Toniolo and Linder, Equation (7)

Percentage Accurate: 33.4% → 85.0%

Time: 17.1s

Alternatives: 8

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: 33.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t_m}^{2}\\ t_3 := t_2 + {l_m}^{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 5.7 \cdot 10^{-223}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{2} \cdot l_m}{t_m \cdot \sqrt{x}}}\\ \mathbf{elif}\;t_m \leq 1.65 \cdot 10^{-146}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{0.5 \cdot \frac{t_3 + t_3}{t_m \cdot \left(\sqrt{2} \cdot x\right)} + t_m \cdot \sqrt{2}}\\ \mathbf{elif}\;t_m \leq 29:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{\sqrt{\left(2 \cdot \frac{{t_m}^{2}}{x} + \left(t_2 + \frac{{l_m}^{2}}{x}\right)\right) + \frac{t_3}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{x + -1}}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 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 5.7e-223)
      (/ (sqrt 2.0) (/ (* (sqrt 2.0) l_m) (* t_m (sqrt x))))
      (if (<= t_m 1.65e-146)
        (*
         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 29.0)
          (*
           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 2.0) (sqrt (* 2.0 (/ (+ x 1.0) (+ x -1.0)))))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double t_3 = t_2 + pow(l_m, 2.0);
	double tmp;
	if (t_m <= 5.7e-223) {
		tmp = sqrt(2.0) / ((sqrt(2.0) * l_m) / (t_m * sqrt(x)));
	} else if (t_m <= 1.65e-146) {
		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 <= 29.0) {
		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(2.0) / sqrt((2.0 * ((x + 1.0) / (x + -1.0))));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    t_3 = t_2 + (l_m ** 2.0d0)
    if (t_m <= 5.7d-223) then
        tmp = sqrt(2.0d0) / ((sqrt(2.0d0) * l_m) / (t_m * sqrt(x)))
    else if (t_m <= 1.65d-146) 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 <= 29.0d0) 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(2.0d0) / sqrt((2.0d0 * ((x + 1.0d0) / (x + (-1.0d0)))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * Math.pow(t_m, 2.0);
	double t_3 = t_2 + Math.pow(l_m, 2.0);
	double tmp;
	if (t_m <= 5.7e-223) {
		tmp = Math.sqrt(2.0) / ((Math.sqrt(2.0) * l_m) / (t_m * Math.sqrt(x)));
	} else if (t_m <= 1.65e-146) {
		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 <= 29.0) {
		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(2.0) / Math.sqrt((2.0 * ((x + 1.0) / (x + -1.0))));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = 2.0 * math.pow(t_m, 2.0)
	t_3 = t_2 + math.pow(l_m, 2.0)
	tmp = 0
	if t_m <= 5.7e-223:
		tmp = math.sqrt(2.0) / ((math.sqrt(2.0) * l_m) / (t_m * math.sqrt(x)))
	elif t_m <= 1.65e-146:
		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 <= 29.0:
		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(2.0) / math.sqrt((2.0 * ((x + 1.0) / (x + -1.0))))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(2.0 * (t_m ^ 2.0))
	t_3 = Float64(t_2 + (l_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 5.7e-223)
		tmp = Float64(sqrt(2.0) / Float64(Float64(sqrt(2.0) * l_m) / Float64(t_m * sqrt(x))));
	elseif (t_m <= 1.65e-146)
		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 <= 29.0)
		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 = Float64(sqrt(2.0) / sqrt(Float64(2.0 * Float64(Float64(x + 1.0) / Float64(x + -1.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = 2.0 * (t_m ^ 2.0);
	t_3 = t_2 + (l_m ^ 2.0);
	tmp = 0.0;
	if (t_m <= 5.7e-223)
		tmp = sqrt(2.0) / ((sqrt(2.0) * l_m) / (t_m * sqrt(x)));
	elseif (t_m <= 1.65e-146)
		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 <= 29.0)
		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(2.0) / sqrt((2.0 * ((x + 1.0) / (x + -1.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, 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, 5.7e-223], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision] / N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.65e-146], 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, 29.0], 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[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(2.0 * N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 5.6999999999999998e-223

   1. Initial program 27.5%

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

   3. Simplified 27.6%

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

   4. Taylor expanded in l around inf 1.6%

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

      1. *-commutative 1.6%

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

      2. associate--l+ 11.5%

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

      3. sub-neg 11.5%

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

      4. metadata-eval 11.5%

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

      5. +-commutative 11.5%

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

      6. sub-neg 11.5%

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

      7. metadata-eval 11.5%

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

      8. +-commutative 11.5%

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

   6. Simplified 11.5%

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

   7. Taylor expanded in x around inf 16.2%

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

   8. Taylor expanded in x around inf 16.2%

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

      1. sqrt-div 16.2%

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

      2. frac-times 18.1%

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

   10. Applied egg-rr 18.1%

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

   if 5.6999999999999998e-223 < t < 1.65e-146

   1. Initial program 8.3%

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

   3. Simplified 8.3%

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

   4. Taylor expanded in x around inf 78.3%

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

   if 1.65e-146 < t < 29

   1. Initial program 49.2%

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

   3. Simplified 49.3%

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

   4. Taylor expanded in x around inf 81.6%

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

   if 29 < t

   1. Initial program 30.0%

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

   3. Simplified 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 94.3%

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

      1. +-commutative 94.3%

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

      2. sub-neg 94.3%

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

      3. metadata-eval 94.3%

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

      4. +-commutative 94.3%

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

   6. Simplified 94.3%

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

      1. *-commutative 94.3%

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

      2. sqrt-unprod 94.2%

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

      3. +-commutative 94.2%

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

   8. Applied egg-rr 94.2%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.7 \cdot 10^{-223}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{2} \cdot \ell}{t \cdot \sqrt{x}}}\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-146}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(\sqrt{2} \cdot x\right)} + t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 29:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{x + -1}}}\\ \end{array} \]

Alternative 2: 82.7% accurate, 0.2× speedup

Math

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

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 (/ (+ x 1.0) (+ x -1.0)))
        (t_3
         (/
          (* t_m (sqrt 2.0))
          (sqrt (- (* t_2 (+ (* l_m l_m) (* 2.0 (* t_m t_m)))) (* l_m l_m)))))
        (t_4 (* 2.0 (pow t_m 2.0))))
   (*
    t_s
    (if (<= t_3 2.0)
      (/ (sqrt 2.0) (sqrt (* 2.0 t_2)))
      (if (<= t_3 INFINITY)
        (*
         t_m
         (/
          (sqrt 2.0)
          (sqrt
           (+
            (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_4 (/ (pow l_m 2.0) x)))
            (/ (+ t_4 (pow l_m 2.0)) x)))))
        (* t_m (/ (sqrt x) 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 t_2 = (x + 1.0) / (x + -1.0);
	double t_3 = (t_m * sqrt(2.0)) / sqrt(((t_2 * ((l_m * l_m) + (2.0 * (t_m * t_m)))) - (l_m * l_m)));
	double t_4 = 2.0 * pow(t_m, 2.0);
	double tmp;
	if (t_3 <= 2.0) {
		tmp = sqrt(2.0) / sqrt((2.0 * t_2));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_m * (sqrt(2.0) / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_4 + (pow(l_m, 2.0) / x))) + ((t_4 + pow(l_m, 2.0)) / x))));
	} else {
		tmp = t_m * (sqrt(x) / l_m);
	}
	return t_s * tmp;
}

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 = (x + 1.0) / (x + -1.0);
	double t_3 = (t_m * Math.sqrt(2.0)) / Math.sqrt(((t_2 * ((l_m * l_m) + (2.0 * (t_m * t_m)))) - (l_m * l_m)));
	double t_4 = 2.0 * Math.pow(t_m, 2.0);
	double tmp;
	if (t_3 <= 2.0) {
		tmp = Math.sqrt(2.0) / Math.sqrt((2.0 * t_2));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_m * (Math.sqrt(2.0) / Math.sqrt((((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_4 + (Math.pow(l_m, 2.0) / x))) + ((t_4 + Math.pow(l_m, 2.0)) / x))));
	} else {
		tmp = t_m * (Math.sqrt(x) / l_m);
	}
	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 = (x + 1.0) / (x + -1.0)
	t_3 = (t_m * math.sqrt(2.0)) / math.sqrt(((t_2 * ((l_m * l_m) + (2.0 * (t_m * t_m)))) - (l_m * l_m)))
	t_4 = 2.0 * math.pow(t_m, 2.0)
	tmp = 0
	if t_3 <= 2.0:
		tmp = math.sqrt(2.0) / math.sqrt((2.0 * t_2))
	elif t_3 <= math.inf:
		tmp = t_m * (math.sqrt(2.0) / math.sqrt((((2.0 * (math.pow(t_m, 2.0) / x)) + (t_4 + (math.pow(l_m, 2.0) / x))) + ((t_4 + math.pow(l_m, 2.0)) / x))))
	else:
		tmp = t_m * (math.sqrt(x) / 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)
	t_2 = Float64(Float64(x + 1.0) / Float64(x + -1.0))
	t_3 = Float64(Float64(t_m * sqrt(2.0)) / sqrt(Float64(Float64(t_2 * Float64(Float64(l_m * l_m) + Float64(2.0 * Float64(t_m * t_m)))) - Float64(l_m * l_m))))
	t_4 = Float64(2.0 * (t_m ^ 2.0))
	tmp = 0.0
	if (t_3 <= 2.0)
		tmp = Float64(sqrt(2.0) / sqrt(Float64(2.0 * t_2)));
	elseif (t_3 <= Inf)
		tmp = Float64(t_m * Float64(sqrt(2.0) / sqrt(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_4 + Float64((l_m ^ 2.0) / x))) + Float64(Float64(t_4 + (l_m ^ 2.0)) / x)))));
	else
		tmp = Float64(t_m * Float64(sqrt(x) / l_m));
	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 = (x + 1.0) / (x + -1.0);
	t_3 = (t_m * sqrt(2.0)) / sqrt(((t_2 * ((l_m * l_m) + (2.0 * (t_m * t_m)))) - (l_m * l_m)));
	t_4 = 2.0 * (t_m ^ 2.0);
	tmp = 0.0;
	if (t_3 <= 2.0)
		tmp = sqrt(2.0) / sqrt((2.0 * t_2));
	elseif (t_3 <= Inf)
		tmp = t_m * (sqrt(2.0) / sqrt((((2.0 * ((t_m ^ 2.0) / x)) + (t_4 + ((l_m ^ 2.0) / x))) + ((t_4 + (l_m ^ 2.0)) / x))));
	else
		tmp = t_m * (sqrt(x) / l_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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

Derivation

1. Split input into 3 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 48.3%

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

   3. Simplified 48.3%

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

   4. Taylor expanded in l around 0 43.1%

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

      1. +-commutative 43.1%

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

      2. sub-neg 43.1%

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

      3. metadata-eval 43.1%

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

      4. +-commutative 43.1%

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

   6. Simplified 43.1%

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

      1. *-commutative 43.1%

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

      2. sqrt-unprod 43.1%

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

      3. +-commutative 43.1%

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

   8. Applied egg-rr 43.1%

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

   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)))) < +inf.0

   1. Initial program 2.7%

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

   3. Simplified 2.7%

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

   4. Taylor expanded in x around inf 61.8%

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

   if +inf.0 < (/.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 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%

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

   4. Taylor expanded in l around inf 1.2%

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

      1. *-commutative 1.2%

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

      2. associate--l+ 27.4%

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

      3. sub-neg 27.4%

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

      4. metadata-eval 27.4%

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

      5. +-commutative 27.4%

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

      6. sub-neg 27.4%

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

      7. metadata-eval 27.4%

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

      8. +-commutative 27.4%

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

   6. Simplified 27.4%

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

   7. Taylor expanded in x around inf 34.9%

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

   8. Taylor expanded in x around inf 34.9%

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

      1. expm1-log1p-u 34.9%

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

      2. expm1-udef 28.5%

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

      3. associate-/r* 28.5%

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

      4. sqrt-undiv 28.5%

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

   10. Applied egg-rr 28.5%

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

       1. expm1-def 34.9%

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

       2. expm1-log1p 35.0%

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

       3. associate-/r/ 42.0%

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

       4. associate-/r/ 42.0%

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

       5. metadata-eval 42.0%

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

       6. *-lft-identity 42.0%

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

   12. Simplified 42.0%

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

4. Final simplification 45.8%

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

Alternative 3: 80.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x + 1}{x + -1}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;\frac{t_m \cdot \sqrt{2}}{\sqrt{t_2 \cdot \left(l_m \cdot l_m + 2 \cdot \left(t_m \cdot t_m\right)\right) - l_m \cdot l_m}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2 \cdot t_2}}\\ \mathbf{else}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{x}}{l_m}\\ \end{array} \end{array} \end{array} \]

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 (/ (+ x 1.0) (+ x -1.0))))
   (*
    t_s
    (if (<=
         (/
          (* t_m (sqrt 2.0))
          (sqrt (- (* t_2 (+ (* l_m l_m) (* 2.0 (* t_m t_m)))) (* l_m l_m))))
         INFINITY)
      (/ (sqrt 2.0) (sqrt (* 2.0 t_2)))
      (* t_m (/ (sqrt x) 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 t_2 = (x + 1.0) / (x + -1.0);
	double tmp;
	if (((t_m * sqrt(2.0)) / sqrt(((t_2 * ((l_m * l_m) + (2.0 * (t_m * t_m)))) - (l_m * l_m)))) <= ((double) INFINITY)) {
		tmp = sqrt(2.0) / sqrt((2.0 * t_2));
	} else {
		tmp = t_m * (sqrt(x) / l_m);
	}
	return t_s * tmp;
}

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 = (x + 1.0) / (x + -1.0);
	double tmp;
	if (((t_m * Math.sqrt(2.0)) / Math.sqrt(((t_2 * ((l_m * l_m) + (2.0 * (t_m * t_m)))) - (l_m * l_m)))) <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(2.0) / Math.sqrt((2.0 * t_2));
	} else {
		tmp = t_m * (Math.sqrt(x) / l_m);
	}
	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 = (x + 1.0) / (x + -1.0)
	tmp = 0
	if ((t_m * math.sqrt(2.0)) / math.sqrt(((t_2 * ((l_m * l_m) + (2.0 * (t_m * t_m)))) - (l_m * l_m)))) <= math.inf:
		tmp = math.sqrt(2.0) / math.sqrt((2.0 * t_2))
	else:
		tmp = t_m * (math.sqrt(x) / 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)
	t_2 = Float64(Float64(x + 1.0) / Float64(x + -1.0))
	tmp = 0.0
	if (Float64(Float64(t_m * sqrt(2.0)) / sqrt(Float64(Float64(t_2 * Float64(Float64(l_m * l_m) + Float64(2.0 * Float64(t_m * t_m)))) - Float64(l_m * l_m)))) <= Inf)
		tmp = Float64(sqrt(2.0) / sqrt(Float64(2.0 * t_2)));
	else
		tmp = Float64(t_m * Float64(sqrt(x) / l_m));
	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 = (x + 1.0) / (x + -1.0);
	tmp = 0.0;
	if (((t_m * sqrt(2.0)) / sqrt(((t_2 * ((l_m * l_m) + (2.0 * (t_m * t_m)))) - (l_m * l_m)))) <= Inf)
		tmp = sqrt(2.0) / sqrt((2.0 * t_2));
	else
		tmp = t_m * (sqrt(x) / l_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(t$95$2 * 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], Infinity], N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(2.0 * t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$m * N[(N[Sqrt[x], $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)))) < +inf.0

   1. Initial program 39.2%

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

   3. Simplified 39.2%

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

   4. Taylor expanded in l around 0 43.2%

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

      1. +-commutative 43.2%

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

      2. sub-neg 43.2%

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

      3. metadata-eval 43.2%

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

      4. +-commutative 43.2%

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

   6. Simplified 43.2%

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

      1. *-commutative 43.2%

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

      2. sqrt-unprod 43.2%

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

      3. +-commutative 43.2%

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

   8. Applied egg-rr 43.2%

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

   if +inf.0 < (/.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 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%

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

   4. Taylor expanded in l around inf 1.2%

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

      1. *-commutative 1.2%

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

      2. associate--l+ 27.4%

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

      3. sub-neg 27.4%

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

      4. metadata-eval 27.4%

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

      5. +-commutative 27.4%

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

      6. sub-neg 27.4%

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

      7. metadata-eval 27.4%

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

      8. +-commutative 27.4%

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

   6. Simplified 27.4%

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

   7. Taylor expanded in x around inf 34.9%

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

   8. Taylor expanded in x around inf 34.9%

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

      1. expm1-log1p-u 34.9%

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

      2. expm1-udef 28.5%

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

      3. associate-/r* 28.5%

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

      4. sqrt-undiv 28.5%

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

   10. Applied egg-rr 28.5%

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

       1. expm1-def 34.9%

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

       2. expm1-log1p 35.0%

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

       3. associate-/r/ 42.0%

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

       4. associate-/r/ 42.0%

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

       5. metadata-eval 42.0%

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

       6. *-lft-identity 42.0%

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

   12. Simplified 42.0%

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

4. Final simplification 42.9%

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

Alternative 4: 80.5% accurate, 2.1× speedup

Math

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

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

Derivation

1. Split input into 2 regimes

2. if l < 1.35000000000000003e154

   1. Initial program 34.3%

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

   3. Simplified 34.3%

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

   4. Taylor expanded in l around 0 40.0%

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

      1. +-commutative 40.0%

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

      2. sub-neg 40.0%

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

      3. metadata-eval 40.0%

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

      4. +-commutative 40.0%

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

   6. Simplified 40.0%

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

      1. add-sqr-sqrt 40.0%

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

      2. sqrt-unprod 40.0%

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

      3. associate-/r* 40.0%

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

      4. sqrt-undiv 40.0%

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

      5. metadata-eval 40.0%

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

      6. metadata-eval 40.0%

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

      7. associate-/r* 40.0%

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

      8. sqrt-undiv 40.0%

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

      9. metadata-eval 40.0%

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

      10. metadata-eval 40.0%

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

      11. frac-times 40.0%

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

      12. metadata-eval 40.0%

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

      13. add-sqr-sqrt 40.0%

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

      14. +-commutative 40.0%

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

   8. Applied egg-rr 40.0%

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

      1. associate-/r/ 39.8%

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

      2. +-commutative 39.8%

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

   10. Simplified 39.8%

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

       1. associate-*l/ 40.0%

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

       2. *-un-lft-identity 40.0%

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

       3. +-commutative 40.0%

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

       4. +-commutative 40.0%

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

   12. Applied egg-rr 40.0%

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

   if 1.35000000000000003e154 < 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%

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

   4. Taylor expanded in l around inf 1.7%

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

      1. *-commutative 1.7%

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

      2. associate--l+ 32.1%

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

      3. sub-neg 32.1%

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

      4. metadata-eval 32.1%

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

      5. +-commutative 32.1%

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

      6. sub-neg 32.1%

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

      7. metadata-eval 32.1%

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

      8. +-commutative 32.1%

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

   6. Simplified 32.1%

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

   7. Taylor expanded in x around inf 46.9%

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

   8. Taylor expanded in x around inf 46.9%

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

      1. expm1-log1p-u 46.9%

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

      2. expm1-udef 33.7%

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

      3. associate-/r* 33.7%

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

      4. sqrt-undiv 33.7%

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

   10. Applied egg-rr 33.7%

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

       1. expm1-def 46.9%

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

       2. expm1-log1p 46.9%

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

       3. associate-/r/ 61.0%

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

       4. associate-/r/ 61.0%

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

       5. metadata-eval 61.0%

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

       6. *-lft-identity 61.0%

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

   12. Simplified 61.0%

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

4. Final simplification 42.4%

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

Alternative 5: 78.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;l_m \leq 3.6 \cdot 10^{+154}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t_m}{l_m}\\ \end{array} \end{array} \]

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 (<= l_m 3.6e+154) (+ 1.0 (/ -1.0 x)) (* (sqrt x) (/ t_m 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 (l_m <= 3.6e+154) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = sqrt(x) * (t_m / l_m);
	}
	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 <= 3.6d+154) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else
        tmp = sqrt(x) * (t_m / l_m)
    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 <= 3.6e+154) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = Math.sqrt(x) * (t_m / l_m);
	}
	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 <= 3.6e+154:
		tmp = 1.0 + (-1.0 / x)
	else:
		tmp = math.sqrt(x) * (t_m / 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 (l_m <= 3.6e+154)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	else
		tmp = Float64(sqrt(x) * Float64(t_m / l_m));
	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 <= 3.6e+154)
		tmp = 1.0 + (-1.0 / x);
	else
		tmp = sqrt(x) * (t_m / l_m);
	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, 3.6e+154], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3.6000000000000001e154

   1. Initial program 34.3%

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

   3. Simplified 34.3%

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

   4. Taylor expanded in l around 0 40.0%

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

      1. +-commutative 40.0%

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

      2. sub-neg 40.0%

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

      3. metadata-eval 40.0%

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

      4. +-commutative 40.0%

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

   6. Simplified 40.0%

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

   7. Taylor expanded in x around inf 39.3%

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

   if 3.6000000000000001e154 < 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%

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

   4. Taylor expanded in l around inf 1.7%

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

      1. *-commutative 1.7%

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

      2. associate--l+ 32.1%

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

      3. sub-neg 32.1%

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

      4. metadata-eval 32.1%

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

      5. +-commutative 32.1%

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

      6. sub-neg 32.1%

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

      7. metadata-eval 32.1%

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

      8. +-commutative 32.1%

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

   6. Simplified 32.1%

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

   7. Taylor expanded in x around inf 46.9%

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

   8. Taylor expanded in x around inf 46.9%

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

   9. Taylor expanded in l around 0 47.8%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.6 \cdot 10^{+154}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \end{array} \]

Alternative 6: 80.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;l_m \leq 1.65 \cdot 10^{+154}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{x}}{l_m}\\ \end{array} \end{array} \]

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 (<= l_m 1.65e+154) (+ 1.0 (/ -1.0 x)) (* t_m (/ (sqrt x) 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 (l_m <= 1.65e+154) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = t_m * (sqrt(x) / l_m);
	}
	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 <= 1.65d+154) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else
        tmp = t_m * (sqrt(x) / l_m)
    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 <= 1.65e+154) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = t_m * (Math.sqrt(x) / l_m);
	}
	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 <= 1.65e+154:
		tmp = 1.0 + (-1.0 / x)
	else:
		tmp = t_m * (math.sqrt(x) / 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 (l_m <= 1.65e+154)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	else
		tmp = Float64(t_m * Float64(sqrt(x) / l_m));
	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 <= 1.65e+154)
		tmp = 1.0 + (-1.0 / x);
	else
		tmp = t_m * (sqrt(x) / l_m);
	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, 1.65e+154], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.65e154

   1. Initial program 34.3%

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

   3. Simplified 34.3%

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

   4. Taylor expanded in l around 0 40.0%

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

      1. +-commutative 40.0%

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

      2. sub-neg 40.0%

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

      3. metadata-eval 40.0%

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

      4. +-commutative 40.0%

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

   6. Simplified 40.0%

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

   7. Taylor expanded in x around inf 39.3%

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

   if 1.65e154 < 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%

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

   4. Taylor expanded in l around inf 1.7%

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

      1. *-commutative 1.7%

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

      2. associate--l+ 32.1%

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

      3. sub-neg 32.1%

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

      4. metadata-eval 32.1%

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

      5. +-commutative 32.1%

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

      6. sub-neg 32.1%

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

      7. metadata-eval 32.1%

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

      8. +-commutative 32.1%

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

   6. Simplified 32.1%

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

   7. Taylor expanded in x around inf 46.9%

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

   8. Taylor expanded in x around inf 46.9%

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

      1. expm1-log1p-u 46.9%

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

      2. expm1-udef 33.7%

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

      3. associate-/r* 33.7%

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

      4. sqrt-undiv 33.7%

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

   10. Applied egg-rr 33.7%

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

       1. expm1-def 46.9%

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

       2. expm1-log1p 46.9%

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

       3. associate-/r/ 61.0%

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

       4. associate-/r/ 61.0%

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

       5. metadata-eval 61.0%

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

       6. *-lft-identity 61.0%

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

   12. Simplified 61.0%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.65 \cdot 10^{+154}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \end{array} \]

Alternative 7: 75.9% 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 30.5%

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

3. Simplified 30.4%

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

4. Taylor expanded in l around 0 38.1%

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

   1. +-commutative 38.1%

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

   2. sub-neg 38.1%

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

   3. metadata-eval 38.1%

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

   4. +-commutative 38.1%

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

6. Simplified 38.1%

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

7. Taylor expanded in x around inf 37.5%

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

8. Final simplification 37.5%

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

Alternative 8: 75.3% 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 30.5%

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

3. Simplified 30.4%

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

4. Taylor expanded in l around 0 38.1%

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

   1. +-commutative 38.1%

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

   2. sub-neg 38.1%

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

   3. metadata-eval 38.1%

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

   4. +-commutative 38.1%

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

6. Simplified 38.1%

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

7. Taylor expanded in x around inf 37.1%

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

8. Final simplification 37.1%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023339 
(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)))))