Toniolo and Linder, Equation (2)

Percentage Accurate: 84.3% → 97.0%

Time: 18.8s

Alternatives: 10

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: 84.3% 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: 97.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t_m}{\ell} \leq -500:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t_m \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t_m}{\ell} \leq 0.1:\\ \;\;\;\;\sin^{-1} \left(\mathsf{fma}\left(-0.5, \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t_m}{\sqrt{0.5}}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (if (<= (/ t_m l) -500.0)
   (asin (/ (- l) (* t_m (sqrt 2.0))))
   (if (<= (/ t_m l) 0.1)
     (asin (fma -0.5 (/ (/ Om Omc) (/ Omc Om)) 1.0))
     (asin (/ l (/ t_m (sqrt 0.5)))))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	double tmp;
	if ((t_m / l) <= -500.0) {
		tmp = asin((-l / (t_m * sqrt(2.0))));
	} else if ((t_m / l) <= 0.1) {
		tmp = asin(fma(-0.5, ((Om / Omc) / (Omc / Om)), 1.0));
	} else {
		tmp = asin((l / (t_m / sqrt(0.5))));
	}
	return tmp;
}

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l) <= -500.0)
		tmp = asin(Float64(Float64(-l) / Float64(t_m * sqrt(2.0))));
	elseif (Float64(t_m / l) <= 0.1)
		tmp = asin(fma(-0.5, Float64(Float64(Om / Omc) / Float64(Omc / Om)), 1.0));
	else
		tmp = asin(Float64(l / Float64(t_m / sqrt(0.5))));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l), $MachinePrecision], -500.0], N[ArcSin[N[((-l) / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t$95$m / l), $MachinePrecision], 0.1], N[ArcSin[N[(-0.5 * N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l / N[(t$95$m / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

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1. Add Preprocessing

Alternative 2: 98.4% 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

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1. Add Preprocessing

Alternative 3: 97.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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1. Add Preprocessing

Alternative 4: 97.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t_m}{\ell} \leq -500:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t_m \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t_m}{\ell} \leq 0.1:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t_m}{\sqrt{0.5}}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (if (<= (/ t_m l) -500.0)
   (asin (/ (- l) (* t_m (sqrt 2.0))))
   (if (<= (/ t_m l) 0.1)
     (asin (sqrt (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))
     (asin (/ l (/ t_m (sqrt 0.5)))))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	double tmp;
	if ((t_m / l) <= -500.0) {
		tmp = asin((-l / (t_m * sqrt(2.0))));
	} else if ((t_m / l) <= 0.1) {
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} else {
		tmp = asin((l / (t_m / sqrt(0.5))));
	}
	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) <= (-500.0d0)) then
        tmp = asin((-l / (t_m * sqrt(2.0d0))))
    else if ((t_m / l) <= 0.1d0) then
        tmp = asin(sqrt((1.0d0 - ((om / omc) / (omc / om)))))
    else
        tmp = asin((l / (t_m / sqrt(0.5d0))))
    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) <= -500.0) {
		tmp = Math.asin((-l / (t_m * Math.sqrt(2.0))));
	} else if ((t_m / l) <= 0.1) {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} else {
		tmp = Math.asin((l / (t_m / Math.sqrt(0.5))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	tmp = 0
	if (t_m / l) <= -500.0:
		tmp = math.asin((-l / (t_m * math.sqrt(2.0))))
	elif (t_m / l) <= 0.1:
		tmp = math.asin(math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))))
	else:
		tmp = math.asin((l / (t_m / math.sqrt(0.5))))
	return tmp

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l) <= -500.0)
		tmp = asin(Float64(Float64(-l) / Float64(t_m * sqrt(2.0))));
	elseif (Float64(t_m / l) <= 0.1)
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))));
	else
		tmp = asin(Float64(l / Float64(t_m / sqrt(0.5))));
	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) <= -500.0)
		tmp = asin((-l / (t_m * sqrt(2.0))));
	elseif ((t_m / l) <= 0.1)
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	else
		tmp = asin((l / (t_m / sqrt(0.5))));
	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], -500.0], N[ArcSin[N[((-l) / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t$95$m / l), $MachinePrecision], 0.1], 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[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

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

Alternative 5: 72.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7.5 \cdot 10^{-5}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{-\ell}{t_m}\right)\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{-80}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t_m}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\ell \leq 8.6 \cdot 10^{-34}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 85000000000000:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t_m \cdot \sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (if (<= l -7.5e-5)
   (asin 1.0)
   (if (<= l -2e-310)
     (asin (* (sqrt 0.5) (/ (- l) t_m)))
     (if (<= l 3.3e-80)
       (asin (/ l (/ t_m (sqrt 0.5))))
       (if (<= l 8.6e-34)
         (asin 1.0)
         (if (<= l 85000000000000.0)
           (asin (/ l (* t_m (sqrt 2.0))))
           (asin 1.0)))))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	double tmp;
	if (l <= -7.5e-5) {
		tmp = asin(1.0);
	} else if (l <= -2e-310) {
		tmp = asin((sqrt(0.5) * (-l / t_m)));
	} else if (l <= 3.3e-80) {
		tmp = asin((l / (t_m / sqrt(0.5))));
	} else if (l <= 8.6e-34) {
		tmp = asin(1.0);
	} else if (l <= 85000000000000.0) {
		tmp = asin((l / (t_m * sqrt(2.0))));
	} else {
		tmp = asin(1.0);
	}
	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 <= (-7.5d-5)) then
        tmp = asin(1.0d0)
    else if (l <= (-2d-310)) then
        tmp = asin((sqrt(0.5d0) * (-l / t_m)))
    else if (l <= 3.3d-80) then
        tmp = asin((l / (t_m / sqrt(0.5d0))))
    else if (l <= 8.6d-34) then
        tmp = asin(1.0d0)
    else if (l <= 85000000000000.0d0) then
        tmp = asin((l / (t_m * sqrt(2.0d0))))
    else
        tmp = asin(1.0d0)
    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 <= -7.5e-5) {
		tmp = Math.asin(1.0);
	} else if (l <= -2e-310) {
		tmp = Math.asin((Math.sqrt(0.5) * (-l / t_m)));
	} else if (l <= 3.3e-80) {
		tmp = Math.asin((l / (t_m / Math.sqrt(0.5))));
	} else if (l <= 8.6e-34) {
		tmp = Math.asin(1.0);
	} else if (l <= 85000000000000.0) {
		tmp = Math.asin((l / (t_m * Math.sqrt(2.0))));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	tmp = 0
	if l <= -7.5e-5:
		tmp = math.asin(1.0)
	elif l <= -2e-310:
		tmp = math.asin((math.sqrt(0.5) * (-l / t_m)))
	elif l <= 3.3e-80:
		tmp = math.asin((l / (t_m / math.sqrt(0.5))))
	elif l <= 8.6e-34:
		tmp = math.asin(1.0)
	elif l <= 85000000000000.0:
		tmp = math.asin((l / (t_m * math.sqrt(2.0))))
	else:
		tmp = math.asin(1.0)
	return tmp

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	tmp = 0.0
	if (l <= -7.5e-5)
		tmp = asin(1.0);
	elseif (l <= -2e-310)
		tmp = asin(Float64(sqrt(0.5) * Float64(Float64(-l) / t_m)));
	elseif (l <= 3.3e-80)
		tmp = asin(Float64(l / Float64(t_m / sqrt(0.5))));
	elseif (l <= 8.6e-34)
		tmp = asin(1.0);
	elseif (l <= 85000000000000.0)
		tmp = asin(Float64(l / Float64(t_m * sqrt(2.0))));
	else
		tmp = asin(1.0);
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(t_m, l, Om, Omc)
	tmp = 0.0;
	if (l <= -7.5e-5)
		tmp = asin(1.0);
	elseif (l <= -2e-310)
		tmp = asin((sqrt(0.5) * (-l / t_m)));
	elseif (l <= 3.3e-80)
		tmp = asin((l / (t_m / sqrt(0.5))));
	elseif (l <= 8.6e-34)
		tmp = asin(1.0);
	elseif (l <= 85000000000000.0)
		tmp = asin((l / (t_m * sqrt(2.0))));
	else
		tmp = asin(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := If[LessEqual[l, -7.5e-5], N[ArcSin[1.0], $MachinePrecision], If[LessEqual[l, -2e-310], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] * N[((-l) / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3.3e-80], N[ArcSin[N[(l / N[(t$95$m / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 8.6e-34], N[ArcSin[1.0], $MachinePrecision], If[LessEqual[l, 85000000000000.0], N[ArcSin[N[(l / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]]]]]

Derivation

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1. Add Preprocessing

Alternative 6: 96.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t_m}{\ell} \leq -500:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{-\ell}{t_m}\right)\\ \mathbf{elif}\;\frac{t_m}{\ell} \leq 0.1:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t_m}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t_m}{\sqrt{0.5}}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (if (<= (/ t_m l) -500.0)
   (asin (* (sqrt 0.5) (/ (- l) t_m)))
   (if (<= (/ t_m l) 0.1)
     (asin (- 1.0 (pow (/ t_m l) 2.0)))
     (asin (/ l (/ t_m (sqrt 0.5)))))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	double tmp;
	if ((t_m / l) <= -500.0) {
		tmp = asin((sqrt(0.5) * (-l / t_m)));
	} else if ((t_m / l) <= 0.1) {
		tmp = asin((1.0 - pow((t_m / l), 2.0)));
	} else {
		tmp = asin((l / (t_m / sqrt(0.5))));
	}
	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) <= (-500.0d0)) then
        tmp = asin((sqrt(0.5d0) * (-l / t_m)))
    else if ((t_m / l) <= 0.1d0) then
        tmp = asin((1.0d0 - ((t_m / l) ** 2.0d0)))
    else
        tmp = asin((l / (t_m / sqrt(0.5d0))))
    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) <= -500.0) {
		tmp = Math.asin((Math.sqrt(0.5) * (-l / t_m)));
	} else if ((t_m / l) <= 0.1) {
		tmp = Math.asin((1.0 - Math.pow((t_m / l), 2.0)));
	} else {
		tmp = Math.asin((l / (t_m / Math.sqrt(0.5))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	tmp = 0
	if (t_m / l) <= -500.0:
		tmp = math.asin((math.sqrt(0.5) * (-l / t_m)))
	elif (t_m / l) <= 0.1:
		tmp = math.asin((1.0 - math.pow((t_m / l), 2.0)))
	else:
		tmp = math.asin((l / (t_m / math.sqrt(0.5))))
	return tmp

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l) <= -500.0)
		tmp = asin(Float64(sqrt(0.5) * Float64(Float64(-l) / t_m)));
	elseif (Float64(t_m / l) <= 0.1)
		tmp = asin(Float64(1.0 - (Float64(t_m / l) ^ 2.0)));
	else
		tmp = asin(Float64(l / Float64(t_m / sqrt(0.5))));
	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) <= -500.0)
		tmp = asin((sqrt(0.5) * (-l / t_m)));
	elseif ((t_m / l) <= 0.1)
		tmp = asin((1.0 - ((t_m / l) ^ 2.0)));
	else
		tmp = asin((l / (t_m / sqrt(0.5))));
	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], -500.0], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] * N[((-l) / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t$95$m / l), $MachinePrecision], 0.1], N[ArcSin[N[(1.0 - N[Power[N[(t$95$m / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l / N[(t$95$m / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

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1. Add Preprocessing

Alternative 7: 96.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t_m}{\ell} \leq -500:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t_m \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t_m}{\ell} \leq 0.1:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t_m}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t_m}{\sqrt{0.5}}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (if (<= (/ t_m l) -500.0)
   (asin (/ (- l) (* t_m (sqrt 2.0))))
   (if (<= (/ t_m l) 0.1)
     (asin (- 1.0 (pow (/ t_m l) 2.0)))
     (asin (/ l (/ t_m (sqrt 0.5)))))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	double tmp;
	if ((t_m / l) <= -500.0) {
		tmp = asin((-l / (t_m * sqrt(2.0))));
	} else if ((t_m / l) <= 0.1) {
		tmp = asin((1.0 - pow((t_m / l), 2.0)));
	} else {
		tmp = asin((l / (t_m / sqrt(0.5))));
	}
	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) <= (-500.0d0)) then
        tmp = asin((-l / (t_m * sqrt(2.0d0))))
    else if ((t_m / l) <= 0.1d0) then
        tmp = asin((1.0d0 - ((t_m / l) ** 2.0d0)))
    else
        tmp = asin((l / (t_m / sqrt(0.5d0))))
    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) <= -500.0) {
		tmp = Math.asin((-l / (t_m * Math.sqrt(2.0))));
	} else if ((t_m / l) <= 0.1) {
		tmp = Math.asin((1.0 - Math.pow((t_m / l), 2.0)));
	} else {
		tmp = Math.asin((l / (t_m / Math.sqrt(0.5))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	tmp = 0
	if (t_m / l) <= -500.0:
		tmp = math.asin((-l / (t_m * math.sqrt(2.0))))
	elif (t_m / l) <= 0.1:
		tmp = math.asin((1.0 - math.pow((t_m / l), 2.0)))
	else:
		tmp = math.asin((l / (t_m / math.sqrt(0.5))))
	return tmp

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l) <= -500.0)
		tmp = asin(Float64(Float64(-l) / Float64(t_m * sqrt(2.0))));
	elseif (Float64(t_m / l) <= 0.1)
		tmp = asin(Float64(1.0 - (Float64(t_m / l) ^ 2.0)));
	else
		tmp = asin(Float64(l / Float64(t_m / sqrt(0.5))));
	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) <= -500.0)
		tmp = asin((-l / (t_m * sqrt(2.0))));
	elseif ((t_m / l) <= 0.1)
		tmp = asin((1.0 - ((t_m / l) ^ 2.0)));
	else
		tmp = asin((l / (t_m / sqrt(0.5))));
	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], -500.0], N[ArcSin[N[((-l) / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t$95$m / l), $MachinePrecision], 0.1], N[ArcSin[N[(1.0 - N[Power[N[(t$95$m / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l / N[(t$95$m / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

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1. Add Preprocessing

Alternative 8: 63.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7.5 \cdot 10^{-178}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{-83} \lor \neg \left(\ell \leq 7 \cdot 10^{-32}\right) \land \ell \leq 85000000000000:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t_m \cdot \sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (if (<= l -7.5e-178)
   (asin 1.0)
   (if (or (<= l 3.4e-83) (and (not (<= l 7e-32)) (<= l 85000000000000.0)))
     (asin (/ l (* t_m (sqrt 2.0))))
     (asin 1.0))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	double tmp;
	if (l <= -7.5e-178) {
		tmp = asin(1.0);
	} else if ((l <= 3.4e-83) || (!(l <= 7e-32) && (l <= 85000000000000.0))) {
		tmp = asin((l / (t_m * sqrt(2.0))));
	} else {
		tmp = asin(1.0);
	}
	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 <= (-7.5d-178)) then
        tmp = asin(1.0d0)
    else if ((l <= 3.4d-83) .or. (.not. (l <= 7d-32)) .and. (l <= 85000000000000.0d0)) then
        tmp = asin((l / (t_m * sqrt(2.0d0))))
    else
        tmp = asin(1.0d0)
    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 <= -7.5e-178) {
		tmp = Math.asin(1.0);
	} else if ((l <= 3.4e-83) || (!(l <= 7e-32) && (l <= 85000000000000.0))) {
		tmp = Math.asin((l / (t_m * Math.sqrt(2.0))));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	tmp = 0
	if l <= -7.5e-178:
		tmp = math.asin(1.0)
	elif (l <= 3.4e-83) or (not (l <= 7e-32) and (l <= 85000000000000.0)):
		tmp = math.asin((l / (t_m * math.sqrt(2.0))))
	else:
		tmp = math.asin(1.0)
	return tmp

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	tmp = 0.0
	if (l <= -7.5e-178)
		tmp = asin(1.0);
	elseif ((l <= 3.4e-83) || (!(l <= 7e-32) && (l <= 85000000000000.0)))
		tmp = asin(Float64(l / Float64(t_m * sqrt(2.0))));
	else
		tmp = asin(1.0);
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(t_m, l, Om, Omc)
	tmp = 0.0;
	if (l <= -7.5e-178)
		tmp = asin(1.0);
	elseif ((l <= 3.4e-83) || (~((l <= 7e-32)) && (l <= 85000000000000.0)))
		tmp = asin((l / (t_m * sqrt(2.0))));
	else
		tmp = asin(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := If[LessEqual[l, -7.5e-178], N[ArcSin[1.0], $MachinePrecision], If[Or[LessEqual[l, 3.4e-83], And[N[Not[LessEqual[l, 7e-32]], $MachinePrecision], LessEqual[l, 85000000000000.0]]], N[ArcSin[N[(l / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]]

Derivation

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1. Add Preprocessing

Alternative 9: 63.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.7 \cdot 10^{-180}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-80}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t_m}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{-34}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 1900000000000:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t_m \cdot \sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (if (<= l -2.7e-180)
   (asin 1.0)
   (if (<= l 2.5e-80)
     (asin (/ l (/ t_m (sqrt 0.5))))
     (if (<= l 2.8e-34)
       (asin 1.0)
       (if (<= l 1900000000000.0)
         (asin (/ l (* t_m (sqrt 2.0))))
         (asin 1.0))))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	double tmp;
	if (l <= -2.7e-180) {
		tmp = asin(1.0);
	} else if (l <= 2.5e-80) {
		tmp = asin((l / (t_m / sqrt(0.5))));
	} else if (l <= 2.8e-34) {
		tmp = asin(1.0);
	} else if (l <= 1900000000000.0) {
		tmp = asin((l / (t_m * sqrt(2.0))));
	} else {
		tmp = asin(1.0);
	}
	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 <= (-2.7d-180)) then
        tmp = asin(1.0d0)
    else if (l <= 2.5d-80) then
        tmp = asin((l / (t_m / sqrt(0.5d0))))
    else if (l <= 2.8d-34) then
        tmp = asin(1.0d0)
    else if (l <= 1900000000000.0d0) then
        tmp = asin((l / (t_m * sqrt(2.0d0))))
    else
        tmp = asin(1.0d0)
    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 <= -2.7e-180) {
		tmp = Math.asin(1.0);
	} else if (l <= 2.5e-80) {
		tmp = Math.asin((l / (t_m / Math.sqrt(0.5))));
	} else if (l <= 2.8e-34) {
		tmp = Math.asin(1.0);
	} else if (l <= 1900000000000.0) {
		tmp = Math.asin((l / (t_m * Math.sqrt(2.0))));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	tmp = 0
	if l <= -2.7e-180:
		tmp = math.asin(1.0)
	elif l <= 2.5e-80:
		tmp = math.asin((l / (t_m / math.sqrt(0.5))))
	elif l <= 2.8e-34:
		tmp = math.asin(1.0)
	elif l <= 1900000000000.0:
		tmp = math.asin((l / (t_m * math.sqrt(2.0))))
	else:
		tmp = math.asin(1.0)
	return tmp

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	tmp = 0.0
	if (l <= -2.7e-180)
		tmp = asin(1.0);
	elseif (l <= 2.5e-80)
		tmp = asin(Float64(l / Float64(t_m / sqrt(0.5))));
	elseif (l <= 2.8e-34)
		tmp = asin(1.0);
	elseif (l <= 1900000000000.0)
		tmp = asin(Float64(l / Float64(t_m * sqrt(2.0))));
	else
		tmp = asin(1.0);
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(t_m, l, Om, Omc)
	tmp = 0.0;
	if (l <= -2.7e-180)
		tmp = asin(1.0);
	elseif (l <= 2.5e-80)
		tmp = asin((l / (t_m / sqrt(0.5))));
	elseif (l <= 2.8e-34)
		tmp = asin(1.0);
	elseif (l <= 1900000000000.0)
		tmp = asin((l / (t_m * sqrt(2.0))));
	else
		tmp = asin(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := If[LessEqual[l, -2.7e-180], N[ArcSin[1.0], $MachinePrecision], If[LessEqual[l, 2.5e-80], N[ArcSin[N[(l / N[(t$95$m / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.8e-34], N[ArcSin[1.0], $MachinePrecision], If[LessEqual[l, 1900000000000.0], N[ArcSin[N[(l / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]]]]

Derivation

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

Alternative 10: 50.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \sin^{-1} 1 \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc) :precision binary64 (asin 1.0))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	return asin(1.0);
}

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(1.0d0)
end function

Java

t_m = Math.abs(t);
public static double code(double t_m, double l, double Om, double Omc) {
	return Math.asin(1.0);
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	return math.asin(1.0)

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	return asin(1.0)
end

MATLAB

t_m = abs(t);
function tmp = code(t_m, l, Om, Omc)
	tmp = asin(1.0);
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := N[ArcSin[1.0], $MachinePrecision]

Derivation

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

Reproduce

herbie shell --seed 2023364 
(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)))))))