Toniolo and Linder, Equation (2)

Percentage Accurate: 83.6% → 98.3%

Time: 16.9s

Alternatives: 17

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: 83.6% 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: 98.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

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))) / Math.hypot(1.0, ((Math.sqrt(2.0) * t) / l))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, Om_, Omc_] := N[ArcSin[N[(N[Sqrt[N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 82.2%

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

   1. sqrt-div 82.1%

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

   2. div-inv 82.1%

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

   3. add-sqr-sqrt 82.1%

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

   4. hypot-1-def 82.1%

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

   5. *-commutative 82.1%

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

   6. sqrt-prod 82.1%

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

   7. unpow2 82.1%

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

   8. sqrt-prod 50.4%

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

   9. add-sqr-sqrt 98.0%

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

3. Applied egg-rr 98.0%

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

   1. unpow2 98.0%

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

   2. times-frac 84.3%

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

   3. unpow2 84.3%

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

   4. unpow2 84.3%

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

   5. associate-*r/ 84.3%

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

   6. *-rgt-identity 84.3%

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

   7. unpow2 84.3%

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

   8. unpow2 84.3%

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

   9. times-frac 98.0%

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

   10. unpow2 98.0%

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

5. Simplified 98.0%

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

6. Taylor expanded in t around 0 98.0%

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

7. Final simplification 98.0%

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

Alternative 2: 98.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

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))) / Math.hypot(1.0, (Math.sqrt(2.0) * (t / l)))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, Om_, Omc_] := N[ArcSin[N[(N[Sqrt[N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 82.2%

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

   1. sqrt-div 82.1%

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

   2. div-inv 82.1%

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

   3. add-sqr-sqrt 82.1%

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

   4. hypot-1-def 82.1%

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

   5. *-commutative 82.1%

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

   6. sqrt-prod 82.1%

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

   7. unpow2 82.1%

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

   8. sqrt-prod 50.4%

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

   9. add-sqr-sqrt 98.0%

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

3. Applied egg-rr 98.0%

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

   1. unpow2 98.0%

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

   2. times-frac 84.3%

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

   3. unpow2 84.3%

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

   4. unpow2 84.3%

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

   5. associate-*r/ 84.3%

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

   6. *-rgt-identity 84.3%

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

   7. unpow2 84.3%

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

   8. unpow2 84.3%

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

   9. times-frac 98.0%

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

   10. unpow2 98.0%

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

5. Simplified 98.0%

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

6. Final simplification 98.0%

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

Alternative 3: 97.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+159}:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5 \cdot \left(1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\right)}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+154}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\mathsf{fma}\left(Om \cdot \frac{Om}{Omc \cdot Omc}, -0.5, 1\right)}{\sqrt{2} \cdot \frac{t}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -5e+159)
   (- (asin (* l (/ (sqrt (* 0.5 (- 1.0 (/ (/ Om Omc) (/ Omc Om))))) t))))
   (if (<= (/ t l) 1e+154)
     (asin
      (sqrt
       (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (* (/ t l) (/ t l)))))))
     (asin
      (/ (fma (* Om (/ Om (* Omc Omc))) -0.5 1.0) (* (sqrt 2.0) (/ t l)))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -5e+159) {
		tmp = -asin((l * (sqrt((0.5 * (1.0 - ((Om / Omc) / (Omc / Om))))) / t)));
	} else if ((t / l) <= 1e+154) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} else {
		tmp = asin((fma((Om * (Om / (Omc * Omc))), -0.5, 1.0) / (sqrt(2.0) * (t / l))));
	}
	return tmp;
}

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -5e+159)
		tmp = Float64(-asin(Float64(l * Float64(sqrt(Float64(0.5 * Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om))))) / t))));
	elseif (Float64(t / l) <= 1e+154)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) * Float64(t / l)))))));
	else
		tmp = asin(Float64(fma(Float64(Om * Float64(Om / Float64(Omc * Omc))), -0.5, 1.0) / Float64(sqrt(2.0) * Float64(t / l))));
	end
	return tmp
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -5e+159], (-N[ArcSin[N[(l * N[(N[Sqrt[N[(0.5 * N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[(t / l), $MachinePrecision], 1e+154], 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 / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[(Om * N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -5.00000000000000003e159

   1. Initial program 32.9%

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

   2. Taylor expanded in t around -inf 89.3%

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

      1. mul-1-neg 89.3%

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

      2. *-commutative 89.3%

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

      3. unpow2 89.3%

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

      4. unpow2 89.3%

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

      5. times-frac 99.4%

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

      6. unpow2 99.4%

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

      7. associate-/l* 99.4%

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

   4. Simplified 99.4%

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

      1. expm1-log1p-u 99.4%

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

      2. expm1-udef 32.9%

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

      3. asin-neg 32.9%

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

      4. associate-*r/ 32.9%

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

      5. pow1/2 32.9%

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

      6. pow1/2 32.9%

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

      7. pow-prod-down 32.9%

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

   6. Applied egg-rr 32.9%

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

      1. expm1-def 99.3%

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

      2. expm1-log1p 99.3%

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

      3. associate-/r/ 99.6%

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

      4. unpow1/2 99.6%

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

      5. *-commutative 99.6%

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

   8. Simplified 99.6%

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

      1. unpow2 3.7%

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

      2. clear-num 3.7%

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

      3. un-div-inv 3.7%

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

   10. Applied egg-rr 99.6%

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

   if -5.00000000000000003e159 < (/.f64 t l) < 1.00000000000000004e154

   1. Initial program 98.3%

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

      1. unpow2 98.3%

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

   3. Applied egg-rr 98.3%

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

   if 1.00000000000000004e154 < (/.f64 t l)

   1. Initial program 52.0%

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

      1. sqrt-div 52.0%

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

      2. div-inv 52.0%

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

      3. add-sqr-sqrt 52.0%

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

      4. hypot-1-def 52.0%

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

      5. *-commutative 52.0%

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

      6. sqrt-prod 52.0%

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

      7. unpow2 52.0%

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

      8. sqrt-prod 95.0%

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

      9. add-sqr-sqrt 95.4%

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

   3. Applied egg-rr 95.4%

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

      1. unpow2 95.4%

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

      2. times-frac 77.8%

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

      3. unpow2 77.8%

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

      4. unpow2 77.8%

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

      5. associate-*r/ 77.8%

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

      6. *-rgt-identity 77.8%

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

      7. unpow2 77.8%

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

      8. unpow2 77.8%

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

      9. times-frac 95.4%

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

      10. unpow2 95.4%

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

   5. Simplified 95.4%

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

   6. Taylor expanded in Om around 0 77.8%

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

      1. unpow2 2.8%

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

      2. unpow2 2.8%

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

   8. Simplified 77.8%

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

      1. times-frac 3.4%

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

      2. clear-num 3.4%

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

      3. div-inv 3.4%

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

      4. associate-/r/ 3.4%

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

   10. Applied egg-rr 95.4%

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

   11. Taylor expanded in t around inf 81.9%

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

       1. associate-/l* 77.8%

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

       2. +-commutative 77.8%

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

       3. *-commutative 77.8%

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

       4. unpow2 77.8%

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

       5. associate-*r/ 89.6%

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

       6. unpow2 89.6%

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

       7. fma-udef 89.6%

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

       8. associate-*r/ 89.6%

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

   13. Simplified 89.6%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+159}:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5 \cdot \left(1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\right)}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+154}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\mathsf{fma}\left(Om \cdot \frac{Om}{Omc \cdot Omc}, -0.5, 1\right)}{\sqrt{2} \cdot \frac{t}{\ell}}\right)\\ \end{array} \]

Alternative 4: 95.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -5.5 \cdot 10^{+25}:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5 \cdot t_1}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+154}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))
   (if (<= (/ t l) -5.5e+25)
     (- (asin (* l (/ (sqrt (* 0.5 t_1)) t))))
     (if (<= (/ t l) 1e+154)
       (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (/ (* t (/ t l)) l))))))
       (asin
        (*
         (sqrt (- 1.0 (/ (* Om Om) (* Omc Omc))))
         (* l (/ (sqrt 0.5) t))))))))

C

double code(double t, double l, double Om, double Omc) {
	double t_1 = 1.0 - ((Om / Omc) / (Omc / Om));
	double tmp;
	if ((t / l) <= -5.5e+25) {
		tmp = -asin((l * (sqrt((0.5 * t_1)) / t)));
	} else if ((t / l) <= 1e+154) {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t * (t / l)) / l))))));
	} else {
		tmp = asin((sqrt((1.0 - ((Om * Om) / (Omc * Omc)))) * (l * (sqrt(0.5) / t))));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 - ((om / omc) / (omc / om))
    if ((t / l) <= (-5.5d+25)) then
        tmp = -asin((l * (sqrt((0.5d0 * t_1)) / t)))
    else if ((t / l) <= 1d+154) then
        tmp = asin(sqrt((t_1 / (1.0d0 + (2.0d0 * ((t * (t / l)) / l))))))
    else
        tmp = asin((sqrt((1.0d0 - ((om * om) / (omc * omc)))) * (l * (sqrt(0.5d0) / t))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double t_1 = 1.0 - ((Om / Omc) / (Omc / Om));
	double tmp;
	if ((t / l) <= -5.5e+25) {
		tmp = -Math.asin((l * (Math.sqrt((0.5 * t_1)) / t)));
	} else if ((t / l) <= 1e+154) {
		tmp = Math.asin(Math.sqrt((t_1 / (1.0 + (2.0 * ((t * (t / l)) / l))))));
	} else {
		tmp = Math.asin((Math.sqrt((1.0 - ((Om * Om) / (Omc * Omc)))) * (l * (Math.sqrt(0.5) / t))));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	t_1 = 1.0 - ((Om / Omc) / (Omc / Om))
	tmp = 0
	if (t / l) <= -5.5e+25:
		tmp = -math.asin((l * (math.sqrt((0.5 * t_1)) / t)))
	elif (t / l) <= 1e+154:
		tmp = math.asin(math.sqrt((t_1 / (1.0 + (2.0 * ((t * (t / l)) / l))))))
	else:
		tmp = math.asin((math.sqrt((1.0 - ((Om * Om) / (Omc * Omc)))) * (l * (math.sqrt(0.5) / t))))
	return tmp

Julia

function code(t, l, Om, Omc)
	t_1 = Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))
	tmp = 0.0
	if (Float64(t / l) <= -5.5e+25)
		tmp = Float64(-asin(Float64(l * Float64(sqrt(Float64(0.5 * t_1)) / t))));
	elseif (Float64(t / l) <= 1e+154)
		tmp = asin(sqrt(Float64(t_1 / Float64(1.0 + Float64(2.0 * Float64(Float64(t * Float64(t / l)) / l))))));
	else
		tmp = asin(Float64(sqrt(Float64(1.0 - Float64(Float64(Om * Om) / Float64(Omc * Omc)))) * Float64(l * Float64(sqrt(0.5) / t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	t_1 = 1.0 - ((Om / Omc) / (Omc / Om));
	tmp = 0.0;
	if ((t / l) <= -5.5e+25)
		tmp = -asin((l * (sqrt((0.5 * t_1)) / t)));
	elseif ((t / l) <= 1e+154)
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t * (t / l)) / l))))));
	else
		tmp = asin((sqrt((1.0 - ((Om * Om) / (Omc * Omc)))) * (l * (sqrt(0.5) / t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -5.5e+25], (-N[ArcSin[N[(l * N[(N[Sqrt[N[(0.5 * t$95$1), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[(t / l), $MachinePrecision], 1e+154], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(1.0 + N[(2.0 * N[(N[(t * N[(t / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[(1.0 - N[(N[(Om * Om), $MachinePrecision] / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l * N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -5.50000000000000018e25

   1. Initial program 58.3%

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

   2. Taylor expanded in t around -inf 88.3%

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

      1. mul-1-neg 88.3%

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

      2. *-commutative 88.3%

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

      3. unpow2 88.3%

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

      4. unpow2 88.3%

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

      5. times-frac 99.3%

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

      6. unpow2 99.3%

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

      7. associate-/l* 99.3%

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

   4. Simplified 99.3%

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

      1. expm1-log1p-u 99.3%

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

      2. expm1-udef 21.8%

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

      3. asin-neg 21.8%

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

      4. associate-*r/ 21.8%

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

      5. pow1/2 21.8%

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

      6. pow1/2 21.8%

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

      7. pow-prod-down 21.8%

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

   6. Applied egg-rr 21.8%

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

      1. expm1-def 99.3%

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

      2. expm1-log1p 99.3%

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

      3. associate-/r/ 99.5%

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

      4. unpow1/2 99.5%

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

      5. *-commutative 99.5%

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

   8. Simplified 99.5%

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

      1. unpow2 4.8%

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

      2. clear-num 4.8%

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

      3. un-div-inv 4.8%

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

   10. Applied egg-rr 99.5%

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

   if -5.50000000000000018e25 < (/.f64 t l) < 1.00000000000000004e154

   1. Initial program 98.1%

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

      1. unpow2 98.1%

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

      2. associate-*r/ 98.1%

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

   3. Applied egg-rr 98.1%

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

      1. unpow2 77.8%

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

      2. clear-num 77.8%

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

      3. un-div-inv 77.8%

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

   5. Applied egg-rr 98.1%

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

   if 1.00000000000000004e154 < (/.f64 t l)

   1. Initial program 52.0%

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

      1. unpow2 52.0%

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

      2. associate-*r/ 52.0%

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

   3. Applied egg-rr 52.0%

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

   4. Taylor expanded in t around inf 82.1%

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

      1. associate-/l* 77.8%

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

      2. *-commutative 77.8%

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

      3. unpow2 77.8%

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

      4. unpow2 77.8%

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

      5. times-frac 95.4%

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

      6. unpow2 95.4%

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

      7. associate-/r/ 99.6%

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

      8. *-commutative 99.6%

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

      9. unpow2 99.6%

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

      10. times-frac 82.0%

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

   6. Simplified 82.0%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5.5 \cdot 10^{+25}:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5 \cdot \left(1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\right)}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+154}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\right)\\ \end{array} \]

Alternative 5: 96.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -5.5 \cdot 10^{+25}:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5 \cdot t_1}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+154}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\mathsf{fma}\left(Om \cdot \frac{Om}{Omc \cdot Omc}, -0.5, 1\right)}{\sqrt{2} \cdot \frac{t}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))
   (if (<= (/ t l) -5.5e+25)
     (- (asin (* l (/ (sqrt (* 0.5 t_1)) t))))
     (if (<= (/ t l) 1e+154)
       (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (/ (* t (/ t l)) l))))))
       (asin
        (/
         (fma (* Om (/ Om (* Omc Omc))) -0.5 1.0)
         (* (sqrt 2.0) (/ t l))))))))

C

double code(double t, double l, double Om, double Omc) {
	double t_1 = 1.0 - ((Om / Omc) / (Omc / Om));
	double tmp;
	if ((t / l) <= -5.5e+25) {
		tmp = -asin((l * (sqrt((0.5 * t_1)) / t)));
	} else if ((t / l) <= 1e+154) {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t * (t / l)) / l))))));
	} else {
		tmp = asin((fma((Om * (Om / (Omc * Omc))), -0.5, 1.0) / (sqrt(2.0) * (t / l))));
	}
	return tmp;
}

Julia

function code(t, l, Om, Omc)
	t_1 = Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))
	tmp = 0.0
	if (Float64(t / l) <= -5.5e+25)
		tmp = Float64(-asin(Float64(l * Float64(sqrt(Float64(0.5 * t_1)) / t))));
	elseif (Float64(t / l) <= 1e+154)
		tmp = asin(sqrt(Float64(t_1 / Float64(1.0 + Float64(2.0 * Float64(Float64(t * Float64(t / l)) / l))))));
	else
		tmp = asin(Float64(fma(Float64(Om * Float64(Om / Float64(Omc * Omc))), -0.5, 1.0) / Float64(sqrt(2.0) * Float64(t / l))));
	end
	return tmp
end

Wolfram

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -5.5e+25], (-N[ArcSin[N[(l * N[(N[Sqrt[N[(0.5 * t$95$1), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[(t / l), $MachinePrecision], 1e+154], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(1.0 + N[(2.0 * N[(N[(t * N[(t / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[(Om * N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -5.50000000000000018e25

   1. Initial program 58.3%

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

   2. Taylor expanded in t around -inf 88.3%

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

      1. mul-1-neg 88.3%

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

      2. *-commutative 88.3%

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

      3. unpow2 88.3%

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

      4. unpow2 88.3%

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

      5. times-frac 99.3%

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

      6. unpow2 99.3%

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

      7. associate-/l* 99.3%

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

   4. Simplified 99.3%

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

      1. expm1-log1p-u 99.3%

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

      2. expm1-udef 21.8%

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

      3. asin-neg 21.8%

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

      4. associate-*r/ 21.8%

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

      5. pow1/2 21.8%

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

      6. pow1/2 21.8%

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

      7. pow-prod-down 21.8%

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

   6. Applied egg-rr 21.8%

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

      1. expm1-def 99.3%

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

      2. expm1-log1p 99.3%

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

      3. associate-/r/ 99.5%

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

      4. unpow1/2 99.5%

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

      5. *-commutative 99.5%

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

   8. Simplified 99.5%

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

      1. unpow2 4.8%

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

      2. clear-num 4.8%

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

      3. un-div-inv 4.8%

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

   10. Applied egg-rr 99.5%

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

   if -5.50000000000000018e25 < (/.f64 t l) < 1.00000000000000004e154

   1. Initial program 98.1%

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

      1. unpow2 98.1%

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

      2. associate-*r/ 98.1%

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

   3. Applied egg-rr 98.1%

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

      1. unpow2 77.8%

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

      2. clear-num 77.8%

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

      3. un-div-inv 77.8%

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

   5. Applied egg-rr 98.1%

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

   if 1.00000000000000004e154 < (/.f64 t l)

   1. Initial program 52.0%

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

      1. sqrt-div 52.0%

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

      2. div-inv 52.0%

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

      3. add-sqr-sqrt 52.0%

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

      4. hypot-1-def 52.0%

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

      5. *-commutative 52.0%

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

      6. sqrt-prod 52.0%

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

      7. unpow2 52.0%

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

      8. sqrt-prod 95.0%

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

      9. add-sqr-sqrt 95.4%

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

   3. Applied egg-rr 95.4%

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

      1. unpow2 95.4%

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

      2. times-frac 77.8%

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

      3. unpow2 77.8%

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

      4. unpow2 77.8%

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

      5. associate-*r/ 77.8%

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

      6. *-rgt-identity 77.8%

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

      7. unpow2 77.8%

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

      8. unpow2 77.8%

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

      9. times-frac 95.4%

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

      10. unpow2 95.4%

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

   5. Simplified 95.4%

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

   6. Taylor expanded in Om around 0 77.8%

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

      1. unpow2 2.8%

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

      2. unpow2 2.8%

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

   8. Simplified 77.8%

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

      1. times-frac 3.4%

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

      2. clear-num 3.4%

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

      3. div-inv 3.4%

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

      4. associate-/r/ 3.4%

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

   10. Applied egg-rr 95.4%

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

   11. Taylor expanded in t around inf 81.9%

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

       1. associate-/l* 77.8%

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

       2. +-commutative 77.8%

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

       3. *-commutative 77.8%

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

       4. unpow2 77.8%

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

       5. associate-*r/ 89.6%

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

       6. unpow2 89.6%

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

       7. fma-udef 89.6%

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

       8. associate-*r/ 89.6%

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

   13. Simplified 89.6%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5.5 \cdot 10^{+25}:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5 \cdot \left(1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\right)}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+154}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\mathsf{fma}\left(Om \cdot \frac{Om}{Omc \cdot Omc}, -0.5, 1\right)}{\sqrt{2} \cdot \frac{t}{\ell}}\right)\\ \end{array} \]

Alternative 6: 97.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} \left(\frac{1 + -0.5 \cdot \left(Om \cdot \frac{\frac{Om}{Omc}}{Omc}\right)}{\mathsf{hypot}\left(1, \sqrt{2} \cdot \frac{t}{\ell}\right)}\right) \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (asin
  (/
   (+ 1.0 (* -0.5 (* Om (/ (/ Om Omc) Omc))))
   (hypot 1.0 (* (sqrt 2.0) (/ t l))))))

C

double code(double t, double l, double Om, double Omc) {
	return asin(((1.0 + (-0.5 * (Om * ((Om / Omc) / Omc)))) / hypot(1.0, (sqrt(2.0) * (t / l)))));
}

Java

public static double code(double t, double l, double Om, double Omc) {
	return Math.asin(((1.0 + (-0.5 * (Om * ((Om / Omc) / Omc)))) / Math.hypot(1.0, (Math.sqrt(2.0) * (t / l)))));
}

Python

def code(t, l, Om, Omc):
	return math.asin(((1.0 + (-0.5 * (Om * ((Om / Omc) / Omc)))) / math.hypot(1.0, (math.sqrt(2.0) * (t / l)))))

Julia

function code(t, l, Om, Omc)
	return asin(Float64(Float64(1.0 + Float64(-0.5 * Float64(Om * Float64(Float64(Om / Omc) / Omc)))) / hypot(1.0, Float64(sqrt(2.0) * Float64(t / l)))))
end

MATLAB

function tmp = code(t, l, Om, Omc)
	tmp = asin(((1.0 + (-0.5 * (Om * ((Om / Omc) / Omc)))) / hypot(1.0, (sqrt(2.0) * (t / l)))));
end

Wolfram

code[t_, l_, Om_, Omc_] := N[ArcSin[N[(N[(1.0 + N[(-0.5 * N[(Om * N[(N[(Om / Omc), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 82.2%

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

   1. sqrt-div 82.1%

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

   2. div-inv 82.1%

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

   3. add-sqr-sqrt 82.1%

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

   4. hypot-1-def 82.1%

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

   5. *-commutative 82.1%

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

   6. sqrt-prod 82.1%

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

   7. unpow2 82.1%

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

   8. sqrt-prod 50.4%

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

   9. add-sqr-sqrt 98.0%

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

3. Applied egg-rr 98.0%

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

   1. unpow2 98.0%

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

   2. times-frac 84.3%

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

   3. unpow2 84.3%

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

   4. unpow2 84.3%

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

   5. associate-*r/ 84.3%

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

   6. *-rgt-identity 84.3%

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

   7. unpow2 84.3%

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

   8. unpow2 84.3%

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

   9. times-frac 98.0%

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

   10. unpow2 98.0%

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

5. Simplified 98.0%

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

6. Taylor expanded in Om around 0 84.2%

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

   1. unpow2 42.4%

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

   2. unpow2 42.4%

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

8. Simplified 84.2%

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

   1. times-frac 49.4%

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

   2. clear-num 49.4%

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

   3. div-inv 49.4%

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

   4. associate-/r/ 49.4%

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

10. Applied egg-rr 97.4%

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

11. Final simplification 97.4%

    \[\leadsto \sin^{-1} \left(\frac{1 + -0.5 \cdot \left(Om \cdot \frac{\frac{Om}{Omc}}{Omc}\right)}{\mathsf{hypot}\left(1, \sqrt{2} \cdot \frac{t}{\ell}\right)}\right) \]

Alternative 7: 97.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} \left(\frac{1 + -0.5 \cdot \left(Om \cdot \frac{\frac{Om}{Omc}}{Omc}\right)}{\mathsf{hypot}\left(1, \frac{\sqrt{2} \cdot t}{\ell}\right)}\right) \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (asin
  (/
   (+ 1.0 (* -0.5 (* Om (/ (/ Om Omc) Omc))))
   (hypot 1.0 (/ (* (sqrt 2.0) t) l)))))

C

double code(double t, double l, double Om, double Omc) {
	return asin(((1.0 + (-0.5 * (Om * ((Om / Omc) / Omc)))) / hypot(1.0, ((sqrt(2.0) * t) / l))));
}

Java

public static double code(double t, double l, double Om, double Omc) {
	return Math.asin(((1.0 + (-0.5 * (Om * ((Om / Omc) / Omc)))) / Math.hypot(1.0, ((Math.sqrt(2.0) * t) / l))));
}

Python

def code(t, l, Om, Omc):
	return math.asin(((1.0 + (-0.5 * (Om * ((Om / Omc) / Omc)))) / math.hypot(1.0, ((math.sqrt(2.0) * t) / l))))

Julia

function code(t, l, Om, Omc)
	return asin(Float64(Float64(1.0 + Float64(-0.5 * Float64(Om * Float64(Float64(Om / Omc) / Omc)))) / hypot(1.0, Float64(Float64(sqrt(2.0) * t) / l))))
end

MATLAB

function tmp = code(t, l, Om, Omc)
	tmp = asin(((1.0 + (-0.5 * (Om * ((Om / Omc) / Omc)))) / hypot(1.0, ((sqrt(2.0) * t) / l))));
end

Wolfram

code[t_, l_, Om_, Omc_] := N[ArcSin[N[(N[(1.0 + N[(-0.5 * N[(Om * N[(N[(Om / Omc), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 82.2%

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

   1. sqrt-div 82.1%

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

   2. div-inv 82.1%

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

   3. add-sqr-sqrt 82.1%

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

   4. hypot-1-def 82.1%

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

   5. *-commutative 82.1%

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

   6. sqrt-prod 82.1%

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

   7. unpow2 82.1%

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

   8. sqrt-prod 50.4%

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

   9. add-sqr-sqrt 98.0%

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

3. Applied egg-rr 98.0%

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

   1. unpow2 98.0%

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

   2. times-frac 84.3%

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

   3. unpow2 84.3%

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

   4. unpow2 84.3%

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

   5. associate-*r/ 84.3%

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

   6. *-rgt-identity 84.3%

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

   7. unpow2 84.3%

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

   8. unpow2 84.3%

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

   9. times-frac 98.0%

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

   10. unpow2 98.0%

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

5. Simplified 98.0%

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

6. Taylor expanded in Om around 0 84.2%

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

   1. unpow2 42.4%

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

   2. unpow2 42.4%

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

8. Simplified 84.2%

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

   1. times-frac 49.4%

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

   2. clear-num 49.4%

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

   3. div-inv 49.4%

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

   4. associate-/r/ 49.4%

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

10. Applied egg-rr 97.4%

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

11. Taylor expanded in t around 0 97.4%

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

12. Final simplification 97.4%

    \[\leadsto \sin^{-1} \left(\frac{1 + -0.5 \cdot \left(Om \cdot \frac{\frac{Om}{Omc}}{Omc}\right)}{\mathsf{hypot}\left(1, \frac{\sqrt{2} \cdot t}{\ell}\right)}\right) \]

Alternative 8: 95.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -5.5 \cdot 10^{+25}:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5 \cdot t_1}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+154}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{1 + -0.5 \cdot \frac{Om \cdot Om}{Omc \cdot Omc}}{\sqrt{2}} \cdot \frac{\ell}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))
   (if (<= (/ t l) -5.5e+25)
     (- (asin (* l (/ (sqrt (* 0.5 t_1)) t))))
     (if (<= (/ t l) 1e+154)
       (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (/ (* t (/ t l)) l))))))
       (asin
        (*
         (/ (+ 1.0 (* -0.5 (/ (* Om Om) (* Omc Omc)))) (sqrt 2.0))
         (/ l t)))))))

C

double code(double t, double l, double Om, double Omc) {
	double t_1 = 1.0 - ((Om / Omc) / (Omc / Om));
	double tmp;
	if ((t / l) <= -5.5e+25) {
		tmp = -asin((l * (sqrt((0.5 * t_1)) / t)));
	} else if ((t / l) <= 1e+154) {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t * (t / l)) / l))))));
	} else {
		tmp = asin((((1.0 + (-0.5 * ((Om * Om) / (Omc * Omc)))) / sqrt(2.0)) * (l / t)));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 - ((om / omc) / (omc / om))
    if ((t / l) <= (-5.5d+25)) then
        tmp = -asin((l * (sqrt((0.5d0 * t_1)) / t)))
    else if ((t / l) <= 1d+154) then
        tmp = asin(sqrt((t_1 / (1.0d0 + (2.0d0 * ((t * (t / l)) / l))))))
    else
        tmp = asin((((1.0d0 + ((-0.5d0) * ((om * om) / (omc * omc)))) / sqrt(2.0d0)) * (l / t)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double t_1 = 1.0 - ((Om / Omc) / (Omc / Om));
	double tmp;
	if ((t / l) <= -5.5e+25) {
		tmp = -Math.asin((l * (Math.sqrt((0.5 * t_1)) / t)));
	} else if ((t / l) <= 1e+154) {
		tmp = Math.asin(Math.sqrt((t_1 / (1.0 + (2.0 * ((t * (t / l)) / l))))));
	} else {
		tmp = Math.asin((((1.0 + (-0.5 * ((Om * Om) / (Omc * Omc)))) / Math.sqrt(2.0)) * (l / t)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	t_1 = 1.0 - ((Om / Omc) / (Omc / Om))
	tmp = 0
	if (t / l) <= -5.5e+25:
		tmp = -math.asin((l * (math.sqrt((0.5 * t_1)) / t)))
	elif (t / l) <= 1e+154:
		tmp = math.asin(math.sqrt((t_1 / (1.0 + (2.0 * ((t * (t / l)) / l))))))
	else:
		tmp = math.asin((((1.0 + (-0.5 * ((Om * Om) / (Omc * Omc)))) / math.sqrt(2.0)) * (l / t)))
	return tmp

Julia

function code(t, l, Om, Omc)
	t_1 = Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))
	tmp = 0.0
	if (Float64(t / l) <= -5.5e+25)
		tmp = Float64(-asin(Float64(l * Float64(sqrt(Float64(0.5 * t_1)) / t))));
	elseif (Float64(t / l) <= 1e+154)
		tmp = asin(sqrt(Float64(t_1 / Float64(1.0 + Float64(2.0 * Float64(Float64(t * Float64(t / l)) / l))))));
	else
		tmp = asin(Float64(Float64(Float64(1.0 + Float64(-0.5 * Float64(Float64(Om * Om) / Float64(Omc * Omc)))) / sqrt(2.0)) * Float64(l / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	t_1 = 1.0 - ((Om / Omc) / (Omc / Om));
	tmp = 0.0;
	if ((t / l) <= -5.5e+25)
		tmp = -asin((l * (sqrt((0.5 * t_1)) / t)));
	elseif ((t / l) <= 1e+154)
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t * (t / l)) / l))))));
	else
		tmp = asin((((1.0 + (-0.5 * ((Om * Om) / (Omc * Omc)))) / sqrt(2.0)) * (l / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -5.5e+25], (-N[ArcSin[N[(l * N[(N[Sqrt[N[(0.5 * t$95$1), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[(t / l), $MachinePrecision], 1e+154], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(1.0 + N[(2.0 * N[(N[(t * N[(t / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[(1.0 + N[(-0.5 * N[(N[(Om * Om), $MachinePrecision] / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -5.50000000000000018e25

   1. Initial program 58.3%

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

   2. Taylor expanded in t around -inf 88.3%

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

      1. mul-1-neg 88.3%

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

      2. *-commutative 88.3%

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

      3. unpow2 88.3%

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

      4. unpow2 88.3%

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

      5. times-frac 99.3%

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

      6. unpow2 99.3%

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

      7. associate-/l* 99.3%

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

   4. Simplified 99.3%

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

      1. expm1-log1p-u 99.3%

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

      2. expm1-udef 21.8%

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

      3. asin-neg 21.8%

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

      4. associate-*r/ 21.8%

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

      5. pow1/2 21.8%

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

      6. pow1/2 21.8%

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

      7. pow-prod-down 21.8%

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

   6. Applied egg-rr 21.8%

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

      1. expm1-def 99.3%

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

      2. expm1-log1p 99.3%

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

      3. associate-/r/ 99.5%

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

      4. unpow1/2 99.5%

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

      5. *-commutative 99.5%

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

   8. Simplified 99.5%

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

      1. unpow2 4.8%

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

      2. clear-num 4.8%

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

      3. un-div-inv 4.8%

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

   10. Applied egg-rr 99.5%

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

   if -5.50000000000000018e25 < (/.f64 t l) < 1.00000000000000004e154

   1. Initial program 98.1%

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

      1. unpow2 98.1%

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

      2. associate-*r/ 98.1%

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

   3. Applied egg-rr 98.1%

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

      1. unpow2 77.8%

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

      2. clear-num 77.8%

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

      3. un-div-inv 77.8%

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

   5. Applied egg-rr 98.1%

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

   if 1.00000000000000004e154 < (/.f64 t l)

   1. Initial program 52.0%

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

      1. sqrt-div 52.0%

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

      2. div-inv 52.0%

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

      3. add-sqr-sqrt 52.0%

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

      4. hypot-1-def 52.0%

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

      5. *-commutative 52.0%

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

      6. sqrt-prod 52.0%

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

      7. unpow2 52.0%

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

      8. sqrt-prod 95.0%

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

      9. add-sqr-sqrt 95.4%

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

   3. Applied egg-rr 95.4%

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

      1. unpow2 95.4%

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

      2. times-frac 77.8%

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

      3. unpow2 77.8%

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

      4. unpow2 77.8%

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

      5. associate-*r/ 77.8%

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

      6. *-rgt-identity 77.8%

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

      7. unpow2 77.8%

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

      8. unpow2 77.8%

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

      9. times-frac 95.4%

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

      10. unpow2 95.4%

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

   5. Simplified 95.4%

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

   6. Taylor expanded in Om around 0 77.8%

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

      1. unpow2 2.8%

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

      2. unpow2 2.8%

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

   8. Simplified 77.8%

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

   9. Taylor expanded in t around inf 81.9%

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

       1. times-frac 82.0%

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

       2. unpow2 82.0%

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

       3. unpow2 82.0%

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

   11. Simplified 82.0%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5.5 \cdot 10^{+25}:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5 \cdot \left(1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\right)}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+154}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{1 + -0.5 \cdot \frac{Om \cdot Om}{Omc \cdot Omc}}{\sqrt{2}} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 9: 94.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -40000:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5 \cdot t_1}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{-7}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{1 + -0.5 \cdot \frac{Om \cdot Om}{Omc \cdot Omc}}{\sqrt{2}} \cdot \frac{\ell}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))
   (if (<= (/ t l) -40000.0)
     (- (asin (* l (/ (sqrt (* 0.5 t_1)) t))))
     (if (<= (/ t l) 1e-7)
       (asin (sqrt t_1))
       (asin
        (*
         (/ (+ 1.0 (* -0.5 (/ (* Om Om) (* Omc Omc)))) (sqrt 2.0))
         (/ l t)))))))

C

double code(double t, double l, double Om, double Omc) {
	double t_1 = 1.0 - ((Om / Omc) / (Omc / Om));
	double tmp;
	if ((t / l) <= -40000.0) {
		tmp = -asin((l * (sqrt((0.5 * t_1)) / t)));
	} else if ((t / l) <= 1e-7) {
		tmp = asin(sqrt(t_1));
	} else {
		tmp = asin((((1.0 + (-0.5 * ((Om * Om) / (Omc * Omc)))) / sqrt(2.0)) * (l / t)));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 - ((om / omc) / (omc / om))
    if ((t / l) <= (-40000.0d0)) then
        tmp = -asin((l * (sqrt((0.5d0 * t_1)) / t)))
    else if ((t / l) <= 1d-7) then
        tmp = asin(sqrt(t_1))
    else
        tmp = asin((((1.0d0 + ((-0.5d0) * ((om * om) / (omc * omc)))) / sqrt(2.0d0)) * (l / t)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double t_1 = 1.0 - ((Om / Omc) / (Omc / Om));
	double tmp;
	if ((t / l) <= -40000.0) {
		tmp = -Math.asin((l * (Math.sqrt((0.5 * t_1)) / t)));
	} else if ((t / l) <= 1e-7) {
		tmp = Math.asin(Math.sqrt(t_1));
	} else {
		tmp = Math.asin((((1.0 + (-0.5 * ((Om * Om) / (Omc * Omc)))) / Math.sqrt(2.0)) * (l / t)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	t_1 = 1.0 - ((Om / Omc) / (Omc / Om))
	tmp = 0
	if (t / l) <= -40000.0:
		tmp = -math.asin((l * (math.sqrt((0.5 * t_1)) / t)))
	elif (t / l) <= 1e-7:
		tmp = math.asin(math.sqrt(t_1))
	else:
		tmp = math.asin((((1.0 + (-0.5 * ((Om * Om) / (Omc * Omc)))) / math.sqrt(2.0)) * (l / t)))
	return tmp

Julia

function code(t, l, Om, Omc)
	t_1 = Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))
	tmp = 0.0
	if (Float64(t / l) <= -40000.0)
		tmp = Float64(-asin(Float64(l * Float64(sqrt(Float64(0.5 * t_1)) / t))));
	elseif (Float64(t / l) <= 1e-7)
		tmp = asin(sqrt(t_1));
	else
		tmp = asin(Float64(Float64(Float64(1.0 + Float64(-0.5 * Float64(Float64(Om * Om) / Float64(Omc * Omc)))) / sqrt(2.0)) * Float64(l / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	t_1 = 1.0 - ((Om / Omc) / (Omc / Om));
	tmp = 0.0;
	if ((t / l) <= -40000.0)
		tmp = -asin((l * (sqrt((0.5 * t_1)) / t)));
	elseif ((t / l) <= 1e-7)
		tmp = asin(sqrt(t_1));
	else
		tmp = asin((((1.0 + (-0.5 * ((Om * Om) / (Omc * Omc)))) / sqrt(2.0)) * (l / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -40000.0], (-N[ArcSin[N[(l * N[(N[Sqrt[N[(0.5 * t$95$1), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[(t / l), $MachinePrecision], 1e-7], N[ArcSin[N[Sqrt[t$95$1], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[(1.0 + N[(-0.5 * N[(N[(Om * Om), $MachinePrecision] / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -4e4

   1. Initial program 62.4%

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

   2. Taylor expanded in t around -inf 87.0%

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

      1. mul-1-neg 87.0%

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

      2. *-commutative 87.0%

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

      3. unpow2 87.0%

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

      4. unpow2 87.0%

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

      5. times-frac 98.4%

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

      6. unpow2 98.4%

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

      7. associate-/l* 98.5%

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

   4. Simplified 98.5%

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

      1. expm1-log1p-u 98.5%

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

      2. expm1-udef 23.8%

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

      3. asin-neg 23.8%

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

      4. associate-*r/ 23.8%

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

      5. pow1/2 23.8%

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

      6. pow1/2 23.8%

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

      7. pow-prod-down 23.8%

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

   6. Applied egg-rr 23.8%

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

      1. expm1-def 98.4%

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

      2. expm1-log1p 98.4%

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

      3. associate-/r/ 98.6%

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

      4. unpow1/2 98.6%

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

      5. *-commutative 98.6%

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

   8. Simplified 98.6%

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

      1. unpow2 5.4%

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

      2. clear-num 5.4%

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

      3. un-div-inv 5.4%

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

   10. Applied egg-rr 98.6%

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

   if -4e4 < (/.f64 t l) < 9.9999999999999995e-8

   1. Initial program 97.7%

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

   2. Taylor expanded in t around 0 82.8%

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

      1. unpow2 82.8%

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

      2. unpow2 82.8%

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

      3. times-frac 97.4%

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

      4. unpow2 97.4%

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

   4. Simplified 97.4%

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

      1. unpow2 97.4%

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

      2. clear-num 97.4%

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

      3. un-div-inv 97.4%

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

   6. Applied egg-rr 97.4%

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

   if 9.9999999999999995e-8 < (/.f64 t l)

   1. Initial program 73.5%

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

      1. sqrt-div 73.4%

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

      2. div-inv 73.4%

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

      3. add-sqr-sqrt 73.4%

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

      4. hypot-1-def 73.4%

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

      5. *-commutative 73.4%

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

      6. sqrt-prod 73.3%

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

      7. unpow2 73.3%

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

      8. sqrt-prod 96.8%

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

      9. add-sqr-sqrt 97.1%

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

   3. Applied egg-rr 97.1%

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

      1. unpow2 97.1%

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

      2. times-frac 82.6%

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

      3. unpow2 82.6%

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

      4. unpow2 82.6%

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

      5. associate-*r/ 82.6%

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

      6. *-rgt-identity 82.6%

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

      7. unpow2 82.6%

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

      8. unpow2 82.6%

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

      9. times-frac 97.1%

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

      10. unpow2 97.1%

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

   5. Simplified 97.1%

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

   6. Taylor expanded in Om around 0 82.7%

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

      1. unpow2 4.8%

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

      2. unpow2 4.8%

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

   8. Simplified 82.7%

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

   9. Taylor expanded in t around inf 83.3%

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

       1. times-frac 83.2%

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

       2. unpow2 83.2%

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

       3. unpow2 83.2%

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

   11. Simplified 83.2%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -40000:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5 \cdot \left(1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\right)}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{-7}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{1 + -0.5 \cdot \frac{Om \cdot Om}{Omc \cdot Omc}}{\sqrt{2}} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 10: 84.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -40000:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -40000.0)
   (- (asin (* l (/ (sqrt 0.5) t))))
   (if (<= (/ t l) 5e-10)
     (asin (sqrt (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))
     (asin (sqrt (/ 1.0 (+ 1.0 (* 2.0 (/ (* t t) (* l l))))))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -40000.0) {
		tmp = -asin((l * (sqrt(0.5) / t)));
	} else if ((t / l) <= 5e-10) {
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} else {
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 * ((t * t) / (l * l)))))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((t / l) <= (-40000.0d0)) then
        tmp = -asin((l * (sqrt(0.5d0) / t)))
    else if ((t / l) <= 5d-10) then
        tmp = asin(sqrt((1.0d0 - ((om / omc) / (omc / om)))))
    else
        tmp = asin(sqrt((1.0d0 / (1.0d0 + (2.0d0 * ((t * t) / (l * l)))))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -40000.0) {
		tmp = -Math.asin((l * (Math.sqrt(0.5) / t)));
	} else if ((t / l) <= 5e-10) {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} else {
		tmp = Math.asin(Math.sqrt((1.0 / (1.0 + (2.0 * ((t * t) / (l * l)))))));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -40000.0:
		tmp = -math.asin((l * (math.sqrt(0.5) / t)))
	elif (t / l) <= 5e-10:
		tmp = math.asin(math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))))
	else:
		tmp = math.asin(math.sqrt((1.0 / (1.0 + (2.0 * ((t * t) / (l * l)))))))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -40000.0)
		tmp = Float64(-asin(Float64(l * Float64(sqrt(0.5) / t))));
	elseif (Float64(t / l) <= 5e-10)
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))));
	else
		tmp = asin(sqrt(Float64(1.0 / Float64(1.0 + Float64(2.0 * Float64(Float64(t * t) / Float64(l * l)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -40000.0)
		tmp = -asin((l * (sqrt(0.5) / t)));
	elseif ((t / l) <= 5e-10)
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	else
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 * ((t * t) / (l * l)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -40000.0], (-N[ArcSin[N[(l * N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[(t / l), $MachinePrecision], 5e-10], N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 / N[(1.0 + N[(2.0 * N[(N[(t * t), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -4e4

   1. Initial program 62.4%

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

   2. Taylor expanded in t around -inf 87.0%

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

      1. mul-1-neg 87.0%

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

      2. *-commutative 87.0%

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

      3. unpow2 87.0%

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

      4. unpow2 87.0%

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

      5. times-frac 98.4%

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

      6. unpow2 98.4%

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

      7. associate-/l* 98.5%

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

   4. Simplified 98.5%

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

      1. expm1-log1p-u 98.5%

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

      2. expm1-udef 23.8%

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

      3. asin-neg 23.8%

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

      4. associate-*r/ 23.8%

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

      5. pow1/2 23.8%

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

      6. pow1/2 23.8%

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

      7. pow-prod-down 23.8%

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

   6. Applied egg-rr 23.8%

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

      1. expm1-def 98.4%

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

      2. expm1-log1p 98.4%

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

      3. associate-/r/ 98.6%

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

      4. unpow1/2 98.6%

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

      5. *-commutative 98.6%

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

   8. Simplified 98.6%

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

   9. Taylor expanded in Om around 0 98.0%

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

   if -4e4 < (/.f64 t l) < 5.00000000000000031e-10

   1. Initial program 97.9%

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

   2. Taylor expanded in t around 0 83.0%

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

      1. unpow2 83.0%

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

      2. unpow2 83.0%

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

      3. times-frac 97.7%

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

      4. unpow2 97.7%

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

   4. Simplified 97.7%

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

      1. unpow2 97.7%

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

      2. clear-num 97.7%

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

      3. un-div-inv 97.7%

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

   6. Applied egg-rr 97.7%

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

   if 5.00000000000000031e-10 < (/.f64 t l)

   1. Initial program 73.4%

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

      1. unpow2 73.4%

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

      2. associate-*r/ 73.4%

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

   3. Applied egg-rr 73.4%

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

   4. Taylor expanded in Om around 0 62.2%

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

      1. associate-*r/ 62.2%

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

      2. rem-square-sqrt 62.0%

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

      3. unpow2 62.0%

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

      4. unpow2 62.0%

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

      5. rem-square-sqrt 62.2%

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

      6. associate-*r/ 62.2%

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

      7. unpow2 62.2%

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

      8. unpow2 62.2%

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

   6. Simplified 62.2%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -40000:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)\\ \end{array} \]

Alternative 11: 85.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -40000:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2}{\ell} \cdot \frac{t \cdot t}{\ell}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -40000.0)
   (- (asin (* l (/ (sqrt 0.5) t))))
   (if (<= (/ t l) 5e-10)
     (asin (sqrt (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))
     (asin (sqrt (/ 1.0 (+ 1.0 (* (/ 2.0 l) (/ (* t t) l)))))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -40000.0) {
		tmp = -asin((l * (sqrt(0.5) / t)));
	} else if ((t / l) <= 5e-10) {
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} else {
		tmp = asin(sqrt((1.0 / (1.0 + ((2.0 / l) * ((t * t) / l))))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((t / l) <= (-40000.0d0)) then
        tmp = -asin((l * (sqrt(0.5d0) / t)))
    else if ((t / l) <= 5d-10) then
        tmp = asin(sqrt((1.0d0 - ((om / omc) / (omc / om)))))
    else
        tmp = asin(sqrt((1.0d0 / (1.0d0 + ((2.0d0 / l) * ((t * t) / l))))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -40000.0) {
		tmp = -Math.asin((l * (Math.sqrt(0.5) / t)));
	} else if ((t / l) <= 5e-10) {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} else {
		tmp = Math.asin(Math.sqrt((1.0 / (1.0 + ((2.0 / l) * ((t * t) / l))))));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -40000.0:
		tmp = -math.asin((l * (math.sqrt(0.5) / t)))
	elif (t / l) <= 5e-10:
		tmp = math.asin(math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))))
	else:
		tmp = math.asin(math.sqrt((1.0 / (1.0 + ((2.0 / l) * ((t * t) / l))))))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -40000.0)
		tmp = Float64(-asin(Float64(l * Float64(sqrt(0.5) / t))));
	elseif (Float64(t / l) <= 5e-10)
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))));
	else
		tmp = asin(sqrt(Float64(1.0 / Float64(1.0 + Float64(Float64(2.0 / l) * Float64(Float64(t * t) / l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -40000.0)
		tmp = -asin((l * (sqrt(0.5) / t)));
	elseif ((t / l) <= 5e-10)
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	else
		tmp = asin(sqrt((1.0 / (1.0 + ((2.0 / l) * ((t * t) / l))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -40000.0], (-N[ArcSin[N[(l * N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[(t / l), $MachinePrecision], 5e-10], N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 / N[(1.0 + N[(N[(2.0 / l), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -4e4

   1. Initial program 62.4%

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

   2. Taylor expanded in t around -inf 87.0%

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

      1. mul-1-neg 87.0%

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

      2. *-commutative 87.0%

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

      3. unpow2 87.0%

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

      4. unpow2 87.0%

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

      5. times-frac 98.4%

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

      6. unpow2 98.4%

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

      7. associate-/l* 98.5%

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

   4. Simplified 98.5%

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

      1. expm1-log1p-u 98.5%

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

      2. expm1-udef 23.8%

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

      3. asin-neg 23.8%

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

      4. associate-*r/ 23.8%

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

      5. pow1/2 23.8%

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

      6. pow1/2 23.8%

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

      7. pow-prod-down 23.8%

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

   6. Applied egg-rr 23.8%

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

      1. expm1-def 98.4%

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

      2. expm1-log1p 98.4%

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

      3. associate-/r/ 98.6%

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

      4. unpow1/2 98.6%

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

      5. *-commutative 98.6%

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

   8. Simplified 98.6%

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

   9. Taylor expanded in Om around 0 98.0%

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

   if -4e4 < (/.f64 t l) < 5.00000000000000031e-10

   1. Initial program 97.9%

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

   2. Taylor expanded in t around 0 83.0%

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

      1. unpow2 83.0%

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

      2. unpow2 83.0%

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

      3. times-frac 97.7%

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

      4. unpow2 97.7%

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

   4. Simplified 97.7%

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

      1. unpow2 97.7%

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

      2. clear-num 97.7%

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

      3. un-div-inv 97.7%

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

   6. Applied egg-rr 97.7%

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

   if 5.00000000000000031e-10 < (/.f64 t l)

   1. Initial program 73.4%

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

      1. sqrt-div 73.4%

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

      2. div-inv 73.4%

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

      3. add-sqr-sqrt 73.4%

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

      4. hypot-1-def 73.4%

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

      5. *-commutative 73.4%

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

      6. sqrt-prod 73.2%

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

      7. unpow2 73.2%

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

      8. sqrt-prod 96.3%

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

      9. add-sqr-sqrt 96.6%

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

   3. Applied egg-rr 96.6%

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

      1. unpow2 96.6%

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

      2. times-frac 82.4%

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

      3. unpow2 82.4%

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

      4. unpow2 82.4%

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

      5. associate-*r/ 82.4%

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

      6. *-rgt-identity 82.4%

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

      7. unpow2 82.4%

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

      8. unpow2 82.4%

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

      9. times-frac 96.6%

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

      10. unpow2 96.6%

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

   5. Simplified 96.6%

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

   6. Taylor expanded in Om around 0 82.4%

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

      1. unpow2 5.5%

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

      2. unpow2 5.5%

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

   8. Simplified 82.4%

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

      1. times-frac 6.2%

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

      2. clear-num 6.2%

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

      3. div-inv 6.2%

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

      4. associate-/r/ 6.2%

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

   10. Applied egg-rr 96.7%

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

   11. Taylor expanded in Om around 0 62.0%

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

       1. associate-/l* 62.0%

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

       2. unpow2 62.0%

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

       3. rem-square-sqrt 62.2%

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

       4. associate-/l* 62.2%

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

       5. unpow2 62.2%

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

       6. times-frac 64.9%

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

       7. unpow2 64.9%

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

   13. Simplified 64.9%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -40000:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2}{\ell} \cdot \frac{t \cdot t}{\ell}}}\right)\\ \end{array} \]

Alternative 12: 86.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -40000:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5 \cdot t_1}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2}{\ell} \cdot \frac{t \cdot t}{\ell}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))
   (if (<= (/ t l) -40000.0)
     (- (asin (* l (/ (sqrt (* 0.5 t_1)) t))))
     (if (<= (/ t l) 5e-10)
       (asin (sqrt t_1))
       (asin (sqrt (/ 1.0 (+ 1.0 (* (/ 2.0 l) (/ (* t t) l))))))))))

C

double code(double t, double l, double Om, double Omc) {
	double t_1 = 1.0 - ((Om / Omc) / (Omc / Om));
	double tmp;
	if ((t / l) <= -40000.0) {
		tmp = -asin((l * (sqrt((0.5 * t_1)) / t)));
	} else if ((t / l) <= 5e-10) {
		tmp = asin(sqrt(t_1));
	} else {
		tmp = asin(sqrt((1.0 / (1.0 + ((2.0 / l) * ((t * t) / l))))));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 - ((om / omc) / (omc / om))
    if ((t / l) <= (-40000.0d0)) then
        tmp = -asin((l * (sqrt((0.5d0 * t_1)) / t)))
    else if ((t / l) <= 5d-10) then
        tmp = asin(sqrt(t_1))
    else
        tmp = asin(sqrt((1.0d0 / (1.0d0 + ((2.0d0 / l) * ((t * t) / l))))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double t_1 = 1.0 - ((Om / Omc) / (Omc / Om));
	double tmp;
	if ((t / l) <= -40000.0) {
		tmp = -Math.asin((l * (Math.sqrt((0.5 * t_1)) / t)));
	} else if ((t / l) <= 5e-10) {
		tmp = Math.asin(Math.sqrt(t_1));
	} else {
		tmp = Math.asin(Math.sqrt((1.0 / (1.0 + ((2.0 / l) * ((t * t) / l))))));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	t_1 = 1.0 - ((Om / Omc) / (Omc / Om))
	tmp = 0
	if (t / l) <= -40000.0:
		tmp = -math.asin((l * (math.sqrt((0.5 * t_1)) / t)))
	elif (t / l) <= 5e-10:
		tmp = math.asin(math.sqrt(t_1))
	else:
		tmp = math.asin(math.sqrt((1.0 / (1.0 + ((2.0 / l) * ((t * t) / l))))))
	return tmp

Julia

function code(t, l, Om, Omc)
	t_1 = Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))
	tmp = 0.0
	if (Float64(t / l) <= -40000.0)
		tmp = Float64(-asin(Float64(l * Float64(sqrt(Float64(0.5 * t_1)) / t))));
	elseif (Float64(t / l) <= 5e-10)
		tmp = asin(sqrt(t_1));
	else
		tmp = asin(sqrt(Float64(1.0 / Float64(1.0 + Float64(Float64(2.0 / l) * Float64(Float64(t * t) / l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	t_1 = 1.0 - ((Om / Omc) / (Omc / Om));
	tmp = 0.0;
	if ((t / l) <= -40000.0)
		tmp = -asin((l * (sqrt((0.5 * t_1)) / t)));
	elseif ((t / l) <= 5e-10)
		tmp = asin(sqrt(t_1));
	else
		tmp = asin(sqrt((1.0 / (1.0 + ((2.0 / l) * ((t * t) / l))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -40000.0], (-N[ArcSin[N[(l * N[(N[Sqrt[N[(0.5 * t$95$1), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[(t / l), $MachinePrecision], 5e-10], N[ArcSin[N[Sqrt[t$95$1], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 / N[(1.0 + N[(N[(2.0 / l), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -4e4

   1. Initial program 62.4%

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

   2. Taylor expanded in t around -inf 87.0%

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

      1. mul-1-neg 87.0%

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

      2. *-commutative 87.0%

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

      3. unpow2 87.0%

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

      4. unpow2 87.0%

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

      5. times-frac 98.4%

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

      6. unpow2 98.4%

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

      7. associate-/l* 98.5%

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

   4. Simplified 98.5%

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

      1. expm1-log1p-u 98.5%

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

      2. expm1-udef 23.8%

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

      3. asin-neg 23.8%

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

      4. associate-*r/ 23.8%

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

      5. pow1/2 23.8%

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

      6. pow1/2 23.8%

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

      7. pow-prod-down 23.8%

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

   6. Applied egg-rr 23.8%

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

      1. expm1-def 98.4%

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

      2. expm1-log1p 98.4%

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

      3. associate-/r/ 98.6%

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

      4. unpow1/2 98.6%

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

      5. *-commutative 98.6%

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

   8. Simplified 98.6%

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

      1. unpow2 5.4%

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

      2. clear-num 5.4%

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

      3. un-div-inv 5.4%

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

   10. Applied egg-rr 98.6%

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

   if -4e4 < (/.f64 t l) < 5.00000000000000031e-10

   1. Initial program 97.9%

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

   2. Taylor expanded in t around 0 83.0%

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

      1. unpow2 83.0%

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

      2. unpow2 83.0%

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

      3. times-frac 97.7%

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

      4. unpow2 97.7%

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

   4. Simplified 97.7%

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

      1. unpow2 97.7%

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

      2. clear-num 97.7%

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

      3. un-div-inv 97.7%

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

   6. Applied egg-rr 97.7%

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

   if 5.00000000000000031e-10 < (/.f64 t l)

   1. Initial program 73.4%

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

      1. sqrt-div 73.4%

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

      2. div-inv 73.4%

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

      3. add-sqr-sqrt 73.4%

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

      4. hypot-1-def 73.4%

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

      5. *-commutative 73.4%

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

      6. sqrt-prod 73.2%

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

      7. unpow2 73.2%

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

      8. sqrt-prod 96.3%

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

      9. add-sqr-sqrt 96.6%

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

   3. Applied egg-rr 96.6%

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

      1. unpow2 96.6%

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

      2. times-frac 82.4%

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

      3. unpow2 82.4%

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

      4. unpow2 82.4%

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

      5. associate-*r/ 82.4%

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

      6. *-rgt-identity 82.4%

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

      7. unpow2 82.4%

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

      8. unpow2 82.4%

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

      9. times-frac 96.6%

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

      10. unpow2 96.6%

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

   5. Simplified 96.6%

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

   6. Taylor expanded in Om around 0 82.4%

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

      1. unpow2 5.5%

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

      2. unpow2 5.5%

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

   8. Simplified 82.4%

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

      1. times-frac 6.2%

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

      2. clear-num 6.2%

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

      3. div-inv 6.2%

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

      4. associate-/r/ 6.2%

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

   10. Applied egg-rr 96.7%

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

   11. Taylor expanded in Om around 0 62.0%

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

       1. associate-/l* 62.0%

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

       2. unpow2 62.0%

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

       3. rem-square-sqrt 62.2%

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

       4. associate-/l* 62.2%

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

       5. unpow2 62.2%

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

       6. times-frac 64.9%

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

       7. unpow2 64.9%

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

   13. Simplified 64.9%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -40000:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5 \cdot \left(1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\right)}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2}{\ell} \cdot \frac{t \cdot t}{\ell}}}\right)\\ \end{array} \]

Alternative 13: 80.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -40000:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+184}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -40000.0)
   (- (asin (* l (/ (sqrt 0.5) t))))
   (if (<= (/ t l) 2e+184)
     (asin (sqrt (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))
     (- (asin (/ (sqrt 0.5) (/ t l)))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -40000.0) {
		tmp = -asin((l * (sqrt(0.5) / t)));
	} else if ((t / l) <= 2e+184) {
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} else {
		tmp = -asin((sqrt(0.5) / (t / l)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((t / l) <= (-40000.0d0)) then
        tmp = -asin((l * (sqrt(0.5d0) / t)))
    else if ((t / l) <= 2d+184) then
        tmp = asin(sqrt((1.0d0 - ((om / omc) / (omc / om)))))
    else
        tmp = -asin((sqrt(0.5d0) / (t / l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -40000.0) {
		tmp = -Math.asin((l * (Math.sqrt(0.5) / t)));
	} else if ((t / l) <= 2e+184) {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} else {
		tmp = -Math.asin((Math.sqrt(0.5) / (t / l)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -40000.0:
		tmp = -math.asin((l * (math.sqrt(0.5) / t)))
	elif (t / l) <= 2e+184:
		tmp = math.asin(math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))))
	else:
		tmp = -math.asin((math.sqrt(0.5) / (t / l)))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -40000.0)
		tmp = Float64(-asin(Float64(l * Float64(sqrt(0.5) / t))));
	elseif (Float64(t / l) <= 2e+184)
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))));
	else
		tmp = Float64(-asin(Float64(sqrt(0.5) / Float64(t / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -40000.0)
		tmp = -asin((l * (sqrt(0.5) / t)));
	elseif ((t / l) <= 2e+184)
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	else
		tmp = -asin((sqrt(0.5) / (t / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -40000.0], (-N[ArcSin[N[(l * N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[(t / l), $MachinePrecision], 2e+184], N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], (-N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -4e4

   1. Initial program 62.4%

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

   2. Taylor expanded in t around -inf 87.0%

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

      1. mul-1-neg 87.0%

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

      2. *-commutative 87.0%

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

      3. unpow2 87.0%

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

      4. unpow2 87.0%

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

      5. times-frac 98.4%

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

      6. unpow2 98.4%

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

      7. associate-/l* 98.5%

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

   4. Simplified 98.5%

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

      1. expm1-log1p-u 98.5%

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

      2. expm1-udef 23.8%

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

      3. asin-neg 23.8%

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

      4. associate-*r/ 23.8%

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

      5. pow1/2 23.8%

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

      6. pow1/2 23.8%

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

      7. pow-prod-down 23.8%

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

   6. Applied egg-rr 23.8%

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

      1. expm1-def 98.4%

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

      2. expm1-log1p 98.4%

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

      3. associate-/r/ 98.6%

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

      4. unpow1/2 98.6%

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

      5. *-commutative 98.6%

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

   8. Simplified 98.6%

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

   9. Taylor expanded in Om around 0 98.0%

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

   if -4e4 < (/.f64 t l) < 2.00000000000000003e184

   1. Initial program 95.1%

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

   2. Taylor expanded in t around 0 66.8%

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

      1. unpow2 66.8%

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

      2. unpow2 66.8%

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

      3. times-frac 78.5%

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

      4. unpow2 78.5%

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

   4. Simplified 78.5%

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

      1. unpow2 78.5%

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

      2. clear-num 78.5%

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

      3. un-div-inv 78.5%

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

   6. Applied egg-rr 78.5%

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

   if 2.00000000000000003e184 < (/.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. Taylor expanded in t around -inf 48.5%

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

      1. mul-1-neg 48.5%

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

      2. *-commutative 48.5%

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

      3. unpow2 48.5%

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

      4. unpow2 48.5%

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

      5. times-frac 59.3%

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

      6. unpow2 59.3%

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

      7. associate-/l* 59.6%

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

   4. Simplified 59.6%

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

      1. expm1-log1p-u 59.6%

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

      2. expm1-udef 60.1%

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

      3. asin-neg 60.1%

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

      4. associate-*r/ 60.1%

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

      5. pow1/2 60.1%

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

      6. pow1/2 60.1%

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

      7. pow-prod-down 60.1%

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

   6. Applied egg-rr 60.1%

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

      1. expm1-def 59.6%

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

      2. expm1-log1p 59.6%

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

      3. associate-/r/ 59.3%

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

      4. unpow1/2 59.3%

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

      5. *-commutative 59.3%

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

   8. Simplified 59.3%

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

   9. Taylor expanded in Om around 0 59.3%

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

       1. associate-/l* 59.6%

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

   11. Simplified 59.6%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -40000:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+184}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \]

Alternative 14: 80.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -40000:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+184}:\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \left(Om \cdot \frac{\frac{Om}{Omc}}{Omc}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -40000.0)
   (- (asin (* l (/ (sqrt 0.5) t))))
   (if (<= (/ t l) 2e+184)
     (asin (+ 1.0 (* -0.5 (* Om (/ (/ Om Omc) Omc)))))
     (- (asin (/ (sqrt 0.5) (/ t l)))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -40000.0) {
		tmp = -asin((l * (sqrt(0.5) / t)));
	} else if ((t / l) <= 2e+184) {
		tmp = asin((1.0 + (-0.5 * (Om * ((Om / Omc) / Omc)))));
	} else {
		tmp = -asin((sqrt(0.5) / (t / l)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((t / l) <= (-40000.0d0)) then
        tmp = -asin((l * (sqrt(0.5d0) / t)))
    else if ((t / l) <= 2d+184) then
        tmp = asin((1.0d0 + ((-0.5d0) * (om * ((om / omc) / omc)))))
    else
        tmp = -asin((sqrt(0.5d0) / (t / l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -40000.0) {
		tmp = -Math.asin((l * (Math.sqrt(0.5) / t)));
	} else if ((t / l) <= 2e+184) {
		tmp = Math.asin((1.0 + (-0.5 * (Om * ((Om / Omc) / Omc)))));
	} else {
		tmp = -Math.asin((Math.sqrt(0.5) / (t / l)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -40000.0:
		tmp = -math.asin((l * (math.sqrt(0.5) / t)))
	elif (t / l) <= 2e+184:
		tmp = math.asin((1.0 + (-0.5 * (Om * ((Om / Omc) / Omc)))))
	else:
		tmp = -math.asin((math.sqrt(0.5) / (t / l)))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -40000.0)
		tmp = Float64(-asin(Float64(l * Float64(sqrt(0.5) / t))));
	elseif (Float64(t / l) <= 2e+184)
		tmp = asin(Float64(1.0 + Float64(-0.5 * Float64(Om * Float64(Float64(Om / Omc) / Omc)))));
	else
		tmp = Float64(-asin(Float64(sqrt(0.5) / Float64(t / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -40000.0)
		tmp = -asin((l * (sqrt(0.5) / t)));
	elseif ((t / l) <= 2e+184)
		tmp = asin((1.0 + (-0.5 * (Om * ((Om / Omc) / Omc)))));
	else
		tmp = -asin((sqrt(0.5) / (t / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -40000.0], (-N[ArcSin[N[(l * N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[(t / l), $MachinePrecision], 2e+184], N[ArcSin[N[(1.0 + N[(-0.5 * N[(Om * N[(N[(Om / Omc), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], (-N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -4e4

   1. Initial program 62.4%

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

   2. Taylor expanded in t around -inf 87.0%

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

      1. mul-1-neg 87.0%

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

      2. *-commutative 87.0%

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

      3. unpow2 87.0%

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

      4. unpow2 87.0%

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

      5. times-frac 98.4%

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

      6. unpow2 98.4%

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

      7. associate-/l* 98.5%

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

   4. Simplified 98.5%

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

      1. expm1-log1p-u 98.5%

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

      2. expm1-udef 23.8%

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

      3. asin-neg 23.8%

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

      4. associate-*r/ 23.8%

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

      5. pow1/2 23.8%

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

      6. pow1/2 23.8%

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

      7. pow-prod-down 23.8%

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

   6. Applied egg-rr 23.8%

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

      1. expm1-def 98.4%

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

      2. expm1-log1p 98.4%

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

      3. associate-/r/ 98.6%

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

      4. unpow1/2 98.6%

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

      5. *-commutative 98.6%

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

   8. Simplified 98.6%

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

   9. Taylor expanded in Om around 0 98.0%

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

   if -4e4 < (/.f64 t l) < 2.00000000000000003e184

   1. Initial program 95.1%

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

   2. Taylor expanded in t around 0 66.8%

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

      1. unpow2 66.8%

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

      2. unpow2 66.8%

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

      3. times-frac 78.5%

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

      4. unpow2 78.5%

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

   4. Simplified 78.5%

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

   5. Taylor expanded in Om around 0 66.5%

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

      1. unpow2 66.5%

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

      2. unpow2 66.5%

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

   7. Simplified 66.5%

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

      1. times-frac 77.5%

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

      2. clear-num 77.5%

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

      3. div-inv 77.5%

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

      4. associate-/r/ 77.5%

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

   9. Applied egg-rr 77.5%

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

   if 2.00000000000000003e184 < (/.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. Taylor expanded in t around -inf 48.5%

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

      1. mul-1-neg 48.5%

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

      2. *-commutative 48.5%

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

      3. unpow2 48.5%

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

      4. unpow2 48.5%

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

      5. times-frac 59.3%

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

      6. unpow2 59.3%

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

      7. associate-/l* 59.6%

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

   4. Simplified 59.6%

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

      1. expm1-log1p-u 59.6%

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

      2. expm1-udef 60.1%

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

      3. asin-neg 60.1%

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

      4. associate-*r/ 60.1%

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

      5. pow1/2 60.1%

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

      6. pow1/2 60.1%

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

      7. pow-prod-down 60.1%

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

   6. Applied egg-rr 60.1%

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

      1. expm1-def 59.6%

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

      2. expm1-log1p 59.6%

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

      3. associate-/r/ 59.3%

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

      4. unpow1/2 59.3%

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

      5. *-commutative 59.3%

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

   8. Simplified 59.3%

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

   9. Taylor expanded in Om around 0 59.3%

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

       1. associate-/l* 59.6%

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

   11. Simplified 59.6%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -40000:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+184}:\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \left(Om \cdot \frac{\frac{Om}{Omc}}{Omc}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \]

Alternative 15: 62.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-43} \lor \neg \left(t \leq 1.35 \cdot 10^{+94}\right):\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\sqrt{2} \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \left(Om \cdot \frac{\frac{Om}{Omc}}{Omc}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (or (<= t -2e-43) (not (<= t 1.35e+94)))
   (asin (/ (- l) (* (sqrt 2.0) t)))
   (asin (+ 1.0 (* -0.5 (* Om (/ (/ Om Omc) Omc)))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t <= -2e-43) || !(t <= 1.35e+94)) {
		tmp = asin((-l / (sqrt(2.0) * t)));
	} else {
		tmp = asin((1.0 + (-0.5 * (Om * ((Om / Omc) / Omc)))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((t <= (-2d-43)) .or. (.not. (t <= 1.35d+94))) then
        tmp = asin((-l / (sqrt(2.0d0) * t)))
    else
        tmp = asin((1.0d0 + ((-0.5d0) * (om * ((om / omc) / omc)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t <= -2e-43) || !(t <= 1.35e+94)) {
		tmp = Math.asin((-l / (Math.sqrt(2.0) * t)));
	} else {
		tmp = Math.asin((1.0 + (-0.5 * (Om * ((Om / Omc) / Omc)))));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t <= -2e-43) or not (t <= 1.35e+94):
		tmp = math.asin((-l / (math.sqrt(2.0) * t)))
	else:
		tmp = math.asin((1.0 + (-0.5 * (Om * ((Om / Omc) / Omc)))))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if ((t <= -2e-43) || !(t <= 1.35e+94))
		tmp = asin(Float64(Float64(-l) / Float64(sqrt(2.0) * t)));
	else
		tmp = asin(Float64(1.0 + Float64(-0.5 * Float64(Om * Float64(Float64(Om / Omc) / Omc)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t <= -2e-43) || ~((t <= 1.35e+94)))
		tmp = asin((-l / (sqrt(2.0) * t)));
	else
		tmp = asin((1.0 + (-0.5 * (Om * ((Om / Omc) / Omc)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[Or[LessEqual[t, -2e-43], N[Not[LessEqual[t, 1.35e+94]], $MachinePrecision]], N[ArcSin[N[((-l) / N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(1.0 + N[(-0.5 * N[(Om * N[(N[(Om / Omc), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.00000000000000015e-43 or 1.3500000000000001e94 < t

   1. Initial program 71.9%

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

      1. sqrt-div 71.8%

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

      2. div-inv 71.8%

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

      3. add-sqr-sqrt 71.8%

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

      4. hypot-1-def 71.8%

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

      5. *-commutative 71.8%

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

      6. sqrt-prod 71.7%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2}}\right)}\right) \]

      7. unpow2 71.7%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}} \cdot \sqrt{2}\right)}\right) \]

      8. sqrt-prod 37.7%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{t}{\ell}} \cdot \sqrt{\frac{t}{\ell}}\right)} \cdot \sqrt{2}\right)}\right) \]

      9. add-sqr-sqrt 97.5%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell}} \cdot \sqrt{2}\right)}\right) \]

   3. Applied egg-rr 97.5%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]
   4. Step-by-step derivation

      1. unpow2 97.5%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]

      2. times-frac 82.0%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om \cdot Om}{Omc \cdot Omc}}} \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]

      3. unpow2 82.0%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{\color{blue}{{Om}^{2}}}{Omc \cdot Omc}} \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]

      4. unpow2 82.0%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{{Om}^{2}}{\color{blue}{{Omc}^{2}}}} \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]

      5. associate-*r/ 82.0%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot 1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]

      6. *-rgt-identity 82.0%

         \[\leadsto \sin^{-1} \left(\frac{\color{blue}{\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]

      7. unpow2 82.0%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]

      8. unpow2 82.0%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]

      9. times-frac 97.5%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]

      10. unpow2 97.5%

          \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]

   5. Simplified 97.5%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]

   6. Taylor expanded in Om around 0 82.0%

      \[\leadsto \sin^{-1} \left(\frac{\color{blue}{1 + -0.5 \cdot \frac{{Om}^{2}}{{Omc}^{2}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
   7. Step-by-step derivation

      1. unpow2 18.6%

         \[\leadsto \sin^{-1} \left(1 + -0.5 \cdot \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}\right) \]

      2. unpow2 18.6%

         \[\leadsto \sin^{-1} \left(1 + -0.5 \cdot \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}\right) \]

   8. Simplified 82.0%

      \[\leadsto \sin^{-1} \left(\frac{\color{blue}{1 + -0.5 \cdot \frac{Om \cdot Om}{Omc \cdot Omc}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]

   9. Taylor expanded in t around -inf 54.4%

      \[\leadsto \sin^{-1} \color{blue}{\left(-1 \cdot \frac{\left(1 + -0.5 \cdot \frac{{Om}^{2}}{{Omc}^{2}}\right) \cdot \ell}{\sqrt{2} \cdot t}\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg 54.4%

          \[\leadsto \sin^{-1} \color{blue}{\left(-\frac{\left(1 + -0.5 \cdot \frac{{Om}^{2}}{{Omc}^{2}}\right) \cdot \ell}{\sqrt{2} \cdot t}\right)} \]

       2. times-frac 54.3%

          \[\leadsto \sin^{-1} \left(-\color{blue}{\frac{1 + -0.5 \cdot \frac{{Om}^{2}}{{Omc}^{2}}}{\sqrt{2}} \cdot \frac{\ell}{t}}\right) \]

       3. unpow2 54.3%

          \[\leadsto \sin^{-1} \left(-\frac{1 + -0.5 \cdot \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}}{\sqrt{2}} \cdot \frac{\ell}{t}\right) \]

       4. unpow2 54.3%

          \[\leadsto \sin^{-1} \left(-\frac{1 + -0.5 \cdot \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}}{\sqrt{2}} \cdot \frac{\ell}{t}\right) \]

   11. Simplified 54.3%

       \[\leadsto \sin^{-1} \color{blue}{\left(-\frac{1 + -0.5 \cdot \frac{Om \cdot Om}{Omc \cdot Omc}}{\sqrt{2}} \cdot \frac{\ell}{t}\right)} \]

   12. Taylor expanded in Om around 0 61.6%

       \[\leadsto \sin^{-1} \left(-\color{blue}{\frac{\ell}{\sqrt{2} \cdot t}}\right) \]

   if -2.00000000000000015e-43 < t < 1.3500000000000001e94

   1. Initial program 91.6%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]

   2. Taylor expanded in t around 0 64.3%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]
   3. Step-by-step derivation

      1. unpow2 64.3%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}}\right) \]

      2. unpow2 64.3%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}}\right) \]

      3. times-frac 74.3%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}}\right) \]

      4. unpow2 74.3%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}}\right) \]

   4. Simplified 74.3%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}\right)} \]

   5. Taylor expanded in Om around 0 64.0%

      \[\leadsto \sin^{-1} \color{blue}{\left(1 + -0.5 \cdot \frac{{Om}^{2}}{{Omc}^{2}}\right)} \]
   6. Step-by-step derivation

      1. unpow2 64.0%

         \[\leadsto \sin^{-1} \left(1 + -0.5 \cdot \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}\right) \]

      2. unpow2 64.0%

         \[\leadsto \sin^{-1} \left(1 + -0.5 \cdot \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}\right) \]

   7. Simplified 64.0%

      \[\leadsto \sin^{-1} \color{blue}{\left(1 + -0.5 \cdot \frac{Om \cdot Om}{Omc \cdot Omc}\right)} \]
   8. Step-by-step derivation

      1. times-frac 73.6%

         \[\leadsto \sin^{-1} \left(1 + -0.5 \cdot \color{blue}{\left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)}\right) \]

      2. clear-num 73.6%

         \[\leadsto \sin^{-1} \left(1 + -0.5 \cdot \left(\frac{Om}{Omc} \cdot \color{blue}{\frac{1}{\frac{Omc}{Om}}}\right)\right) \]

      3. div-inv 73.6%

         \[\leadsto \sin^{-1} \left(1 + -0.5 \cdot \color{blue}{\frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right) \]

      4. associate-/r/ 73.6%

         \[\leadsto \sin^{-1} \left(1 + -0.5 \cdot \color{blue}{\left(\frac{\frac{Om}{Omc}}{Omc} \cdot Om\right)}\right) \]

   9. Applied egg-rr 73.6%

      \[\leadsto \sin^{-1} \left(1 + -0.5 \cdot \color{blue}{\left(\frac{\frac{Om}{Omc}}{Omc} \cdot Om\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-43} \lor \neg \left(t \leq 1.35 \cdot 10^{+94}\right):\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\sqrt{2} \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \left(Om \cdot \frac{\frac{Om}{Omc}}{Omc}\right)\right)\\ \end{array} \]

Alternative 16: 62.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{-43}:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;t \leq 3.45 \cdot 10^{+95}:\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \left(Om \cdot \frac{\frac{Om}{Omc}}{Omc}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\sqrt{2} \cdot t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= t -9.5e-43)
   (- (asin (* l (/ (sqrt 0.5) t))))
   (if (<= t 3.45e+95)
     (asin (+ 1.0 (* -0.5 (* Om (/ (/ Om Omc) Omc)))))
     (asin (/ (- l) (* (sqrt 2.0) t))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (t <= -9.5e-43) {
		tmp = -asin((l * (sqrt(0.5) / t)));
	} else if (t <= 3.45e+95) {
		tmp = asin((1.0 + (-0.5 * (Om * ((Om / Omc) / Omc)))));
	} else {
		tmp = asin((-l / (sqrt(2.0) * t)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (t <= (-9.5d-43)) then
        tmp = -asin((l * (sqrt(0.5d0) / t)))
    else if (t <= 3.45d+95) then
        tmp = asin((1.0d0 + ((-0.5d0) * (om * ((om / omc) / omc)))))
    else
        tmp = asin((-l / (sqrt(2.0d0) * t)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (t <= -9.5e-43) {
		tmp = -Math.asin((l * (Math.sqrt(0.5) / t)));
	} else if (t <= 3.45e+95) {
		tmp = Math.asin((1.0 + (-0.5 * (Om * ((Om / Omc) / Omc)))));
	} else {
		tmp = Math.asin((-l / (Math.sqrt(2.0) * t)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if t <= -9.5e-43:
		tmp = -math.asin((l * (math.sqrt(0.5) / t)))
	elif t <= 3.45e+95:
		tmp = math.asin((1.0 + (-0.5 * (Om * ((Om / Omc) / Omc)))))
	else:
		tmp = math.asin((-l / (math.sqrt(2.0) * t)))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (t <= -9.5e-43)
		tmp = Float64(-asin(Float64(l * Float64(sqrt(0.5) / t))));
	elseif (t <= 3.45e+95)
		tmp = asin(Float64(1.0 + Float64(-0.5 * Float64(Om * Float64(Float64(Om / Omc) / Omc)))));
	else
		tmp = asin(Float64(Float64(-l) / Float64(sqrt(2.0) * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (t <= -9.5e-43)
		tmp = -asin((l * (sqrt(0.5) / t)));
	elseif (t <= 3.45e+95)
		tmp = asin((1.0 + (-0.5 * (Om * ((Om / Omc) / Omc)))));
	else
		tmp = asin((-l / (sqrt(2.0) * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[t, -9.5e-43], (-N[ArcSin[N[(l * N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, 3.45e+95], N[ArcSin[N[(1.0 + N[(-0.5 * N[(Om * N[(N[(Om / Omc), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[((-l) / N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.50000000000000044e-43

   1. Initial program 67.1%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]

   2. Taylor expanded in t around -inf 55.1%

      \[\leadsto \sin^{-1} \color{blue}{\left(-1 \cdot \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 55.1%

         \[\leadsto \sin^{-1} \color{blue}{\left(-\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]

      2. *-commutative 55.1%

         \[\leadsto \sin^{-1} \left(-\color{blue}{\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}}\right) \]

      3. unpow2 55.1%

         \[\leadsto \sin^{-1} \left(-\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]

      4. unpow2 55.1%

         \[\leadsto \sin^{-1} \left(-\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]

      5. times-frac 61.7%

         \[\leadsto \sin^{-1} \left(-\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]

      6. unpow2 61.7%

         \[\leadsto \sin^{-1} \left(-\sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]

      7. associate-/l* 61.7%

         \[\leadsto \sin^{-1} \left(-\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \color{blue}{\frac{\sqrt{0.5}}{\frac{t}{\ell}}}\right) \]

   4. Simplified 61.7%

      \[\leadsto \sin^{-1} \color{blue}{\left(-\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 61.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(-\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\right)\right)} \]

      2. expm1-udef 21.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin^{-1} \left(-\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\right)} - 1} \]

      3. asin-neg 21.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{-\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)}\right)} - 1 \]

      4. associate-*r/ 21.9%

         \[\leadsto e^{\mathsf{log1p}\left(-\sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \sqrt{0.5}}{\frac{t}{\ell}}\right)}\right)} - 1 \]

      5. pow1/2 21.9%

         \[\leadsto e^{\mathsf{log1p}\left(-\sin^{-1} \left(\frac{\color{blue}{{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}^{0.5}} \cdot \sqrt{0.5}}{\frac{t}{\ell}}\right)\right)} - 1 \]

      6. pow1/2 21.9%

         \[\leadsto e^{\mathsf{log1p}\left(-\sin^{-1} \left(\frac{{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}^{0.5} \cdot \color{blue}{{0.5}^{0.5}}}{\frac{t}{\ell}}\right)\right)} - 1 \]

      7. pow-prod-down 21.9%

         \[\leadsto e^{\mathsf{log1p}\left(-\sin^{-1} \left(\frac{\color{blue}{{\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5\right)}^{0.5}}}{\frac{t}{\ell}}\right)\right)} - 1 \]

   6. Applied egg-rr 21.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\sin^{-1} \left(\frac{{\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5\right)}^{0.5}}{\frac{t}{\ell}}\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 61.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-\sin^{-1} \left(\frac{{\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5\right)}^{0.5}}{\frac{t}{\ell}}\right)\right)\right)} \]

      2. expm1-log1p 61.7%

         \[\leadsto \color{blue}{-\sin^{-1} \left(\frac{{\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5\right)}^{0.5}}{\frac{t}{\ell}}\right)} \]

      3. associate-/r/ 61.8%

         \[\leadsto -\sin^{-1} \color{blue}{\left(\frac{{\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5\right)}^{0.5}}{t} \cdot \ell\right)} \]

      4. unpow1/2 61.8%

         \[\leadsto -\sin^{-1} \left(\frac{\color{blue}{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}}}{t} \cdot \ell\right) \]

      5. *-commutative 61.8%

         \[\leadsto -\sin^{-1} \left(\frac{\sqrt{\color{blue}{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}}{t} \cdot \ell\right) \]

   8. Simplified 61.8%

      \[\leadsto \color{blue}{-\sin^{-1} \left(\frac{\sqrt{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}{t} \cdot \ell\right)} \]

   9. Taylor expanded in Om around 0 61.8%

      \[\leadsto -\sin^{-1} \left(\color{blue}{\frac{\sqrt{0.5}}{t}} \cdot \ell\right) \]

   if -9.50000000000000044e-43 < t < 3.45000000000000011e95

   1. Initial program 91.6%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]

   2. Taylor expanded in t around 0 64.3%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]
   3. Step-by-step derivation

      1. unpow2 64.3%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}}\right) \]

      2. unpow2 64.3%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}}\right) \]

      3. times-frac 74.3%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}}\right) \]

      4. unpow2 74.3%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}}\right) \]

   4. Simplified 74.3%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}\right)} \]

   5. Taylor expanded in Om around 0 64.0%

      \[\leadsto \sin^{-1} \color{blue}{\left(1 + -0.5 \cdot \frac{{Om}^{2}}{{Omc}^{2}}\right)} \]
   6. Step-by-step derivation

      1. unpow2 64.0%

         \[\leadsto \sin^{-1} \left(1 + -0.5 \cdot \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}\right) \]

      2. unpow2 64.0%

         \[\leadsto \sin^{-1} \left(1 + -0.5 \cdot \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}\right) \]

   7. Simplified 64.0%

      \[\leadsto \sin^{-1} \color{blue}{\left(1 + -0.5 \cdot \frac{Om \cdot Om}{Omc \cdot Omc}\right)} \]
   8. Step-by-step derivation

      1. times-frac 73.6%

         \[\leadsto \sin^{-1} \left(1 + -0.5 \cdot \color{blue}{\left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)}\right) \]

      2. clear-num 73.6%

         \[\leadsto \sin^{-1} \left(1 + -0.5 \cdot \left(\frac{Om}{Omc} \cdot \color{blue}{\frac{1}{\frac{Omc}{Om}}}\right)\right) \]

      3. div-inv 73.6%

         \[\leadsto \sin^{-1} \left(1 + -0.5 \cdot \color{blue}{\frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right) \]

      4. associate-/r/ 73.6%

         \[\leadsto \sin^{-1} \left(1 + -0.5 \cdot \color{blue}{\left(\frac{\frac{Om}{Omc}}{Omc} \cdot Om\right)}\right) \]

   9. Applied egg-rr 73.6%

      \[\leadsto \sin^{-1} \left(1 + -0.5 \cdot \color{blue}{\left(\frac{\frac{Om}{Omc}}{Omc} \cdot Om\right)}\right) \]

   if 3.45000000000000011e95 < t

   1. Initial program 80.0%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. sqrt-div 80.0%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]

      2. div-inv 80.0%

         \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]

      3. add-sqr-sqrt 80.0%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\sqrt{1 + \color{blue}{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]

      4. hypot-1-def 80.0%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right) \]

      5. *-commutative 80.0%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right)}\right) \]

      6. sqrt-prod 80.0%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2}}\right)}\right) \]

      7. unpow2 80.0%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}} \cdot \sqrt{2}\right)}\right) \]

      8. sqrt-prod 39.3%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{t}{\ell}} \cdot \sqrt{\frac{t}{\ell}}\right)} \cdot \sqrt{2}\right)}\right) \]

      9. add-sqr-sqrt 96.4%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell}} \cdot \sqrt{2}\right)}\right) \]

   3. Applied egg-rr 96.4%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]
   4. Step-by-step derivation

      1. unpow2 96.4%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]

      2. times-frac 76.5%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om \cdot Om}{Omc \cdot Omc}}} \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]

      3. unpow2 76.5%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{\color{blue}{{Om}^{2}}}{Omc \cdot Omc}} \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]

      4. unpow2 76.5%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{{Om}^{2}}{\color{blue}{{Omc}^{2}}}} \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]

      5. associate-*r/ 76.5%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot 1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]

      6. *-rgt-identity 76.5%

         \[\leadsto \sin^{-1} \left(\frac{\color{blue}{\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]

      7. unpow2 76.5%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]

      8. unpow2 76.5%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]

      9. times-frac 96.4%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]

      10. unpow2 96.4%

          \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]

   5. Simplified 96.4%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]

   6. Taylor expanded in Om around 0 76.5%

      \[\leadsto \sin^{-1} \left(\frac{\color{blue}{1 + -0.5 \cdot \frac{{Om}^{2}}{{Omc}^{2}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
   7. Step-by-step derivation

      1. unpow2 15.3%

         \[\leadsto \sin^{-1} \left(1 + -0.5 \cdot \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}\right) \]

      2. unpow2 15.3%

         \[\leadsto \sin^{-1} \left(1 + -0.5 \cdot \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}\right) \]

   8. Simplified 76.5%

      \[\leadsto \sin^{-1} \left(\frac{\color{blue}{1 + -0.5 \cdot \frac{Om \cdot Om}{Omc \cdot Omc}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]

   9. Taylor expanded in t around -inf 53.2%

      \[\leadsto \sin^{-1} \color{blue}{\left(-1 \cdot \frac{\left(1 + -0.5 \cdot \frac{{Om}^{2}}{{Omc}^{2}}\right) \cdot \ell}{\sqrt{2} \cdot t}\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg 53.2%

          \[\leadsto \sin^{-1} \color{blue}{\left(-\frac{\left(1 + -0.5 \cdot \frac{{Om}^{2}}{{Omc}^{2}}\right) \cdot \ell}{\sqrt{2} \cdot t}\right)} \]

       2. times-frac 53.2%

          \[\leadsto \sin^{-1} \left(-\color{blue}{\frac{1 + -0.5 \cdot \frac{{Om}^{2}}{{Omc}^{2}}}{\sqrt{2}} \cdot \frac{\ell}{t}}\right) \]

       3. unpow2 53.2%

          \[\leadsto \sin^{-1} \left(-\frac{1 + -0.5 \cdot \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}}{\sqrt{2}} \cdot \frac{\ell}{t}\right) \]

       4. unpow2 53.2%

          \[\leadsto \sin^{-1} \left(-\frac{1 + -0.5 \cdot \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}}{\sqrt{2}} \cdot \frac{\ell}{t}\right) \]

   11. Simplified 53.2%

       \[\leadsto \sin^{-1} \color{blue}{\left(-\frac{1 + -0.5 \cdot \frac{Om \cdot Om}{Omc \cdot Omc}}{\sqrt{2}} \cdot \frac{\ell}{t}\right)} \]

   12. Taylor expanded in Om around 0 61.4%

       \[\leadsto \sin^{-1} \left(-\color{blue}{\frac{\ell}{\sqrt{2} \cdot t}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{-43}:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;t \leq 3.45 \cdot 10^{+95}:\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \left(Om \cdot \frac{\frac{Om}{Omc}}{Omc}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\sqrt{2} \cdot t}\right)\\ \end{array} \]

Alternative 17: 51.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} \left(1 + -0.5 \cdot \left(Om \cdot \frac{\frac{Om}{Omc}}{Omc}\right)\right) \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (asin (+ 1.0 (* -0.5 (* Om (/ (/ Om Omc) Omc))))))

C

double code(double t, double l, double Om, double Omc) {
	return asin((1.0 + (-0.5 * (Om * ((Om / Omc) / Omc)))));
}

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((1.0d0 + ((-0.5d0) * (om * ((om / omc) / omc)))))
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	return Math.asin((1.0 + (-0.5 * (Om * ((Om / Omc) / Omc)))));
}

Python

def code(t, l, Om, Omc):
	return math.asin((1.0 + (-0.5 * (Om * ((Om / Omc) / Omc)))))

Julia

function code(t, l, Om, Omc)
	return asin(Float64(1.0 + Float64(-0.5 * Float64(Om * Float64(Float64(Om / Omc) / Omc)))))
end

MATLAB

function tmp = code(t, l, Om, Omc)
	tmp = asin((1.0 + (-0.5 * (Om * ((Om / Omc) / Omc)))));
end

Wolfram

code[t_, l_, Om_, Omc_] := N[ArcSin[N[(1.0 + N[(-0.5 * N[(Om * N[(N[(Om / Omc), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 82.2%

   \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]

2. Taylor expanded in t around 0 42.5%

   \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]
3. Step-by-step derivation

   1. unpow2 42.5%

      \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}}\right) \]

   2. unpow2 42.5%

      \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}}\right) \]

   3. times-frac 50.0%

      \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}}\right) \]

   4. unpow2 50.0%

      \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}}\right) \]

4. Simplified 50.0%

   \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}\right)} \]

5. Taylor expanded in Om around 0 42.4%

   \[\leadsto \sin^{-1} \color{blue}{\left(1 + -0.5 \cdot \frac{{Om}^{2}}{{Omc}^{2}}\right)} \]
6. Step-by-step derivation

   1. unpow2 42.4%

      \[\leadsto \sin^{-1} \left(1 + -0.5 \cdot \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}\right) \]

   2. unpow2 42.4%

      \[\leadsto \sin^{-1} \left(1 + -0.5 \cdot \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}\right) \]

7. Simplified 42.4%

   \[\leadsto \sin^{-1} \color{blue}{\left(1 + -0.5 \cdot \frac{Om \cdot Om}{Omc \cdot Omc}\right)} \]
8. Step-by-step derivation

   1. times-frac 49.4%

      \[\leadsto \sin^{-1} \left(1 + -0.5 \cdot \color{blue}{\left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)}\right) \]

   2. clear-num 49.4%

      \[\leadsto \sin^{-1} \left(1 + -0.5 \cdot \left(\frac{Om}{Omc} \cdot \color{blue}{\frac{1}{\frac{Omc}{Om}}}\right)\right) \]

   3. div-inv 49.4%

      \[\leadsto \sin^{-1} \left(1 + -0.5 \cdot \color{blue}{\frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right) \]

   4. associate-/r/ 49.4%

      \[\leadsto \sin^{-1} \left(1 + -0.5 \cdot \color{blue}{\left(\frac{\frac{Om}{Omc}}{Omc} \cdot Om\right)}\right) \]

9. Applied egg-rr 49.4%

   \[\leadsto \sin^{-1} \left(1 + -0.5 \cdot \color{blue}{\left(\frac{\frac{Om}{Omc}}{Omc} \cdot Om\right)}\right) \]

10. Final simplification 49.4%

    \[\leadsto \sin^{-1} \left(1 + -0.5 \cdot \left(Om \cdot \frac{\frac{Om}{Omc}}{Omc}\right)\right) \]

Reproduce

herbie shell --seed 2023196 
(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)))))))