Toniolo and Linder, Equation (2)

Percentage Accurate: 84.1% → 97.9%

Time: 23.0s

Alternatives: 11

Speedup: 1.3×

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: 84.1% 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: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t_m}{\ell} \leq -1 \cdot 10^{+170}:\\ \;\;\;\;\left|\sin^{-1} \left(\frac{\ell}{\frac{t_m}{\sqrt{0.5}}}\right)\right|\\ \mathbf{elif}\;\frac{t_m}{\ell} \leq 10^{+118}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \frac{\frac{t_m}{\ell}}{\frac{\ell}{t_m}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{t_1} \cdot \frac{\ell}{t_m \cdot \sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (let* ((t_1 (- 1.0 (pow (/ Om Omc) 2.0))))
   (if (<= (/ t_m l) -1e+170)
     (fabs (asin (/ l (/ t_m (sqrt 0.5)))))
     (if (<= (/ t_m l) 1e+118)
       (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (/ (/ t_m l) (/ l t_m)))))))
       (asin (* (sqrt t_1) (/ l (* t_m (sqrt 2.0)))))))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	double t_1 = 1.0 - pow((Om / Omc), 2.0);
	double tmp;
	if ((t_m / l) <= -1e+170) {
		tmp = fabs(asin((l / (t_m / sqrt(0.5)))));
	} else if ((t_m / l) <= 1e+118) {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))));
	} else {
		tmp = asin((sqrt(t_1) * (l / (t_m * sqrt(2.0)))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(t_m, l, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 - ((om / omc) ** 2.0d0)
    if ((t_m / l) <= (-1d+170)) then
        tmp = abs(asin((l / (t_m / sqrt(0.5d0)))))
    else if ((t_m / l) <= 1d+118) then
        tmp = asin(sqrt((t_1 / (1.0d0 + (2.0d0 * ((t_m / l) / (l / t_m)))))))
    else
        tmp = asin((sqrt(t_1) * (l / (t_m * sqrt(2.0d0)))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double t_m, double l, double Om, double Omc) {
	double t_1 = 1.0 - Math.pow((Om / Omc), 2.0);
	double tmp;
	if ((t_m / l) <= -1e+170) {
		tmp = Math.abs(Math.asin((l / (t_m / Math.sqrt(0.5)))));
	} else if ((t_m / l) <= 1e+118) {
		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) * (l / (t_m * Math.sqrt(2.0)))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	t_1 = 1.0 - math.pow((Om / Omc), 2.0)
	tmp = 0
	if (t_m / l) <= -1e+170:
		tmp = math.fabs(math.asin((l / (t_m / math.sqrt(0.5)))))
	elif (t_m / l) <= 1e+118:
		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) * (l / (t_m * math.sqrt(2.0)))))
	return tmp

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	t_1 = Float64(1.0 - (Float64(Om / Omc) ^ 2.0))
	tmp = 0.0
	if (Float64(t_m / l) <= -1e+170)
		tmp = abs(asin(Float64(l / Float64(t_m / sqrt(0.5)))));
	elseif (Float64(t_m / l) <= 1e+118)
		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(l / Float64(t_m * sqrt(2.0)))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(t_m, l, Om, Omc)
	t_1 = 1.0 - ((Om / Omc) ^ 2.0);
	tmp = 0.0;
	if ((t_m / l) <= -1e+170)
		tmp = abs(asin((l / (t_m / sqrt(0.5)))));
	elseif ((t_m / l) <= 1e+118)
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))));
	else
		tmp = asin((sqrt(t_1) * (l / (t_m * sqrt(2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := Block[{t$95$1 = N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$m / l), $MachinePrecision], -1e+170], N[Abs[N[ArcSin[N[(l / N[(t$95$m / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t$95$m / l), $MachinePrecision], 1e+118], 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[(l / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -1.00000000000000003e170

   1. Initial program 68.1%

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

      1. unpow2 68.1%

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

      2. frac-2neg 68.1%

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

      3. frac-times 68.1%

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

   3. Applied egg-rr 68.1%

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

   4. Taylor expanded in t around inf 59.3%

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

      1. *-commutative 59.3%

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

      2. associate-*l/ 59.3%

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

      3. *-commutative 59.3%

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

      4. associate-*l* 59.3%

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

      5. unpow2 59.3%

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

      6. unpow2 59.3%

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

      7. times-frac 67.6%

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

      8. unpow2 67.6%

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

   6. Simplified 67.6%

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

   7. Taylor expanded in Om around 0 67.6%

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

      1. associate-/l* 67.6%

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

   9. Simplified 67.6%

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

       1. add-sqr-sqrt 65.8%

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

       2. sqrt-unprod 68.1%

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

       3. pow2 68.1%

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

       4. associate-/r/ 68.1%

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

   11. Applied egg-rr 68.1%

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

       1. unpow2 68.1%

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

       2. rem-sqrt-square 99.9%

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

       3. associate-*l/ 99.8%

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

       4. associate-/l* 99.8%

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

   13. Simplified 99.8%

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

   if -1.00000000000000003e170 < (/.f64 t l) < 9.99999999999999967e117

   1. Initial program 97.7%

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

      1. unpow2 97.7%

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

      2. clear-num 97.7%

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

      3. un-div-inv 97.8%

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

   3. Applied egg-rr 97.8%

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

   if 9.99999999999999967e117 < (/.f64 t l)

   1. Initial program 54.4%

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

      1. sqrt-div 54.3%

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

      2. div-inv 54.3%

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

      3. add-sqr-sqrt 54.3%

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

      4. hypot-1-def 54.3%

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

      5. *-commutative 54.3%

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

      6. sqrt-prod 54.4%

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

      7. unpow2 54.4%

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

      8. sqrt-prod 99.2%

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

      9. add-sqr-sqrt 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.6%

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

      2. *-rgt-identity 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in t around inf 81.9%

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

      1. *-commutative 81.9%

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

      2. unpow2 81.9%

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

      3. unpow2 81.9%

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

      4. times-frac 99.8%

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

      5. unpow2 99.8%

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

   8. Simplified 99.8%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+170}:\\ \;\;\;\;\left|\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\right|\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+118}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\ell}{t \cdot \sqrt{2}}\right)\\ \end{array} \]

Alternative 2: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\frac{\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}}{\mathsf{hypot}\left(1, \frac{t_m \cdot \sqrt{2}}{\ell}\right)}\right)\right)\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (expm1
  (log1p
   (asin
    (/
     (sqrt (- 1.0 (/ (/ Om Omc) (/ Omc Om))))
     (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 expm1(log1p(asin((sqrt((1.0 - ((Om / Omc) / (Omc / Om)))) / 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.expm1(Math.log1p(Math.asin((Math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))) / 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.expm1(math.log1p(math.asin((math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))) / math.hypot(1.0, ((t_m * math.sqrt(2.0)) / l))))))

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	return expm1(log1p(asin(Float64(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))) / hypot(1.0, Float64(Float64(t_m * sqrt(2.0)) / l))))))
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := N[(Exp[N[Log[1 + 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 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]

Derivation

1. Initial program 86.7%

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

   1. sqrt-div 86.6%

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

   2. div-inv 86.6%

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

   3. add-sqr-sqrt 86.6%

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

   4. hypot-1-def 86.6%

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

   5. *-commutative 86.6%

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

   6. sqrt-prod 86.6%

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

   7. unpow2 86.6%

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

   8. sqrt-prod 58.3%

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

   9. add-sqr-sqrt 98.5%

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

3. Applied egg-rr 98.5%

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

   1. associate-*r/ 98.5%

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

   2. *-rgt-identity 98.5%

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

5. Simplified 98.5%

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

   1. unpow2 50.1%

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

   2. clear-num 50.1%

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

   3. un-div-inv 50.1%

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

7. Applied egg-rr 98.5%

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

   1. expm1-log1p-u 98.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\frac{\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)\right)\right)} \]

   2. sub-neg 98.5%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\frac{\sqrt{\color{blue}{1 + \left(-\frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\right)}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)\right)\right) \]

   3. div-inv 98.5%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\frac{\sqrt{1 + \left(-\color{blue}{\frac{Om}{Omc} \cdot \frac{1}{\frac{Omc}{Om}}}\right)}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)\right)\right) \]

   4. clear-num 98.5%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\frac{\sqrt{1 + \left(-\frac{Om}{Omc} \cdot \color{blue}{\frac{Om}{Omc}}\right)}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)\right)\right) \]

   5. unpow2 98.5%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\frac{\sqrt{1 + \left(-\color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}\right)}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)\right)\right) \]

   6. sub-neg 98.5%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\frac{\sqrt{\color{blue}{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)\right)\right) \]

   7. associate-*l/ 98.5%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\frac{t \cdot \sqrt{2}}{\ell}}\right)}\right)\right)\right) \]

9. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t \cdot \sqrt{2}}{\ell}\right)}\right)\right)\right)} \]
10. Step-by-step derivation

    1. unpow2 50.1%

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

    2. clear-num 50.1%

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

    3. un-div-inv 50.1%

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

11. Applied egg-rr 98.5%

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\frac{\sqrt{1 - \color{blue}{\frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}}}{\mathsf{hypot}\left(1, \frac{t \cdot \sqrt{2}}{\ell}\right)}\right)\right)\right) \]

12. Final simplification 98.5%

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\frac{\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}}{\mathsf{hypot}\left(1, \frac{t \cdot \sqrt{2}}{\ell}\right)}\right)\right)\right) \]

Alternative 3: 98.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;{\left(\frac{t_m}{\ell}\right)}^{2} \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \frac{\frac{t_m}{\ell}}{\frac{\ell}{t_m}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\sin^{-1} \left(\ell \cdot \frac{\sqrt{t_1 \cdot 0.5}}{t_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (let* ((t_1 (- 1.0 (pow (/ Om Omc) 2.0))))
   (if (<= (pow (/ t_m l) 2.0) 2e+302)
     (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (/ (/ t_m l) (/ l t_m)))))))
     (fabs (asin (* l (/ (sqrt (* t_1 0.5)) t_m)))))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	double t_1 = 1.0 - pow((Om / Omc), 2.0);
	double tmp;
	if (pow((t_m / l), 2.0) <= 2e+302) {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))));
	} else {
		tmp = fabs(asin((l * (sqrt((t_1 * 0.5)) / t_m))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(t_m, l, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 - ((om / omc) ** 2.0d0)
    if (((t_m / l) ** 2.0d0) <= 2d+302) then
        tmp = asin(sqrt((t_1 / (1.0d0 + (2.0d0 * ((t_m / l) / (l / t_m)))))))
    else
        tmp = abs(asin((l * (sqrt((t_1 * 0.5d0)) / t_m))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double t_m, double l, double Om, double Omc) {
	double t_1 = 1.0 - Math.pow((Om / Omc), 2.0);
	double tmp;
	if (Math.pow((t_m / l), 2.0) <= 2e+302) {
		tmp = Math.asin(Math.sqrt((t_1 / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))));
	} else {
		tmp = Math.abs(Math.asin((l * (Math.sqrt((t_1 * 0.5)) / t_m))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	t_1 = 1.0 - math.pow((Om / Omc), 2.0)
	tmp = 0
	if math.pow((t_m / l), 2.0) <= 2e+302:
		tmp = math.asin(math.sqrt((t_1 / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))))
	else:
		tmp = math.fabs(math.asin((l * (math.sqrt((t_1 * 0.5)) / t_m))))
	return tmp

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	t_1 = Float64(1.0 - (Float64(Om / Omc) ^ 2.0))
	tmp = 0.0
	if ((Float64(t_m / l) ^ 2.0) <= 2e+302)
		tmp = asin(sqrt(Float64(t_1 / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m / l) / Float64(l / t_m)))))));
	else
		tmp = abs(asin(Float64(l * Float64(sqrt(Float64(t_1 * 0.5)) / t_m))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(t_m, l, Om, Omc)
	t_1 = 1.0 - ((Om / Omc) ^ 2.0);
	tmp = 0.0;
	if (((t_m / l) ^ 2.0) <= 2e+302)
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))));
	else
		tmp = abs(asin((l * (sqrt((t_1 * 0.5)) / t_m))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := Block[{t$95$1 = N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[N[(t$95$m / l), $MachinePrecision], 2.0], $MachinePrecision], 2e+302], 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[Abs[N[ArcSin[N[(l * N[(N[Sqrt[N[(t$95$1 * 0.5), $MachinePrecision]], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 t l) 2) < 2.0000000000000002e302

   1. Initial program 98.3%

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

      1. unpow2 98.3%

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

      2. clear-num 98.3%

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

      3. un-div-inv 98.3%

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

   3. Applied egg-rr 98.3%

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

   if 2.0000000000000002e302 < (pow.f64 (/.f64 t l) 2)

   1. Initial program 56.0%

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

      1. unpow2 56.0%

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

      2. frac-2neg 56.0%

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

      3. frac-times 55.2%

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

   3. Applied egg-rr 55.2%

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

   4. Taylor expanded in t around inf 66.4%

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

      1. *-commutative 66.4%

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

      2. associate-*l/ 66.4%

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

      3. *-commutative 66.4%

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

      4. associate-*l* 66.4%

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

      5. unpow2 66.4%

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

      6. unpow2 66.4%

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

      7. times-frac 80.9%

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

      8. unpow2 80.9%

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

   6. Simplified 80.9%

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

      1. add-sqr-sqrt 79.6%

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

      2. sqrt-unprod 56.4%

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

      3. pow2 56.4%

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

      4. associate-*l/ 56.4%

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

      5. sqrt-unprod 56.4%

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

   8. Applied egg-rr 56.4%

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

      1. unpow2 56.4%

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

      2. rem-sqrt-square 99.7%

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

   10. Simplified 99.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{t}{\ell}\right)}^{2} \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\sin^{-1} \left(\ell \cdot \frac{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}}{t}\right)\right|\\ \end{array} \]

Alternative 4: 98.4% accurate, 1.0× speedup

Math

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

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 (* (sqrt 2.0) (/ t_m l))))))

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, (sqrt(2.0) * (t_m / l)))));
}

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, (Math.sqrt(2.0) * (t_m / l)))));
}

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, (math.sqrt(2.0) * (t_m / l)))))

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(sqrt(2.0) * Float64(t_m / l)))))
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, (sqrt(2.0) * (t_m / l)))));
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[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 86.7%

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

   1. sqrt-div 86.6%

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

   2. div-inv 86.6%

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

   3. add-sqr-sqrt 86.6%

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

   4. hypot-1-def 86.6%

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

   5. *-commutative 86.6%

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

   6. sqrt-prod 86.6%

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

   7. unpow2 86.6%

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

   8. sqrt-prod 58.3%

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

   9. add-sqr-sqrt 98.5%

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

3. Applied egg-rr 98.5%

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

   1. associate-*r/ 98.5%

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

   2. *-rgt-identity 98.5%

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

5. Simplified 98.5%

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

   1. unpow2 50.1%

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

   2. clear-num 50.1%

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

   3. un-div-inv 50.1%

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

7. Applied egg-rr 98.5%

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

8. Final simplification 98.5%

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

Alternative 5: 97.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t_m}{\ell} \leq -1 \cdot 10^{+170}:\\ \;\;\;\;\left|\sin^{-1} \left(\frac{\ell}{\frac{t_m}{\sqrt{0.5}}}\right)\right|\\ \mathbf{elif}\;\frac{t_m}{\ell} \leq 2 \cdot 10^{+128}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(\frac{t_m}{\ell} \cdot \frac{t_m}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}} \cdot \frac{\sqrt{0.5}}{t_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (if (<= (/ t_m l) -1e+170)
   (fabs (asin (/ l (/ t_m (sqrt 0.5)))))
   (if (<= (/ t_m l) 2e+128)
     (asin
      (sqrt
       (/
        (- 1.0 (pow (/ Om Omc) 2.0))
        (+ 1.0 (* 2.0 (* (/ t_m l) (/ t_m l)))))))
     (asin
      (* l (* (sqrt (- 1.0 (/ (/ Om Omc) (/ Omc Om)))) (/ (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) <= -1e+170) {
		tmp = fabs(asin((l / (t_m / sqrt(0.5)))));
	} else if ((t_m / l) <= 2e+128) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * ((t_m / l) * (t_m / l)))))));
	} else {
		tmp = asin((l * (sqrt((1.0 - ((Om / Omc) / (Omc / Om)))) * (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) <= (-1d+170)) then
        tmp = abs(asin((l / (t_m / sqrt(0.5d0)))))
    else if ((t_m / l) <= 2d+128) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t_m / l) * (t_m / l)))))))
    else
        tmp = asin((l * (sqrt((1.0d0 - ((om / omc) / (omc / om)))) * (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) <= -1e+170) {
		tmp = Math.abs(Math.asin((l / (t_m / Math.sqrt(0.5)))));
	} else if ((t_m / l) <= 2e+128) {
		tmp = Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * ((t_m / l) * (t_m / l)))))));
	} else {
		tmp = Math.asin((l * (Math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))) * (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) <= -1e+170:
		tmp = math.fabs(math.asin((l / (t_m / math.sqrt(0.5)))))
	elif (t_m / l) <= 2e+128:
		tmp = math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * ((t_m / l) * (t_m / l)))))))
	else:
		tmp = math.asin((l * (math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))) * (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) <= -1e+170)
		tmp = abs(asin(Float64(l / Float64(t_m / sqrt(0.5)))));
	elseif (Float64(t_m / l) <= 2e+128)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m / l) * Float64(t_m / l)))))));
	else
		tmp = asin(Float64(l * Float64(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))) * 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 ((t_m / l) <= -1e+170)
		tmp = abs(asin((l / (t_m / sqrt(0.5)))));
	elseif ((t_m / l) <= 2e+128)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t_m / l) * (t_m / l)))))));
	else
		tmp = asin((l * (sqrt((1.0 - ((Om / Omc) / (Omc / Om)))) * (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], -1e+170], N[Abs[N[ArcSin[N[(l / N[(t$95$m / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t$95$m / l), $MachinePrecision], 2e+128], 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[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l * N[(N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -1.00000000000000003e170

   1. Initial program 68.1%

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

      1. unpow2 68.1%

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

      2. frac-2neg 68.1%

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

      3. frac-times 68.1%

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

   3. Applied egg-rr 68.1%

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

   4. Taylor expanded in t around inf 59.3%

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

      1. *-commutative 59.3%

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

      2. associate-*l/ 59.3%

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

      3. *-commutative 59.3%

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

      4. associate-*l* 59.3%

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

      5. unpow2 59.3%

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

      6. unpow2 59.3%

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

      7. times-frac 67.6%

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

      8. unpow2 67.6%

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

   6. Simplified 67.6%

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

   7. Taylor expanded in Om around 0 67.6%

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

      1. associate-/l* 67.6%

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

   9. Simplified 67.6%

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

       1. add-sqr-sqrt 65.8%

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

       2. sqrt-unprod 68.1%

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

       3. pow2 68.1%

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

       4. associate-/r/ 68.1%

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

   11. Applied egg-rr 68.1%

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

       1. unpow2 68.1%

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

       2. rem-sqrt-square 99.9%

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

       3. associate-*l/ 99.8%

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

       4. associate-/l* 99.8%

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

   13. Simplified 99.8%

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

   if -1.00000000000000003e170 < (/.f64 t l) < 2.0000000000000002e128

   1. Initial program 97.7%

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

      1. unpow2 97.7%

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

   3. Applied egg-rr 97.7%

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

   if 2.0000000000000002e128 < (/.f64 t l)

   1. Initial program 52.0%

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

      1. unpow2 52.0%

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

      2. frac-2neg 52.0%

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

      3. frac-times 37.6%

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

   3. Applied egg-rr 37.6%

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

   4. Taylor expanded in t around inf 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-*l/ 80.6%

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

      3. *-commutative 80.6%

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

      4. associate-*l* 80.6%

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

      5. unpow2 80.6%

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

      6. unpow2 80.6%

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

      7. times-frac 99.5%

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

      8. unpow2 99.5%

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

   6. Simplified 99.5%

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

      1. unpow2 3.8%

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

      2. clear-num 3.8%

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

      3. un-div-inv 3.8%

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

   8. Applied egg-rr 99.5%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+170}:\\ \;\;\;\;\left|\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\right|\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+128}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}} \cdot \frac{\sqrt{0.5}}{t}\right)\right)\\ \end{array} \]

Alternative 6: 98.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t_m}{\ell} \leq -1 \cdot 10^{+170}:\\ \;\;\;\;\left|\sin^{-1} \left(\frac{\ell}{\frac{t_m}{\sqrt{0.5}}}\right)\right|\\ \mathbf{elif}\;\frac{t_m}{\ell} \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \frac{\frac{t_m}{\ell}}{\frac{\ell}{t_m}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{t_1 \cdot 0.5}}{t_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (let* ((t_1 (- 1.0 (pow (/ Om Omc) 2.0))))
   (if (<= (/ t_m l) -1e+170)
     (fabs (asin (/ l (/ t_m (sqrt 0.5)))))
     (if (<= (/ t_m l) 2e+139)
       (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (/ (/ t_m l) (/ l t_m)))))))
       (asin (* l (/ (sqrt (* t_1 0.5)) t_m)))))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	double t_1 = 1.0 - pow((Om / Omc), 2.0);
	double tmp;
	if ((t_m / l) <= -1e+170) {
		tmp = fabs(asin((l / (t_m / sqrt(0.5)))));
	} else if ((t_m / l) <= 2e+139) {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))));
	} else {
		tmp = asin((l * (sqrt((t_1 * 0.5)) / t_m)));
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(t_m, l, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 - ((om / omc) ** 2.0d0)
    if ((t_m / l) <= (-1d+170)) then
        tmp = abs(asin((l / (t_m / sqrt(0.5d0)))))
    else if ((t_m / l) <= 2d+139) then
        tmp = asin(sqrt((t_1 / (1.0d0 + (2.0d0 * ((t_m / l) / (l / t_m)))))))
    else
        tmp = asin((l * (sqrt((t_1 * 0.5d0)) / t_m)))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double t_m, double l, double Om, double Omc) {
	double t_1 = 1.0 - Math.pow((Om / Omc), 2.0);
	double tmp;
	if ((t_m / l) <= -1e+170) {
		tmp = Math.abs(Math.asin((l / (t_m / Math.sqrt(0.5)))));
	} else if ((t_m / l) <= 2e+139) {
		tmp = Math.asin(Math.sqrt((t_1 / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))));
	} else {
		tmp = Math.asin((l * (Math.sqrt((t_1 * 0.5)) / t_m)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	t_1 = 1.0 - math.pow((Om / Omc), 2.0)
	tmp = 0
	if (t_m / l) <= -1e+170:
		tmp = math.fabs(math.asin((l / (t_m / math.sqrt(0.5)))))
	elif (t_m / l) <= 2e+139:
		tmp = math.asin(math.sqrt((t_1 / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))))
	else:
		tmp = math.asin((l * (math.sqrt((t_1 * 0.5)) / t_m)))
	return tmp

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	t_1 = Float64(1.0 - (Float64(Om / Omc) ^ 2.0))
	tmp = 0.0
	if (Float64(t_m / l) <= -1e+170)
		tmp = abs(asin(Float64(l / Float64(t_m / sqrt(0.5)))));
	elseif (Float64(t_m / l) <= 2e+139)
		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(l * Float64(sqrt(Float64(t_1 * 0.5)) / t_m)));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(t_m, l, Om, Omc)
	t_1 = 1.0 - ((Om / Omc) ^ 2.0);
	tmp = 0.0;
	if ((t_m / l) <= -1e+170)
		tmp = abs(asin((l / (t_m / sqrt(0.5)))));
	elseif ((t_m / l) <= 2e+139)
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))));
	else
		tmp = asin((l * (sqrt((t_1 * 0.5)) / t_m)));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := Block[{t$95$1 = N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$m / l), $MachinePrecision], -1e+170], N[Abs[N[ArcSin[N[(l / N[(t$95$m / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t$95$m / l), $MachinePrecision], 2e+139], 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[(l * N[(N[Sqrt[N[(t$95$1 * 0.5), $MachinePrecision]], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -1.00000000000000003e170

   1. Initial program 68.1%

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

      1. unpow2 68.1%

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

      2. frac-2neg 68.1%

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

      3. frac-times 68.1%

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

   3. Applied egg-rr 68.1%

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

   4. Taylor expanded in t around inf 59.3%

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

      1. *-commutative 59.3%

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

      2. associate-*l/ 59.3%

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

      3. *-commutative 59.3%

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

      4. associate-*l* 59.3%

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

      5. unpow2 59.3%

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

      6. unpow2 59.3%

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

      7. times-frac 67.6%

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

      8. unpow2 67.6%

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

   6. Simplified 67.6%

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

   7. Taylor expanded in Om around 0 67.6%

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

      1. associate-/l* 67.6%

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

   9. Simplified 67.6%

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

       1. add-sqr-sqrt 65.8%

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

       2. sqrt-unprod 68.1%

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

       3. pow2 68.1%

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

       4. associate-/r/ 68.1%

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

   11. Applied egg-rr 68.1%

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

       1. unpow2 68.1%

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

       2. rem-sqrt-square 99.9%

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

       3. associate-*l/ 99.8%

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

       4. associate-/l* 99.8%

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

   13. Simplified 99.8%

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

   if -1.00000000000000003e170 < (/.f64 t l) < 2.00000000000000007e139

   1. Initial program 97.8%

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

      1. unpow2 97.8%

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

      2. clear-num 97.7%

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

      3. un-div-inv 97.8%

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

   3. Applied egg-rr 97.8%

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

   if 2.00000000000000007e139 < (/.f64 t l)

   1. Initial program 49.3%

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

      1. unpow2 49.3%

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

      2. frac-2neg 49.3%

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

      3. frac-times 39.5%

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

   3. Applied egg-rr 39.5%

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

   4. Taylor expanded in t around inf 79.7%

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

      1. *-commutative 79.7%

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

      2. associate-*l/ 79.6%

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

      3. *-commutative 79.6%

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

      4. associate-*l* 79.6%

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

      5. unpow2 79.6%

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

      6. unpow2 79.6%

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

      7. times-frac 99.5%

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

      8. unpow2 99.5%

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

   6. Simplified 99.5%

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

      1. expm1-log1p-u 99.5%

         \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot \left(\frac{\sqrt{0.5}}{t} \cdot \sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}\right)\right)\right)\right)} \]

      2. expm1-udef 35.8%

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

      3. associate-*l/ 35.8%

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

      4. sqrt-unprod 35.8%

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

   8. Applied egg-rr 35.8%

      \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\ell \cdot \frac{\sqrt{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}{t}\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-def 99.5%

         \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot \frac{\sqrt{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}{t}\right)\right)\right)} \]

      2. expm1-log1p 99.5%

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

   10. Simplified 99.5%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+170}:\\ \;\;\;\;\left|\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\right|\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}}{t}\right)\\ \end{array} \]

Alternative 7: 84.1% accurate, 1.9× speedup

Math

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

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 (/ 1.0 (* (/ l t_m) (/ 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 * (1.0 / ((l / t_m) * (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 * (1.0d0 / ((l / t_m) * (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 * (1.0 / ((l / t_m) * (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 * (1.0 / ((l / t_m) * (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(1.0 / Float64(Float64(l / t_m) * 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 * (1.0 / ((l / t_m) * (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[(1.0 / N[(N[(l / t$95$m), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 86.7%

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

   1. unpow2 86.7%

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

   2. clear-num 86.7%

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

   3. clear-num 86.7%

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

   4. frac-times 86.7%

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

   5. metadata-eval 86.7%

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

3. Applied egg-rr 86.7%

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

   1. unpow2 50.1%

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

   2. clear-num 50.1%

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

   3. un-div-inv 50.1%

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

5. Applied egg-rr 86.7%

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

6. Final simplification 86.7%

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

Alternative 8: 64.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.2 \cdot 10^{-101} \lor \neg \left(\ell \leq 6.5 \cdot 10^{+72}\right):\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t_m \cdot {0.5}^{-0.5}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (if (or (<= l -1.2e-101) (not (<= l 6.5e+72)))
   (asin (sqrt (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))
   (asin (/ l (* t_m (pow 0.5 -0.5))))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	double tmp;
	if ((l <= -1.2e-101) || !(l <= 6.5e+72)) {
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} else {
		tmp = asin((l / (t_m * pow(0.5, -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 ((l <= (-1.2d-101)) .or. (.not. (l <= 6.5d+72))) then
        tmp = asin(sqrt((1.0d0 - ((om / omc) / (omc / om)))))
    else
        tmp = asin((l / (t_m * (0.5d0 ** (-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 ((l <= -1.2e-101) || !(l <= 6.5e+72)) {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} else {
		tmp = Math.asin((l / (t_m * Math.pow(0.5, -0.5))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	tmp = 0
	if (l <= -1.2e-101) or not (l <= 6.5e+72):
		tmp = math.asin(math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))))
	else:
		tmp = math.asin((l / (t_m * math.pow(0.5, -0.5))))
	return tmp

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	tmp = 0.0
	if ((l <= -1.2e-101) || !(l <= 6.5e+72))
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))));
	else
		tmp = asin(Float64(l / Float64(t_m * (0.5 ^ -0.5))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(t_m, l, Om, Omc)
	tmp = 0.0;
	if ((l <= -1.2e-101) || ~((l <= 6.5e+72)))
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	else
		tmp = asin((l / (t_m * (0.5 ^ -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := If[Or[LessEqual[l, -1.2e-101], N[Not[LessEqual[l, 6.5e+72]], $MachinePrecision]], 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[Power[0.5, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.2e-101 or 6.5000000000000001e72 < l

   1. Initial program 93.1%

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

   2. Taylor expanded in t around 0 65.2%

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

      1. unpow2 65.2%

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

      2. unpow2 65.2%

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

      3. times-frac 72.6%

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

      4. unpow2 72.6%

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

   4. Simplified 72.6%

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

      1. unpow2 72.6%

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

      2. clear-num 72.6%

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

      3. un-div-inv 72.6%

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

   6. Applied egg-rr 72.6%

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

   if -1.2e-101 < l < 6.5000000000000001e72

   1. Initial program 79.2%

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

      1. unpow2 79.2%

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

      2. frac-2neg 79.2%

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

      3. frac-times 49.8%

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

   3. Applied egg-rr 49.8%

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

   4. Taylor expanded in t around inf 49.2%

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

      1. *-commutative 49.2%

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

      2. associate-*l/ 49.1%

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

      3. *-commutative 49.1%

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

      4. associate-*l* 49.1%

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

      5. unpow2 49.1%

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

      6. unpow2 49.1%

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

      7. times-frac 57.7%

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

      8. unpow2 57.7%

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

   6. Simplified 57.7%

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

   7. Taylor expanded in Om around 0 57.3%

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

      1. associate-/l* 57.2%

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

   9. Simplified 57.2%

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

       1. div-inv 57.2%

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

       2. pow1/2 57.2%

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

       3. pow-flip 57.4%

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

       4. metadata-eval 57.4%

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

   11. Applied egg-rr 57.4%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.2 \cdot 10^{-101} \lor \neg \left(\ell \leq 6.5 \cdot 10^{+72}\right):\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot {0.5}^{-0.5}}\right)\\ \end{array} \]

Alternative 9: 31.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	return asin((l / (t_m * pow(0.5, -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 * (0.5d0 ** (-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.pow(0.5, -0.5))));
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 86.7%

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

   1. unpow2 86.7%

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

   2. frac-2neg 86.7%

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

   3. frac-times 59.0%

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

3. Applied egg-rr 59.0%

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

4. Taylor expanded in t around inf 28.6%

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

   1. *-commutative 28.6%

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

   2. associate-*l/ 28.6%

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

   3. *-commutative 28.6%

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

   4. associate-*l* 28.6%

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

   5. unpow2 28.6%

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

   6. unpow2 28.6%

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

   7. times-frac 34.1%

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

   8. unpow2 34.1%

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

6. Simplified 34.1%

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

7. Taylor expanded in Om around 0 33.9%

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

   1. associate-/l* 33.8%

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

9. Simplified 33.8%

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

    1. div-inv 33.8%

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

    2. pow1/2 33.8%

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

    3. pow-flip 33.9%

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

    4. metadata-eval 33.9%

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

11. Applied egg-rr 33.9%

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

12. Final simplification 33.9%

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

Alternative 10: 31.3% accurate, 2.0× speedup

Math

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

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 / 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[(N[(l / t$95$m), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 86.7%

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

   1. unpow2 86.7%

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

   2. frac-2neg 86.7%

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

   3. frac-times 59.0%

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

3. Applied egg-rr 59.0%

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

4. Taylor expanded in t around inf 28.6%

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

   1. *-commutative 28.6%

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

   2. associate-*l/ 28.6%

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

   3. *-commutative 28.6%

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

   4. associate-*l* 28.6%

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

   5. unpow2 28.6%

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

   6. unpow2 28.6%

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

   7. times-frac 34.1%

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

   8. unpow2 34.1%

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

6. Simplified 34.1%

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

7. Taylor expanded in Om around 0 33.9%

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

   1. associate-/l* 33.8%

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

9. Simplified 33.8%

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

    1. associate-/r/ 33.9%

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

11. Applied egg-rr 33.9%

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

12. Final simplification 33.9%

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

Alternative 11: 31.3% 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 86.7%

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

   1. unpow2 86.7%

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

   2. frac-2neg 86.7%

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

   3. frac-times 59.0%

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

3. Applied egg-rr 59.0%

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

4. Taylor expanded in t around inf 28.6%

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

   1. *-commutative 28.6%

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

   2. associate-*l/ 28.6%

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

   3. *-commutative 28.6%

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

   4. associate-*l* 28.6%

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

   5. unpow2 28.6%

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

   6. unpow2 28.6%

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

   7. times-frac 34.1%

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

   8. unpow2 34.1%

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

6. Simplified 34.1%

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

7. Taylor expanded in Om around 0 33.9%

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

8. Final simplification 33.9%

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

Reproduce

herbie shell --seed 2023332 
(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)))))))