Toniolo and Linder, Equation (2)

Percentage Accurate: 84.8% → 98.4%

Time: 12.9s

Alternatives: 12

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: 84.8% 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: 98.4% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (asin
  (/ (sqrt (- 1.0 (pow (/ Om Omc) 2.0))) (hypot 1.0 (/ (* t (sqrt 2.0)) 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, ((t * sqrt(2.0)) / 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, ((t * Math.sqrt(2.0)) / 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, ((t * math.sqrt(2.0)) / 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(t * sqrt(2.0)) / l))))
end

MATLAB

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

Derivation

1. Initial program 83.1%

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

   1. sqrt-div 83.0%

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

   2. div-inv 83.0%

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

   3. add-sqr-sqrt 83.1%

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

   4. hypot-1-def 83.1%

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

   5. *-commutative 83.1%

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

   6. sqrt-prod 83.0%

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

   7. unpow2 83.0%

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

   8. sqrt-prod 57.1%

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

   9. add-sqr-sqrt 99.0%

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

3. Applied egg-rr 99.0%

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

   1. associate-*r/ 99.0%

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

   2. *-rgt-identity 99.0%

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

5. Simplified 99.0%

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

6. Taylor expanded in t around 0 99.0%

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

7. Final simplification 99.0%

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

Alternative 2: 98.4% 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 83.1%

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

   1. sqrt-div 83.0%

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

   2. div-inv 83.0%

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

   3. add-sqr-sqrt 83.1%

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

   4. hypot-1-def 83.1%

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

   5. *-commutative 83.1%

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

   6. sqrt-prod 83.0%

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

   7. unpow2 83.0%

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

   8. sqrt-prod 57.1%

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

   9. add-sqr-sqrt 99.0%

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

3. Applied egg-rr 99.0%

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

   1. associate-*r/ 99.0%

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

   2. *-rgt-identity 99.0%

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

5. Simplified 99.0%

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

6. Final simplification 99.0%

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

Alternative 3: 97.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.1%

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

   1. sqrt-div 83.0%

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

   2. div-inv 83.0%

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

   3. add-sqr-sqrt 83.1%

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

   4. hypot-1-def 83.1%

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

   5. *-commutative 83.1%

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

   6. sqrt-prod 83.0%

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

   7. unpow2 83.0%

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

   8. sqrt-prod 57.1%

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

   9. add-sqr-sqrt 99.0%

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

3. Applied egg-rr 99.0%

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

   1. associate-*r/ 99.0%

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

   2. *-rgt-identity 99.0%

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

5. Simplified 99.0%

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

6. Taylor expanded in t around 0 99.0%

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

7. Taylor expanded in Om around 0 66.0%

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

   1. unpow2 66.0%

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

   2. unpow2 66.0%

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

   3. swap-sqr 66.1%

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

   4. unpow2 66.1%

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

   5. times-frac 82.0%

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

   6. associate-*l/ 82.0%

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

   7. associate-*l/ 82.0%

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

   8. associate-/r/ 82.0%

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

   9. associate-/r/ 82.0%

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

   10. unpow2 82.0%

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

   11. associate-/r/ 82.0%

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

   12. *-commutative 82.0%

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

9. Simplified 82.0%

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

    1. sqrt-div 82.0%

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

    2. metadata-eval 82.0%

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

    3. unpow2 82.0%

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

    4. hypot-1-def 98.0%

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

11. Applied egg-rr 98.0%

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

12. Final simplification 98.0%

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

Alternative 4: 97.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -500:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{-\ell}{t}}{\sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.1:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -500.0)
   (asin (/ (/ (- l) t) (sqrt 2.0)))
   (if (<= (/ t l) 0.1)
     (asin (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc)))))
     (asin (/ l (/ t (sqrt 0.5)))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -500.0) {
		tmp = asin(((-l / t) / sqrt(2.0)));
	} else if ((t / l) <= 0.1) {
		tmp = asin(sqrt((1.0 - ((Om / Omc) * (Om / Omc)))));
	} else {
		tmp = asin((l / (t / sqrt(0.5))));
	}
	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) <= (-500.0d0)) then
        tmp = asin(((-l / t) / sqrt(2.0d0)))
    else if ((t / l) <= 0.1d0) then
        tmp = asin(sqrt((1.0d0 - ((om / omc) * (om / omc)))))
    else
        tmp = asin((l / (t / sqrt(0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -500.0) {
		tmp = Math.asin(((-l / t) / Math.sqrt(2.0)));
	} else if ((t / l) <= 0.1) {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om / Omc) * (Om / Omc)))));
	} else {
		tmp = Math.asin((l / (t / Math.sqrt(0.5))));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -500.0:
		tmp = math.asin(((-l / t) / math.sqrt(2.0)))
	elif (t / l) <= 0.1:
		tmp = math.asin(math.sqrt((1.0 - ((Om / Omc) * (Om / Omc)))))
	else:
		tmp = math.asin((l / (t / math.sqrt(0.5))))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -500.0)
		tmp = asin(Float64(Float64(Float64(-l) / t) / sqrt(2.0)));
	elseif (Float64(t / l) <= 0.1)
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc)))));
	else
		tmp = asin(Float64(l / Float64(t / sqrt(0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -500.0)
		tmp = asin(((-l / t) / sqrt(2.0)));
	elseif ((t / l) <= 0.1)
		tmp = asin(sqrt((1.0 - ((Om / Omc) * (Om / Omc)))));
	else
		tmp = asin((l / (t / sqrt(0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -500.0], N[ArcSin[N[(N[((-l) / t), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 0.1], N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l / N[(t / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -500

   1. Initial program 73.3%

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

      1. sqrt-div 73.3%

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

      2. div-inv 73.3%

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

      3. add-sqr-sqrt 73.3%

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

      4. hypot-1-def 73.3%

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

      5. *-commutative 73.3%

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

      6. sqrt-prod 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 0.0%

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

      9. add-sqr-sqrt 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. associate-*r/ 99.4%

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

      2. *-rgt-identity 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in t around 0 99.5%

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

   7. Taylor expanded in Om around 0 52.6%

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

      1. unpow2 52.6%

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

      2. unpow2 52.6%

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

      3. swap-sqr 52.8%

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

      4. unpow2 52.8%

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

      5. times-frac 72.5%

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

      6. associate-*l/ 72.5%

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

      7. associate-*l/ 72.4%

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

      8. associate-/r/ 72.5%

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

      9. associate-/r/ 72.4%

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

      10. unpow2 72.4%

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

      11. associate-/r/ 72.4%

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

      12. *-commutative 72.4%

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

   9. Simplified 72.4%

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

   10. Taylor expanded in t around -inf 97.9%

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

       1. mul-1-neg 97.9%

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

       2. associate-/r* 98.0%

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

       3. distribute-neg-frac 98.0%

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

   12. Simplified 98.0%

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

   if -500 < (/.f64 t l) < 0.10000000000000001

   1. Initial program 99.2%

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

   2. Taylor expanded in t around 0 83.2%

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

      1. unpow2 83.2%

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

      2. unpow2 83.2%

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

      3. times-frac 98.1%

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

      4. unpow2 98.1%

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

   4. Simplified 98.1%

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

      1. unpow2 98.1%

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

   6. Applied egg-rr 98.1%

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

   if 0.10000000000000001 < (/.f64 t l)

   1. Initial program 64.9%

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

   2. Taylor expanded in t around inf 43.6%

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

      1. associate-/l* 43.6%

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

      2. unpow2 43.6%

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

      3. unpow2 43.6%

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

      4. unpow2 43.6%

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

      5. unpow2 43.6%

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

   4. Simplified 43.6%

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

   5. Taylor expanded in Om around 0 98.9%

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

      1. associate-/l* 98.9%

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

   7. Simplified 98.9%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -500:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{-\ell}{t}}{\sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.1:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \]

Alternative 5: 96.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -500:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{-\ell}{t}}{\sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.1:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -500.0)
   (asin (/ (/ (- l) t) (sqrt 2.0)))
   (if (<= (/ t l) 0.1)
     (asin (- 1.0 (pow (/ t l) 2.0)))
     (asin (/ l (/ t (sqrt 0.5)))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -500.0) {
		tmp = asin(((-l / t) / sqrt(2.0)));
	} else if ((t / l) <= 0.1) {
		tmp = asin((1.0 - pow((t / l), 2.0)));
	} else {
		tmp = asin((l / (t / sqrt(0.5))));
	}
	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) <= (-500.0d0)) then
        tmp = asin(((-l / t) / sqrt(2.0d0)))
    else if ((t / l) <= 0.1d0) then
        tmp = asin((1.0d0 - ((t / l) ** 2.0d0)))
    else
        tmp = asin((l / (t / sqrt(0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -500.0) {
		tmp = Math.asin(((-l / t) / Math.sqrt(2.0)));
	} else if ((t / l) <= 0.1) {
		tmp = Math.asin((1.0 - Math.pow((t / l), 2.0)));
	} else {
		tmp = Math.asin((l / (t / Math.sqrt(0.5))));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -500.0:
		tmp = math.asin(((-l / t) / math.sqrt(2.0)))
	elif (t / l) <= 0.1:
		tmp = math.asin((1.0 - math.pow((t / l), 2.0)))
	else:
		tmp = math.asin((l / (t / math.sqrt(0.5))))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -500.0)
		tmp = asin(Float64(Float64(Float64(-l) / t) / sqrt(2.0)));
	elseif (Float64(t / l) <= 0.1)
		tmp = asin(Float64(1.0 - (Float64(t / l) ^ 2.0)));
	else
		tmp = asin(Float64(l / Float64(t / sqrt(0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -500.0)
		tmp = asin(((-l / t) / sqrt(2.0)));
	elseif ((t / l) <= 0.1)
		tmp = asin((1.0 - ((t / l) ^ 2.0)));
	else
		tmp = asin((l / (t / sqrt(0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -500.0], N[ArcSin[N[(N[((-l) / t), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 0.1], N[ArcSin[N[(1.0 - N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l / N[(t / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -500

   1. Initial program 73.3%

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

      1. sqrt-div 73.3%

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

      2. div-inv 73.3%

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

      3. add-sqr-sqrt 73.3%

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

      4. hypot-1-def 73.3%

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

      5. *-commutative 73.3%

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

      6. sqrt-prod 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 0.0%

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

      9. add-sqr-sqrt 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. associate-*r/ 99.4%

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

      2. *-rgt-identity 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in t around 0 99.5%

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

   7. Taylor expanded in Om around 0 52.6%

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

      1. unpow2 52.6%

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

      2. unpow2 52.6%

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

      3. swap-sqr 52.8%

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

      4. unpow2 52.8%

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

      5. times-frac 72.5%

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

      6. associate-*l/ 72.5%

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

      7. associate-*l/ 72.4%

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

      8. associate-/r/ 72.5%

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

      9. associate-/r/ 72.4%

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

      10. unpow2 72.4%

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

      11. associate-/r/ 72.4%

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

      12. *-commutative 72.4%

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

   9. Simplified 72.4%

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

   10. Taylor expanded in t around -inf 97.9%

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

       1. mul-1-neg 97.9%

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

       2. associate-/r* 98.0%

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

       3. distribute-neg-frac 98.0%

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

   12. Simplified 98.0%

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

   if -500 < (/.f64 t l) < 0.10000000000000001

   1. Initial program 99.2%

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

      1. sqrt-div 99.2%

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

      2. div-inv 99.2%

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

      3. add-sqr-sqrt 99.2%

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

      4. hypot-1-def 99.2%

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

      5. *-commutative 99.2%

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

      6. sqrt-prod 99.2%

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

      7. unpow2 99.2%

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

      8. sqrt-prod 63.0%

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

      9. add-sqr-sqrt 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. associate-*r/ 99.2%

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

      2. *-rgt-identity 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in t around 0 99.2%

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

   7. Taylor expanded in Om around 0 83.7%

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

      1. unpow2 83.7%

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

      2. unpow2 83.7%

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

      3. swap-sqr 83.7%

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

      4. unpow2 83.7%

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

      5. times-frac 97.5%

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

      6. associate-*l/ 97.5%

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

      7. associate-*l/ 97.5%

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

      8. associate-/r/ 97.5%

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

      9. associate-/r/ 97.5%

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

      10. unpow2 97.5%

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

      11. associate-/r/ 97.5%

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

      12. *-commutative 97.5%

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

   9. Simplified 97.5%

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

   10. Taylor expanded in t around 0 83.0%

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

       1. unpow2 83.0%

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

       2. *-commutative 83.0%

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

       3. unpow2 83.0%

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

       4. unpow2 83.0%

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

       5. swap-sqr 83.0%

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

       6. times-frac 96.8%

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

       7. associate-*r/ 96.8%

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

       8. associate-*r/ 96.8%

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

       9. swap-sqr 96.8%

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

       10. rem-square-sqrt 96.8%

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

       11. unpow2 96.8%

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

       12. associate-*r* 96.8%

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

       13. metadata-eval 96.8%

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

   12. Simplified 96.8%

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

   if 0.10000000000000001 < (/.f64 t l)

   1. Initial program 64.9%

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

   2. Taylor expanded in t around inf 43.6%

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

      1. associate-/l* 43.6%

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

      2. unpow2 43.6%

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

      3. unpow2 43.6%

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

      4. unpow2 43.6%

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

      5. unpow2 43.6%

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

   4. Simplified 43.6%

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

   5. Taylor expanded in Om around 0 98.9%

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

      1. associate-/l* 98.9%

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

   7. Simplified 98.9%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -500:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{-\ell}{t}}{\sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.1:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \]

Alternative 6: 80.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+208}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.1:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -2e+208)
   (asin (/ (sqrt 0.5) (/ t l)))
   (if (<= (/ t l) 0.1) (asin 1.0) (asin (/ l (/ t (sqrt 0.5)))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -2e+208) {
		tmp = asin((sqrt(0.5) / (t / l)));
	} else if ((t / l) <= 0.1) {
		tmp = asin(1.0);
	} else {
		tmp = asin((l / (t / sqrt(0.5))));
	}
	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) <= (-2d+208)) then
        tmp = asin((sqrt(0.5d0) / (t / l)))
    else if ((t / l) <= 0.1d0) then
        tmp = asin(1.0d0)
    else
        tmp = asin((l / (t / sqrt(0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -2e+208) {
		tmp = Math.asin((Math.sqrt(0.5) / (t / l)));
	} else if ((t / l) <= 0.1) {
		tmp = Math.asin(1.0);
	} else {
		tmp = Math.asin((l / (t / Math.sqrt(0.5))));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -2e+208:
		tmp = math.asin((math.sqrt(0.5) / (t / l)))
	elif (t / l) <= 0.1:
		tmp = math.asin(1.0)
	else:
		tmp = math.asin((l / (t / math.sqrt(0.5))))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -2e+208)
		tmp = asin(Float64(sqrt(0.5) / Float64(t / l)));
	elseif (Float64(t / l) <= 0.1)
		tmp = asin(1.0);
	else
		tmp = asin(Float64(l / Float64(t / sqrt(0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -2e+208)
		tmp = asin((sqrt(0.5) / (t / l)));
	elseif ((t / l) <= 0.1)
		tmp = asin(1.0);
	else
		tmp = asin((l / (t / sqrt(0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -2e+208], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 0.1], N[ArcSin[1.0], $MachinePrecision], N[ArcSin[N[(l / N[(t / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -2e208

   1. Initial program 66.8%

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

   2. Taylor expanded in t around inf 62.8%

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

      1. associate-/l* 62.8%

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

      2. unpow2 62.8%

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

      3. unpow2 62.8%

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

      4. unpow2 62.8%

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

      5. unpow2 62.8%

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

   4. Simplified 62.8%

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

   5. Taylor expanded in Om around 0 66.2%

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

      1. associate-/l* 66.2%

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

      2. associate-/r/ 66.2%

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

   7. Simplified 66.2%

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

      1. *-commutative 66.2%

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

      2. clear-num 66.2%

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

      3. un-div-inv 66.2%

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

   9. Applied egg-rr 66.2%

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

   if -2e208 < (/.f64 t l) < 0.10000000000000001

   1. Initial program 93.8%

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

      1. sqrt-div 93.8%

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

      2. div-inv 93.8%

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

      3. add-sqr-sqrt 93.8%

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

      4. hypot-1-def 93.8%

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

      5. *-commutative 93.8%

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

      6. sqrt-prod 93.8%

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

      7. unpow2 93.8%

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

      8. sqrt-prod 47.5%

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

      9. add-sqr-sqrt 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. associate-*r/ 99.2%

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

      2. *-rgt-identity 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in t around 0 99.2%

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

   7. Taylor expanded in Om around 0 73.9%

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

      1. unpow2 73.9%

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

      2. unpow2 73.9%

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

      3. swap-sqr 73.9%

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

      4. unpow2 73.9%

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

      5. times-frac 92.3%

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

      6. associate-*l/ 92.3%

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

      7. associate-*l/ 92.2%

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

      8. associate-/r/ 92.3%

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

      9. associate-/r/ 92.3%

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

      10. unpow2 92.3%

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

      11. associate-/r/ 92.2%

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

      12. *-commutative 92.2%

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

   9. Simplified 92.2%

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

   10. Taylor expanded in t around 0 74.3%

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

   if 0.10000000000000001 < (/.f64 t l)

   1. Initial program 64.9%

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

   2. Taylor expanded in t around inf 43.6%

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

      1. associate-/l* 43.6%

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

      2. unpow2 43.6%

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

      3. unpow2 43.6%

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

      4. unpow2 43.6%

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

      5. unpow2 43.6%

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

   4. Simplified 43.6%

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

   5. Taylor expanded in Om around 0 98.9%

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

      1. associate-/l* 98.9%

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

   7. Simplified 98.9%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+208}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.1:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \]

Alternative 7: 96.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -500:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.1:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -500.0)
   (asin (/ (- l) (* t (sqrt 2.0))))
   (if (<= (/ t l) 0.1) (asin 1.0) (asin (/ l (/ t (sqrt 0.5)))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -500.0) {
		tmp = asin((-l / (t * sqrt(2.0))));
	} else if ((t / l) <= 0.1) {
		tmp = asin(1.0);
	} else {
		tmp = asin((l / (t / sqrt(0.5))));
	}
	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) <= (-500.0d0)) then
        tmp = asin((-l / (t * sqrt(2.0d0))))
    else if ((t / l) <= 0.1d0) then
        tmp = asin(1.0d0)
    else
        tmp = asin((l / (t / sqrt(0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -500.0) {
		tmp = Math.asin((-l / (t * Math.sqrt(2.0))));
	} else if ((t / l) <= 0.1) {
		tmp = Math.asin(1.0);
	} else {
		tmp = Math.asin((l / (t / Math.sqrt(0.5))));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -500.0:
		tmp = math.asin((-l / (t * math.sqrt(2.0))))
	elif (t / l) <= 0.1:
		tmp = math.asin(1.0)
	else:
		tmp = math.asin((l / (t / math.sqrt(0.5))))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -500.0)
		tmp = asin(Float64(Float64(-l) / Float64(t * sqrt(2.0))));
	elseif (Float64(t / l) <= 0.1)
		tmp = asin(1.0);
	else
		tmp = asin(Float64(l / Float64(t / sqrt(0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -500.0)
		tmp = asin((-l / (t * sqrt(2.0))));
	elseif ((t / l) <= 0.1)
		tmp = asin(1.0);
	else
		tmp = asin((l / (t / sqrt(0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -500.0], N[ArcSin[N[((-l) / N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 0.1], N[ArcSin[1.0], $MachinePrecision], N[ArcSin[N[(l / N[(t / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -500

   1. Initial program 73.3%

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

      1. sqrt-div 73.3%

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

      2. div-inv 73.3%

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

      3. add-sqr-sqrt 73.3%

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

      4. hypot-1-def 73.3%

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

      5. *-commutative 73.3%

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

      6. sqrt-prod 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 0.0%

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

      9. add-sqr-sqrt 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. associate-*r/ 99.4%

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

      2. *-rgt-identity 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in t around 0 99.5%

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

   7. Taylor expanded in Om around 0 52.6%

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

      1. unpow2 52.6%

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

      2. unpow2 52.6%

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

      3. swap-sqr 52.8%

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

      4. unpow2 52.8%

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

      5. times-frac 72.5%

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

      6. associate-*l/ 72.5%

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

      7. associate-*l/ 72.4%

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

      8. associate-/r/ 72.5%

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

      9. associate-/r/ 72.4%

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

      10. unpow2 72.4%

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

      11. associate-/r/ 72.4%

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

      12. *-commutative 72.4%

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

   9. Simplified 72.4%

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

   10. Taylor expanded in t around -inf 97.9%

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

       1. mul-1-neg 97.9%

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

   12. Simplified 97.9%

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

   if -500 < (/.f64 t l) < 0.10000000000000001

   1. Initial program 99.2%

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

      1. sqrt-div 99.2%

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

      2. div-inv 99.2%

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

      3. add-sqr-sqrt 99.2%

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

      4. hypot-1-def 99.2%

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

      5. *-commutative 99.2%

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

      6. sqrt-prod 99.2%

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

      7. unpow2 99.2%

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

      8. sqrt-prod 63.0%

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

      9. add-sqr-sqrt 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. associate-*r/ 99.2%

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

      2. *-rgt-identity 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in t around 0 99.2%

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

   7. Taylor expanded in Om around 0 83.7%

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

      1. unpow2 83.7%

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

      2. unpow2 83.7%

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

      3. swap-sqr 83.7%

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

      4. unpow2 83.7%

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

      5. times-frac 97.5%

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

      6. associate-*l/ 97.5%

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

      7. associate-*l/ 97.5%

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

      8. associate-/r/ 97.5%

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

      9. associate-/r/ 97.5%

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

      10. unpow2 97.5%

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

      11. associate-/r/ 97.5%

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

      12. *-commutative 97.5%

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

   9. Simplified 97.5%

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

   10. Taylor expanded in t around 0 96.4%

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

   if 0.10000000000000001 < (/.f64 t l)

   1. Initial program 64.9%

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

   2. Taylor expanded in t around inf 43.6%

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

      1. associate-/l* 43.6%

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

      2. unpow2 43.6%

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

      3. unpow2 43.6%

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

      4. unpow2 43.6%

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

      5. unpow2 43.6%

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

   4. Simplified 43.6%

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

   5. Taylor expanded in Om around 0 98.9%

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

      1. associate-/l* 98.9%

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

   7. Simplified 98.9%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -500:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.1:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \]

Alternative 8: 96.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -500:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{-\ell}{t}}{\sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.1:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -500.0)
   (asin (/ (/ (- l) t) (sqrt 2.0)))
   (if (<= (/ t l) 0.1) (asin 1.0) (asin (/ l (/ t (sqrt 0.5)))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -500.0) {
		tmp = asin(((-l / t) / sqrt(2.0)));
	} else if ((t / l) <= 0.1) {
		tmp = asin(1.0);
	} else {
		tmp = asin((l / (t / sqrt(0.5))));
	}
	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) <= (-500.0d0)) then
        tmp = asin(((-l / t) / sqrt(2.0d0)))
    else if ((t / l) <= 0.1d0) then
        tmp = asin(1.0d0)
    else
        tmp = asin((l / (t / sqrt(0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -500.0) {
		tmp = Math.asin(((-l / t) / Math.sqrt(2.0)));
	} else if ((t / l) <= 0.1) {
		tmp = Math.asin(1.0);
	} else {
		tmp = Math.asin((l / (t / Math.sqrt(0.5))));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -500.0:
		tmp = math.asin(((-l / t) / math.sqrt(2.0)))
	elif (t / l) <= 0.1:
		tmp = math.asin(1.0)
	else:
		tmp = math.asin((l / (t / math.sqrt(0.5))))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -500.0)
		tmp = asin(Float64(Float64(Float64(-l) / t) / sqrt(2.0)));
	elseif (Float64(t / l) <= 0.1)
		tmp = asin(1.0);
	else
		tmp = asin(Float64(l / Float64(t / sqrt(0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -500.0)
		tmp = asin(((-l / t) / sqrt(2.0)));
	elseif ((t / l) <= 0.1)
		tmp = asin(1.0);
	else
		tmp = asin((l / (t / sqrt(0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -500.0], N[ArcSin[N[(N[((-l) / t), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 0.1], N[ArcSin[1.0], $MachinePrecision], N[ArcSin[N[(l / N[(t / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -500

   1. Initial program 73.3%

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

      1. sqrt-div 73.3%

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

      2. div-inv 73.3%

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

      3. add-sqr-sqrt 73.3%

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

      4. hypot-1-def 73.3%

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

      5. *-commutative 73.3%

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

      6. sqrt-prod 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 0.0%

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

      9. add-sqr-sqrt 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. associate-*r/ 99.4%

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

      2. *-rgt-identity 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in t around 0 99.5%

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

   7. Taylor expanded in Om around 0 52.6%

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

      1. unpow2 52.6%

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

      2. unpow2 52.6%

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

      3. swap-sqr 52.8%

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

      4. unpow2 52.8%

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

      5. times-frac 72.5%

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

      6. associate-*l/ 72.5%

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

      7. associate-*l/ 72.4%

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

      8. associate-/r/ 72.5%

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

      9. associate-/r/ 72.4%

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

      10. unpow2 72.4%

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

      11. associate-/r/ 72.4%

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

      12. *-commutative 72.4%

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

   9. Simplified 72.4%

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

   10. Taylor expanded in t around -inf 97.9%

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

       1. mul-1-neg 97.9%

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

       2. associate-/r* 98.0%

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

       3. distribute-neg-frac 98.0%

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

   12. Simplified 98.0%

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

   if -500 < (/.f64 t l) < 0.10000000000000001

   1. Initial program 99.2%

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

      1. sqrt-div 99.2%

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

      2. div-inv 99.2%

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

      3. add-sqr-sqrt 99.2%

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

      4. hypot-1-def 99.2%

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

      5. *-commutative 99.2%

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

      6. sqrt-prod 99.2%

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

      7. unpow2 99.2%

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

      8. sqrt-prod 63.0%

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

      9. add-sqr-sqrt 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. associate-*r/ 99.2%

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

      2. *-rgt-identity 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in t around 0 99.2%

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

   7. Taylor expanded in Om around 0 83.7%

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

      1. unpow2 83.7%

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

      2. unpow2 83.7%

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

      3. swap-sqr 83.7%

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

      4. unpow2 83.7%

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

      5. times-frac 97.5%

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

      6. associate-*l/ 97.5%

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

      7. associate-*l/ 97.5%

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

      8. associate-/r/ 97.5%

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

      9. associate-/r/ 97.5%

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

      10. unpow2 97.5%

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

      11. associate-/r/ 97.5%

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

      12. *-commutative 97.5%

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

   9. Simplified 97.5%

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

   10. Taylor expanded in t around 0 96.4%

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

   if 0.10000000000000001 < (/.f64 t l)

   1. Initial program 64.9%

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

   2. Taylor expanded in t around inf 43.6%

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

      1. associate-/l* 43.6%

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

      2. unpow2 43.6%

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

      3. unpow2 43.6%

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

      4. unpow2 43.6%

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

      5. unpow2 43.6%

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

   4. Simplified 43.6%

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

   5. Taylor expanded in Om around 0 98.9%

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

      1. associate-/l* 98.9%

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

   7. Simplified 98.9%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -500:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{-\ell}{t}}{\sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.1:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \]

Alternative 9: 57.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.05 \cdot 10^{+60}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= t 3.05e+60) (asin 1.0) (asin (* l (/ (sqrt 0.5) t)))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (t <= 3.05e+60) {
		tmp = asin(1.0);
	} else {
		tmp = asin((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) :: tmp
    if (t <= 3.05d+60) then
        tmp = asin(1.0d0)
    else
        tmp = asin((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 tmp;
	if (t <= 3.05e+60) {
		tmp = Math.asin(1.0);
	} else {
		tmp = Math.asin((l * (Math.sqrt(0.5) / t)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if t <= 3.05e+60:
		tmp = math.asin(1.0)
	else:
		tmp = math.asin((l * (math.sqrt(0.5) / t)))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (t <= 3.05e+60)
		tmp = asin(1.0);
	else
		tmp = asin(Float64(l * Float64(sqrt(0.5) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (t <= 3.05e+60)
		tmp = asin(1.0);
	else
		tmp = asin((l * (sqrt(0.5) / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[t, 3.05e+60], N[ArcSin[1.0], $MachinePrecision], N[ArcSin[N[(l * N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.05e60

   1. Initial program 85.0%

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

      1. sqrt-div 85.0%

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

      2. div-inv 85.0%

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

      3. add-sqr-sqrt 85.0%

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

      4. hypot-1-def 85.0%

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

      5. *-commutative 85.0%

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

      6. sqrt-prod 84.9%

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

      7. unpow2 84.9%

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

      8. sqrt-prod 56.4%

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

      9. add-sqr-sqrt 99.1%

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

   3. Applied egg-rr 99.1%

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

      1. associate-*r/ 99.1%

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

      2. *-rgt-identity 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in t around 0 99.1%

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

   7. Taylor expanded in Om around 0 67.8%

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

      1. unpow2 67.8%

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

      2. unpow2 67.8%

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

      3. swap-sqr 67.9%

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

      4. unpow2 67.9%

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

      5. times-frac 83.7%

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

      6. associate-*l/ 83.7%

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

      7. associate-*l/ 83.7%

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

      8. associate-/r/ 83.7%

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

      9. associate-/r/ 83.7%

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

      10. unpow2 83.7%

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

      11. associate-/r/ 83.7%

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

      12. *-commutative 83.7%

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

   9. Simplified 83.7%

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

   10. Taylor expanded in t around 0 57.0%

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

   if 3.05e60 < t

   1. Initial program 76.5%

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

   2. Taylor expanded in t around inf 49.7%

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

      1. associate-/l* 49.7%

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

      2. unpow2 49.7%

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

      3. unpow2 49.7%

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

      4. unpow2 49.7%

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

      5. unpow2 49.7%

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

   4. Simplified 49.7%

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

   5. Taylor expanded in Om around 0 71.0%

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

      1. associate-/l* 71.0%

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

      2. associate-/r/ 70.9%

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

   7. Simplified 70.9%

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

   8. Taylor expanded in l around 0 71.0%

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

      1. associate-*r/ 71.0%

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

   10. Simplified 71.0%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.05 \cdot 10^{+60}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 10: 57.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 6.8 \cdot 10^{+60}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= t 6.8e+60) (asin 1.0) (asin (* (/ l t) (sqrt 0.5)))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (t <= 6.8e+60) {
		tmp = asin(1.0);
	} else {
		tmp = asin(((l / t) * sqrt(0.5)));
	}
	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 <= 6.8d+60) then
        tmp = asin(1.0d0)
    else
        tmp = asin(((l / t) * sqrt(0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (t <= 6.8e+60) {
		tmp = Math.asin(1.0);
	} else {
		tmp = Math.asin(((l / t) * Math.sqrt(0.5)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if t <= 6.8e+60:
		tmp = math.asin(1.0)
	else:
		tmp = math.asin(((l / t) * math.sqrt(0.5)))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (t <= 6.8e+60)
		tmp = asin(1.0);
	else
		tmp = asin(Float64(Float64(l / t) * sqrt(0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (t <= 6.8e+60)
		tmp = asin(1.0);
	else
		tmp = asin(((l / t) * sqrt(0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[t, 6.8e+60], N[ArcSin[1.0], $MachinePrecision], N[ArcSin[N[(N[(l / t), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 6.7999999999999999e60

   1. Initial program 85.0%

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

      1. sqrt-div 85.0%

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

      2. div-inv 85.0%

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

      3. add-sqr-sqrt 85.0%

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

      4. hypot-1-def 85.0%

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

      5. *-commutative 85.0%

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

      6. sqrt-prod 84.9%

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

      7. unpow2 84.9%

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

      8. sqrt-prod 56.4%

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

      9. add-sqr-sqrt 99.1%

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

   3. Applied egg-rr 99.1%

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

      1. associate-*r/ 99.1%

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

      2. *-rgt-identity 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in t around 0 99.1%

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

   7. Taylor expanded in Om around 0 67.8%

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

      1. unpow2 67.8%

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

      2. unpow2 67.8%

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

      3. swap-sqr 67.9%

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

      4. unpow2 67.9%

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

      5. times-frac 83.7%

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

      6. associate-*l/ 83.7%

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

      7. associate-*l/ 83.7%

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

      8. associate-/r/ 83.7%

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

      9. associate-/r/ 83.7%

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

      10. unpow2 83.7%

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

      11. associate-/r/ 83.7%

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

      12. *-commutative 83.7%

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

   9. Simplified 83.7%

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

   10. Taylor expanded in t around 0 57.0%

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

   if 6.7999999999999999e60 < t

   1. Initial program 76.5%

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

   2. Taylor expanded in t around inf 49.7%

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

      1. associate-/l* 49.7%

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

      2. unpow2 49.7%

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

      3. unpow2 49.7%

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

      4. unpow2 49.7%

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

      5. unpow2 49.7%

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

   4. Simplified 49.7%

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

   5. Taylor expanded in Om around 0 71.0%

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

      1. associate-/l* 71.0%

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

      2. associate-/r/ 70.9%

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

   7. Simplified 70.9%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.8 \cdot 10^{+60}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 11: 57.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{+60}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= t 7.2e+60) (asin 1.0) (asin (/ l (* t (sqrt 2.0))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (t <= 7.2e+60) {
		tmp = asin(1.0);
	} else {
		tmp = asin((l / (t * sqrt(2.0))));
	}
	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 (t <= 7.2d+60) then
        tmp = asin(1.0d0)
    else
        tmp = asin((l / (t * sqrt(2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (t <= 7.2e+60) {
		tmp = Math.asin(1.0);
	} else {
		tmp = Math.asin((l / (t * Math.sqrt(2.0))));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if t <= 7.2e+60:
		tmp = math.asin(1.0)
	else:
		tmp = math.asin((l / (t * math.sqrt(2.0))))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (t <= 7.2e+60)
		tmp = asin(1.0);
	else
		tmp = asin(Float64(l / Float64(t * sqrt(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (t <= 7.2e+60)
		tmp = asin(1.0);
	else
		tmp = asin((l / (t * sqrt(2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[t, 7.2e+60], N[ArcSin[1.0], $MachinePrecision], N[ArcSin[N[(l / N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 7.19999999999999935e60

   1. Initial program 85.0%

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

      1. sqrt-div 85.0%

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

      2. div-inv 85.0%

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

      3. add-sqr-sqrt 85.0%

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

      4. hypot-1-def 85.0%

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

      5. *-commutative 85.0%

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

      6. sqrt-prod 84.9%

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

      7. unpow2 84.9%

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

      8. sqrt-prod 56.4%

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

      9. add-sqr-sqrt 99.1%

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

   3. Applied egg-rr 99.1%

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

      1. associate-*r/ 99.1%

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

      2. *-rgt-identity 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in t around 0 99.1%

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

   7. Taylor expanded in Om around 0 67.8%

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

      1. unpow2 67.8%

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

      2. unpow2 67.8%

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

      3. swap-sqr 67.9%

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

      4. unpow2 67.9%

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

      5. times-frac 83.7%

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

      6. associate-*l/ 83.7%

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

      7. associate-*l/ 83.7%

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

      8. associate-/r/ 83.7%

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

      9. associate-/r/ 83.7%

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

      10. unpow2 83.7%

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

      11. associate-/r/ 83.7%

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

      12. *-commutative 83.7%

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

   9. Simplified 83.7%

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

   10. Taylor expanded in t around 0 57.0%

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

   if 7.19999999999999935e60 < t

   1. Initial program 76.5%

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

      1. sqrt-div 76.4%

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

      2. div-inv 76.4%

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

      3. add-sqr-sqrt 76.4%

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

      4. hypot-1-def 76.4%

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

      5. *-commutative 76.4%

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

      6. sqrt-prod 76.4%

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

      7. unpow2 76.4%

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

      8. sqrt-prod 59.7%

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

      9. add-sqr-sqrt 98.7%

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

   3. Applied egg-rr 98.7%

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

      1. associate-*r/ 98.7%

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

      2. *-rgt-identity 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in t around 0 98.7%

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

   7. Taylor expanded in Om around 0 59.8%

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

      1. unpow2 59.8%

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

      2. unpow2 59.8%

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

      3. swap-sqr 59.9%

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

      4. unpow2 59.9%

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

      5. times-frac 76.4%

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

      6. associate-*l/ 76.4%

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

      7. associate-*l/ 76.3%

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

      8. associate-/r/ 76.4%

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

      9. associate-/r/ 76.4%

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

      10. unpow2 76.4%

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

      11. associate-/r/ 76.3%

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

      12. *-commutative 76.3%

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

   9. Simplified 76.3%

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

   10. Taylor expanded in t around inf 71.1%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{+60}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \end{array} \]

Alternative 12: 51.2% 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 83.1%

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

   1. sqrt-div 83.0%

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

   2. div-inv 83.0%

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

   3. add-sqr-sqrt 83.1%

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

   4. hypot-1-def 83.1%

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

   5. *-commutative 83.1%

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

   6. sqrt-prod 83.0%

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

   7. unpow2 83.0%

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

   8. sqrt-prod 57.1%

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

   9. add-sqr-sqrt 99.0%

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

3. Applied egg-rr 99.0%

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

   1. associate-*r/ 99.0%

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

   2. *-rgt-identity 99.0%

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

5. Simplified 99.0%

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

6. Taylor expanded in t around 0 99.0%

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

7. Taylor expanded in Om around 0 66.0%

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

   1. unpow2 66.0%

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

   2. unpow2 66.0%

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

   3. swap-sqr 66.1%

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

   4. unpow2 66.1%

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

   5. times-frac 82.0%

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

   6. associate-*l/ 82.0%

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

   7. associate-*l/ 82.0%

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

   8. associate-/r/ 82.0%

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

   9. associate-/r/ 82.0%

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

   10. unpow2 82.0%

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

   11. associate-/r/ 82.0%

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

   12. *-commutative 82.0%

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

9. Simplified 82.0%

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

10. Taylor expanded in t around 0 48.0%

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

11. Final simplification 48.0%

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

Reproduce

herbie shell --seed 2023271 
(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)))))))