Toniolo and Linder, Equation (2)

Percentage Accurate: 83.9% → 98.9%

Time: 16.9s

Alternatives: 12

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{t\_m}{l\_m}\right)}^{2} \leq 10^{+227}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{\frac{t\_m}{l\_m}}{\frac{l\_m}{t\_m}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right) \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (pow (/ t_m l_m) 2.0) 1e+227)
   (asin
    (sqrt
     (/
      (- 1.0 (pow (/ Om Omc) 2.0))
      (+ 1.0 (* 2.0 (/ (/ t_m l_m) (/ l_m t_m)))))))
   (asin
    (* (* l_m (/ (sqrt 0.5) t_m)) (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc))))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (pow((t_m / l_m), 2.0) <= 1e+227) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * ((t_m / l_m) / (l_m / t_m)))))));
	} else {
		tmp = asin(((l_m * (sqrt(0.5) / t_m)) * sqrt((1.0 - ((Om / Omc) * (Om / Omc))))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (((t_m / l_m) ** 2.0d0) <= 1d+227) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t_m / l_m) / (l_m / t_m)))))))
    else
        tmp = asin(((l_m * (sqrt(0.5d0) / t_m)) * sqrt((1.0d0 - ((om / omc) * (om / omc))))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (Math.pow((t_m / l_m), 2.0) <= 1e+227) {
		tmp = Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * ((t_m / l_m) / (l_m / t_m)))))));
	} else {
		tmp = Math.asin(((l_m * (Math.sqrt(0.5) / t_m)) * Math.sqrt((1.0 - ((Om / Omc) * (Om / Omc))))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if math.pow((t_m / l_m), 2.0) <= 1e+227:
		tmp = math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * ((t_m / l_m) / (l_m / t_m)))))))
	else:
		tmp = math.asin(((l_m * (math.sqrt(0.5) / t_m)) * math.sqrt((1.0 - ((Om / Omc) * (Om / Omc))))))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if ((Float64(t_m / l_m) ^ 2.0) <= 1e+227)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m / l_m) / Float64(l_m / t_m)))))));
	else
		tmp = asin(Float64(Float64(l_m * Float64(sqrt(0.5) / t_m)) * sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc))))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if (((t_m / l_m) ^ 2.0) <= 1e+227)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t_m / l_m) / (l_m / t_m)))))));
	else
		tmp = asin(((l_m * (sqrt(0.5) / t_m)) * sqrt((1.0 - ((Om / Omc) * (Om / Omc))))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision], 1e+227], 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$95$m), $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 t l) #s(literal 2 binary64)) < 1.0000000000000001e227

   1. Initial program 98.9%

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

      1. unpow2 98.9%

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

      2. clear-num 99.0%

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

      3. un-div-inv 99.0%

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

   4. Applied egg-rr 99.0%

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

   if 1.0000000000000001e227 < (pow.f64 (/.f64 t l) #s(literal 2 binary64))

   1. Initial program 47.3%

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

   3. Taylor expanded in t around inf 60.1%

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

      1. unpow2 60.1%

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

      2. unpow2 60.1%

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

      3. frac-times 64.9%

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

   5. Applied egg-rr 64.9%

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

      1. *-un-lft-identity 64.9%

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

      2. associate-/l* 65.0%

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

   7. Applied egg-rr 65.0%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{t}{\ell}\right)}^{2} \leq 10^{+227}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left(\ell \cdot \frac{\sqrt{0.5}}{t}\right) \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.4%

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

   1. sqrt-div 84.4%

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

   2. add-sqr-sqrt 84.4%

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

   3. hypot-1-def 84.4%

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

   4. *-commutative 84.4%

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

   5. sqrt-prod 84.4%

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

   6. sqrt-pow1 97.9%

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

   7. metadata-eval 97.9%

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

   8. pow1 97.9%

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

4. Applied egg-rr 97.9%

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

Alternative 3: 98.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.4%

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

   1. sqrt-div 84.4%

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

   2. div-inv 84.4%

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

   3. add-sqr-sqrt 84.4%

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

   4. hypot-1-def 84.4%

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

   5. *-commutative 84.4%

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

   6. sqrt-prod 84.4%

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

   7. sqrt-pow1 97.9%

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

   8. metadata-eval 97.9%

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

   9. pow1 97.9%

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

4. Applied egg-rr 97.9%

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

   1. associate-*r/ 97.9%

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

   2. *-rgt-identity 97.9%

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

   3. associate-*l/ 97.8%

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

   4. associate-/l* 97.9%

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

6. Simplified 97.9%

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

Alternative 4: 98.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 10^{+34}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\left(t\_m \cdot \frac{t\_m}{l\_m}\right) \cdot \frac{1}{l\_m}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right) \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 1e+34)
   (asin
    (sqrt
     (/
      (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
      (+ 1.0 (* 2.0 (* (* t_m (/ t_m l_m)) (/ 1.0 l_m)))))))
   (asin
    (* (* l_m (/ (sqrt 0.5) t_m)) (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc))))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 1e+34) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m * (t_m / l_m)) * (1.0 / l_m)))))));
	} else {
		tmp = asin(((l_m * (sqrt(0.5) / t_m)) * sqrt((1.0 - ((Om / Omc) * (Om / Omc))))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 1d+34) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * ((t_m * (t_m / l_m)) * (1.0d0 / l_m)))))))
    else
        tmp = asin(((l_m * (sqrt(0.5d0) / t_m)) * sqrt((1.0d0 - ((om / omc) * (om / omc))))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 1e+34) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m * (t_m / l_m)) * (1.0 / l_m)))))));
	} else {
		tmp = Math.asin(((l_m * (Math.sqrt(0.5) / t_m)) * Math.sqrt((1.0 - ((Om / Omc) * (Om / Omc))))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 1e+34:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m * (t_m / l_m)) * (1.0 / l_m)))))))
	else:
		tmp = math.asin(((l_m * (math.sqrt(0.5) / t_m)) * math.sqrt((1.0 - ((Om / Omc) * (Om / Omc))))))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 1e+34)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om))) / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m * Float64(t_m / l_m)) * Float64(1.0 / l_m)))))));
	else
		tmp = asin(Float64(Float64(l_m * Float64(sqrt(0.5) / t_m)) * sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc))))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 1e+34)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m * (t_m / l_m)) * (1.0 / l_m)))))));
	else
		tmp = asin(((l_m * (sqrt(0.5) / t_m)) * sqrt((1.0 - ((Om / Omc) * (Om / Omc))))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 1e+34], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t$95$m * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 9.99999999999999946e33

   1. Initial program 89.2%

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

      1. unpow2 89.2%

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

      2. div-inv 89.2%

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

      3. associate-*r* 88.8%

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

   4. Applied egg-rr 88.8%

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

      1. unpow2 88.8%

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

      2. clear-num 88.8%

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

      3. un-div-inv 88.8%

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

   6. Applied egg-rr 88.8%

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

   if 9.99999999999999946e33 < (/.f64 t l)

   1. Initial program 61.8%

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

   3. Taylor expanded in t around inf 88.3%

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

      1. unpow2 88.3%

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

      2. unpow2 88.3%

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

      3. frac-times 99.3%

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

   5. Applied egg-rr 99.3%

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

      1. *-un-lft-identity 99.3%

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

      2. associate-/l* 99.6%

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

   7. Applied egg-rr 99.6%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 10^{+34}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\left(t \cdot \frac{t}{\ell}\right) \cdot \frac{1}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left(\ell \cdot \frac{\sqrt{0.5}}{t}\right) \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 2 \cdot 10^{+35}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\left(t\_m \cdot \frac{t\_m}{l\_m}\right) \cdot \frac{1}{l\_m}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 2e+35)
   (asin
    (sqrt
     (/
      (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
      (+ 1.0 (* 2.0 (* (* t_m (/ t_m l_m)) (/ 1.0 l_m)))))))
   (asin
    (* (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc)))) (/ (* l_m (sqrt 0.5)) t_m)))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 2e+35) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m * (t_m / l_m)) * (1.0 / l_m)))))));
	} else {
		tmp = asin((sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * ((l_m * sqrt(0.5)) / t_m)));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 2d+35) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * ((t_m * (t_m / l_m)) * (1.0d0 / l_m)))))))
    else
        tmp = asin((sqrt((1.0d0 - ((om / omc) * (om / omc)))) * ((l_m * sqrt(0.5d0)) / t_m)))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 2e+35) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m * (t_m / l_m)) * (1.0 / l_m)))))));
	} else {
		tmp = Math.asin((Math.sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * ((l_m * Math.sqrt(0.5)) / t_m)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 2e+35:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m * (t_m / l_m)) * (1.0 / l_m)))))))
	else:
		tmp = math.asin((math.sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * ((l_m * math.sqrt(0.5)) / t_m)))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 2e+35)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om))) / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m * Float64(t_m / l_m)) * Float64(1.0 / l_m)))))));
	else
		tmp = asin(Float64(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc)))) * Float64(Float64(l_m * sqrt(0.5)) / t_m)));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 2e+35)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m * (t_m / l_m)) * (1.0 / l_m)))))));
	else
		tmp = asin((sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * ((l_m * sqrt(0.5)) / t_m)));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 2e+35], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t$95$m * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 1.9999999999999999e35

   1. Initial program 89.3%

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

      1. unpow2 89.3%

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

      2. div-inv 89.3%

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

      3. associate-*r* 88.9%

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

   4. Applied egg-rr 88.9%

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

      1. unpow2 88.9%

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

      2. clear-num 88.9%

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

      3. un-div-inv 88.9%

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

   6. Applied egg-rr 88.9%

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

   if 1.9999999999999999e35 < (/.f64 t l)

   1. Initial program 60.1%

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

   3. Taylor expanded in t around inf 87.8%

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

      1. unpow2 87.8%

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

      2. unpow2 87.8%

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

      3. frac-times 99.3%

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

   5. Applied egg-rr 99.3%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 2 \cdot 10^{+35}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\left(t \cdot \frac{t}{\ell}\right) \cdot \frac{1}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 10^{+34}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\left(t\_m \cdot \frac{t\_m}{l\_m}\right) \cdot \frac{1}{l\_m}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \left(\sqrt{0.5} \cdot \frac{1}{t\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 1e+34)
   (asin
    (sqrt
     (/
      (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
      (+ 1.0 (* 2.0 (* (* t_m (/ t_m l_m)) (/ 1.0 l_m)))))))
   (asin (* l_m (* (sqrt 0.5) (/ 1.0 t_m))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 1e+34) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m * (t_m / l_m)) * (1.0 / l_m)))))));
	} else {
		tmp = asin((l_m * (sqrt(0.5) * (1.0 / t_m))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 1d+34) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * ((t_m * (t_m / l_m)) * (1.0d0 / l_m)))))))
    else
        tmp = asin((l_m * (sqrt(0.5d0) * (1.0d0 / t_m))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 1e+34) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m * (t_m / l_m)) * (1.0 / l_m)))))));
	} else {
		tmp = Math.asin((l_m * (Math.sqrt(0.5) * (1.0 / t_m))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 1e+34:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m * (t_m / l_m)) * (1.0 / l_m)))))))
	else:
		tmp = math.asin((l_m * (math.sqrt(0.5) * (1.0 / t_m))))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 1e+34)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om))) / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m * Float64(t_m / l_m)) * Float64(1.0 / l_m)))))));
	else
		tmp = asin(Float64(l_m * Float64(sqrt(0.5) * Float64(1.0 / t_m))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 1e+34)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m * (t_m / l_m)) * (1.0 / l_m)))))));
	else
		tmp = asin((l_m * (sqrt(0.5) * (1.0 / t_m))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 1e+34], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t$95$m * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] * N[(1.0 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 9.99999999999999946e33

   1. Initial program 89.2%

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

      1. unpow2 89.2%

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

      2. div-inv 89.2%

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

      3. associate-*r* 88.8%

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

   4. Applied egg-rr 88.8%

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

      1. unpow2 88.8%

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

      2. clear-num 88.8%

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

      3. un-div-inv 88.8%

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

   6. Applied egg-rr 88.8%

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

   if 9.99999999999999946e33 < (/.f64 t l)

   1. Initial program 61.8%

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

   3. Taylor expanded in t around inf 88.3%

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

   4. Taylor expanded in Om around 0 99.2%

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

      1. associate-*r/ 99.5%

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

   6. Simplified 99.5%

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

      1. div-inv 99.6%

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

   8. Applied egg-rr 99.6%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 10^{+34}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\left(t \cdot \frac{t}{\ell}\right) \cdot \frac{1}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \left(\sqrt{0.5} \cdot \frac{1}{t}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+113}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{\frac{t\_m}{l\_m}}{\frac{l\_m}{t\_m}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \left(\sqrt{0.5} \cdot \frac{1}{t\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 5e+113)
   (asin (sqrt (/ 1.0 (+ 1.0 (* 2.0 (/ (/ t_m l_m) (/ l_m t_m)))))))
   (asin (* l_m (* (sqrt 0.5) (/ 1.0 t_m))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+113) {
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 * ((t_m / l_m) / (l_m / t_m)))))));
	} else {
		tmp = asin((l_m * (sqrt(0.5) * (1.0 / t_m))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 5d+113) then
        tmp = asin(sqrt((1.0d0 / (1.0d0 + (2.0d0 * ((t_m / l_m) / (l_m / t_m)))))))
    else
        tmp = asin((l_m * (sqrt(0.5d0) * (1.0d0 / t_m))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+113) {
		tmp = Math.asin(Math.sqrt((1.0 / (1.0 + (2.0 * ((t_m / l_m) / (l_m / t_m)))))));
	} else {
		tmp = Math.asin((l_m * (Math.sqrt(0.5) * (1.0 / t_m))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 5e+113:
		tmp = math.asin(math.sqrt((1.0 / (1.0 + (2.0 * ((t_m / l_m) / (l_m / t_m)))))))
	else:
		tmp = math.asin((l_m * (math.sqrt(0.5) * (1.0 / t_m))))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 5e+113)
		tmp = asin(sqrt(Float64(1.0 / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m / l_m) / Float64(l_m / t_m)))))));
	else
		tmp = asin(Float64(l_m * Float64(sqrt(0.5) * Float64(1.0 / t_m))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 5e+113)
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 * ((t_m / l_m) / (l_m / t_m)))))));
	else
		tmp = asin((l_m * (sqrt(0.5) * (1.0 / t_m))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 5e+113], N[ArcSin[N[Sqrt[N[(1.0 / N[(1.0 + N[(2.0 * N[(N[(t$95$m / l$95$m), $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] * N[(1.0 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 5e113

   1. Initial program 89.9%

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

   3. Taylor expanded in Om around 0 88.9%

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

      1. unpow2 89.9%

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

      2. clear-num 89.9%

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

      3. un-div-inv 89.9%

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

   5. Applied egg-rr 89.0%

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

   if 5e113 < (/.f64 t l)

   1. Initial program 44.9%

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

   3. Taylor expanded in t around inf 93.1%

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

   4. Taylor expanded in Om around 0 99.5%

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

      1. associate-*r/ 99.7%

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

   6. Simplified 99.7%

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

      1. div-inv 99.7%

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

   8. Applied egg-rr 99.7%

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

Alternative 8: 97.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.002:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \left(\sqrt{0.5} \cdot \frac{1}{t\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 0.002)
   (asin (sqrt (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))
   (asin (* l_m (* (sqrt 0.5) (/ 1.0 t_m))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.002) {
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} else {
		tmp = asin((l_m * (sqrt(0.5) * (1.0 / t_m))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 0.002d0) then
        tmp = asin(sqrt((1.0d0 - ((om / omc) / (omc / om)))))
    else
        tmp = asin((l_m * (sqrt(0.5d0) * (1.0d0 / t_m))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.002) {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} else {
		tmp = Math.asin((l_m * (Math.sqrt(0.5) * (1.0 / t_m))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 0.002:
		tmp = math.asin(math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))))
	else:
		tmp = math.asin((l_m * (math.sqrt(0.5) * (1.0 / t_m))))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 0.002)
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))));
	else
		tmp = asin(Float64(l_m * Float64(sqrt(0.5) * Float64(1.0 / t_m))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 0.002)
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	else
		tmp = asin((l_m * (sqrt(0.5) * (1.0 / t_m))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 0.002], N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] * N[(1.0 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 2e-3

   1. Initial program 88.5%

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

   3. Taylor expanded in t around 0 63.6%

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

      1. unpow2 63.6%

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

      2. unpow2 63.6%

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

      3. times-frac 70.6%

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

      4. unpow2 70.6%

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

   5. Simplified 70.6%

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

      1. unpow2 88.1%

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

      2. clear-num 88.1%

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

      3. un-div-inv 88.1%

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

   7. Applied egg-rr 70.6%

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

   if 2e-3 < (/.f64 t l)

   1. Initial program 70.7%

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

   3. Taylor expanded in t around inf 83.4%

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

   4. Taylor expanded in Om around 0 96.7%

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

      1. associate-*r/ 96.9%

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

   6. Simplified 96.9%

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

      1. div-inv 97.0%

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

   8. Applied egg-rr 97.0%

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

Alternative 9: 97.1% accurate, 1.9× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 1.0)
   (asin (/ 1.0 (+ 1.0 (* (/ t_m l_m) (/ t_m l_m)))))
   (asin (* l_m (* (sqrt 0.5) (/ 1.0 t_m))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 1.0) {
		tmp = asin((1.0 / (1.0 + ((t_m / l_m) * (t_m / l_m)))));
	} else {
		tmp = asin((l_m * (sqrt(0.5) * (1.0 / t_m))));
	}
	return tmp;
}

Fortran

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

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 1.0) {
		tmp = Math.asin((1.0 / (1.0 + ((t_m / l_m) * (t_m / l_m)))));
	} else {
		tmp = Math.asin((l_m * (Math.sqrt(0.5) * (1.0 / t_m))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 1.0:
		tmp = math.asin((1.0 / (1.0 + ((t_m / l_m) * (t_m / l_m)))))
	else:
		tmp = math.asin((l_m * (math.sqrt(0.5) * (1.0 / t_m))))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 1.0)
		tmp = asin(Float64(1.0 / Float64(1.0 + Float64(Float64(t_m / l_m) * Float64(t_m / l_m)))));
	else
		tmp = asin(Float64(l_m * Float64(sqrt(0.5) * Float64(1.0 / t_m))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 1.0)
		tmp = asin((1.0 / (1.0 + ((t_m / l_m) * (t_m / l_m)))));
	else
		tmp = asin((l_m * (sqrt(0.5) * (1.0 / t_m))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 1.0], N[ArcSin[N[(1.0 / N[(1.0 + N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] * N[(1.0 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 1

   1. Initial program 88.5%

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

      1. sqrt-div 88.5%

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

      2. add-sqr-sqrt 88.5%

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

      3. hypot-1-def 88.5%

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

      4. *-commutative 88.5%

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

      5. sqrt-prod 88.5%

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

      6. sqrt-pow1 98.1%

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

      7. metadata-eval 98.1%

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

      8. pow1 98.1%

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

   4. Applied egg-rr 98.1%

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

   5. Taylor expanded in Om around 0 96.7%

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

   6. Taylor expanded in t around 0 70.5%

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

      1. *-commutative 70.5%

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

      2. associate-/l* 70.5%

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

      3. associate-*r* 70.5%

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

      4. unpow2 70.5%

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

      5. /-rgt-identity 70.5%

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

      6. associate-/r/ 70.5%

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

      7. *-commutative 70.5%

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

      8. associate-*r/ 70.5%

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

      9. unpow2 70.5%

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

      10. rem-square-sqrt 70.5%

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

      11. metadata-eval 70.5%

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

      12. *-commutative 70.5%

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

      13. associate-*l/ 70.5%

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

      14. unpow2 70.5%

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

      15. *-lft-identity 70.5%

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

      16. associate-/r/ 70.5%

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

      17. /-rgt-identity 70.5%

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

      18. times-frac 78.0%

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

      19. unpow2 78.0%

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

   8. Simplified 78.0%

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

      1. unpow2 78.0%

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

   10. Applied egg-rr 78.0%

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

   if 1 < (/.f64 t l)

   1. Initial program 70.7%

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

   3. Taylor expanded in t around inf 83.4%

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

   4. Taylor expanded in Om around 0 96.7%

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

      1. associate-*r/ 96.9%

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

   6. Simplified 96.9%

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

      1. div-inv 97.0%

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

   8. Applied egg-rr 97.0%

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

Alternative 10: 97.1% accurate, 2.0× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 1.0)
   (asin (/ 1.0 (+ 1.0 (* (/ t_m l_m) (/ t_m l_m)))))
   (asin (* l_m (/ (sqrt 0.5) t_m)))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 1.0) {
		tmp = asin((1.0 / (1.0 + ((t_m / l_m) * (t_m / l_m)))));
	} else {
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	}
	return tmp;
}

Fortran

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

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 1.0) {
		tmp = Math.asin((1.0 / (1.0 + ((t_m / l_m) * (t_m / l_m)))));
	} else {
		tmp = Math.asin((l_m * (Math.sqrt(0.5) / t_m)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 1.0:
		tmp = math.asin((1.0 / (1.0 + ((t_m / l_m) * (t_m / l_m)))))
	else:
		tmp = math.asin((l_m * (math.sqrt(0.5) / t_m)))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 1.0)
		tmp = asin(Float64(1.0 / Float64(1.0 + Float64(Float64(t_m / l_m) * Float64(t_m / l_m)))));
	else
		tmp = asin(Float64(l_m * Float64(sqrt(0.5) / t_m)));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 1.0)
		tmp = asin((1.0 / (1.0 + ((t_m / l_m) * (t_m / l_m)))));
	else
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 1.0], N[ArcSin[N[(1.0 / N[(1.0 + N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 1

   1. Initial program 88.5%

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

      1. sqrt-div 88.5%

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

      2. add-sqr-sqrt 88.5%

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

      3. hypot-1-def 88.5%

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

      4. *-commutative 88.5%

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

      5. sqrt-prod 88.5%

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

      6. sqrt-pow1 98.1%

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

      7. metadata-eval 98.1%

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

      8. pow1 98.1%

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

   4. Applied egg-rr 98.1%

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

   5. Taylor expanded in Om around 0 96.7%

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

   6. Taylor expanded in t around 0 70.5%

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

      1. *-commutative 70.5%

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

      2. associate-/l* 70.5%

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

      3. associate-*r* 70.5%

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

      4. unpow2 70.5%

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

      5. /-rgt-identity 70.5%

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

      6. associate-/r/ 70.5%

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

      7. *-commutative 70.5%

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

      8. associate-*r/ 70.5%

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

      9. unpow2 70.5%

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

      10. rem-square-sqrt 70.5%

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

      11. metadata-eval 70.5%

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

      12. *-commutative 70.5%

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

      13. associate-*l/ 70.5%

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

      14. unpow2 70.5%

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

      15. *-lft-identity 70.5%

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

      16. associate-/r/ 70.5%

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

      17. /-rgt-identity 70.5%

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

      18. times-frac 78.0%

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

      19. unpow2 78.0%

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

   8. Simplified 78.0%

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

      1. unpow2 78.0%

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

   10. Applied egg-rr 78.0%

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

   if 1 < (/.f64 t l)

   1. Initial program 70.7%

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

   3. Taylor expanded in t around inf 83.4%

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

   4. Taylor expanded in Om around 0 96.7%

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

      1. associate-*r/ 96.9%

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

   6. Simplified 96.9%

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

Alternative 11: 62.9% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    code = asin((1.0d0 / (1.0d0 + ((t_m / l_m) * (t_m / l_m)))))
end function

Java

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

Python

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

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	return asin(Float64(1.0 / Float64(1.0 + Float64(Float64(t_m / l_m) * Float64(t_m / l_m)))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.4%

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

   1. sqrt-div 84.4%

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

   2. add-sqr-sqrt 84.4%

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

   3. hypot-1-def 84.4%

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

   4. *-commutative 84.4%

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

   5. sqrt-prod 84.4%

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

   6. sqrt-pow1 97.9%

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

   7. metadata-eval 97.9%

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

   8. pow1 97.9%

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

4. Applied egg-rr 97.9%

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

5. Taylor expanded in Om around 0 96.8%

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

6. Taylor expanded in t around 0 59.8%

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

   1. *-commutative 59.8%

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

   2. associate-/l* 59.8%

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

   3. associate-*r* 59.8%

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

   4. unpow2 59.8%

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

   5. /-rgt-identity 59.8%

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

   6. associate-/r/ 59.8%

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

   7. *-commutative 59.8%

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

   8. associate-*r/ 59.8%

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

   9. unpow2 59.8%

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

   10. rem-square-sqrt 59.8%

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

   11. metadata-eval 59.8%

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

   12. *-commutative 59.8%

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

   13. associate-*l/ 59.8%

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

   14. unpow2 59.8%

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

   15. *-lft-identity 59.8%

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

   16. associate-/r/ 59.8%

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

   17. /-rgt-identity 59.8%

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

   18. times-frac 66.0%

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

   19. unpow2 66.0%

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

8. Simplified 66.0%

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

   1. unpow2 66.0%

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

10. Applied egg-rr 66.0%

    \[\leadsto \sin^{-1} \left(\frac{1}{1 + \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}}\right) \]
11. Add Preprocessing

Alternative 12: 50.2% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    code = asin(1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.4%

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

3. Taylor expanded in t around 0 50.1%

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

   1. unpow2 50.1%

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

   2. unpow2 50.1%

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

   3. times-frac 55.7%

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

   4. unpow2 55.7%

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

5. Simplified 55.7%

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

6. Taylor expanded in Om around 0 55.2%

   \[\leadsto \sin^{-1} \color{blue}{1} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024130 
(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)))))))