Toniolo and Linder, Equation (2)

Percentage Accurate: 84.0% → 98.8%

Time: 16.2s

Alternatives: 10

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.0% 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: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}} \leq 3 \cdot 10^{-307}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m} \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\frac{t\_m}{l\_m} \cdot \frac{t\_m}{l\_m}\right)}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<=
      (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))
      3e-307)
   (asin (* (/ l_m t_m) (sqrt 0.5)))
   (asin
    (sqrt
     (/
      (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
      (+ 1.0 (* 2.0 (* (/ 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) {
	double tmp;
	if (((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0)))) <= 3e-307) {
		tmp = asin(((l_m / t_m) * sqrt(0.5)));
	} else {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m / l_m) * (t_m / l_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 (((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t_m / l_m) ** 2.0d0)))) <= 3d-307) then
        tmp = asin(((l_m / t_m) * sqrt(0.5d0)))
    else
        tmp = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * ((t_m / l_m) * (t_m / l_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 (((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * Math.pow((t_m / l_m), 2.0)))) <= 3e-307) {
		tmp = Math.asin(((l_m / t_m) * Math.sqrt(0.5)));
	} else {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m / l_m) * (t_m / l_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 ((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * math.pow((t_m / l_m), 2.0)))) <= 3e-307:
		tmp = math.asin(((l_m / t_m) * math.sqrt(0.5)))
	else:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m / l_m) * (t_m / l_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(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0)))) <= 3e-307)
		tmp = asin(Float64(Float64(l_m / t_m) * sqrt(0.5)));
	else
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om))) / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m / l_m) * Float64(t_m / l_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 (((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t_m / l_m) ^ 2.0)))) <= 3e-307)
		tmp = asin(((l_m / t_m) * sqrt(0.5)));
	else
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m / l_m) * (t_m / l_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[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3e-307], N[ArcSin[N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))) < 2.9999999999999999e-307

   1. Initial program 51.4%

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

   3. Taylor expanded in t around inf 63.7%

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

      1. *-commutative 63.7%

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

      2. associate-/l* 63.7%

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

      3. unpow2 63.7%

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

      4. unpow2 63.7%

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

      5. times-frac 69.5%

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

      6. unpow2 69.5%

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

   5. Simplified 69.5%

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

      1. clear-num 69.5%

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

      2. un-div-inv 69.5%

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

   7. Applied egg-rr 69.5%

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

      1. associate-/r/ 69.5%

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

   9. Simplified 69.5%

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

   10. Taylor expanded in Om around 0 69.5%

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

   if 2.9999999999999999e-307 < (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))

   1. Initial program 99.1%

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

      1. unpow2 99.1%

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

      2. associate-*r/ 93.6%

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

   4. Applied egg-rr 93.6%

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

      1. unpow2 93.6%

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

      2. clear-num 93.6%

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

      3. un-div-inv 93.6%

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

   6. Applied egg-rr 93.6%

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

      1. associate-/l* 99.1%

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

   8. Applied egg-rr 99.1%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \leq 3 \cdot 10^{-307}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.4% 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 83.8%

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

   1. sqrt-div 83.7%

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

   2. add-sqr-sqrt 83.7%

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

   3. hypot-1-def 83.7%

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

   4. *-commutative 83.7%

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

   5. sqrt-prod 83.7%

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

   6. sqrt-pow1 97.8%

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

   7. metadata-eval 97.8%

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

   8. pow1 97.8%

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

4. Applied egg-rr 97.8%

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

5. Final simplification 97.8%

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

Alternative 3: 98.4% 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 83.8%

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

   1. sqrt-div 83.7%

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

   2. div-inv 83.7%

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

   3. add-sqr-sqrt 83.7%

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

   4. hypot-1-def 83.7%

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

   5. *-commutative 83.7%

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

   6. sqrt-prod 83.7%

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

   7. sqrt-pow1 97.8%

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

   8. metadata-eval 97.8%

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

   9. pow1 97.8%

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

4. Applied egg-rr 97.8%

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

   1. associate-*r/ 97.8%

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

   2. *-rgt-identity 97.8%

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

   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* 97.8%

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

6. Simplified 97.8%

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

7. Final simplification 97.8%

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

Alternative 4: 97.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

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) 5e-6)
   (asin (sqrt (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))
   (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) <= 5e-6) {
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} 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) <= 5d-6) then
        tmp = asin(sqrt((1.0d0 - ((om / omc) / (omc / om)))))
    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) <= 5e-6) {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} 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) <= 5e-6:
		tmp = math.asin(math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))))
	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) <= 5e-6)
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))));
	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) <= 5e-6)
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	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], 5e-6], 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] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 5.00000000000000041e-6

   1. Initial program 87.3%

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

   3. Taylor expanded in t around 0 55.1%

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

      1. unpow2 55.1%

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

      2. unpow2 55.1%

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

      3. times-frac 63.1%

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

      4. unpow2 63.1%

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

   5. Simplified 63.1%

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

      1. unpow2 84.3%

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

      2. clear-num 84.3%

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

      3. un-div-inv 84.3%

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

   7. Applied egg-rr 63.1%

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

   if 5.00000000000000041e-6 < (/.f64 t l)

   1. Initial program 72.5%

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

   3. Taylor expanded in t around inf 84.4%

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

      1. *-commutative 84.4%

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

      2. associate-/l* 84.5%

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

      3. unpow2 84.5%

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

      4. unpow2 84.5%

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

      5. times-frac 96.7%

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

      6. unpow2 96.7%

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

   5. Simplified 96.7%

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

   6. Taylor expanded in Om around 0 95.5%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\sin^{-1} \left(1 + {\left(\frac{Om}{Omc}\right)}^{2} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

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) 5e-6)
   (asin (+ 1.0 (* (pow (/ Om Omc) 2.0) -0.5)))
   (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) <= 5e-6) {
		tmp = asin((1.0 + (pow((Om / Omc), 2.0) * -0.5)));
	} 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) <= 5d-6) then
        tmp = asin((1.0d0 + (((om / omc) ** 2.0d0) * (-0.5d0))))
    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) <= 5e-6) {
		tmp = Math.asin((1.0 + (Math.pow((Om / Omc), 2.0) * -0.5)));
	} 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) <= 5e-6:
		tmp = math.asin((1.0 + (math.pow((Om / Omc), 2.0) * -0.5)))
	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) <= 5e-6)
		tmp = asin(Float64(1.0 + Float64((Float64(Om / Omc) ^ 2.0) * -0.5)));
	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) <= 5e-6)
		tmp = asin((1.0 + (((Om / Omc) ^ 2.0) * -0.5)));
	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], 5e-6], N[ArcSin[N[(1.0 + N[(N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $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) < 5.00000000000000041e-6

   1. Initial program 87.3%

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

   3. Taylor expanded in t around 0 55.1%

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

      1. unpow2 55.1%

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

      2. unpow2 55.1%

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

      3. times-frac 63.1%

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

      4. unpow2 63.1%

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

   5. Simplified 63.1%

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

   6. Taylor expanded in Om around 0 55.1%

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

      1. unpow2 55.1%

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

      2. unpow2 55.1%

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

      3. times-frac 63.1%

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

      4. unpow2 63.1%

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

   8. Simplified 63.1%

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

   if 5.00000000000000041e-6 < (/.f64 t l)

   1. Initial program 72.5%

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

   3. Taylor expanded in t around inf 84.4%

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

      1. *-commutative 84.4%

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

      2. associate-/l* 84.5%

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

      3. unpow2 84.5%

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

      4. unpow2 84.5%

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

      5. times-frac 96.7%

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

      6. unpow2 96.7%

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

   5. Simplified 96.7%

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

   6. Taylor expanded in Om around 0 95.5%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\sin^{-1} \left(1 + {\left(\frac{Om}{Omc}\right)}^{2} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t\_m}{l\_m}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

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) 5e-6)
   (asin (- 1.0 (pow (/ t_m l_m) 2.0)))
   (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) <= 5e-6) {
		tmp = asin((1.0 - pow((t_m / l_m), 2.0)));
	} 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) <= 5d-6) then
        tmp = asin((1.0d0 - ((t_m / l_m) ** 2.0d0)))
    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) <= 5e-6) {
		tmp = Math.asin((1.0 - Math.pow((t_m / l_m), 2.0)));
	} 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) <= 5e-6:
		tmp = math.asin((1.0 - math.pow((t_m / l_m), 2.0)))
	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) <= 5e-6)
		tmp = asin(Float64(1.0 - (Float64(t_m / l_m) ^ 2.0)));
	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) <= 5e-6)
		tmp = asin((1.0 - ((t_m / l_m) ^ 2.0)));
	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], 5e-6], N[ArcSin[N[(1.0 - N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $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) < 5.00000000000000041e-6

   1. Initial program 87.3%

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

      1. sqrt-div 87.2%

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

      2. add-sqr-sqrt 87.2%

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

      3. hypot-1-def 87.2%

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

      4. *-commutative 87.2%

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

      5. sqrt-prod 87.1%

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

      6. sqrt-pow1 98.0%

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

      7. metadata-eval 98.0%

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

      8. pow1 98.0%

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

   4. Applied egg-rr 98.0%

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

   5. Taylor expanded in Om around 0 97.5%

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

   6. Taylor expanded in t around 0 54.8%

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

      1. associate-*r/ 54.8%

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

      2. *-commutative 54.8%

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

      3. unpow2 54.8%

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

      4. rem-square-sqrt 54.8%

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

      5. associate-*r* 54.8%

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

      6. metadata-eval 54.8%

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

      7. associate-*r/ 54.8%

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

      8. unpow2 54.8%

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

      9. unpow2 54.8%

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

      10. times-frac 61.3%

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

      11. unpow2 61.3%

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

      12. mul-1-neg 61.3%

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

   8. Simplified 61.3%

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

   if 5.00000000000000041e-6 < (/.f64 t l)

   1. Initial program 72.5%

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

   3. Taylor expanded in t around inf 84.4%

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

      1. *-commutative 84.4%

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

      2. associate-/l* 84.5%

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

      3. unpow2 84.5%

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

      4. unpow2 84.5%

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

      5. times-frac 96.7%

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

      6. unpow2 96.7%

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

   5. Simplified 96.7%

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

   6. Taylor expanded in Om around 0 95.5%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.85 \cdot 10^{-71}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m \cdot \sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 1.85e-71) (asin (/ l_m (* t_m (sqrt 2.0)))) (asin 1.0)))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (l_m <= 1.85e-71) {
		tmp = asin((l_m / (t_m * sqrt(2.0))));
	} else {
		tmp = asin(1.0);
	}
	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 (l_m <= 1.85d-71) then
        tmp = asin((l_m / (t_m * sqrt(2.0d0))))
    else
        tmp = asin(1.0d0)
    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 (l_m <= 1.85e-71) {
		tmp = Math.asin((l_m / (t_m * Math.sqrt(2.0))));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if l_m <= 1.85e-71:
		tmp = math.asin((l_m / (t_m * math.sqrt(2.0))))
	else:
		tmp = math.asin(1.0)
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (l_m <= 1.85e-71)
		tmp = asin(Float64(l_m / Float64(t_m * sqrt(2.0))));
	else
		tmp = asin(1.0);
	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 (l_m <= 1.85e-71)
		tmp = asin((l_m / (t_m * sqrt(2.0))));
	else
		tmp = asin(1.0);
	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[l$95$m, 1.85e-71], N[ArcSin[N[(l$95$m / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.8499999999999998e-71

   1. Initial program 81.6%

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

      1. sqrt-div 81.5%

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

      2. add-sqr-sqrt 81.5%

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

      3. hypot-1-def 81.5%

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

      4. *-commutative 81.5%

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

      5. sqrt-prod 81.4%

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

      6. sqrt-pow1 97.6%

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

      7. metadata-eval 97.6%

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

      8. pow1 97.6%

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

   4. Applied egg-rr 97.6%

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

   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 inf 37.8%

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

   if 1.8499999999999998e-71 < l

   1. Initial program 89.5%

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

   3. Taylor expanded in t around 0 59.9%

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

      1. unpow2 59.9%

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

      2. unpow2 59.9%

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

      3. times-frac 69.4%

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

      4. unpow2 69.4%

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

   5. Simplified 69.4%

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

   6. Taylor expanded in Om around 0 69.4%

      \[\leadsto \sin^{-1} \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.85 \cdot 10^{-71}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.85 \cdot 10^{-71}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 1.85e-71) (asin (* l_m (/ (sqrt 0.5) t_m))) (asin 1.0)))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (l_m <= 1.85e-71) {
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	} else {
		tmp = asin(1.0);
	}
	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 (l_m <= 1.85d-71) then
        tmp = asin((l_m * (sqrt(0.5d0) / t_m)))
    else
        tmp = asin(1.0d0)
    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 (l_m <= 1.85e-71) {
		tmp = Math.asin((l_m * (Math.sqrt(0.5) / t_m)));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if l_m <= 1.85e-71:
		tmp = math.asin((l_m * (math.sqrt(0.5) / t_m)))
	else:
		tmp = math.asin(1.0)
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (l_m <= 1.85e-71)
		tmp = asin(Float64(l_m * Float64(sqrt(0.5) / t_m)));
	else
		tmp = asin(1.0);
	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 (l_m <= 1.85e-71)
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	else
		tmp = asin(1.0);
	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[l$95$m, 1.85e-71], N[ArcSin[N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.8499999999999998e-71

   1. Initial program 81.6%

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

   3. Taylor expanded in t around inf 33.9%

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

      1. *-commutative 33.9%

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

      2. associate-/l* 33.9%

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

      3. unpow2 33.9%

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

      4. unpow2 33.9%

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

      5. times-frac 38.3%

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

      6. unpow2 38.3%

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

   5. Simplified 38.3%

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

   6. Taylor expanded in Om around 0 37.9%

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

   if 1.8499999999999998e-71 < l

   1. Initial program 89.5%

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

   3. Taylor expanded in t around 0 59.9%

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

      1. unpow2 59.9%

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

      2. unpow2 59.9%

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

      3. times-frac 69.4%

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

      4. unpow2 69.4%

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

   5. Simplified 69.4%

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

   6. Taylor expanded in Om around 0 69.4%

      \[\leadsto \sin^{-1} \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.85 \cdot 10^{-71}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.85 \cdot 10^{-71}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m} \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 1.85e-71) (asin (* (/ l_m t_m) (sqrt 0.5))) (asin 1.0)))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (l_m <= 1.85e-71) {
		tmp = asin(((l_m / t_m) * sqrt(0.5)));
	} else {
		tmp = asin(1.0);
	}
	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 (l_m <= 1.85d-71) then
        tmp = asin(((l_m / t_m) * sqrt(0.5d0)))
    else
        tmp = asin(1.0d0)
    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 (l_m <= 1.85e-71) {
		tmp = Math.asin(((l_m / t_m) * Math.sqrt(0.5)));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if l_m <= 1.85e-71:
		tmp = math.asin(((l_m / t_m) * math.sqrt(0.5)))
	else:
		tmp = math.asin(1.0)
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (l_m <= 1.85e-71)
		tmp = asin(Float64(Float64(l_m / t_m) * sqrt(0.5)));
	else
		tmp = asin(1.0);
	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 (l_m <= 1.85e-71)
		tmp = asin(((l_m / t_m) * sqrt(0.5)));
	else
		tmp = asin(1.0);
	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[l$95$m, 1.85e-71], N[ArcSin[N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.8499999999999998e-71

   1. Initial program 81.6%

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

   3. Taylor expanded in t around inf 33.9%

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

      1. *-commutative 33.9%

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

      2. associate-/l* 33.9%

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

      3. unpow2 33.9%

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

      4. unpow2 33.9%

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

      5. times-frac 38.3%

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

      6. unpow2 38.3%

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

   5. Simplified 38.3%

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

      1. clear-num 38.3%

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

      2. un-div-inv 38.3%

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

   7. Applied egg-rr 38.3%

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

      1. associate-/r/ 38.3%

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

   9. Simplified 38.3%

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

   10. Taylor expanded in Om around 0 37.9%

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

   if 1.8499999999999998e-71 < l

   1. Initial program 89.5%

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

   3. Taylor expanded in t around 0 59.9%

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

      1. unpow2 59.9%

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

      2. unpow2 59.9%

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

      3. times-frac 69.4%

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

      4. unpow2 69.4%

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

   5. Simplified 69.4%

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

   6. Taylor expanded in Om around 0 69.4%

      \[\leadsto \sin^{-1} \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.85 \cdot 10^{-71}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.3% 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 83.8%

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

3. Taylor expanded in t around 0 43.2%

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

   1. unpow2 43.2%

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

   2. unpow2 43.2%

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

   3. times-frac 49.5%

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

   4. unpow2 49.5%

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

5. Simplified 49.5%

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

6. Taylor expanded in Om around 0 49.4%

   \[\leadsto \sin^{-1} \color{blue}{1} \]

7. Final simplification 49.4%

   \[\leadsto \sin^{-1} 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024080 
(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)))))))