate-/l* 5.4%

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

      2. associate--l+ 29.4%

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

      3. sub-neg 29.4%

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

      4. metadata-eval 29.4%

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

      5. +-commutative 29.4%

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

      6. sub-neg 29.4%

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

      7. metadata-eval 29.4%

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

      8. +-commutative 29.4%

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

   7. Simplified 29.4%

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

   8. Taylor expanded in x around inf 51.6%

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

      1. *-commutative 51.6%

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

   10. Simplified 51.6%

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

       1. pow1 51.6%

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

   12. Applied egg-rr 51.6%

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

       1. unpow1 51.6%

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

       2. associate-*l* 54.3%

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

   14. Simplified 54.3%

       \[\leadsto \color{blue}{t \cdot \left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{x \cdot 0.5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.22 \cdot 10^{+126}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{0.5 \cdot x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.2% accurate, 2.1× speedup

Math

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

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 (sqrt (/ (+ x -1.0) (+ x 1.0)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * sqrt(((x + -1.0) / (x + 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 * sqrt(((x + (-1.0d0)) / (x + 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 * Math.sqrt(((x + -1.0) / (x + 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 * math.sqrt(((x + -1.0) / (x + 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 * sqrt(Float64(Float64(x + -1.0) / Float64(x + 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 * sqrt(((x + -1.0) / (x + 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 * N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.4%

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

3. Simplified 29.8%

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

5. Taylor expanded in t around inf 36.5%

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

6. Taylor expanded in t around 0 36.6%

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

7. Final simplification 36.6%

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

Alternative 10: 76.8% accurate, 25.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.4%

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

3. Simplified 29.8%

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

5. Taylor expanded in t around inf 36.5%

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

6. Taylor expanded in t around 0 36.6%

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

   1. div-inv 36.5%

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

   2. sub-neg 36.5%

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

   3. metadata-eval 36.5%

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

   4. +-commutative 36.5%

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

8. Applied egg-rr 36.5%

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

9. Taylor expanded in x around inf 36.6%

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

    1. associate--l+ 36.6%

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

    2. metadata-eval 36.6%

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

    3. metadata-eval 36.6%

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

    4. metadata-eval 36.6%

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

    5. rem-square-sqrt 0.0%

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

    6. unpow2 0.0%

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

    7. unpow2 0.0%

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

    8. associate-/l/ 0.0%

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

    9. associate-*r/ 0.0%

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

    10. div-sub 0.0%

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

11. Simplified 36.6%

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

12. Final simplification 36.6%

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

Alternative 11: 76.6% accurate, 45.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.4%

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

3. Simplified 29.8%

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

5. Taylor expanded in t around inf 36.5%

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

6. Taylor expanded in x around inf 36.6%

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

Alternative 12: 75.9% accurate, 225.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.4%

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

3. Simplified 29.8%

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

5. Taylor expanded in t around inf 36.5%

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

6. Taylor expanded in x around inf 36.5%

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

Reproduce

herbie shell --seed 2024185 
(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)))))