Toniolo and Linder, Equation (2)

Percentage Accurate: 84.7% → 98.5%

Time: 12.1s

Alternatives: 9

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.7% 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: 98.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

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

Python

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

Julia

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

MATLAB

function tmp = code(t, l, Om, Omc)
	tmp = asin((sqrt((1.0 - ((Om / Omc) ^ 2.0))) / hypot(1.0, ((t / l) * sqrt(2.0)))));
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[(t / l), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 85.2%

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

   1. sqrt-div 85.2%

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

   2. div-inv 85.2%

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

   3. add-sqr-sqrt 85.2%

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

   4. hypot-1-def 85.2%

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

   5. *-commutative 85.2%

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

   6. sqrt-prod 85.1%

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

   7. unpow2 85.1%

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

   8. sqrt-prod 58.5%

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

   9. add-sqr-sqrt 98.5%

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

3. Applied egg-rr 98.5%

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

   1. unpow2 98.5%

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

   2. times-frac 83.8%

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

   3. unpow2 83.8%

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

   4. unpow2 83.8%

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

   5. associate-*r/ 83.8%

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

   6. *-rgt-identity 83.8%

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

   7. unpow2 83.8%

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

   8. unpow2 83.8%

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

   9. times-frac 98.5%

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

   10. unpow2 98.5%

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

5. Simplified 98.5%

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

6. Final simplification 98.5%

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

Alternative 2: 97.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.2%

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

   1. sqrt-div 85.2%

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

   2. div-inv 85.2%

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

   3. add-sqr-sqrt 85.2%

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

   4. hypot-1-def 85.2%

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

   5. *-commutative 85.2%

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

   6. sqrt-prod 85.1%

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

   7. unpow2 85.1%

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

   8. sqrt-prod 58.5%

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

   9. add-sqr-sqrt 98.5%

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

3. Applied egg-rr 98.5%

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

   1. unpow2 98.5%

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

   2. times-frac 83.8%

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

   3. unpow2 83.8%

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

   4. unpow2 83.8%

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

   5. associate-*r/ 83.8%

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

   6. *-rgt-identity 83.8%

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

   7. unpow2 83.8%

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

   8. unpow2 83.8%

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

   9. times-frac 98.5%

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

   10. unpow2 98.5%

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

5. Simplified 98.5%

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

6. Taylor expanded in Om around 0 97.2%

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

7. Final simplification 97.2%

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

Alternative 3: 74.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)\\ t_2 := \sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{if}\;\ell \leq -5.5 \cdot 10^{+118}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -1.85 \cdot 10^{-158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-161}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}\right)\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+71}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (asin (sqrt (/ 1.0 (+ 1.0 (* 2.0 (/ (* t t) (* l l))))))))
        (t_2 (asin (sqrt (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))))
   (if (<= l -5.5e+118)
     t_2
     (if (<= l -1.85e-158)
       t_1
       (if (<= l 2.5e-161)
         (asin (sqrt (* 0.5 (* (/ l t) (/ l t)))))
         (if (<= l 2.6e+71) t_1 t_2))))))

C

double code(double t, double l, double Om, double Omc) {
	double t_1 = asin(sqrt((1.0 / (1.0 + (2.0 * ((t * t) / (l * l)))))));
	double t_2 = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	double tmp;
	if (l <= -5.5e+118) {
		tmp = t_2;
	} else if (l <= -1.85e-158) {
		tmp = t_1;
	} else if (l <= 2.5e-161) {
		tmp = asin(sqrt((0.5 * ((l / t) * (l / t)))));
	} else if (l <= 2.6e+71) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = asin(sqrt((1.0d0 / (1.0d0 + (2.0d0 * ((t * t) / (l * l)))))))
    t_2 = asin(sqrt((1.0d0 - ((om / omc) / (omc / om)))))
    if (l <= (-5.5d+118)) then
        tmp = t_2
    else if (l <= (-1.85d-158)) then
        tmp = t_1
    else if (l <= 2.5d-161) then
        tmp = asin(sqrt((0.5d0 * ((l / t) * (l / t)))))
    else if (l <= 2.6d+71) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double t_1 = Math.asin(Math.sqrt((1.0 / (1.0 + (2.0 * ((t * t) / (l * l)))))));
	double t_2 = Math.asin(Math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	double tmp;
	if (l <= -5.5e+118) {
		tmp = t_2;
	} else if (l <= -1.85e-158) {
		tmp = t_1;
	} else if (l <= 2.5e-161) {
		tmp = Math.asin(Math.sqrt((0.5 * ((l / t) * (l / t)))));
	} else if (l <= 2.6e+71) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	t_1 = math.asin(math.sqrt((1.0 / (1.0 + (2.0 * ((t * t) / (l * l)))))))
	t_2 = math.asin(math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))))
	tmp = 0
	if l <= -5.5e+118:
		tmp = t_2
	elif l <= -1.85e-158:
		tmp = t_1
	elif l <= 2.5e-161:
		tmp = math.asin(math.sqrt((0.5 * ((l / t) * (l / t)))))
	elif l <= 2.6e+71:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(t, l, Om, Omc)
	t_1 = asin(sqrt(Float64(1.0 / Float64(1.0 + Float64(2.0 * Float64(Float64(t * t) / Float64(l * l)))))))
	t_2 = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))))
	tmp = 0.0
	if (l <= -5.5e+118)
		tmp = t_2;
	elseif (l <= -1.85e-158)
		tmp = t_1;
	elseif (l <= 2.5e-161)
		tmp = asin(sqrt(Float64(0.5 * Float64(Float64(l / t) * Float64(l / t)))));
	elseif (l <= 2.6e+71)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	t_1 = asin(sqrt((1.0 / (1.0 + (2.0 * ((t * t) / (l * l)))))));
	t_2 = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	tmp = 0.0;
	if (l <= -5.5e+118)
		tmp = t_2;
	elseif (l <= -1.85e-158)
		tmp = t_1;
	elseif (l <= 2.5e-161)
		tmp = asin(sqrt((0.5 * ((l / t) * (l / t)))));
	elseif (l <= 2.6e+71)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = 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]}, Block[{t$95$2 = N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -5.5e+118], t$95$2, If[LessEqual[l, -1.85e-158], t$95$1, If[LessEqual[l, 2.5e-161], N[ArcSin[N[Sqrt[N[(0.5 * N[(N[(l / t), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.6e+71], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.5000000000000003e118 or 2.59999999999999991e71 < l

   1. Initial program 98.2%

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

   2. Taylor expanded in t around 0 69.4%

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

      1. unpow2 69.4%

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

      2. unpow2 69.4%

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

      3. times-frac 84.0%

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

      4. unpow2 84.0%

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

   4. Simplified 84.0%

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

      1. unpow2 84.0%

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

      2. clear-num 84.0%

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

      3. un-div-inv 84.0%

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

   6. Applied egg-rr 84.0%

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

   if -5.5000000000000003e118 < l < -1.85e-158 or 2.5e-161 < l < 2.59999999999999991e71

   1. Initial program 77.0%

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

      1. unpow2 77.0%

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

      2. clear-num 77.0%

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

      3. un-div-inv 77.0%

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

   3. Applied egg-rr 77.0%

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

   4. Taylor expanded in Om around 0 73.0%

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

      1. *-commutative 73.0%

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

      2. associate-*l/ 73.0%

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

      3. rem-square-sqrt 72.9%

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

      4. unpow2 72.9%

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

      5. *-commutative 72.9%

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

      6. *-commutative 72.9%

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

      7. unpow2 72.9%

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

      8. rem-square-sqrt 73.0%

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

      9. associate-*l/ 73.0%

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

      10. *-commutative 73.0%

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

      11. unpow2 73.0%

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

      12. unpow2 73.0%

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

   6. Simplified 73.0%

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

   if -1.85e-158 < l < 2.5e-161

   1. Initial program 78.4%

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

      1. unpow2 78.4%

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

      2. clear-num 78.5%

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

      3. un-div-inv 78.5%

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

   3. Applied egg-rr 78.5%

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

   4. Taylor expanded in t around inf 35.3%

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

      1. associate-/l* 35.3%

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

      2. unpow2 35.3%

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

      3. unpow2 35.3%

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

      4. unpow2 35.3%

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

      5. unpow2 35.3%

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

   6. Simplified 35.3%

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

   7. Taylor expanded in Om around 0 40.7%

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

      1. unpow2 40.7%

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

      2. unpow2 40.7%

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

      3. times-frac 72.4%

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

   9. Simplified 72.4%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.5 \cdot 10^{+118}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{elif}\;\ell \leq -1.85 \cdot 10^{-158}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-161}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}\right)\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+71}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \end{array} \]

Alternative 4: 84.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

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) / (omc / om))) / (1.0d0 + (2.0d0 * ((t / l) / (l / t)))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.2%

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

   1. unpow2 85.2%

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

   2. clear-num 85.3%

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

   3. un-div-inv 85.2%

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

3. Applied egg-rr 85.2%

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

   1. unpow2 52.0%

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

   2. clear-num 52.0%

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

   3. un-div-inv 52.0%

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

5. Applied egg-rr 85.2%

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

6. Final simplification 85.2%

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

Alternative 5: 64.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.75 \cdot 10^{-93} \lor \neg \left(\ell \leq 1.22 \cdot 10^{-104}\right):\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (or (<= l -2.75e-93) (not (<= l 1.22e-104)))
   (asin (sqrt (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))
   (asin (sqrt (* 0.5 (* (/ l t) (/ l t)))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((l <= -2.75e-93) || !(l <= 1.22e-104)) {
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} else {
		tmp = asin(sqrt((0.5 * ((l / t) * (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) :: tmp
    if ((l <= (-2.75d-93)) .or. (.not. (l <= 1.22d-104))) then
        tmp = asin(sqrt((1.0d0 - ((om / omc) / (omc / om)))))
    else
        tmp = asin(sqrt((0.5d0 * ((l / t) * (l / t)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((l <= -2.75e-93) || !(l <= 1.22e-104)) {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} else {
		tmp = Math.asin(Math.sqrt((0.5 * ((l / t) * (l / t)))));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (l <= -2.75e-93) or not (l <= 1.22e-104):
		tmp = math.asin(math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))))
	else:
		tmp = math.asin(math.sqrt((0.5 * ((l / t) * (l / t)))))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if ((l <= -2.75e-93) || !(l <= 1.22e-104))
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))));
	else
		tmp = asin(sqrt(Float64(0.5 * Float64(Float64(l / t) * Float64(l / t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((l <= -2.75e-93) || ~((l <= 1.22e-104)))
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	else
		tmp = asin(sqrt((0.5 * ((l / t) * (l / t)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[Or[LessEqual[l, -2.75e-93], N[Not[LessEqual[l, 1.22e-104]], $MachinePrecision]], 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[(0.5 * N[(N[(l / t), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.74999999999999984e-93 or 1.21999999999999997e-104 < l

   1. Initial program 88.8%

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

   2. Taylor expanded in t around 0 62.3%

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

      1. unpow2 62.3%

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

      2. unpow2 62.3%

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

      3. times-frac 72.5%

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

      4. unpow2 72.5%

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

   4. Simplified 72.5%

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

      1. unpow2 72.5%

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

      2. clear-num 72.5%

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

      3. un-div-inv 72.5%

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

   6. Applied egg-rr 72.5%

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

   if -2.74999999999999984e-93 < l < 1.21999999999999997e-104

   1. Initial program 78.5%

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

      1. unpow2 78.5%

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

      2. clear-num 78.5%

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

      3. un-div-inv 78.5%

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

   3. Applied egg-rr 78.5%

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

   4. Taylor expanded in t around inf 39.4%

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

      1. associate-/l* 39.4%

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

      2. unpow2 39.4%

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

      3. unpow2 39.4%

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

      4. unpow2 39.4%

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

      5. unpow2 39.4%

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

   6. Simplified 39.4%

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

   7. Taylor expanded in Om around 0 44.7%

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

      1. unpow2 44.7%

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

      2. unpow2 44.7%

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

      3. times-frac 68.8%

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

   9. Simplified 68.8%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.75 \cdot 10^{-93} \lor \neg \left(\ell \leq 1.22 \cdot 10^{-104}\right):\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}\right)\\ \end{array} \]

Alternative 6: 64.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.7 \cdot 10^{-96}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{-103}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= l -5.7e-96)
   (asin 1.0)
   (if (<= l 1.3e-103) (asin (sqrt (* 0.5 (* (/ l t) (/ l t))))) (asin 1.0))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= -5.7e-96) {
		tmp = asin(1.0);
	} else if (l <= 1.3e-103) {
		tmp = asin(sqrt((0.5 * ((l / t) * (l / t)))));
	} else {
		tmp = asin(1.0);
	}
	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 (l <= (-5.7d-96)) then
        tmp = asin(1.0d0)
    else if (l <= 1.3d-103) then
        tmp = asin(sqrt((0.5d0 * ((l / t) * (l / t)))))
    else
        tmp = asin(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= -5.7e-96) {
		tmp = Math.asin(1.0);
	} else if (l <= 1.3e-103) {
		tmp = Math.asin(Math.sqrt((0.5 * ((l / t) * (l / t)))));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if l <= -5.7e-96:
		tmp = math.asin(1.0)
	elif l <= 1.3e-103:
		tmp = math.asin(math.sqrt((0.5 * ((l / t) * (l / t)))))
	else:
		tmp = math.asin(1.0)
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (l <= -5.7e-96)
		tmp = asin(1.0);
	elseif (l <= 1.3e-103)
		tmp = asin(sqrt(Float64(0.5 * Float64(Float64(l / t) * Float64(l / t)))));
	else
		tmp = asin(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (l <= -5.7e-96)
		tmp = asin(1.0);
	elseif (l <= 1.3e-103)
		tmp = asin(sqrt((0.5 * ((l / t) * (l / t)))));
	else
		tmp = asin(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[l, -5.7e-96], N[ArcSin[1.0], $MachinePrecision], If[LessEqual[l, 1.3e-103], N[ArcSin[N[Sqrt[N[(0.5 * N[(N[(l / t), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -5.70000000000000009e-96 or 1.29999999999999998e-103 < l

   1. Initial program 88.8%

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

   2. Taylor expanded in t around 0 62.3%

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

      1. unpow2 62.3%

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

      2. unpow2 62.3%

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

      3. times-frac 72.5%

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

      4. unpow2 72.5%

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

   4. Simplified 72.5%

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

   5. Taylor expanded in Om around 0 71.4%

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

   if -5.70000000000000009e-96 < l < 1.29999999999999998e-103

   1. Initial program 78.5%

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

      1. unpow2 78.5%

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

      2. clear-num 78.5%

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

      3. un-div-inv 78.5%

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

   3. Applied egg-rr 78.5%

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

   4. Taylor expanded in t around inf 39.4%

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

      1. associate-/l* 39.4%

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

      2. unpow2 39.4%

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

      3. unpow2 39.4%

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

      4. unpow2 39.4%

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

      5. unpow2 39.4%

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

   6. Simplified 39.4%

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

   7. Taylor expanded in Om around 0 44.7%

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

      1. unpow2 44.7%

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

      2. unpow2 44.7%

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

      3. times-frac 68.8%

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

   9. Simplified 68.8%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.7 \cdot 10^{-96}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{-103}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 7: 63.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.8 \cdot 10^{-96}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 2.15 \cdot 10^{-116}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= l -3.8e-96)
   (asin 1.0)
   (if (<= l 2.15e-116) (asin (* (/ l t) (sqrt 0.5))) (asin 1.0))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= -3.8e-96) {
		tmp = asin(1.0);
	} else if (l <= 2.15e-116) {
		tmp = asin(((l / t) * sqrt(0.5)));
	} else {
		tmp = asin(1.0);
	}
	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 (l <= (-3.8d-96)) then
        tmp = asin(1.0d0)
    else if (l <= 2.15d-116) then
        tmp = asin(((l / t) * sqrt(0.5d0)))
    else
        tmp = asin(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= -3.8e-96) {
		tmp = Math.asin(1.0);
	} else if (l <= 2.15e-116) {
		tmp = Math.asin(((l / t) * Math.sqrt(0.5)));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if l <= -3.8e-96:
		tmp = math.asin(1.0)
	elif l <= 2.15e-116:
		tmp = math.asin(((l / t) * math.sqrt(0.5)))
	else:
		tmp = math.asin(1.0)
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (l <= -3.8e-96)
		tmp = asin(1.0);
	elseif (l <= 2.15e-116)
		tmp = asin(Float64(Float64(l / t) * sqrt(0.5)));
	else
		tmp = asin(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (l <= -3.8e-96)
		tmp = asin(1.0);
	elseif (l <= 2.15e-116)
		tmp = asin(((l / t) * sqrt(0.5)));
	else
		tmp = asin(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[l, -3.8e-96], N[ArcSin[1.0], $MachinePrecision], If[LessEqual[l, 2.15e-116], N[ArcSin[N[(N[(l / t), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -3.8000000000000001e-96 or 2.1499999999999999e-116 < l

   1. Initial program 89.0%

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

   2. Taylor expanded in t around 0 61.8%

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

      1. unpow2 61.8%

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

      2. unpow2 61.8%

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

      3. times-frac 71.9%

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

      4. unpow2 71.9%

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

   4. Simplified 71.9%

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

   5. Taylor expanded in Om around 0 70.8%

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

   if -3.8000000000000001e-96 < l < 2.1499999999999999e-116

   1. Initial program 77.7%

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

      1. unpow2 77.7%

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

      2. clear-num 77.8%

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

      3. un-div-inv 77.8%

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

   3. Applied egg-rr 77.8%

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

   4. Taylor expanded in t around inf 39.6%

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

      1. associate-/l* 39.6%

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

      2. unpow2 39.6%

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

      3. unpow2 39.6%

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

      4. unpow2 39.6%

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

      5. unpow2 39.6%

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

   6. Simplified 39.6%

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

   7. Taylor expanded in Om around 0 67.4%

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

      1. associate-*r/ 67.4%

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

   9. Simplified 67.4%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.8 \cdot 10^{-96}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 2.15 \cdot 10^{-116}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 8: 63.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{-94}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{-114}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= l -2.1e-94)
   (asin 1.0)
   (if (<= l 2.6e-114) (asin (/ l (* t (sqrt 2.0)))) (asin 1.0))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= -2.1e-94) {
		tmp = asin(1.0);
	} else if (l <= 2.6e-114) {
		tmp = asin((l / (t * sqrt(2.0))));
	} else {
		tmp = asin(1.0);
	}
	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 (l <= (-2.1d-94)) then
        tmp = asin(1.0d0)
    else if (l <= 2.6d-114) then
        tmp = asin((l / (t * sqrt(2.0d0))))
    else
        tmp = asin(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= -2.1e-94) {
		tmp = Math.asin(1.0);
	} else if (l <= 2.6e-114) {
		tmp = Math.asin((l / (t * Math.sqrt(2.0))));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if l <= -2.1e-94:
		tmp = math.asin(1.0)
	elif l <= 2.6e-114:
		tmp = math.asin((l / (t * math.sqrt(2.0))))
	else:
		tmp = math.asin(1.0)
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (l <= -2.1e-94)
		tmp = asin(1.0);
	elseif (l <= 2.6e-114)
		tmp = asin(Float64(l / Float64(t * sqrt(2.0))));
	else
		tmp = asin(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (l <= -2.1e-94)
		tmp = asin(1.0);
	elseif (l <= 2.6e-114)
		tmp = asin((l / (t * sqrt(2.0))));
	else
		tmp = asin(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[l, -2.1e-94], N[ArcSin[1.0], $MachinePrecision], If[LessEqual[l, 2.6e-114], N[ArcSin[N[(l / N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -2.1000000000000001e-94 or 2.60000000000000013e-114 < l

   1. Initial program 89.0%

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

   2. Taylor expanded in t around 0 61.8%

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

      1. unpow2 61.8%

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

      2. unpow2 61.8%

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

      3. times-frac 71.9%

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

      4. unpow2 71.9%

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

   4. Simplified 71.9%

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

   5. Taylor expanded in Om around 0 70.8%

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

   if -2.1000000000000001e-94 < l < 2.60000000000000013e-114

   1. Initial program 77.7%

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

      1. sqrt-div 77.7%

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

      2. div-inv 77.7%

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

      3. add-sqr-sqrt 77.7%

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

      4. hypot-1-def 77.7%

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

      5. *-commutative 77.7%

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

      6. sqrt-prod 77.6%

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

      7. unpow2 77.6%

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

      8. sqrt-prod 57.0%

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

      9. add-sqr-sqrt 97.9%

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

   3. Applied egg-rr 97.9%

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

      1. unpow2 97.9%

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

      2. times-frac 84.0%

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

      3. unpow2 84.0%

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

      4. unpow2 84.0%

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

      5. associate-*r/ 84.0%

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

      6. *-rgt-identity 84.0%

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

      7. unpow2 84.0%

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

      8. unpow2 84.0%

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

      9. times-frac 97.9%

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

      10. unpow2 97.9%

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

   5. Simplified 97.9%

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

   6. Taylor expanded in Om around 0 97.2%

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

   7. Taylor expanded in t around inf 67.4%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{-94}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{-114}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 9: 50.9% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} 1 \end{array} \]

FPCore

(FPCore (t l Om Omc) :precision binary64 (asin 1.0))

C

double code(double t, double l, double Om, double Omc) {
	return asin(1.0);
}

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(1.0d0)
end function

Java

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

Python

def code(t, l, Om, Omc):
	return math.asin(1.0)

Julia

function code(t, l, Om, Omc)
	return asin(1.0)
end

MATLAB

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

Wolfram

code[t_, l_, Om_, Omc_] := N[ArcSin[1.0], $MachinePrecision]

Derivation

1. Initial program 85.2%

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

2. Taylor expanded in t around 0 45.0%

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

   1. unpow2 45.0%

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

   2. unpow2 45.0%

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

   3. times-frac 52.0%

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

   4. unpow2 52.0%

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

4. Simplified 52.0%

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

5. Taylor expanded in Om around 0 51.2%

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

6. Final simplification 51.2%

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

Reproduce

herbie shell --seed 2023189 
(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)))))))