Toniolo and Linder, Equation (7)

Percentage Accurate: 33.9% → 85.2%

Time: 23.1s

Alternatives: 11

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t_m}^{2}\\ t_3 := t_2 + {l_m}^{2}\\ t_4 := t_3 + t_3\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 9.8 \cdot 10^{-254}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t_m}{l_m \cdot \sqrt{\frac{1}{-1 + x} + \frac{1}{x}}}\\ \mathbf{elif}\;t_m \leq 4 \cdot 10^{-169}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{0.5 \cdot \frac{t_4}{t_m \cdot \left(\sqrt{2} \cdot x\right)} + t_m \cdot \sqrt{2}}\\ \mathbf{elif}\;t_m \leq 6.4 \cdot 10^{+84}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{\sqrt{\left(\frac{t_4}{{x}^{2}} + \left(2 \cdot \frac{{t_m}^{2}}{x} + \left(t_2 + \frac{{l_m}^{2}}{x}\right)\right)\right) + \frac{t_3}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow t_m 2.0)))
        (t_3 (+ t_2 (pow l_m 2.0)))
        (t_4 (+ t_3 t_3)))
   (*
    t_s
    (if (<= t_m 9.8e-254)
      (* (sqrt 2.0) (/ t_m (* l_m (sqrt (+ (/ 1.0 (+ -1.0 x)) (/ 1.0 x))))))
      (if (<= t_m 4e-169)
        (*
         t_m
         (/
          (sqrt 2.0)
          (+ (* 0.5 (/ t_4 (* t_m (* (sqrt 2.0) x)))) (* t_m (sqrt 2.0)))))
        (if (<= t_m 6.4e+84)
          (*
           t_m
           (/
            (sqrt 2.0)
            (sqrt
             (+
              (+
               (/ t_4 (pow x 2.0))
               (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l_m 2.0) x))))
              (/ t_3 x)))))
          (sqrt (/ (+ -1.0 x) (+ 1.0 x)))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double t_3 = t_2 + pow(l_m, 2.0);
	double t_4 = t_3 + t_3;
	double tmp;
	if (t_m <= 9.8e-254) {
		tmp = sqrt(2.0) * (t_m / (l_m * sqrt(((1.0 / (-1.0 + x)) + (1.0 / x)))));
	} else if (t_m <= 4e-169) {
		tmp = t_m * (sqrt(2.0) / ((0.5 * (t_4 / (t_m * (sqrt(2.0) * x)))) + (t_m * sqrt(2.0))));
	} else if (t_m <= 6.4e+84) {
		tmp = t_m * (sqrt(2.0) / sqrt((((t_4 / pow(x, 2.0)) + ((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l_m, 2.0) / x)))) + (t_3 / x))));
	} else {
		tmp = sqrt(((-1.0 + x) / (1.0 + x)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    t_3 = t_2 + (l_m ** 2.0d0)
    t_4 = t_3 + t_3
    if (t_m <= 9.8d-254) then
        tmp = sqrt(2.0d0) * (t_m / (l_m * sqrt(((1.0d0 / ((-1.0d0) + x)) + (1.0d0 / x)))))
    else if (t_m <= 4d-169) then
        tmp = t_m * (sqrt(2.0d0) / ((0.5d0 * (t_4 / (t_m * (sqrt(2.0d0) * x)))) + (t_m * sqrt(2.0d0))))
    else if (t_m <= 6.4d+84) then
        tmp = t_m * (sqrt(2.0d0) / sqrt((((t_4 / (x ** 2.0d0)) + ((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_2 + ((l_m ** 2.0d0) / x)))) + (t_3 / x))))
    else
        tmp = sqrt((((-1.0d0) + x) / (1.0d0 + x)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * Math.pow(t_m, 2.0);
	double t_3 = t_2 + Math.pow(l_m, 2.0);
	double t_4 = t_3 + t_3;
	double tmp;
	if (t_m <= 9.8e-254) {
		tmp = Math.sqrt(2.0) * (t_m / (l_m * Math.sqrt(((1.0 / (-1.0 + x)) + (1.0 / x)))));
	} else if (t_m <= 4e-169) {
		tmp = t_m * (Math.sqrt(2.0) / ((0.5 * (t_4 / (t_m * (Math.sqrt(2.0) * x)))) + (t_m * Math.sqrt(2.0))));
	} else if (t_m <= 6.4e+84) {
		tmp = t_m * (Math.sqrt(2.0) / Math.sqrt((((t_4 / Math.pow(x, 2.0)) + ((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_2 + (Math.pow(l_m, 2.0) / x)))) + (t_3 / x))));
	} else {
		tmp = Math.sqrt(((-1.0 + x) / (1.0 + x)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = 2.0 * math.pow(t_m, 2.0)
	t_3 = t_2 + math.pow(l_m, 2.0)
	t_4 = t_3 + t_3
	tmp = 0
	if t_m <= 9.8e-254:
		tmp = math.sqrt(2.0) * (t_m / (l_m * math.sqrt(((1.0 / (-1.0 + x)) + (1.0 / x)))))
	elif t_m <= 4e-169:
		tmp = t_m * (math.sqrt(2.0) / ((0.5 * (t_4 / (t_m * (math.sqrt(2.0) * x)))) + (t_m * math.sqrt(2.0))))
	elif t_m <= 6.4e+84:
		tmp = t_m * (math.sqrt(2.0) / math.sqrt((((t_4 / math.pow(x, 2.0)) + ((2.0 * (math.pow(t_m, 2.0) / x)) + (t_2 + (math.pow(l_m, 2.0) / x)))) + (t_3 / x))))
	else:
		tmp = math.sqrt(((-1.0 + x) / (1.0 + x)))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(2.0 * (t_m ^ 2.0))
	t_3 = Float64(t_2 + (l_m ^ 2.0))
	t_4 = Float64(t_3 + t_3)
	tmp = 0.0
	if (t_m <= 9.8e-254)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(l_m * sqrt(Float64(Float64(1.0 / Float64(-1.0 + x)) + Float64(1.0 / x))))));
	elseif (t_m <= 4e-169)
		tmp = Float64(t_m * Float64(sqrt(2.0) / Float64(Float64(0.5 * Float64(t_4 / Float64(t_m * Float64(sqrt(2.0) * x)))) + Float64(t_m * sqrt(2.0)))));
	elseif (t_m <= 6.4e+84)
		tmp = Float64(t_m * Float64(sqrt(2.0) / sqrt(Float64(Float64(Float64(t_4 / (x ^ 2.0)) + Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l_m ^ 2.0) / x)))) + Float64(t_3 / x)))));
	else
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = 2.0 * (t_m ^ 2.0);
	t_3 = t_2 + (l_m ^ 2.0);
	t_4 = t_3 + t_3;
	tmp = 0.0;
	if (t_m <= 9.8e-254)
		tmp = sqrt(2.0) * (t_m / (l_m * sqrt(((1.0 / (-1.0 + x)) + (1.0 / x)))));
	elseif (t_m <= 4e-169)
		tmp = t_m * (sqrt(2.0) / ((0.5 * (t_4 / (t_m * (sqrt(2.0) * x)))) + (t_m * sqrt(2.0))));
	elseif (t_m <= 6.4e+84)
		tmp = t_m * (sqrt(2.0) / sqrt((((t_4 / (x ^ 2.0)) + ((2.0 * ((t_m ^ 2.0) / x)) + (t_2 + ((l_m ^ 2.0) / x)))) + (t_3 / x))));
	else
		tmp = sqrt(((-1.0 + x) / (1.0 + x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 + t$95$3), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 9.8e-254], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(l$95$m * N[Sqrt[N[(N[(1.0 / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4e-169], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(0.5 * N[(t$95$4 / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.4e+84], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(N[(t$95$4 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < 9.79999999999999959e-254

   1. Initial program 28.2%

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

   3. Simplified 28.1%

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

   4. Taylor expanded in l around inf 2.7%

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

      1. associate--l+ 9.8%

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

      2. sub-neg 9.8%

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

      3. metadata-eval 9.8%

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

      4. +-commutative 9.8%

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

      5. sub-neg 9.8%

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

      6. metadata-eval 9.8%

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

      7. +-commutative 9.8%

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

   6. Simplified 9.8%

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

   7. Taylor expanded in x around inf 20.9%

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

   if 9.79999999999999959e-254 < t < 4.00000000000000008e-169

   1. Initial program 2.6%

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

   3. Simplified 2.6%

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

   4. Taylor expanded in x around inf 67.3%

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

   if 4.00000000000000008e-169 < t < 6.4000000000000002e84

   1. Initial program 45.6%

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

   3. Simplified 45.8%

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

   4. Taylor expanded in x around -inf 78.4%

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

   if 6.4000000000000002e84 < t

   1. Initial program 23.3%

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

   3. Simplified 23.3%

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

   4. Taylor expanded in l around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

   6. Simplified 100.0%

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

      1. sqrt-unprod 100.0%

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

      2. sqrt-undiv 100.0%

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

      3. +-commutative 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. associate-/r* 100.0%

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

      2. metadata-eval 100.0%

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

      3. +-commutative 100.0%

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

   10. Simplified 100.0%

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

       1. expm1-log1p-u 100.0%

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

       2. expm1-udef 100.0%

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

       3. associate-/r/ 100.0%

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

       4. +-commutative 100.0%

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

       5. +-commutative 100.0%

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

   12. Applied egg-rr 100.0%

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

       1. expm1-def 100.0%

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

       2. expm1-log1p 99.5%

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

       3. metadata-eval 99.5%

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

       4. sub-neg 99.5%

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

       5. associate-*l/ 100.0%

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

       6. *-lft-identity 100.0%

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

       7. sub-neg 100.0%

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

       8. metadata-eval 100.0%

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

       9. +-commutative 100.0%

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

       10. +-commutative 100.0%

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

   14. Simplified 100.0%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.8 \cdot 10^{-254}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{1}{-1 + x} + \frac{1}{x}}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-169}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(\sqrt{2} \cdot x\right)} + t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{+84}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\left(\frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{{x}^{2}} + \left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \end{array} \]

Alternative 2: 83.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t_m}^{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 1.6 \cdot 10^{-247}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t_m}{l_m \cdot \sqrt{\frac{1}{-1 + x} + \frac{1}{x}}}\\ \mathbf{elif}\;t_m \leq 2.35 \cdot 10^{-166}:\\ \;\;\;\;1\\ \mathbf{elif}\;t_m \leq 3 \cdot 10^{+84}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{\sqrt{\left(2 \cdot \frac{{t_m}^{2}}{x} + \left(t_2 + \frac{{l_m}^{2}}{x}\right)\right) + \frac{t_2 + {l_m}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow t_m 2.0))))
   (*
    t_s
    (if (<= t_m 1.6e-247)
      (* (sqrt 2.0) (/ t_m (* l_m (sqrt (+ (/ 1.0 (+ -1.0 x)) (/ 1.0 x))))))
      (if (<= t_m 2.35e-166)
        1.0
        (if (<= t_m 3e+84)
          (*
           t_m
           (/
            (sqrt 2.0)
            (sqrt
             (+
              (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l_m 2.0) x)))
              (/ (+ t_2 (pow l_m 2.0)) x)))))
          (sqrt (/ (+ -1.0 x) (+ 1.0 x)))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double tmp;
	if (t_m <= 1.6e-247) {
		tmp = sqrt(2.0) * (t_m / (l_m * sqrt(((1.0 / (-1.0 + x)) + (1.0 / x)))));
	} else if (t_m <= 2.35e-166) {
		tmp = 1.0;
	} else if (t_m <= 3e+84) {
		tmp = t_m * (sqrt(2.0) / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l_m, 2.0) / x))) + ((t_2 + pow(l_m, 2.0)) / x))));
	} else {
		tmp = sqrt(((-1.0 + x) / (1.0 + x)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    if (t_m <= 1.6d-247) then
        tmp = sqrt(2.0d0) * (t_m / (l_m * sqrt(((1.0d0 / ((-1.0d0) + x)) + (1.0d0 / x)))))
    else if (t_m <= 2.35d-166) then
        tmp = 1.0d0
    else if (t_m <= 3d+84) then
        tmp = t_m * (sqrt(2.0d0) / sqrt((((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_2 + ((l_m ** 2.0d0) / x))) + ((t_2 + (l_m ** 2.0d0)) / x))))
    else
        tmp = sqrt((((-1.0d0) + x) / (1.0d0 + x)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * Math.pow(t_m, 2.0);
	double tmp;
	if (t_m <= 1.6e-247) {
		tmp = Math.sqrt(2.0) * (t_m / (l_m * Math.sqrt(((1.0 / (-1.0 + x)) + (1.0 / x)))));
	} else if (t_m <= 2.35e-166) {
		tmp = 1.0;
	} else if (t_m <= 3e+84) {
		tmp = t_m * (Math.sqrt(2.0) / Math.sqrt((((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_2 + (Math.pow(l_m, 2.0) / x))) + ((t_2 + Math.pow(l_m, 2.0)) / x))));
	} else {
		tmp = Math.sqrt(((-1.0 + x) / (1.0 + x)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = 2.0 * math.pow(t_m, 2.0)
	tmp = 0
	if t_m <= 1.6e-247:
		tmp = math.sqrt(2.0) * (t_m / (l_m * math.sqrt(((1.0 / (-1.0 + x)) + (1.0 / x)))))
	elif t_m <= 2.35e-166:
		tmp = 1.0
	elif t_m <= 3e+84:
		tmp = t_m * (math.sqrt(2.0) / math.sqrt((((2.0 * (math.pow(t_m, 2.0) / x)) + (t_2 + (math.pow(l_m, 2.0) / x))) + ((t_2 + math.pow(l_m, 2.0)) / x))))
	else:
		tmp = math.sqrt(((-1.0 + x) / (1.0 + x)))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(2.0 * (t_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 1.6e-247)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(l_m * sqrt(Float64(Float64(1.0 / Float64(-1.0 + x)) + Float64(1.0 / x))))));
	elseif (t_m <= 2.35e-166)
		tmp = 1.0;
	elseif (t_m <= 3e+84)
		tmp = Float64(t_m * Float64(sqrt(2.0) / sqrt(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l_m ^ 2.0) / x))) + Float64(Float64(t_2 + (l_m ^ 2.0)) / x)))));
	else
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = 2.0 * (t_m ^ 2.0);
	tmp = 0.0;
	if (t_m <= 1.6e-247)
		tmp = sqrt(2.0) * (t_m / (l_m * sqrt(((1.0 / (-1.0 + x)) + (1.0 / x)))));
	elseif (t_m <= 2.35e-166)
		tmp = 1.0;
	elseif (t_m <= 3e+84)
		tmp = t_m * (sqrt(2.0) / sqrt((((2.0 * ((t_m ^ 2.0) / x)) + (t_2 + ((l_m ^ 2.0) / x))) + ((t_2 + (l_m ^ 2.0)) / x))));
	else
		tmp = sqrt(((-1.0 + x) / (1.0 + x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.6e-247], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(l$95$m * N[Sqrt[N[(N[(1.0 / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.35e-166], 1.0, If[LessEqual[t$95$m, 3e+84], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 1.59999999999999997e-247

   1. Initial program 27.8%

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

   3. Simplified 27.7%

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

   4. Taylor expanded in l around inf 2.6%

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

      1. associate--l+ 10.4%

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

      2. sub-neg 10.4%

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

      3. metadata-eval 10.4%

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

      4. +-commutative 10.4%

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

      5. sub-neg 10.4%

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

      6. metadata-eval 10.4%

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

      7. +-commutative 10.4%

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

   6. Simplified 10.4%

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

   7. Taylor expanded in x around inf 21.4%

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

   if 1.59999999999999997e-247 < t < 2.35000000000000007e-166

   1. Initial program 2.8%

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

   3. Simplified 2.8%

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

   4. Taylor expanded in l around 0 59.9%

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

      1. +-commutative 59.9%

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

      2. sub-neg 59.9%

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

      3. metadata-eval 59.9%

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

      4. +-commutative 59.9%

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

   6. Simplified 59.9%

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

   7. Taylor expanded in x around inf 59.9%

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

   if 2.35000000000000007e-166 < t < 2.99999999999999996e84

   1. Initial program 45.6%

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

   3. Simplified 45.8%

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

   4. Taylor expanded in x around inf 78.7%

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

   if 2.99999999999999996e84 < t

   1. Initial program 23.3%

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

   3. Simplified 23.3%

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

   4. Taylor expanded in l around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

   6. Simplified 100.0%

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

      1. sqrt-unprod 100.0%

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

      2. sqrt-undiv 100.0%

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

      3. +-commutative 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. associate-/r* 100.0%

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

      2. metadata-eval 100.0%

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

      3. +-commutative 100.0%

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

   10. Simplified 100.0%

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

       1. expm1-log1p-u 100.0%

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

       2. expm1-udef 100.0%

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

       3. associate-/r/ 100.0%

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

       4. +-commutative 100.0%

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

       5. +-commutative 100.0%

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

   12. Applied egg-rr 100.0%

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

       1. expm1-def 100.0%

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

       2. expm1-log1p 99.5%

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

       3. metadata-eval 99.5%

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

       4. sub-neg 99.5%

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

       5. associate-*l/ 100.0%

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

       6. *-lft-identity 100.0%

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

       7. sub-neg 100.0%

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

       8. metadata-eval 100.0%

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

       9. +-commutative 100.0%

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

       10. +-commutative 100.0%

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

   14. Simplified 100.0%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{-247}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{1}{-1 + x} + \frac{1}{x}}}\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{-166}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+84}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \end{array} \]

Alternative 3: 85.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t_m}^{2}\\ t_3 := t_2 + {l_m}^{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 7.1 \cdot 10^{-258}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t_m}{l_m \cdot \sqrt{\frac{1}{-1 + x} + \frac{1}{x}}}\\ \mathbf{elif}\;t_m \leq 1.08 \cdot 10^{-165}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{0.5 \cdot \frac{t_3 + t_3}{t_m \cdot \left(\sqrt{2} \cdot x\right)} + t_m \cdot \sqrt{2}}\\ \mathbf{elif}\;t_m \leq 4.1 \cdot 10^{+84}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{\sqrt{\left(2 \cdot \frac{{t_m}^{2}}{x} + \left(t_2 + \frac{{l_m}^{2}}{x}\right)\right) + \frac{t_3}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow t_m 2.0))) (t_3 (+ t_2 (pow l_m 2.0))))
   (*
    t_s
    (if (<= t_m 7.1e-258)
      (* (sqrt 2.0) (/ t_m (* l_m (sqrt (+ (/ 1.0 (+ -1.0 x)) (/ 1.0 x))))))
      (if (<= t_m 1.08e-165)
        (*
         t_m
         (/
          (sqrt 2.0)
          (+
           (* 0.5 (/ (+ t_3 t_3) (* t_m (* (sqrt 2.0) x))))
           (* t_m (sqrt 2.0)))))
        (if (<= t_m 4.1e+84)
          (*
           t_m
           (/
            (sqrt 2.0)
            (sqrt
             (+
              (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l_m 2.0) x)))
              (/ t_3 x)))))
          (sqrt (/ (+ -1.0 x) (+ 1.0 x)))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double t_3 = t_2 + pow(l_m, 2.0);
	double tmp;
	if (t_m <= 7.1e-258) {
		tmp = sqrt(2.0) * (t_m / (l_m * sqrt(((1.0 / (-1.0 + x)) + (1.0 / x)))));
	} else if (t_m <= 1.08e-165) {
		tmp = t_m * (sqrt(2.0) / ((0.5 * ((t_3 + t_3) / (t_m * (sqrt(2.0) * x)))) + (t_m * sqrt(2.0))));
	} else if (t_m <= 4.1e+84) {
		tmp = t_m * (sqrt(2.0) / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l_m, 2.0) / x))) + (t_3 / x))));
	} else {
		tmp = sqrt(((-1.0 + x) / (1.0 + x)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    t_3 = t_2 + (l_m ** 2.0d0)
    if (t_m <= 7.1d-258) then
        tmp = sqrt(2.0d0) * (t_m / (l_m * sqrt(((1.0d0 / ((-1.0d0) + x)) + (1.0d0 / x)))))
    else if (t_m <= 1.08d-165) then
        tmp = t_m * (sqrt(2.0d0) / ((0.5d0 * ((t_3 + t_3) / (t_m * (sqrt(2.0d0) * x)))) + (t_m * sqrt(2.0d0))))
    else if (t_m <= 4.1d+84) then
        tmp = t_m * (sqrt(2.0d0) / sqrt((((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_2 + ((l_m ** 2.0d0) / x))) + (t_3 / x))))
    else
        tmp = sqrt((((-1.0d0) + x) / (1.0d0 + x)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * Math.pow(t_m, 2.0);
	double t_3 = t_2 + Math.pow(l_m, 2.0);
	double tmp;
	if (t_m <= 7.1e-258) {
		tmp = Math.sqrt(2.0) * (t_m / (l_m * Math.sqrt(((1.0 / (-1.0 + x)) + (1.0 / x)))));
	} else if (t_m <= 1.08e-165) {
		tmp = t_m * (Math.sqrt(2.0) / ((0.5 * ((t_3 + t_3) / (t_m * (Math.sqrt(2.0) * x)))) + (t_m * Math.sqrt(2.0))));
	} else if (t_m <= 4.1e+84) {
		tmp = t_m * (Math.sqrt(2.0) / Math.sqrt((((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_2 + (Math.pow(l_m, 2.0) / x))) + (t_3 / x))));
	} else {
		tmp = Math.sqrt(((-1.0 + x) / (1.0 + x)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = 2.0 * math.pow(t_m, 2.0)
	t_3 = t_2 + math.pow(l_m, 2.0)
	tmp = 0
	if t_m <= 7.1e-258:
		tmp = math.sqrt(2.0) * (t_m / (l_m * math.sqrt(((1.0 / (-1.0 + x)) + (1.0 / x)))))
	elif t_m <= 1.08e-165:
		tmp = t_m * (math.sqrt(2.0) / ((0.5 * ((t_3 + t_3) / (t_m * (math.sqrt(2.0) * x)))) + (t_m * math.sqrt(2.0))))
	elif t_m <= 4.1e+84:
		tmp = t_m * (math.sqrt(2.0) / math.sqrt((((2.0 * (math.pow(t_m, 2.0) / x)) + (t_2 + (math.pow(l_m, 2.0) / x))) + (t_3 / x))))
	else:
		tmp = math.sqrt(((-1.0 + x) / (1.0 + x)))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(2.0 * (t_m ^ 2.0))
	t_3 = Float64(t_2 + (l_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 7.1e-258)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(l_m * sqrt(Float64(Float64(1.0 / Float64(-1.0 + x)) + Float64(1.0 / x))))));
	elseif (t_m <= 1.08e-165)
		tmp = Float64(t_m * Float64(sqrt(2.0) / Float64(Float64(0.5 * Float64(Float64(t_3 + t_3) / Float64(t_m * Float64(sqrt(2.0) * x)))) + Float64(t_m * sqrt(2.0)))));
	elseif (t_m <= 4.1e+84)
		tmp = Float64(t_m * Float64(sqrt(2.0) / sqrt(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l_m ^ 2.0) / x))) + Float64(t_3 / x)))));
	else
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = 2.0 * (t_m ^ 2.0);
	t_3 = t_2 + (l_m ^ 2.0);
	tmp = 0.0;
	if (t_m <= 7.1e-258)
		tmp = sqrt(2.0) * (t_m / (l_m * sqrt(((1.0 / (-1.0 + x)) + (1.0 / x)))));
	elseif (t_m <= 1.08e-165)
		tmp = t_m * (sqrt(2.0) / ((0.5 * ((t_3 + t_3) / (t_m * (sqrt(2.0) * x)))) + (t_m * sqrt(2.0))));
	elseif (t_m <= 4.1e+84)
		tmp = t_m * (sqrt(2.0) / sqrt((((2.0 * ((t_m ^ 2.0) / x)) + (t_2 + ((l_m ^ 2.0) / x))) + (t_3 / x))));
	else
		tmp = sqrt(((-1.0 + x) / (1.0 + x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.1e-258], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(l$95$m * N[Sqrt[N[(N[(1.0 / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.08e-165], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(0.5 * N[(N[(t$95$3 + t$95$3), $MachinePrecision] / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.1e+84], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 7.0999999999999998e-258

   1. Initial program 28.2%

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

   3. Simplified 28.1%

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

   4. Taylor expanded in l around inf 2.7%

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

      1. associate--l+ 9.8%

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

      2. sub-neg 9.8%

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

      3. metadata-eval 9.8%

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

      4. +-commutative 9.8%

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

      5. sub-neg 9.8%

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

      6. metadata-eval 9.8%

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

      7. +-commutative 9.8%

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

   6. Simplified 9.8%

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

   7. Taylor expanded in x around inf 20.9%

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

   if 7.0999999999999998e-258 < t < 1.08000000000000003e-165

   1. Initial program 2.6%

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

   3. Simplified 2.6%

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

   4. Taylor expanded in x around inf 67.3%

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

   if 1.08000000000000003e-165 < t < 4.1000000000000003e84

   1. Initial program 45.6%

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

   3. Simplified 45.8%

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

   4. Taylor expanded in x around inf 78.7%

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

   if 4.1000000000000003e84 < t

   1. Initial program 23.3%

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

   3. Simplified 23.3%

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

   4. Taylor expanded in l around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

   6. Simplified 100.0%

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

      1. sqrt-unprod 100.0%

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

      2. sqrt-undiv 100.0%

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

      3. +-commutative 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. associate-/r* 100.0%

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

      2. metadata-eval 100.0%

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

      3. +-commutative 100.0%

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

   10. Simplified 100.0%

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

       1. expm1-log1p-u 100.0%

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

       2. expm1-udef 100.0%

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

       3. associate-/r/ 100.0%

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

       4. +-commutative 100.0%

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

       5. +-commutative 100.0%

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

   12. Applied egg-rr 100.0%

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

       1. expm1-def 100.0%

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

       2. expm1-log1p 99.5%

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

       3. metadata-eval 99.5%

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

       4. sub-neg 99.5%

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

       5. associate-*l/ 100.0%

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

       6. *-lft-identity 100.0%

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

       7. sub-neg 100.0%

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

       8. metadata-eval 100.0%

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

       9. +-commutative 100.0%

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

       10. +-commutative 100.0%

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

   14. Simplified 100.0%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.1 \cdot 10^{-258}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{1}{-1 + x} + \frac{1}{x}}}\\ \mathbf{elif}\;t \leq 1.08 \cdot 10^{-165}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(\sqrt{2} \cdot x\right)} + t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{+84}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \end{array} \]

Alternative 4: 79.3% accurate, 0.4× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<=
       (/
        (* t_m (sqrt 2.0))
        (sqrt
         (-
          (* (/ (+ 1.0 x) (+ -1.0 x)) (+ (* l_m l_m) (* 2.0 (* t_m t_m))))
          (* l_m l_m))))
       2.0)
    (sqrt (/ (+ -1.0 x) (+ 1.0 x)))
    (* (sqrt 2.0) (/ (* t_m (sqrt (fma x 0.5 -0.5))) l_m)))))

C

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

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (Float64(Float64(t_m * sqrt(2.0)) / sqrt(Float64(Float64(Float64(Float64(1.0 + x) / Float64(-1.0 + x)) * Float64(Float64(l_m * l_m) + Float64(2.0 * Float64(t_m * t_m)))) - Float64(l_m * l_m)))) <= 2.0)
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(1.0 + x)));
	else
		tmp = Float64(sqrt(2.0) * Float64(Float64(t_m * sqrt(fma(x, 0.5, -0.5))) / l_m));
	end
	return Float64(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[N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(1.0 + x), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] * N[(N[(l$95$m * l$95$m), $MachinePrecision] + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(t$95$m * N[Sqrt[N[(x * 0.5 + -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l)))) < 2

   1. Initial program 45.0%

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

   3. Simplified 45.0%

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

   4. Taylor expanded in l around 0 41.4%

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

      1. +-commutative 41.4%

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

      2. sub-neg 41.4%

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

      3. metadata-eval 41.4%

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

      4. +-commutative 41.4%

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

   6. Simplified 41.4%

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

      1. sqrt-unprod 41.5%

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

      2. sqrt-undiv 41.5%

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

      3. +-commutative 41.5%

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

   8. Applied egg-rr 41.5%

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

      1. associate-/r* 41.5%

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

      2. metadata-eval 41.5%

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

      3. +-commutative 41.5%

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

   10. Simplified 41.5%

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

       1. expm1-log1p-u 41.5%

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

       2. expm1-udef 41.5%

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

       3. associate-/r/ 41.5%

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

       4. +-commutative 41.5%

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

       5. +-commutative 41.5%

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

   12. Applied egg-rr 41.5%

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

       1. expm1-def 41.5%

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

       2. expm1-log1p 41.3%

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

       3. metadata-eval 41.3%

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

       4. sub-neg 41.3%

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

       5. associate-*l/ 41.5%

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

       6. *-lft-identity 41.5%

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

       7. sub-neg 41.5%

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

       8. metadata-eval 41.5%

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

       9. +-commutative 41.5%

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

       10. +-commutative 41.5%

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

   14. Simplified 41.5%

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

   if 2 < (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l))))

   1. Initial program 1.2%

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

   3. Simplified 1.2%

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

   4. Taylor expanded in l around inf 1.5%

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

      1. *-commutative 1.5%

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

      2. associate--l+ 17.3%

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

      3. sub-neg 17.3%

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

      4. metadata-eval 17.3%

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

      5. +-commutative 17.3%

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

      6. sub-neg 17.3%

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

      7. metadata-eval 17.3%

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

      8. +-commutative 17.3%

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

   6. Simplified 17.3%

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

   7. Taylor expanded in x around 0 37.3%

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

      1. associate-*r/ 38.7%

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

      2. *-commutative 38.7%

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

      3. fma-neg 38.7%

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

      4. metadata-eval 38.7%

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

   9. Applied egg-rr 38.7%

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

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{1 + x}{-1 + x} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \leq 2:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t \cdot \sqrt{\mathsf{fma}\left(x, 0.5, -0.5\right)}}{\ell}\\ \end{array} \]

Alternative 5: 79.1% accurate, 0.5× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<=
       (/
        (* t_m (sqrt 2.0))
        (sqrt
         (-
          (* (/ (+ 1.0 x) (+ -1.0 x)) (+ (* l_m l_m) (* 2.0 (* t_m t_m))))
          (* l_m l_m))))
       2.0)
    (sqrt (/ (+ -1.0 x) (+ 1.0 x)))
    (* (sqrt 2.0) (/ t_m (* l_m (sqrt (+ (/ 1.0 (+ -1.0 x)) (/ 1.0 x)))))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l)))) < 2

   1. Initial program 45.0%

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

   3. Simplified 45.0%

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

   4. Taylor expanded in l around 0 41.4%

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

      1. +-commutative 41.4%

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

      2. sub-neg 41.4%

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

      3. metadata-eval 41.4%

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

      4. +-commutative 41.4%

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

   6. Simplified 41.4%

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

      1. sqrt-unprod 41.5%

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

      2. sqrt-undiv 41.5%

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

      3. +-commutative 41.5%

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

   8. Applied egg-rr 41.5%

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

      1. associate-/r* 41.5%

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

      2. metadata-eval 41.5%

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

      3. +-commutative 41.5%

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

   10. Simplified 41.5%

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

       1. expm1-log1p-u 41.5%

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

       2. expm1-udef 41.5%

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

       3. associate-/r/ 41.5%

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

       4. +-commutative 41.5%

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

       5. +-commutative 41.5%

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

   12. Applied egg-rr 41.5%

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

       1. expm1-def 41.5%

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

       2. expm1-log1p 41.3%

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

       3. metadata-eval 41.3%

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

       4. sub-neg 41.3%

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

       5. associate-*l/ 41.5%

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

       6. *-lft-identity 41.5%

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

       7. sub-neg 41.5%

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

       8. metadata-eval 41.5%

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

       9. +-commutative 41.5%

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

       10. +-commutative 41.5%

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

   14. Simplified 41.5%

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

   if 2 < (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l))))

   1. Initial program 1.2%

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

   3. Simplified 1.2%

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

   4. Taylor expanded in l around inf 1.8%

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

      1. associate--l+ 17.3%

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

      2. sub-neg 17.3%

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

      3. metadata-eval 17.3%

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

      4. +-commutative 17.3%

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

      5. sub-neg 17.3%

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

      6. metadata-eval 17.3%

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

      7. +-commutative 17.3%

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

   6. Simplified 17.3%

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

   7. Taylor expanded in x around inf 38.6%

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

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{1 + x}{-1 + x} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \leq 2:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{1}{-1 + x} + \frac{1}{x}}}\\ \end{array} \]

Alternative 6: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;l_m \leq 1.82 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \mathbf{elif}\;l_m \leq 4.2 \cdot 10^{+108}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{x \cdot 0.5} \cdot \frac{t_m}{l_m}\right)\\ \mathbf{elif}\;l_m \leq 4.8 \cdot 10^{+178}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_m \cdot \sqrt{2}}{l_m} \cdot \sqrt{x \cdot 0.5 - 0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 1.82e+93)
    (sqrt (/ (+ -1.0 x) (+ 1.0 x)))
    (if (<= l_m 4.2e+108)
      (* (sqrt 2.0) (* (sqrt (* x 0.5)) (/ t_m l_m)))
      (if (<= l_m 4.8e+178)
        1.0
        (* (/ (* t_m (sqrt 2.0)) l_m) (sqrt (- (* x 0.5) 0.5))))))))

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 <= 1.82e+93) {
		tmp = sqrt(((-1.0 + x) / (1.0 + x)));
	} else if (l_m <= 4.2e+108) {
		tmp = sqrt(2.0) * (sqrt((x * 0.5)) * (t_m / l_m));
	} else if (l_m <= 4.8e+178) {
		tmp = 1.0;
	} else {
		tmp = ((t_m * sqrt(2.0)) / l_m) * sqrt(((x * 0.5) - 0.5));
	}
	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 <= 1.82d+93) then
        tmp = sqrt((((-1.0d0) + x) / (1.0d0 + x)))
    else if (l_m <= 4.2d+108) then
        tmp = sqrt(2.0d0) * (sqrt((x * 0.5d0)) * (t_m / l_m))
    else if (l_m <= 4.8d+178) then
        tmp = 1.0d0
    else
        tmp = ((t_m * sqrt(2.0d0)) / l_m) * sqrt(((x * 0.5d0) - 0.5d0))
    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 <= 1.82e+93) {
		tmp = Math.sqrt(((-1.0 + x) / (1.0 + x)));
	} else if (l_m <= 4.2e+108) {
		tmp = Math.sqrt(2.0) * (Math.sqrt((x * 0.5)) * (t_m / l_m));
	} else if (l_m <= 4.8e+178) {
		tmp = 1.0;
	} else {
		tmp = ((t_m * Math.sqrt(2.0)) / l_m) * Math.sqrt(((x * 0.5) - 0.5));
	}
	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 <= 1.82e+93:
		tmp = math.sqrt(((-1.0 + x) / (1.0 + x)))
	elif l_m <= 4.2e+108:
		tmp = math.sqrt(2.0) * (math.sqrt((x * 0.5)) * (t_m / l_m))
	elif l_m <= 4.8e+178:
		tmp = 1.0
	else:
		tmp = ((t_m * math.sqrt(2.0)) / l_m) * math.sqrt(((x * 0.5) - 0.5))
	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 <= 1.82e+93)
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(1.0 + x)));
	elseif (l_m <= 4.2e+108)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(x * 0.5)) * Float64(t_m / l_m)));
	elseif (l_m <= 4.8e+178)
		tmp = 1.0;
	else
		tmp = Float64(Float64(Float64(t_m * sqrt(2.0)) / l_m) * sqrt(Float64(Float64(x * 0.5) - 0.5)));
	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 <= 1.82e+93)
		tmp = sqrt(((-1.0 + x) / (1.0 + x)));
	elseif (l_m <= 4.2e+108)
		tmp = sqrt(2.0) * (sqrt((x * 0.5)) * (t_m / l_m));
	elseif (l_m <= 4.8e+178)
		tmp = 1.0;
	else
		tmp = ((t_m * sqrt(2.0)) / l_m) * sqrt(((x * 0.5) - 0.5));
	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, 1.82e+93], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 4.2e+108], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l$95$m, 4.8e+178], 1.0, N[(N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] * N[Sqrt[N[(N[(x * 0.5), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if l < 1.82000000000000009e93

   1. Initial program 34.2%

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

   3. Simplified 34.2%

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

   4. Taylor expanded in l around 0 41.5%

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

      1. +-commutative 41.5%

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

      2. sub-neg 41.5%

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

      3. metadata-eval 41.5%

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

      4. +-commutative 41.5%

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

   6. Simplified 41.5%

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

      1. sqrt-unprod 41.5%

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

      2. sqrt-undiv 41.5%

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

      3. +-commutative 41.5%

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

   8. Applied egg-rr 41.5%

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

      1. associate-/r* 41.5%

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

      2. metadata-eval 41.5%

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

      3. +-commutative 41.5%

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

   10. Simplified 41.5%

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

       1. expm1-log1p-u 41.5%

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

       2. expm1-udef 41.5%

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

       3. associate-/r/ 41.5%

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

       4. +-commutative 41.5%

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

       5. +-commutative 41.5%

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

   12. Applied egg-rr 41.5%

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

       1. expm1-def 41.5%

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

       2. expm1-log1p 41.4%

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

       3. metadata-eval 41.4%

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

       4. sub-neg 41.4%

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

       5. associate-*l/ 41.5%

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

       6. *-lft-identity 41.5%

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

       7. sub-neg 41.5%

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

       8. metadata-eval 41.5%

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

       9. +-commutative 41.5%

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

       10. +-commutative 41.5%

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

   14. Simplified 41.5%

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

   if 1.82000000000000009e93 < l < 4.20000000000000019e108

   1. Initial program 2.6%

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

   3. Simplified 2.6%

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

   4. Taylor expanded in l around inf 2.6%

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

      1. *-commutative 2.6%

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

      2. associate--l+ 20.2%

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

      3. sub-neg 20.2%

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

      4. metadata-eval 20.2%

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

      5. +-commutative 20.2%

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

      6. sub-neg 20.2%

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

      7. metadata-eval 20.2%

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

      8. +-commutative 20.2%

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

   6. Simplified 20.2%

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

   7. Taylor expanded in x around inf 100.0%

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

      1. *-commutative 100.0%

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

   9. Simplified 100.0%

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

   if 4.20000000000000019e108 < l < 4.8e178

   1. Initial program 9.5%

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

   3. Simplified 9.6%

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

   4. Taylor expanded in l around 0 18.1%

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

      1. +-commutative 18.1%

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

      2. sub-neg 18.1%

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

      3. metadata-eval 18.1%

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

      4. +-commutative 18.1%

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

   6. Simplified 18.1%

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

   7. Taylor expanded in x around inf 18.1%

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

   if 4.8e178 < l

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in l around inf 2.3%

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

      1. *-commutative 2.3%

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

      2. associate--l+ 22.7%

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

      3. sub-neg 22.7%

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

      4. metadata-eval 22.7%

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

      5. +-commutative 22.7%

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

      6. sub-neg 22.7%

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

      7. metadata-eval 22.7%

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

      8. +-commutative 22.7%

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

   6. Simplified 22.7%

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

   7. Taylor expanded in x around 0 66.2%

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

   8. Taylor expanded in t around 0 66.2%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\ell} \cdot \sqrt{0.5 \cdot x - 0.5}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.82 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{+108}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{x \cdot 0.5} \cdot \frac{t}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{+178}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell} \cdot \sqrt{x \cdot 0.5 - 0.5}\\ \end{array} \]

Alternative 7: 78.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;l_m \leq 1.95 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \mathbf{elif}\;l_m \leq 1.2 \cdot 10^{+109} \lor \neg \left(l_m \leq 4.8 \cdot 10^{+171}\right):\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{x \cdot 0.5} \cdot \frac{t_m}{l_m}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 1.95e+93)
    (sqrt (/ (+ -1.0 x) (+ 1.0 x)))
    (if (or (<= l_m 1.2e+109) (not (<= l_m 4.8e+171)))
      (* (sqrt 2.0) (* (sqrt (* x 0.5)) (/ t_m l_m)))
      1.0))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 1.95e+93) {
		tmp = sqrt(((-1.0 + x) / (1.0 + x)));
	} else if ((l_m <= 1.2e+109) || !(l_m <= 4.8e+171)) {
		tmp = sqrt(2.0) * (sqrt((x * 0.5)) * (t_m / l_m));
	} else {
		tmp = 1.0;
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l_m <= 1.95d+93) then
        tmp = sqrt((((-1.0d0) + x) / (1.0d0 + x)))
    else if ((l_m <= 1.2d+109) .or. (.not. (l_m <= 4.8d+171))) then
        tmp = sqrt(2.0d0) * (sqrt((x * 0.5d0)) * (t_m / l_m))
    else
        tmp = 1.0d0
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 1.95e+93) {
		tmp = Math.sqrt(((-1.0 + x) / (1.0 + x)));
	} else if ((l_m <= 1.2e+109) || !(l_m <= 4.8e+171)) {
		tmp = Math.sqrt(2.0) * (Math.sqrt((x * 0.5)) * (t_m / l_m));
	} else {
		tmp = 1.0;
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if l_m <= 1.95e+93:
		tmp = math.sqrt(((-1.0 + x) / (1.0 + x)))
	elif (l_m <= 1.2e+109) or not (l_m <= 4.8e+171):
		tmp = math.sqrt(2.0) * (math.sqrt((x * 0.5)) * (t_m / l_m))
	else:
		tmp = 1.0
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (l_m <= 1.95e+93)
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(1.0 + x)));
	elseif ((l_m <= 1.2e+109) || !(l_m <= 4.8e+171))
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(x * 0.5)) * Float64(t_m / l_m)));
	else
		tmp = 1.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (l_m <= 1.95e+93)
		tmp = sqrt(((-1.0 + x) / (1.0 + x)));
	elseif ((l_m <= 1.2e+109) || ~((l_m <= 4.8e+171)))
		tmp = sqrt(2.0) * (sqrt((x * 0.5)) * (t_m / l_m));
	else
		tmp = 1.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 1.95e+93], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[l$95$m, 1.2e+109], N[Not[LessEqual[l$95$m, 4.8e+171]], $MachinePrecision]], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if l < 1.9500000000000001e93

   1. Initial program 34.2%

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

   3. Simplified 34.2%

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

   4. Taylor expanded in l around 0 41.5%

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

      1. +-commutative 41.5%

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

      2. sub-neg 41.5%

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

      3. metadata-eval 41.5%

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

      4. +-commutative 41.5%

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

   6. Simplified 41.5%

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

      1. sqrt-unprod 41.5%

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

      2. sqrt-undiv 41.5%

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

      3. +-commutative 41.5%

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

   8. Applied egg-rr 41.5%

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

      1. associate-/r* 41.5%

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

      2. metadata-eval 41.5%

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

      3. +-commutative 41.5%

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

   10. Simplified 41.5%

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

       1. expm1-log1p-u 41.5%

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

       2. expm1-udef 41.5%

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

       3. associate-/r/ 41.5%

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

       4. +-commutative 41.5%

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

       5. +-commutative 41.5%

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

   12. Applied egg-rr 41.5%

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

       1. expm1-def 41.5%

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

       2. expm1-log1p 41.4%

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

       3. metadata-eval 41.4%

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

       4. sub-neg 41.4%

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

       5. associate-*l/ 41.5%

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

       6. *-lft-identity 41.5%

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

       7. sub-neg 41.5%

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

       8. metadata-eval 41.5%

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

       9. +-commutative 41.5%

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

       10. +-commutative 41.5%

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

   14. Simplified 41.5%

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

   if 1.9500000000000001e93 < l < 1.19999999999999994e109 or 4.79999999999999995e171 < l

   1. Initial program 0.3%

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

   3. Simplified 0.3%

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

   4. Taylor expanded in l around inf 2.3%

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

      1. *-commutative 2.3%

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

      2. associate--l+ 22.3%

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

      3. sub-neg 22.3%

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

      4. metadata-eval 22.3%

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

      5. +-commutative 22.3%

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

      6. sub-neg 22.3%

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

      7. metadata-eval 22.3%

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

      8. +-commutative 22.3%

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

   6. Simplified 22.3%

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

   7. Taylor expanded in x around inf 70.6%

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

      1. *-commutative 70.6%

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

   9. Simplified 70.6%

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

   if 1.19999999999999994e109 < l < 4.79999999999999995e171

   1. Initial program 9.5%

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

   3. Simplified 9.6%

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

   4. Taylor expanded in l around 0 18.1%

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

      1. +-commutative 18.1%

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

      2. sub-neg 18.1%

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

      3. metadata-eval 18.1%

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

      4. +-commutative 18.1%

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

   6. Simplified 18.1%

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

   7. Taylor expanded in x around inf 18.1%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.95 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+109} \lor \neg \left(\ell \leq 4.8 \cdot 10^{+171}\right):\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{x \cdot 0.5} \cdot \frac{t}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;l_m \leq 4 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \mathbf{elif}\;l_m \leq 1.1 \cdot 10^{+109}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{x \cdot 0.5} \cdot \frac{t_m}{l_m}\right)\\ \mathbf{elif}\;l_m \leq 1.26 \cdot 10^{+169}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_m}{l_m} \cdot \sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 4e+93)
    (sqrt (/ (+ -1.0 x) (+ 1.0 x)))
    (if (<= l_m 1.1e+109)
      (* (sqrt 2.0) (* (sqrt (* x 0.5)) (/ t_m l_m)))
      (if (<= l_m 1.26e+169)
        1.0
        (* (/ t_m l_m) (sqrt (* 2.0 (fma x 0.5 -0.5)))))))))

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 <= 4e+93) {
		tmp = sqrt(((-1.0 + x) / (1.0 + x)));
	} else if (l_m <= 1.1e+109) {
		tmp = sqrt(2.0) * (sqrt((x * 0.5)) * (t_m / l_m));
	} else if (l_m <= 1.26e+169) {
		tmp = 1.0;
	} else {
		tmp = (t_m / l_m) * sqrt((2.0 * fma(x, 0.5, -0.5)));
	}
	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 <= 4e+93)
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(1.0 + x)));
	elseif (l_m <= 1.1e+109)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(x * 0.5)) * Float64(t_m / l_m)));
	elseif (l_m <= 1.26e+169)
		tmp = 1.0;
	else
		tmp = Float64(Float64(t_m / l_m) * sqrt(Float64(2.0 * fma(x, 0.5, -0.5))));
	end
	return Float64(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, 4e+93], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 1.1e+109], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l$95$m, 1.26e+169], 1.0, N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(x * 0.5 + -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if l < 4.00000000000000017e93

   1. Initial program 34.2%

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

   3. Simplified 34.2%

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

   4. Taylor expanded in l around 0 41.5%

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

      1. +-commutative 41.5%

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

      2. sub-neg 41.5%

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

      3. metadata-eval 41.5%

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

      4. +-commutative 41.5%

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

   6. Simplified 41.5%

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

      1. sqrt-unprod 41.5%

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

      2. sqrt-undiv 41.5%

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

      3. +-commutative 41.5%

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

   8. Applied egg-rr 41.5%

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

      1. associate-/r* 41.5%

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

      2. metadata-eval 41.5%

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

      3. +-commutative 41.5%

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

   10. Simplified 41.5%

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

       1. expm1-log1p-u 41.5%

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

       2. expm1-udef 41.5%

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

       3. associate-/r/ 41.5%

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

       4. +-commutative 41.5%

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

       5. +-commutative 41.5%

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

   12. Applied egg-rr 41.5%

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

       1. expm1-def 41.5%

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

       2. expm1-log1p 41.4%

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

       3. metadata-eval 41.4%

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

       4. sub-neg 41.4%

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

       5. associate-*l/ 41.5%

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

       6. *-lft-identity 41.5%

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

       7. sub-neg 41.5%

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

       8. metadata-eval 41.5%

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

       9. +-commutative 41.5%

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

       10. +-commutative 41.5%

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

   14. Simplified 41.5%

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

   if 4.00000000000000017e93 < l < 1.1e109

   1. Initial program 2.6%

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

   3. Simplified 2.6%

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

   4. Taylor expanded in l around inf 2.6%

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

      1. *-commutative 2.6%

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

      2. associate--l+ 20.2%

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

      3. sub-neg 20.2%

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

      4. metadata-eval 20.2%

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

      5. +-commutative 20.2%

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

      6. sub-neg 20.2%

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

      7. metadata-eval 20.2%

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

      8. +-commutative 20.2%

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

   6. Simplified 20.2%

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

   7. Taylor expanded in x around inf 100.0%

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

      1. *-commutative 100.0%

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

   9. Simplified 100.0%

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

   if 1.1e109 < l < 1.2599999999999999e169

   1. Initial program 9.5%

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

   3. Simplified 9.6%

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

   4. Taylor expanded in l around 0 18.1%

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

      1. +-commutative 18.1%

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

      2. sub-neg 18.1%

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

      3. metadata-eval 18.1%

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

      4. +-commutative 18.1%

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

   6. Simplified 18.1%

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

   7. Taylor expanded in x around inf 18.1%

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

   if 1.2599999999999999e169 < l

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in l around inf 2.3%

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

      1. *-commutative 2.3%

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

      2. associate--l+ 22.7%

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

      3. sub-neg 22.7%

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

      4. metadata-eval 22.7%

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

      5. +-commutative 22.7%

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

      6. sub-neg 22.7%

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

      7. metadata-eval 22.7%

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

      8. +-commutative 22.7%

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

   6. Simplified 22.7%

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

   7. Taylor expanded in x around 0 66.2%

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

      1. expm1-log1p-u 65.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2} \cdot \left(\sqrt{0.5 \cdot x - 0.5} \cdot \frac{t}{\ell}\right)\right)\right)} \]

      2. expm1-udef 13.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{2} \cdot \left(\sqrt{0.5 \cdot x - 0.5} \cdot \frac{t}{\ell}\right)\right)} - 1} \]

      3. associate-*r* 13.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot x - 0.5}\right) \cdot \frac{t}{\ell}}\right)} - 1 \]

      4. sqrt-unprod 13.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{2 \cdot \left(0.5 \cdot x - 0.5\right)}} \cdot \frac{t}{\ell}\right)} - 1 \]

      5. *-commutative 13.3%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(\color{blue}{x \cdot 0.5} - 0.5\right)} \cdot \frac{t}{\ell}\right)} - 1 \]

      6. fma-neg 13.3%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(x, 0.5, -0.5\right)}} \cdot \frac{t}{\ell}\right)} - 1 \]

      7. metadata-eval 13.3%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, \color{blue}{-0.5}\right)} \cdot \frac{t}{\ell}\right)} - 1 \]

   9. Applied egg-rr 13.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)} \cdot \frac{t}{\ell}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 65.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)} \cdot \frac{t}{\ell}\right)\right)} \]

       2. expm1-log1p 66.3%

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

       3. *-commutative 66.3%

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

   11. Simplified 66.3%

       \[\leadsto \color{blue}{\frac{t}{\ell} \cdot \sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+109}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{x \cdot 0.5} \cdot \frac{t}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 1.26 \cdot 10^{+169}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}\\ \end{array} \]

Alternative 9: 77.0% accurate, 2.1× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (* t_s (sqrt (/ (+ -1.0 x) (+ 1.0 x)))))

C

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

Fortran

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

Java

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

Python

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

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * sqrt(Float64(Float64(-1.0 + x) / 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 * sqrt(((-1.0 + x) / (1.0 + x)));
end

Wolfram

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

Derivation

1. Initial program 29.9%

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

3. Simplified 29.9%

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

4. Taylor expanded in l around 0 38.0%

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

   1. +-commutative 38.0%

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

   2. sub-neg 38.0%

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

   3. metadata-eval 38.0%

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

   4. +-commutative 38.0%

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

6. Simplified 38.0%

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

   1. sqrt-unprod 38.0%

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

   2. sqrt-undiv 38.0%

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

   3. +-commutative 38.0%

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

8. Applied egg-rr 38.0%

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

   1. associate-/r* 38.0%

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

   2. metadata-eval 38.0%

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

   3. +-commutative 38.0%

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

10. Simplified 38.0%

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

    1. expm1-log1p-u 38.0%

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

    2. expm1-udef 38.0%

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

    3. associate-/r/ 38.0%

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

    4. +-commutative 38.0%

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

    5. +-commutative 38.0%

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

12. Applied egg-rr 38.0%

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

    1. expm1-def 38.0%

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

    2. expm1-log1p 37.9%

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

    3. metadata-eval 37.9%

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

    4. sub-neg 37.9%

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

    5. associate-*l/ 38.0%

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

    6. *-lft-identity 38.0%

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

    7. sub-neg 38.0%

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

    8. metadata-eval 38.0%

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

    9. +-commutative 38.0%

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

    10. +-commutative 38.0%

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

14. Simplified 38.0%

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

15. Final simplification 38.0%

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

Alternative 10: 76.4% accurate, 45.0× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 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 29.9%

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

3. Simplified 29.9%

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

4. Taylor expanded in l around 0 38.0%

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

   1. +-commutative 38.0%

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

   2. sub-neg 38.0%

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

   3. metadata-eval 38.0%

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

   4. +-commutative 38.0%

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

6. Simplified 38.0%

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

7. Taylor expanded in x around inf 37.9%

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

8. Final simplification 37.9%

   \[\leadsto 1 + \frac{-1}{x} \]

Alternative 11: 75.8% accurate, 225.0× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 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 29.9%

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

3. Simplified 29.9%

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

4. Taylor expanded in l around 0 38.0%

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

   1. +-commutative 38.0%

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

   2. sub-neg 38.0%

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

   3. metadata-eval 38.0%

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

   4. +-commutative 38.0%

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

6. Simplified 38.0%

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

7. Taylor expanded in x around inf 37.3%

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

8. Final simplification 37.3%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023322 
(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)))))