Toniolo and Linder, Equation (2)

Percentage Accurate: 83.6% → 98.5%

Time: 18.3s

Alternatives: 12

Speedup: 1.9×

Specification

Math

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

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (asin
  (sqrt (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t l) 2.0)))))))

C

double code(double t, double l, double Om, double Omc) {
	return asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t / l), 2.0))))));
}

Fortran

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

Java

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

Python

def code(t, l, Om, Omc):
	return math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * math.pow((t / l), 2.0))))))

Julia

function code(t, l, Om, Omc)
	return asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t / l) ^ 2.0))))))
end

MATLAB

function tmp = code(t, l, Om, Omc)
	tmp = asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t / l) ^ 2.0))))));
end

Wolfram

code[t_, l_, Om_, Omc_] := N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Initial Program: 83.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (asin
  (sqrt (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t l) 2.0)))))))

C

double code(double t, double l, double Om, double Omc) {
	return asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t / l), 2.0))))));
}

Fortran

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

Java

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

Python

def code(t, l, Om, Omc):
	return math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * math.pow((t / l), 2.0))))))

Julia

function code(t, l, Om, Omc)
	return asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t / l) ^ 2.0))))))
end

MATLAB

function tmp = code(t, l, Om, Omc)
	tmp = asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t / l) ^ 2.0))))));
end

Wolfram

code[t_, l_, Om_, Omc_] := N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Alternative 1: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t_m}{\ell} \leq -1 \cdot 10^{+107}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{-\ell}{\sqrt{2}}}{t_m}\right)\\ \mathbf{elif}\;\frac{t_m}{\ell} \leq 10^{+150}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \frac{1}{\frac{\ell}{t_m} \cdot \frac{\ell}{t_m}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{t_1} \cdot \frac{\sqrt{0.5}}{\frac{t_m}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (let* ((t_1 (- 1.0 (pow (/ Om Omc) 2.0))))
   (if (<= (/ t_m l) -1e+107)
     (asin (/ (/ (- l) (sqrt 2.0)) t_m))
     (if (<= (/ t_m l) 1e+150)
       (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (/ 1.0 (* (/ l t_m) (/ l t_m))))))))
       (asin (* (sqrt t_1) (/ (sqrt 0.5) (/ t_m l))))))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	double t_1 = 1.0 - pow((Om / Omc), 2.0);
	double tmp;
	if ((t_m / l) <= -1e+107) {
		tmp = asin(((-l / sqrt(2.0)) / t_m));
	} else if ((t_m / l) <= 1e+150) {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * (1.0 / ((l / t_m) * (l / t_m))))))));
	} else {
		tmp = asin((sqrt(t_1) * (sqrt(0.5) / (t_m / l))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(t_m, l, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 - ((om / omc) ** 2.0d0)
    if ((t_m / l) <= (-1d+107)) then
        tmp = asin(((-l / sqrt(2.0d0)) / t_m))
    else if ((t_m / l) <= 1d+150) then
        tmp = asin(sqrt((t_1 / (1.0d0 + (2.0d0 * (1.0d0 / ((l / t_m) * (l / t_m))))))))
    else
        tmp = asin((sqrt(t_1) * (sqrt(0.5d0) / (t_m / l))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double t_m, double l, double Om, double Omc) {
	double t_1 = 1.0 - Math.pow((Om / Omc), 2.0);
	double tmp;
	if ((t_m / l) <= -1e+107) {
		tmp = Math.asin(((-l / Math.sqrt(2.0)) / t_m));
	} else if ((t_m / l) <= 1e+150) {
		tmp = Math.asin(Math.sqrt((t_1 / (1.0 + (2.0 * (1.0 / ((l / t_m) * (l / t_m))))))));
	} else {
		tmp = Math.asin((Math.sqrt(t_1) * (Math.sqrt(0.5) / (t_m / l))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	t_1 = 1.0 - math.pow((Om / Omc), 2.0)
	tmp = 0
	if (t_m / l) <= -1e+107:
		tmp = math.asin(((-l / math.sqrt(2.0)) / t_m))
	elif (t_m / l) <= 1e+150:
		tmp = math.asin(math.sqrt((t_1 / (1.0 + (2.0 * (1.0 / ((l / t_m) * (l / t_m))))))))
	else:
		tmp = math.asin((math.sqrt(t_1) * (math.sqrt(0.5) / (t_m / l))))
	return tmp

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	t_1 = Float64(1.0 - (Float64(Om / Omc) ^ 2.0))
	tmp = 0.0
	if (Float64(t_m / l) <= -1e+107)
		tmp = asin(Float64(Float64(Float64(-l) / sqrt(2.0)) / t_m));
	elseif (Float64(t_m / l) <= 1e+150)
		tmp = asin(sqrt(Float64(t_1 / Float64(1.0 + Float64(2.0 * Float64(1.0 / Float64(Float64(l / t_m) * Float64(l / t_m))))))));
	else
		tmp = asin(Float64(sqrt(t_1) * Float64(sqrt(0.5) / Float64(t_m / l))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(t_m, l, Om, Omc)
	t_1 = 1.0 - ((Om / Omc) ^ 2.0);
	tmp = 0.0;
	if ((t_m / l) <= -1e+107)
		tmp = asin(((-l / sqrt(2.0)) / t_m));
	elseif ((t_m / l) <= 1e+150)
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * (1.0 / ((l / t_m) * (l / t_m))))))));
	else
		tmp = asin((sqrt(t_1) * (sqrt(0.5) / (t_m / l))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := Block[{t$95$1 = N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$m / l), $MachinePrecision], -1e+107], N[ArcSin[N[(N[((-l) / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t$95$m / l), $MachinePrecision], 1e+150], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(1.0 + N[(2.0 * N[(1.0 / N[(N[(l / t$95$m), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[t$95$1], $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 2: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t_m}{\ell} \cdot \sqrt{2}\right)}\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (asin
  (/
   (sqrt (- 1.0 (pow (/ Om Omc) 2.0)))
   (hypot 1.0 (* (/ t_m l) (sqrt 2.0))))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	return asin((sqrt((1.0 - pow((Om / Omc), 2.0))) / hypot(1.0, ((t_m / l) * sqrt(2.0)))));
}

Java

t_m = Math.abs(t);
public static double code(double t_m, double l, double Om, double Omc) {
	return Math.asin((Math.sqrt((1.0 - Math.pow((Om / Omc), 2.0))) / Math.hypot(1.0, ((t_m / l) * Math.sqrt(2.0)))));
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	return math.asin((math.sqrt((1.0 - math.pow((Om / Omc), 2.0))) / math.hypot(1.0, ((t_m / l) * math.sqrt(2.0)))))

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	return asin(Float64(sqrt(Float64(1.0 - (Float64(Om / Omc) ^ 2.0))) / hypot(1.0, Float64(Float64(t_m / l) * sqrt(2.0)))))
end

MATLAB

t_m = abs(t);
function tmp = code(t_m, l, Om, Omc)
	tmp = asin((sqrt((1.0 - ((Om / Omc) ^ 2.0))) / hypot(1.0, ((t_m / l) * sqrt(2.0)))));
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := N[ArcSin[N[(N[Sqrt[N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(N[(t$95$m / l), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 3: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t_m}{\ell} \leq -1 \cdot 10^{+107}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{-\ell}{\sqrt{2}}}{t_m}\right)\\ \mathbf{elif}\;\frac{t_m}{\ell} \leq 10^{+150}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \frac{1}{\frac{\ell}{t_m} \cdot \frac{\ell}{t_m}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{t_1} \cdot \left(\ell \cdot \frac{\sqrt{0.5}}{t_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (let* ((t_1 (- 1.0 (pow (/ Om Omc) 2.0))))
   (if (<= (/ t_m l) -1e+107)
     (asin (/ (/ (- l) (sqrt 2.0)) t_m))
     (if (<= (/ t_m l) 1e+150)
       (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (/ 1.0 (* (/ l t_m) (/ l t_m))))))))
       (asin (* (sqrt t_1) (* l (/ (sqrt 0.5) t_m))))))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	double t_1 = 1.0 - pow((Om / Omc), 2.0);
	double tmp;
	if ((t_m / l) <= -1e+107) {
		tmp = asin(((-l / sqrt(2.0)) / t_m));
	} else if ((t_m / l) <= 1e+150) {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * (1.0 / ((l / t_m) * (l / t_m))))))));
	} else {
		tmp = asin((sqrt(t_1) * (l * (sqrt(0.5) / t_m))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(t_m, l, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 - ((om / omc) ** 2.0d0)
    if ((t_m / l) <= (-1d+107)) then
        tmp = asin(((-l / sqrt(2.0d0)) / t_m))
    else if ((t_m / l) <= 1d+150) then
        tmp = asin(sqrt((t_1 / (1.0d0 + (2.0d0 * (1.0d0 / ((l / t_m) * (l / t_m))))))))
    else
        tmp = asin((sqrt(t_1) * (l * (sqrt(0.5d0) / t_m))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double t_m, double l, double Om, double Omc) {
	double t_1 = 1.0 - Math.pow((Om / Omc), 2.0);
	double tmp;
	if ((t_m / l) <= -1e+107) {
		tmp = Math.asin(((-l / Math.sqrt(2.0)) / t_m));
	} else if ((t_m / l) <= 1e+150) {
		tmp = Math.asin(Math.sqrt((t_1 / (1.0 + (2.0 * (1.0 / ((l / t_m) * (l / t_m))))))));
	} else {
		tmp = Math.asin((Math.sqrt(t_1) * (l * (Math.sqrt(0.5) / t_m))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	t_1 = 1.0 - math.pow((Om / Omc), 2.0)
	tmp = 0
	if (t_m / l) <= -1e+107:
		tmp = math.asin(((-l / math.sqrt(2.0)) / t_m))
	elif (t_m / l) <= 1e+150:
		tmp = math.asin(math.sqrt((t_1 / (1.0 + (2.0 * (1.0 / ((l / t_m) * (l / t_m))))))))
	else:
		tmp = math.asin((math.sqrt(t_1) * (l * (math.sqrt(0.5) / t_m))))
	return tmp

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	t_1 = Float64(1.0 - (Float64(Om / Omc) ^ 2.0))
	tmp = 0.0
	if (Float64(t_m / l) <= -1e+107)
		tmp = asin(Float64(Float64(Float64(-l) / sqrt(2.0)) / t_m));
	elseif (Float64(t_m / l) <= 1e+150)
		tmp = asin(sqrt(Float64(t_1 / Float64(1.0 + Float64(2.0 * Float64(1.0 / Float64(Float64(l / t_m) * Float64(l / t_m))))))));
	else
		tmp = asin(Float64(sqrt(t_1) * Float64(l * Float64(sqrt(0.5) / t_m))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(t_m, l, Om, Omc)
	t_1 = 1.0 - ((Om / Omc) ^ 2.0);
	tmp = 0.0;
	if ((t_m / l) <= -1e+107)
		tmp = asin(((-l / sqrt(2.0)) / t_m));
	elseif ((t_m / l) <= 1e+150)
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * (1.0 / ((l / t_m) * (l / t_m))))))));
	else
		tmp = asin((sqrt(t_1) * (l * (sqrt(0.5) / t_m))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := Block[{t$95$1 = N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$m / l), $MachinePrecision], -1e+107], N[ArcSin[N[(N[((-l) / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t$95$m / l), $MachinePrecision], 1e+150], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(1.0 + N[(2.0 * N[(1.0 / N[(N[(l / t$95$m), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[t$95$1], $MachinePrecision] * N[(l * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 4: 98.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t_m}{\ell} \leq -1 \cdot 10^{+107}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{-\ell}{\sqrt{2}}}{t_m}\right)\\ \mathbf{elif}\;\frac{t_m}{\ell} \leq 10^{+150}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{1}{\frac{\ell}{t_m} \cdot \frac{\ell}{t_m}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t_m \cdot \sqrt{2}} \cdot \sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (if (<= (/ t_m l) -1e+107)
   (asin (/ (/ (- l) (sqrt 2.0)) t_m))
   (if (<= (/ t_m l) 1e+150)
     (asin
      (sqrt
       (/
        (- 1.0 (pow (/ Om Omc) 2.0))
        (+ 1.0 (* 2.0 (/ 1.0 (* (/ l t_m) (/ l t_m))))))))
     (asin
      (* (/ l (* t_m (sqrt 2.0))) (sqrt (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	double tmp;
	if ((t_m / l) <= -1e+107) {
		tmp = asin(((-l / sqrt(2.0)) / t_m));
	} else if ((t_m / l) <= 1e+150) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * (1.0 / ((l / t_m) * (l / t_m))))))));
	} else {
		tmp = asin(((l / (t_m * sqrt(2.0))) * sqrt((1.0 - ((Om / Omc) / (Omc / Om))))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(t_m, l, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l) <= (-1d+107)) then
        tmp = asin(((-l / sqrt(2.0d0)) / t_m))
    else if ((t_m / l) <= 1d+150) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * (1.0d0 / ((l / t_m) * (l / t_m))))))))
    else
        tmp = asin(((l / (t_m * sqrt(2.0d0))) * sqrt((1.0d0 - ((om / omc) / (omc / om))))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double t_m, double l, double Om, double Omc) {
	double tmp;
	if ((t_m / l) <= -1e+107) {
		tmp = Math.asin(((-l / Math.sqrt(2.0)) / t_m));
	} else if ((t_m / l) <= 1e+150) {
		tmp = Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * (1.0 / ((l / t_m) * (l / t_m))))))));
	} else {
		tmp = Math.asin(((l / (t_m * Math.sqrt(2.0))) * Math.sqrt((1.0 - ((Om / Omc) / (Omc / Om))))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	tmp = 0
	if (t_m / l) <= -1e+107:
		tmp = math.asin(((-l / math.sqrt(2.0)) / t_m))
	elif (t_m / l) <= 1e+150:
		tmp = math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * (1.0 / ((l / t_m) * (l / t_m))))))))
	else:
		tmp = math.asin(((l / (t_m * math.sqrt(2.0))) * math.sqrt((1.0 - ((Om / Omc) / (Omc / Om))))))
	return tmp

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l) <= -1e+107)
		tmp = asin(Float64(Float64(Float64(-l) / sqrt(2.0)) / t_m));
	elseif (Float64(t_m / l) <= 1e+150)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * Float64(1.0 / Float64(Float64(l / t_m) * Float64(l / t_m))))))));
	else
		tmp = asin(Float64(Float64(l / Float64(t_m * sqrt(2.0))) * sqrt(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om))))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(t_m, l, Om, Omc)
	tmp = 0.0;
	if ((t_m / l) <= -1e+107)
		tmp = asin(((-l / sqrt(2.0)) / t_m));
	elseif ((t_m / l) <= 1e+150)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * (1.0 / ((l / t_m) * (l / t_m))))))));
	else
		tmp = asin(((l / (t_m * sqrt(2.0))) * sqrt((1.0 - ((Om / Omc) / (Omc / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l), $MachinePrecision], -1e+107], N[ArcSin[N[(N[((-l) / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t$95$m / l), $MachinePrecision], 1e+150], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(1.0 / N[(N[(l / t$95$m), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 5: 98.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := 1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\\ \mathbf{if}\;\frac{t_m}{\ell} \leq -1 \cdot 10^{+107}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{-\ell}{\sqrt{2}}}{t_m}\right)\\ \mathbf{elif}\;\frac{t_m}{\ell} \leq 10^{+150}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \frac{\frac{t_m}{\ell}}{\frac{\ell}{t_m}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t_m \cdot \sqrt{2}} \cdot \sqrt{t_1}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (let* ((t_1 (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))
   (if (<= (/ t_m l) -1e+107)
     (asin (/ (/ (- l) (sqrt 2.0)) t_m))
     (if (<= (/ t_m l) 1e+150)
       (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (/ (/ t_m l) (/ l t_m)))))))
       (asin (* (/ l (* t_m (sqrt 2.0))) (sqrt t_1)))))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	double t_1 = 1.0 - ((Om / Omc) / (Omc / Om));
	double tmp;
	if ((t_m / l) <= -1e+107) {
		tmp = asin(((-l / sqrt(2.0)) / t_m));
	} else if ((t_m / l) <= 1e+150) {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))));
	} else {
		tmp = asin(((l / (t_m * sqrt(2.0))) * sqrt(t_1)));
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(t_m, l, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 - ((om / omc) / (omc / om))
    if ((t_m / l) <= (-1d+107)) then
        tmp = asin(((-l / sqrt(2.0d0)) / t_m))
    else if ((t_m / l) <= 1d+150) then
        tmp = asin(sqrt((t_1 / (1.0d0 + (2.0d0 * ((t_m / l) / (l / t_m)))))))
    else
        tmp = asin(((l / (t_m * sqrt(2.0d0))) * sqrt(t_1)))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double t_m, double l, double Om, double Omc) {
	double t_1 = 1.0 - ((Om / Omc) / (Omc / Om));
	double tmp;
	if ((t_m / l) <= -1e+107) {
		tmp = Math.asin(((-l / Math.sqrt(2.0)) / t_m));
	} else if ((t_m / l) <= 1e+150) {
		tmp = Math.asin(Math.sqrt((t_1 / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))));
	} else {
		tmp = Math.asin(((l / (t_m * Math.sqrt(2.0))) * Math.sqrt(t_1)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	t_1 = 1.0 - ((Om / Omc) / (Omc / Om))
	tmp = 0
	if (t_m / l) <= -1e+107:
		tmp = math.asin(((-l / math.sqrt(2.0)) / t_m))
	elif (t_m / l) <= 1e+150:
		tmp = math.asin(math.sqrt((t_1 / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))))
	else:
		tmp = math.asin(((l / (t_m * math.sqrt(2.0))) * math.sqrt(t_1)))
	return tmp

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	t_1 = Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))
	tmp = 0.0
	if (Float64(t_m / l) <= -1e+107)
		tmp = asin(Float64(Float64(Float64(-l) / sqrt(2.0)) / t_m));
	elseif (Float64(t_m / l) <= 1e+150)
		tmp = asin(sqrt(Float64(t_1 / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m / l) / Float64(l / t_m)))))));
	else
		tmp = asin(Float64(Float64(l / Float64(t_m * sqrt(2.0))) * sqrt(t_1)));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(t_m, l, Om, Omc)
	t_1 = 1.0 - ((Om / Omc) / (Omc / Om));
	tmp = 0.0;
	if ((t_m / l) <= -1e+107)
		tmp = asin(((-l / sqrt(2.0)) / t_m));
	elseif ((t_m / l) <= 1e+150)
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))));
	else
		tmp = asin(((l / (t_m * sqrt(2.0))) * sqrt(t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := Block[{t$95$1 = N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$m / l), $MachinePrecision], -1e+107], N[ArcSin[N[(N[((-l) / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t$95$m / l), $MachinePrecision], 1e+150], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(1.0 + N[(2.0 * N[(N[(t$95$m / l), $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 6: 97.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t_m}{\ell} \leq -2 \cdot 10^{+102}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{-\ell}{\sqrt{2}}}{t_m}\right)\\ \mathbf{elif}\;\frac{t_m}{\ell} \leq 2 \cdot 10^{+84}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(t_m \cdot \frac{\frac{t_m}{\ell}}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\frac{1}{t_m}}{\sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (if (<= (/ t_m l) -2e+102)
   (asin (/ (/ (- l) (sqrt 2.0)) t_m))
   (if (<= (/ t_m l) 2e+84)
     (asin
      (sqrt
       (/
        (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
        (+ 1.0 (* 2.0 (* t_m (/ (/ t_m l) l)))))))
     (asin (* l (/ (/ 1.0 t_m) (sqrt 2.0)))))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	double tmp;
	if ((t_m / l) <= -2e+102) {
		tmp = asin(((-l / sqrt(2.0)) / t_m));
	} else if ((t_m / l) <= 2e+84) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * (t_m * ((t_m / l) / l)))))));
	} else {
		tmp = asin((l * ((1.0 / t_m) / sqrt(2.0))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(t_m, l, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l) <= (-2d+102)) then
        tmp = asin(((-l / sqrt(2.0d0)) / t_m))
    else if ((t_m / l) <= 2d+84) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * (t_m * ((t_m / l) / l)))))))
    else
        tmp = asin((l * ((1.0d0 / t_m) / sqrt(2.0d0))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double t_m, double l, double Om, double Omc) {
	double tmp;
	if ((t_m / l) <= -2e+102) {
		tmp = Math.asin(((-l / Math.sqrt(2.0)) / t_m));
	} else if ((t_m / l) <= 2e+84) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * (t_m * ((t_m / l) / l)))))));
	} else {
		tmp = Math.asin((l * ((1.0 / t_m) / Math.sqrt(2.0))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	tmp = 0
	if (t_m / l) <= -2e+102:
		tmp = math.asin(((-l / math.sqrt(2.0)) / t_m))
	elif (t_m / l) <= 2e+84:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * (t_m * ((t_m / l) / l)))))))
	else:
		tmp = math.asin((l * ((1.0 / t_m) / math.sqrt(2.0))))
	return tmp

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l) <= -2e+102)
		tmp = asin(Float64(Float64(Float64(-l) / sqrt(2.0)) / t_m));
	elseif (Float64(t_m / l) <= 2e+84)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om))) / Float64(1.0 + Float64(2.0 * Float64(t_m * Float64(Float64(t_m / l) / l)))))));
	else
		tmp = asin(Float64(l * Float64(Float64(1.0 / t_m) / sqrt(2.0))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(t_m, l, Om, Omc)
	tmp = 0.0;
	if ((t_m / l) <= -2e+102)
		tmp = asin(((-l / sqrt(2.0)) / t_m));
	elseif ((t_m / l) <= 2e+84)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * (t_m * ((t_m / l) / l)))))));
	else
		tmp = asin((l * ((1.0 / t_m) / sqrt(2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l), $MachinePrecision], -2e+102], N[ArcSin[N[(N[((-l) / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t$95$m / l), $MachinePrecision], 2e+84], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l * N[(N[(1.0 / t$95$m), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 7: 98.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t_m}{\ell} \leq -1 \cdot 10^{+107}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{-\ell}{\sqrt{2}}}{t_m}\right)\\ \mathbf{elif}\;\frac{t_m}{\ell} \leq 10^{+150}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t_m}{\ell}}{\frac{\ell}{t_m}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\frac{1}{t_m}}{\sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (if (<= (/ t_m l) -1e+107)
   (asin (/ (/ (- l) (sqrt 2.0)) t_m))
   (if (<= (/ t_m l) 1e+150)
     (asin
      (sqrt
       (/
        (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
        (+ 1.0 (* 2.0 (/ (/ t_m l) (/ l t_m)))))))
     (asin (* l (/ (/ 1.0 t_m) (sqrt 2.0)))))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	double tmp;
	if ((t_m / l) <= -1e+107) {
		tmp = asin(((-l / sqrt(2.0)) / t_m));
	} else if ((t_m / l) <= 1e+150) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))));
	} else {
		tmp = asin((l * ((1.0 / t_m) / sqrt(2.0))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(t_m, l, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l) <= (-1d+107)) then
        tmp = asin(((-l / sqrt(2.0d0)) / t_m))
    else if ((t_m / l) <= 1d+150) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * ((t_m / l) / (l / t_m)))))))
    else
        tmp = asin((l * ((1.0d0 / t_m) / sqrt(2.0d0))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double t_m, double l, double Om, double Omc) {
	double tmp;
	if ((t_m / l) <= -1e+107) {
		tmp = Math.asin(((-l / Math.sqrt(2.0)) / t_m));
	} else if ((t_m / l) <= 1e+150) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))));
	} else {
		tmp = Math.asin((l * ((1.0 / t_m) / Math.sqrt(2.0))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	tmp = 0
	if (t_m / l) <= -1e+107:
		tmp = math.asin(((-l / math.sqrt(2.0)) / t_m))
	elif (t_m / l) <= 1e+150:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))))
	else:
		tmp = math.asin((l * ((1.0 / t_m) / math.sqrt(2.0))))
	return tmp

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l) <= -1e+107)
		tmp = asin(Float64(Float64(Float64(-l) / sqrt(2.0)) / t_m));
	elseif (Float64(t_m / l) <= 1e+150)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om))) / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m / l) / Float64(l / t_m)))))));
	else
		tmp = asin(Float64(l * Float64(Float64(1.0 / t_m) / sqrt(2.0))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(t_m, l, Om, Omc)
	tmp = 0.0;
	if ((t_m / l) <= -1e+107)
		tmp = asin(((-l / sqrt(2.0)) / t_m));
	elseif ((t_m / l) <= 1e+150)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))));
	else
		tmp = asin((l * ((1.0 / t_m) / sqrt(2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l), $MachinePrecision], -1e+107], N[ArcSin[N[(N[((-l) / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t$95$m / l), $MachinePrecision], 1e+150], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t$95$m / l), $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l * N[(N[(1.0 / t$95$m), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 8: 97.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t_m}{\ell} \leq -1000:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{-\ell}{\sqrt{2}}}{t_m}\right)\\ \mathbf{elif}\;\frac{t_m}{\ell} \leq 0.005:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t_m \cdot \sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (if (<= (/ t_m l) -1000.0)
   (asin (/ (/ (- l) (sqrt 2.0)) t_m))
   (if (<= (/ t_m l) 0.005)
     (asin (sqrt (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))
     (asin (/ l (* t_m (sqrt 2.0)))))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	double tmp;
	if ((t_m / l) <= -1000.0) {
		tmp = asin(((-l / sqrt(2.0)) / t_m));
	} else if ((t_m / l) <= 0.005) {
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} else {
		tmp = asin((l / (t_m * sqrt(2.0))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(t_m, l, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l) <= (-1000.0d0)) then
        tmp = asin(((-l / sqrt(2.0d0)) / t_m))
    else if ((t_m / l) <= 0.005d0) then
        tmp = asin(sqrt((1.0d0 - ((om / omc) / (omc / om)))))
    else
        tmp = asin((l / (t_m * sqrt(2.0d0))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double t_m, double l, double Om, double Omc) {
	double tmp;
	if ((t_m / l) <= -1000.0) {
		tmp = Math.asin(((-l / Math.sqrt(2.0)) / t_m));
	} else if ((t_m / l) <= 0.005) {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} else {
		tmp = Math.asin((l / (t_m * Math.sqrt(2.0))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	tmp = 0
	if (t_m / l) <= -1000.0:
		tmp = math.asin(((-l / math.sqrt(2.0)) / t_m))
	elif (t_m / l) <= 0.005:
		tmp = math.asin(math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))))
	else:
		tmp = math.asin((l / (t_m * math.sqrt(2.0))))
	return tmp

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l) <= -1000.0)
		tmp = asin(Float64(Float64(Float64(-l) / sqrt(2.0)) / t_m));
	elseif (Float64(t_m / l) <= 0.005)
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))));
	else
		tmp = asin(Float64(l / Float64(t_m * sqrt(2.0))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(t_m, l, Om, Omc)
	tmp = 0.0;
	if ((t_m / l) <= -1000.0)
		tmp = asin(((-l / sqrt(2.0)) / t_m));
	elseif ((t_m / l) <= 0.005)
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	else
		tmp = asin((l / (t_m * sqrt(2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l), $MachinePrecision], -1000.0], N[ArcSin[N[(N[((-l) / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t$95$m / l), $MachinePrecision], 0.005], N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 9: 97.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t_m}{\ell} \leq -1000:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{-\ell}{\sqrt{2}}}{t_m}\right)\\ \mathbf{elif}\;\frac{t_m}{\ell} \leq 0.005:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\frac{1}{t_m}}{\sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (if (<= (/ t_m l) -1000.0)
   (asin (/ (/ (- l) (sqrt 2.0)) t_m))
   (if (<= (/ t_m l) 0.005)
     (asin (sqrt (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))
     (asin (* l (/ (/ 1.0 t_m) (sqrt 2.0)))))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	double tmp;
	if ((t_m / l) <= -1000.0) {
		tmp = asin(((-l / sqrt(2.0)) / t_m));
	} else if ((t_m / l) <= 0.005) {
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} else {
		tmp = asin((l * ((1.0 / t_m) / sqrt(2.0))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(t_m, l, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l) <= (-1000.0d0)) then
        tmp = asin(((-l / sqrt(2.0d0)) / t_m))
    else if ((t_m / l) <= 0.005d0) then
        tmp = asin(sqrt((1.0d0 - ((om / omc) / (omc / om)))))
    else
        tmp = asin((l * ((1.0d0 / t_m) / sqrt(2.0d0))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double t_m, double l, double Om, double Omc) {
	double tmp;
	if ((t_m / l) <= -1000.0) {
		tmp = Math.asin(((-l / Math.sqrt(2.0)) / t_m));
	} else if ((t_m / l) <= 0.005) {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} else {
		tmp = Math.asin((l * ((1.0 / t_m) / Math.sqrt(2.0))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	tmp = 0
	if (t_m / l) <= -1000.0:
		tmp = math.asin(((-l / math.sqrt(2.0)) / t_m))
	elif (t_m / l) <= 0.005:
		tmp = math.asin(math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))))
	else:
		tmp = math.asin((l * ((1.0 / t_m) / math.sqrt(2.0))))
	return tmp

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l) <= -1000.0)
		tmp = asin(Float64(Float64(Float64(-l) / sqrt(2.0)) / t_m));
	elseif (Float64(t_m / l) <= 0.005)
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))));
	else
		tmp = asin(Float64(l * Float64(Float64(1.0 / t_m) / sqrt(2.0))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(t_m, l, Om, Omc)
	tmp = 0.0;
	if ((t_m / l) <= -1000.0)
		tmp = asin(((-l / sqrt(2.0)) / t_m));
	elseif ((t_m / l) <= 0.005)
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	else
		tmp = asin((l * ((1.0 / t_m) / sqrt(2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l), $MachinePrecision], -1000.0], N[ArcSin[N[(N[((-l) / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t$95$m / l), $MachinePrecision], 0.005], N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l * N[(N[(1.0 / t$95$m), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 10: 49.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := t_m \cdot \sqrt{2}\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t_1}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (let* ((t_1 (* t_m (sqrt 2.0))))
   (if (<= l -2e-310) (asin (/ (- l) t_1)) (asin (/ l t_1)))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	double t_1 = t_m * sqrt(2.0);
	double tmp;
	if (l <= -2e-310) {
		tmp = asin((-l / t_1));
	} else {
		tmp = asin((l / t_1));
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(t_m, l, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t_m * sqrt(2.0d0)
    if (l <= (-2d-310)) then
        tmp = asin((-l / t_1))
    else
        tmp = asin((l / t_1))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double t_m, double l, double Om, double Omc) {
	double t_1 = t_m * Math.sqrt(2.0);
	double tmp;
	if (l <= -2e-310) {
		tmp = Math.asin((-l / t_1));
	} else {
		tmp = Math.asin((l / t_1));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	t_1 = t_m * math.sqrt(2.0)
	tmp = 0
	if l <= -2e-310:
		tmp = math.asin((-l / t_1))
	else:
		tmp = math.asin((l / t_1))
	return tmp

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	t_1 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (l <= -2e-310)
		tmp = asin(Float64(Float64(-l) / t_1));
	else
		tmp = asin(Float64(l / t_1));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(t_m, l, Om, Omc)
	t_1 = t_m * sqrt(2.0);
	tmp = 0.0;
	if (l <= -2e-310)
		tmp = asin((-l / t_1));
	else
		tmp = asin((l / t_1));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := Block[{t$95$1 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2e-310], N[ArcSin[N[((-l) / t$95$1), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l / t$95$1), $MachinePrecision]], $MachinePrecision]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 11: 49.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{-\ell}{\sqrt{2}}}{t_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t_m \cdot \sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (if (<= l -2e-310)
   (asin (/ (/ (- l) (sqrt 2.0)) t_m))
   (asin (/ l (* t_m (sqrt 2.0))))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	double tmp;
	if (l <= -2e-310) {
		tmp = asin(((-l / sqrt(2.0)) / t_m));
	} else {
		tmp = asin((l / (t_m * sqrt(2.0))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(t_m, l, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (l <= (-2d-310)) then
        tmp = asin(((-l / sqrt(2.0d0)) / t_m))
    else
        tmp = asin((l / (t_m * sqrt(2.0d0))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double t_m, double l, double Om, double Omc) {
	double tmp;
	if (l <= -2e-310) {
		tmp = Math.asin(((-l / Math.sqrt(2.0)) / t_m));
	} else {
		tmp = Math.asin((l / (t_m * Math.sqrt(2.0))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	tmp = 0
	if l <= -2e-310:
		tmp = math.asin(((-l / math.sqrt(2.0)) / t_m))
	else:
		tmp = math.asin((l / (t_m * math.sqrt(2.0))))
	return tmp

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	tmp = 0.0
	if (l <= -2e-310)
		tmp = asin(Float64(Float64(Float64(-l) / sqrt(2.0)) / t_m));
	else
		tmp = asin(Float64(l / Float64(t_m * sqrt(2.0))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(t_m, l, Om, Omc)
	tmp = 0.0;
	if (l <= -2e-310)
		tmp = asin(((-l / sqrt(2.0)) / t_m));
	else
		tmp = asin((l / (t_m * sqrt(2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := If[LessEqual[l, -2e-310], N[ArcSin[N[(N[((-l) / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 12: 31.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \sin^{-1} \left(\frac{\ell}{t_m \cdot \sqrt{2}}\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc) :precision binary64 (asin (/ l (* t_m (sqrt 2.0)))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	return asin((l / (t_m * sqrt(2.0))));
}

Fortran

t_m = abs(t)
real(8) function code(t_m, l, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    code = asin((l / (t_m * sqrt(2.0d0))))
end function

Java

t_m = Math.abs(t);
public static double code(double t_m, double l, double Om, double Omc) {
	return Math.asin((l / (t_m * Math.sqrt(2.0))));
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	return math.asin((l / (t_m * math.sqrt(2.0))))

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	return asin(Float64(l / Float64(t_m * sqrt(2.0))))
end

MATLAB

t_m = abs(t);
function tmp = code(t_m, l, Om, Omc)
	tmp = asin((l / (t_m * sqrt(2.0))));
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := N[ArcSin[N[(l / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Reproduce

herbie shell --seed 2023340 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  :precision binary64
  (asin (sqrt (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t l) 2.0)))))))