Toniolo and Linder, Equation (2)

Percentage Accurate: 84.2% → 98.2%

Time: 11.8s

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: 84.2% 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: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t_m}{\ell} \leq -5 \cdot 10^{+163}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t_m}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t_m}{\ell} \leq 2 \cdot 10^{+131}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t_m}{\ell}}{\frac{\ell}{t_m}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (if (<= (/ t_m l) -5e+163)
   (asin (/ (- l) (/ t_m (sqrt 0.5))))
   (if (<= (/ t_m l) 2e+131)
     (asin
      (sqrt
       (/
        (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
        (+ 1.0 (* 2.0 (/ (/ t_m l) (/ l t_m)))))))
     (asin (* (sqrt (- 1.0 (pow (/ Om Omc) 2.0))) (/ (* l (sqrt 0.5)) t_m))))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	double tmp;
	if ((t_m / l) <= -5e+163) {
		tmp = asin((-l / (t_m / sqrt(0.5))));
	} else if ((t_m / l) <= 2e+131) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))));
	} else {
		tmp = asin((sqrt((1.0 - pow((Om / Omc), 2.0))) * ((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) :: tmp
    if ((t_m / l) <= (-5d+163)) then
        tmp = asin((-l / (t_m / sqrt(0.5d0))))
    else if ((t_m / l) <= 2d+131) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * ((t_m / l) / (l / t_m)))))))
    else
        tmp = asin((sqrt((1.0d0 - ((om / omc) ** 2.0d0))) * ((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 tmp;
	if ((t_m / l) <= -5e+163) {
		tmp = Math.asin((-l / (t_m / Math.sqrt(0.5))));
	} else if ((t_m / l) <= 2e+131) {
		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((Math.sqrt((1.0 - Math.pow((Om / Omc), 2.0))) * ((l * Math.sqrt(0.5)) / t_m)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	tmp = 0
	if (t_m / l) <= -5e+163:
		tmp = math.asin((-l / (t_m / math.sqrt(0.5))))
	elif (t_m / l) <= 2e+131:
		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((math.sqrt((1.0 - math.pow((Om / Omc), 2.0))) * ((l * math.sqrt(0.5)) / t_m)))
	return tmp

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l) <= -5e+163)
		tmp = asin(Float64(Float64(-l) / Float64(t_m / sqrt(0.5))));
	elseif (Float64(t_m / l) <= 2e+131)
		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(sqrt(Float64(1.0 - (Float64(Om / Omc) ^ 2.0))) * Float64(Float64(l * sqrt(0.5)) / t_m)));
	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) <= -5e+163)
		tmp = asin((-l / (t_m / sqrt(0.5))));
	elseif ((t_m / l) <= 2e+131)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))));
	else
		tmp = asin((sqrt((1.0 - ((Om / Omc) ^ 2.0))) * ((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_] := If[LessEqual[N[(t$95$m / l), $MachinePrecision], -5e+163], N[ArcSin[N[((-l) / N[(t$95$m / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t$95$m / l), $MachinePrecision], 2e+131], 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[(N[Sqrt[N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -5e163

   1. Initial program 48.3%

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

   2. Taylor expanded in t around -inf 87.9%

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

      1. mul-1-neg 87.9%

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

      2. *-commutative 87.9%

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

      3. distribute-rgt-neg-in 87.9%

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

      4. unpow2 87.9%

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

      5. unpow2 87.9%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\right) \]

      6. times-frac 99.7%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\right) \]

      7. unpow2 99.7%

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

      8. associate-/l* 99.7%

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

      9. associate-/r/ 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in Om around 0 99.7%

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

      1. associate-*l/ 99.6%

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

      2. neg-mul-1 99.6%

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

      3. distribute-rgt-neg-in 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in l around 0 99.7%

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

      1. mul-1-neg 99.7%

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

      2. associate-/l* 99.7%

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

      3. distribute-neg-frac 99.7%

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

   10. Simplified 99.7%

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

   if -5e163 < (/.f64 t l) < 1.9999999999999998e131

   1. Initial program 99.2%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. unpow2 99.2%

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

      2. clear-num 99.2%

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

      3. un-div-inv 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. unpow2 99.2%

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

      2. clear-num 99.2%

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

      3. un-div-inv 99.2%

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

   5. Applied egg-rr 99.2%

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

   if 1.9999999999999998e131 < (/.f64 t l)

   1. Initial program 55.7%

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

   2. Taylor expanded in t around inf 86.8%

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

      1. *-commutative 86.8%

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

      2. unpow2 86.8%

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

      3. unpow2 86.8%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right) \]

      4. times-frac 99.8%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right) \]

      5. unpow2 99.8%

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

      6. associate-/l* 99.7%

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

      7. associate-/r/ 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in l around 0 99.8%

      \[\leadsto \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \color{blue}{\frac{\ell \cdot \sqrt{0.5}}{t}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+163}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+131}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 2: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-307} \lor \neg \left(\ell \leq 1.65 \cdot 10^{-165}\right):\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t_m}{\ell}}{\frac{\ell}{t_m}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(\ell \cdot \frac{\sqrt{0.5}}{t_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (if (or (<= l 5e-307) (not (<= l 1.65e-165)))
   (asin
    (sqrt
     (/
      (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
      (+ 1.0 (* 2.0 (/ (/ t_m l) (/ l t_m)))))))
   (asin (* (sqrt (- 1.0 (pow (/ Om Omc) 2.0))) (* l (/ (sqrt 0.5) t_m))))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	double tmp;
	if ((l <= 5e-307) || !(l <= 1.65e-165)) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))));
	} else {
		tmp = asin((sqrt((1.0 - pow((Om / Omc), 2.0))) * (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) :: tmp
    if ((l <= 5d-307) .or. (.not. (l <= 1.65d-165))) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * ((t_m / l) / (l / t_m)))))))
    else
        tmp = asin((sqrt((1.0d0 - ((om / omc) ** 2.0d0))) * (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 tmp;
	if ((l <= 5e-307) || !(l <= 1.65e-165)) {
		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((Math.sqrt((1.0 - Math.pow((Om / Omc), 2.0))) * (l * (Math.sqrt(0.5) / t_m))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	tmp = 0
	if (l <= 5e-307) or not (l <= 1.65e-165):
		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((math.sqrt((1.0 - math.pow((Om / Omc), 2.0))) * (l * (math.sqrt(0.5) / t_m))))
	return tmp

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	tmp = 0.0
	if ((l <= 5e-307) || !(l <= 1.65e-165))
		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(sqrt(Float64(1.0 - (Float64(Om / Omc) ^ 2.0))) * 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)
	tmp = 0.0;
	if ((l <= 5e-307) || ~((l <= 1.65e-165)))
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))));
	else
		tmp = asin((sqrt((1.0 - ((Om / Omc) ^ 2.0))) * (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_] := If[Or[LessEqual[l, 5e-307], N[Not[LessEqual[l, 1.65e-165]], $MachinePrecision]], 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[(N[Sqrt[N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.00000000000000014e-307 or 1.6499999999999999e-165 < l

   1. Initial program 89.5%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. unpow2 89.5%

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

      2. clear-num 89.5%

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

      3. un-div-inv 89.5%

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

   3. Applied egg-rr 89.5%

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

      1. unpow2 89.5%

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

      2. clear-num 89.5%

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

      3. un-div-inv 89.5%

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

   5. Applied egg-rr 89.5%

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

   if 5.00000000000000014e-307 < l < 1.6499999999999999e-165

   1. Initial program 70.6%

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

   2. Taylor expanded in t around inf 51.3%

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

      1. *-commutative 51.3%

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

      2. unpow2 51.3%

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

      3. unpow2 51.3%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right) \]

      4. times-frac 57.4%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right) \]

      5. unpow2 57.4%

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

      6. associate-/l* 57.4%

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

      7. associate-/r/ 57.3%

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

   4. Simplified 57.3%

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

   5. Taylor expanded in l around 0 57.4%

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

      1. associate-*r/ 57.4%

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

   7. Simplified 57.4%

      \[\leadsto \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \color{blue}{\left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-307} \lor \neg \left(\ell \leq 1.65 \cdot 10^{-165}\right):\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\right)\\ \end{array} \]

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (asin
  (/
   (sqrt (- 1.0 (/ (/ Om Omc) (/ Omc Om))))
   (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 - ((Om / Omc) / (Omc / Om)))) / 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 - ((Om / Omc) / (Omc / Om)))) / 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 - ((Om / Omc) / (Omc / Om)))) / 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(Float64(Om / Omc) / Float64(Omc / Om)))) / 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) / (Omc / Om)))) / 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[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $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

1. Initial program 86.9%

   \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
2. Step-by-step derivation

   1. sqrt-div 86.9%

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

   2. div-inv 86.9%

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

   3. add-sqr-sqrt 86.9%

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

   4. hypot-1-def 86.9%

      \[\leadsto \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right) \]

   5. *-commutative 86.9%

      \[\leadsto \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right)}\right) \]

   6. sqrt-prod 86.9%

      \[\leadsto \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2}}\right)}\right) \]

   7. unpow2 86.9%

      \[\leadsto \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}} \cdot \sqrt{2}\right)}\right) \]

   8. sqrt-prod 48.9%

      \[\leadsto \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{t}{\ell}} \cdot \sqrt{\frac{t}{\ell}}\right)} \cdot \sqrt{2}\right)}\right) \]

   9. add-sqr-sqrt 99.1%

      \[\leadsto \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell}} \cdot \sqrt{2}\right)}\right) \]

3. Applied egg-rr 99.1%

   \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]
4. Step-by-step derivation

   1. associate-*r/ 99.1%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot 1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]

   2. *-rgt-identity 99.1%

      \[\leadsto \sin^{-1} \left(\frac{\color{blue}{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]

5. Simplified 99.1%

   \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]
6. Step-by-step derivation

   1. unpow2 86.9%

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

   2. clear-num 86.9%

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

   3. un-div-inv 86.9%

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

7. Applied egg-rr 99.1%

   \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - \color{blue}{\frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]

8. Final simplification 99.1%

   \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]

Alternative 4: 85.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := 1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\\ \mathbf{if}\;\ell \leq 5 \cdot 10^{-307} \lor \neg \left(\ell \leq 1.4 \cdot 10^{-165}\right):\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \frac{\frac{t_m}{\ell}}{\frac{\ell}{t_m}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{t_1} \cdot \left(\sqrt{0.5} \cdot \frac{\ell}{t_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (let* ((t_1 (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))
   (if (or (<= l 5e-307) (not (<= l 1.4e-165)))
     (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (/ (/ t_m l) (/ l t_m)))))))
     (asin (* (sqrt t_1) (* (sqrt 0.5) (/ l t_m)))))))

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 ((l <= 5e-307) || !(l <= 1.4e-165)) {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))));
	} else {
		tmp = asin((sqrt(t_1) * (sqrt(0.5) * (l / 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) / (omc / om))
    if ((l <= 5d-307) .or. (.not. (l <= 1.4d-165))) then
        tmp = asin(sqrt((t_1 / (1.0d0 + (2.0d0 * ((t_m / l) / (l / t_m)))))))
    else
        tmp = asin((sqrt(t_1) * (sqrt(0.5d0) * (l / 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 - ((Om / Omc) / (Omc / Om));
	double tmp;
	if ((l <= 5e-307) || !(l <= 1.4e-165)) {
		tmp = Math.asin(Math.sqrt((t_1 / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))));
	} else {
		tmp = Math.asin((Math.sqrt(t_1) * (Math.sqrt(0.5) * (l / t_m))));
	}
	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 (l <= 5e-307) or not (l <= 1.4e-165):
		tmp = math.asin(math.sqrt((t_1 / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))))
	else:
		tmp = math.asin((math.sqrt(t_1) * (math.sqrt(0.5) * (l / t_m))))
	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 ((l <= 5e-307) || !(l <= 1.4e-165))
		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(sqrt(t_1) * Float64(sqrt(0.5) * Float64(l / 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) / (Omc / Om));
	tmp = 0.0;
	if ((l <= 5e-307) || ~((l <= 1.4e-165)))
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))));
	else
		tmp = asin((sqrt(t_1) * (sqrt(0.5) * (l / 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[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[l, 5e-307], N[Not[LessEqual[l, 1.4e-165]], $MachinePrecision]], 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[Sqrt[t$95$1], $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 5.00000000000000014e-307 or 1.4e-165 < l

   1. Initial program 89.5%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. unpow2 89.5%

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

      2. clear-num 89.5%

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

      3. un-div-inv 89.5%

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

   3. Applied egg-rr 89.5%

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

      1. unpow2 89.5%

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

      2. clear-num 89.5%

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

      3. un-div-inv 89.5%

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

   5. Applied egg-rr 89.5%

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

   if 5.00000000000000014e-307 < l < 1.4e-165

   1. Initial program 70.6%

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

   2. Taylor expanded in t around inf 51.3%

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

      1. *-commutative 51.3%

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

      2. unpow2 51.3%

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

      3. unpow2 51.3%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right) \]

      4. times-frac 57.4%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right) \]

      5. unpow2 57.4%

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

      6. associate-/l* 57.4%

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

      7. associate-/r/ 57.3%

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

   4. Simplified 57.3%

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

      1. unpow2 70.6%

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

      2. clear-num 70.6%

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

      3. un-div-inv 70.6%

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

   6. Applied egg-rr 57.3%

      \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}} \cdot \left(\frac{\ell}{t} \cdot \sqrt{0.5}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-307} \lor \neg \left(\ell \leq 1.4 \cdot 10^{-165}\right):\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}} \cdot \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\right)\\ \end{array} \]

Alternative 5: 84.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

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(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * ((t_m / l) / (l / t_m)))))))
end function

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 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.9%

   \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
2. Step-by-step derivation

   1. unpow2 86.9%

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

   2. clear-num 86.9%

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

   3. un-div-inv 86.9%

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

3. Applied egg-rr 86.9%

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

   1. unpow2 86.9%

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

   2. clear-num 86.9%

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

   3. un-div-inv 86.9%

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

5. Applied egg-rr 86.9%

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

6. Final simplification 86.9%

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

Alternative 6: 64.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;t_m \leq 4.5 \cdot 10^{+29}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t_m}{\sqrt{0.5}}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (if (<= t_m 4.5e+29)
   (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 <= 4.5e+29) {
		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 <= 4.5d+29) 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 <= 4.5e+29) {
		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 <= 4.5e+29:
		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 (t_m <= 4.5e+29)
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))));
	else
		tmp = asin(Float64(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 <= 4.5e+29)
		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[t$95$m, 4.5e+29], 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

1. Split input into 2 regimes

2. if t < 4.5000000000000002e29

   1. Initial program 90.9%

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

   2. Taylor expanded in t around 0 51.9%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]
   3. Step-by-step derivation

      1. unpow2 51.9%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}}\right) \]

      2. unpow2 51.9%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}}\right) \]

      3. times-frac 61.6%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}}\right) \]

      4. unpow2 61.6%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}}\right) \]

   4. Simplified 61.6%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}\right)} \]
   5. Step-by-step derivation

      1. unpow2 90.9%

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

      2. clear-num 90.9%

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

      3. un-div-inv 90.9%

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

   6. Applied egg-rr 61.6%

      \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}}\right) \]

   if 4.5000000000000002e29 < t

   1. Initial program 70.0%

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

   2. Taylor expanded in t around -inf 61.6%

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

      1. mul-1-neg 61.6%

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

      2. *-commutative 61.6%

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

      3. distribute-rgt-neg-in 61.6%

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

      4. unpow2 61.6%

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

      5. unpow2 61.6%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\right) \]

      6. times-frac 65.8%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\right) \]

      7. unpow2 65.8%

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

      8. associate-/l* 65.9%

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

      9. associate-/r/ 65.8%

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

   4. Simplified 65.8%

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

   5. Taylor expanded in Om around 0 65.8%

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

      1. associate-*l/ 65.8%

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

      2. neg-mul-1 65.8%

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

      3. distribute-rgt-neg-in 65.8%

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

   7. Simplified 65.8%

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

   8. Taylor expanded in l around 0 65.8%

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

      1. mul-1-neg 65.8%

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

      2. associate-/l* 65.9%

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

      3. distribute-neg-frac 65.9%

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

   10. Simplified 65.9%

       \[\leadsto \sin^{-1} \color{blue}{\left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{+29}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \]

Alternative 7: 31.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \sin^{-1} \left(\frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t_m}\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc) :precision binary64 (asin (/ (* l (- (sqrt 0.5))) t_m)))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	return asin(((l * -sqrt(0.5)) / t_m));
}

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 * -sqrt(0.5d0)) / t_m))
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 * -Math.sqrt(0.5)) / t_m));
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	return math.asin(((l * -math.sqrt(0.5)) / t_m))

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	return asin(Float64(Float64(l * Float64(-sqrt(0.5))) / t_m))
end

MATLAB

t_m = abs(t);
function tmp = code(t_m, l, Om, Omc)
	tmp = asin(((l * -sqrt(0.5)) / t_m));
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := N[ArcSin[N[(N[(l * (-N[Sqrt[0.5], $MachinePrecision])), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 86.9%

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

2. Taylor expanded in t around -inf 30.2%

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

   1. mul-1-neg 30.2%

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

   2. *-commutative 30.2%

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

   3. associate-/l* 30.2%

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

4. Simplified 30.2%

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

   1. unpow2 30.2%

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

   2. unpow2 30.2%

      \[\leadsto \sin^{-1} \left(-\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \frac{\ell}{\frac{t}{\sqrt{0.5}}}\right) \]

   3. frac-times 33.7%

      \[\leadsto \sin^{-1} \left(-\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \frac{\ell}{\frac{t}{\sqrt{0.5}}}\right) \]

6. Applied egg-rr 33.7%

   \[\leadsto \sin^{-1} \left(-\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \frac{\ell}{\frac{t}{\sqrt{0.5}}}\right) \]

7. Taylor expanded in Om around 0 33.7%

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

8. Final simplification 33.7%

   \[\leadsto \sin^{-1} \left(\frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right) \]

Alternative 8: 31.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \sin^{-1} \left(\sqrt{0.5} \cdot \frac{-\ell}{t_m}\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc) :precision binary64 (asin (* (sqrt 0.5) (/ (- l) t_m))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	return asin((sqrt(0.5) * (-l / t_m)));
}

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((sqrt(0.5d0) * (-l / t_m)))
end function

Java

t_m = Math.abs(t);
public static double code(double t_m, double l, double Om, double Omc) {
	return Math.asin((Math.sqrt(0.5) * (-l / t_m)));
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	return math.asin((math.sqrt(0.5) * (-l / t_m)))

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	return asin(Float64(sqrt(0.5) * Float64(Float64(-l) / t_m)))
end

MATLAB

t_m = abs(t);
function tmp = code(t_m, l, Om, Omc)
	tmp = asin((sqrt(0.5) * (-l / t_m)));
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] * N[((-l) / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 86.9%

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

2. Taylor expanded in t around -inf 30.2%

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

   1. mul-1-neg 30.2%

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

   2. *-commutative 30.2%

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

   3. distribute-rgt-neg-in 30.2%

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

   4. unpow2 30.2%

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

   5. unpow2 30.2%

      \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\right) \]

   6. times-frac 33.7%

      \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\right) \]

   7. unpow2 33.7%

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

   8. associate-/l* 33.7%

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

   9. associate-/r/ 33.7%

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

4. Simplified 33.7%

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

5. Taylor expanded in Om around 0 33.7%

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

   1. associate-*l/ 33.7%

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

   2. neg-mul-1 33.7%

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

   3. distribute-rgt-neg-in 33.7%

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

7. Simplified 33.7%

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

8. Final simplification 33.7%

   \[\leadsto \sin^{-1} \left(\sqrt{0.5} \cdot \frac{-\ell}{t}\right) \]

Alternative 9: 31.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \sin^{-1} \left(\frac{-\ell}{\frac{t_m}{\sqrt{0.5}}}\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc) :precision binary64 (asin (/ (- l) (/ t_m (sqrt 0.5)))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	return asin((-l / (t_m / sqrt(0.5))));
}

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(0.5d0))))
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(0.5))));
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	return math.asin((-l / (t_m / math.sqrt(0.5))))

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	return asin(Float64(Float64(-l) / Float64(t_m / sqrt(0.5))))
end

MATLAB

t_m = abs(t);
function tmp = code(t_m, l, Om, Omc)
	tmp = asin((-l / (t_m / sqrt(0.5))));
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[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 86.9%

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

2. Taylor expanded in t around -inf 30.2%

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

   1. mul-1-neg 30.2%

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

   2. *-commutative 30.2%

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

   3. distribute-rgt-neg-in 30.2%

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

   4. unpow2 30.2%

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

   5. unpow2 30.2%

      \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\right) \]

   6. times-frac 33.7%

      \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\right) \]

   7. unpow2 33.7%

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

   8. associate-/l* 33.7%

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

   9. associate-/r/ 33.7%

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

4. Simplified 33.7%

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

5. Taylor expanded in Om around 0 33.7%

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

   1. associate-*l/ 33.7%

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

   2. neg-mul-1 33.7%

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

   3. distribute-rgt-neg-in 33.7%

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

7. Simplified 33.7%

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

8. Taylor expanded in l around 0 33.7%

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

   1. mul-1-neg 33.7%

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

   2. associate-/l* 33.7%

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

   3. distribute-neg-frac 33.7%

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

10. Simplified 33.7%

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

11. Final simplification 33.7%

    \[\leadsto \sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right) \]

Alternative 10: 0.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \sin^{-1} \left(\frac{\ell \cdot \sqrt{-0.5}}{t_m}\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc) :precision binary64 (asin (/ (* l (sqrt -0.5)) t_m)))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	return asin(((l * sqrt(-0.5)) / t_m));
}

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 * sqrt((-0.5d0))) / t_m))
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 * Math.sqrt(-0.5)) / t_m));
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	return math.asin(((l * math.sqrt(-0.5)) / t_m))

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	return asin(Float64(Float64(l * sqrt(-0.5)) / t_m))
end

MATLAB

t_m = abs(t);
function tmp = code(t_m, l, Om, Omc)
	tmp = asin(((l * sqrt(-0.5)) / t_m));
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := N[ArcSin[N[(N[(l * N[Sqrt[-0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 86.9%

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

2. Taylor expanded in t around -inf 30.2%

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

   1. mul-1-neg 30.2%

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

   2. *-commutative 30.2%

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

   3. distribute-rgt-neg-in 30.2%

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

   4. unpow2 30.2%

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

   5. unpow2 30.2%

      \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\right) \]

   6. times-frac 33.7%

      \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\right) \]

   7. unpow2 33.7%

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

   8. associate-/l* 33.7%

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

   9. associate-/r/ 33.7%

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

4. Simplified 33.7%

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

5. Taylor expanded in Om around 0 33.7%

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

   1. associate-*l/ 33.7%

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

   2. neg-mul-1 33.7%

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

   3. distribute-rgt-neg-in 33.7%

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

7. Simplified 33.7%

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

9. Applied egg-rr 0.0%

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

10. Final simplification 0.0%

    \[\leadsto \sin^{-1} \left(\frac{\ell \cdot \sqrt{-0.5}}{t}\right) \]

Reproduce

herbie shell --seed 2023318 
(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)))))))