Toniolo and Linder, Equation (2)

Percentage Accurate: 85.0% → 98.4%

Time: 17.3s

Alternatives: 11

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: 85.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.3%

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

   1. sqrt-div 82.2%

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

   2. div-inv 82.2%

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

   3. add-sqr-sqrt 82.2%

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

   4. hypot-1-def 82.2%

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

   5. *-commutative 82.2%

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

   6. sqrt-prod 82.2%

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

   7. unpow2 82.2%

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

   8. sqrt-prod 49.9%

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

   9. add-sqr-sqrt 99.2%

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

4. Applied egg-rr 99.2%

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

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

   3. associate-*l/ 99.2%

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

   4. associate-/l* 99.2%

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

6. Simplified 99.2%

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

7. Final simplification 99.2%

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

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 82.3%

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

   1. sqrt-div 82.2%

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

   2. add-sqr-sqrt 82.2%

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

   3. hypot-1-def 82.2%

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

   4. *-commutative 82.2%

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

   5. sqrt-prod 82.2%

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

   6. unpow2 82.2%

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

   7. sqrt-prod 49.9%

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

   8. add-sqr-sqrt 99.2%

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

4. Applied egg-rr 99.2%

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

5. Final simplification 99.2%

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

Alternative 3: 97.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.3%

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

   1. sqrt-div 82.2%

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

   2. div-inv 82.2%

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

   3. add-sqr-sqrt 82.2%

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

   4. hypot-1-def 82.2%

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

   5. *-commutative 82.2%

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

   6. sqrt-prod 82.2%

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

   7. unpow2 82.2%

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

   8. sqrt-prod 49.9%

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

   9. add-sqr-sqrt 99.2%

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

4. Applied egg-rr 99.2%

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

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

   3. associate-*l/ 99.2%

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

   4. associate-/l* 99.2%

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

6. Simplified 99.2%

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

7. Taylor expanded in Om around 0 98.6%

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

8. Final simplification 98.6%

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

Alternative 4: 98.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -8 \cdot 10^{+143}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \left(-\ell\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+147}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -8e+143)
   (asin (/ (* (sqrt 0.5) (- l)) t))
   (if (<= (/ t l) 2e+147)
     (asin (sqrt (/ 1.0 (+ 1.0 (* 2.0 (* (/ t l) (/ t l)))))))
     (asin (/ (* l (sqrt 0.5)) t)))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -8e+143) {
		tmp = asin(((sqrt(0.5) * -l) / t));
	} else if ((t / l) <= 2e+147) {
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} 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 / l) <= (-8d+143)) then
        tmp = asin(((sqrt(0.5d0) * -l) / t))
    else if ((t / l) <= 2d+147) then
        tmp = asin(sqrt((1.0d0 / (1.0d0 + (2.0d0 * ((t / l) * (t / l)))))))
    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 / l) <= -8e+143) {
		tmp = Math.asin(((Math.sqrt(0.5) * -l) / t));
	} else if ((t / l) <= 2e+147) {
		tmp = Math.asin(Math.sqrt((1.0 / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} else {
		tmp = Math.asin(((l * Math.sqrt(0.5)) / t));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -8e+143:
		tmp = math.asin(((math.sqrt(0.5) * -l) / t))
	elif (t / l) <= 2e+147:
		tmp = math.asin(math.sqrt((1.0 / (1.0 + (2.0 * ((t / l) * (t / l)))))))
	else:
		tmp = math.asin(((l * math.sqrt(0.5)) / t))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -8e+143)
		tmp = asin(Float64(Float64(sqrt(0.5) * Float64(-l)) / t));
	elseif (Float64(t / l) <= 2e+147)
		tmp = asin(sqrt(Float64(1.0 / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) * Float64(t / l)))))));
	else
		tmp = asin(Float64(Float64(l * sqrt(0.5)) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -8e+143)
		tmp = asin(((sqrt(0.5) * -l) / t));
	elseif ((t / l) <= 2e+147)
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	else
		tmp = asin(((l * sqrt(0.5)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -8e+143], N[ArcSin[N[(N[(N[Sqrt[0.5], $MachinePrecision] * (-l)), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 2e+147], N[ArcSin[N[Sqrt[N[(1.0 / N[(1.0 + N[(2.0 * N[(N[(t / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -8.0000000000000002e143

   1. Initial program 45.2%

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

   3. Taylor expanded in Om around 0 45.2%

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

   4. Taylor expanded in t around -inf 99.7%

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

      1. associate-*r/ 99.7%

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

      2. neg-mul-1 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

   6. Simplified 99.7%

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

   if -8.0000000000000002e143 < (/.f64 t l) < 2e147

   1. Initial program 99.4%

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

   3. Taylor expanded in Om around 0 74.6%

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

      1. unpow2 74.6%

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

      2. unpow2 74.6%

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

      3. frac-times 98.4%

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

   5. Applied egg-rr 98.4%

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

   if 2e147 < (/.f64 t l)

   1. Initial program 47.1%

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

   3. Taylor expanded in Om around 0 47.1%

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

   4. Taylor expanded in t around inf 99.7%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -8 \cdot 10^{+143}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \left(-\ell\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+147}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -20:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \left(-\ell\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -20.0)
   (asin (/ (* (sqrt 0.5) (- l)) t))
   (if (<= (/ t l) 2e-8)
     (asin (sqrt (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))
     (asin (/ l (* t (sqrt 2.0)))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -20.0) {
		tmp = asin(((sqrt(0.5) * -l) / t));
	} else if ((t / l) <= 2e-8) {
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} 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 / l) <= (-20.0d0)) then
        tmp = asin(((sqrt(0.5d0) * -l) / t))
    else if ((t / l) <= 2d-8) then
        tmp = asin(sqrt((1.0d0 - ((om / omc) / (omc / om)))))
    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 / l) <= -20.0) {
		tmp = Math.asin(((Math.sqrt(0.5) * -l) / t));
	} else if ((t / l) <= 2e-8) {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} else {
		tmp = Math.asin((l / (t * Math.sqrt(2.0))));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -20.0:
		tmp = math.asin(((math.sqrt(0.5) * -l) / t))
	elif (t / l) <= 2e-8:
		tmp = math.asin(math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))))
	else:
		tmp = math.asin((l / (t * math.sqrt(2.0))))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -20.0)
		tmp = asin(Float64(Float64(sqrt(0.5) * Float64(-l)) / t));
	elseif (Float64(t / l) <= 2e-8)
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))));
	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 / l) <= -20.0)
		tmp = asin(((sqrt(0.5) * -l) / t));
	elseif ((t / l) <= 2e-8)
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	else
		tmp = asin((l / (t * sqrt(2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -20.0], N[ArcSin[N[(N[(N[Sqrt[0.5], $MachinePrecision] * (-l)), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 2e-8], N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l / N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -20

   1. Initial program 68.9%

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

   3. Taylor expanded in Om around 0 40.2%

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

   4. Taylor expanded in t around -inf 97.7%

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

      1. associate-*r/ 97.7%

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

      2. neg-mul-1 97.7%

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

      3. distribute-rgt-neg-in 97.7%

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

   6. Simplified 97.7%

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

   if -20 < (/.f64 t l) < 2e-8

   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. Add Preprocessing

   3. Taylor expanded in t around 0 83.7%

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

      1. unpow2 83.7%

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

      2. unpow2 83.7%

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

      3. times-frac 97.5%

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

      4. unpow2 97.5%

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

   5. Simplified 97.5%

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

      1. unpow2 97.5%

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

      2. clear-num 97.5%

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

      3. un-div-inv 97.5%

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

   7. Applied egg-rr 97.5%

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

   if 2e-8 < (/.f64 t l)

   1. Initial program 67.7%

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

      1. sqrt-div 67.7%

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

      2. div-inv 67.7%

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

      3. add-sqr-sqrt 67.7%

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

      4. hypot-1-def 67.7%

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

      5. *-commutative 67.7%

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

      6. sqrt-prod 67.6%

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

      7. unpow2 67.6%

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

      8. sqrt-prod 98.6%

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

      9. add-sqr-sqrt 98.7%

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

   4. Applied egg-rr 98.7%

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

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

      3. associate-*l/ 98.7%

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

      4. associate-/l* 98.8%

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

   6. Simplified 98.8%

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

   7. Taylor expanded in Om around 0 98.8%

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

   8. Taylor expanded in t around inf 98.8%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -20:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \left(-\ell\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{+136}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq -2.8 \cdot 10^{-308}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{-13}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= l -5e+136)
   (asin 1.0)
   (if (<= l -2.8e-308)
     (asin (/ (- l) (/ t (sqrt 0.5))))
     (if (<= l 9.5e-13) (asin (/ (* l (sqrt 0.5)) t)) (asin 1.0)))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= -5e+136) {
		tmp = asin(1.0);
	} else if (l <= -2.8e-308) {
		tmp = asin((-l / (t / sqrt(0.5))));
	} else if (l <= 9.5e-13) {
		tmp = asin(((l * sqrt(0.5)) / t));
	} else {
		tmp = asin(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (l <= (-5d+136)) then
        tmp = asin(1.0d0)
    else if (l <= (-2.8d-308)) then
        tmp = asin((-l / (t / sqrt(0.5d0))))
    else if (l <= 9.5d-13) then
        tmp = asin(((l * sqrt(0.5d0)) / t))
    else
        tmp = asin(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= -5e+136) {
		tmp = Math.asin(1.0);
	} else if (l <= -2.8e-308) {
		tmp = Math.asin((-l / (t / Math.sqrt(0.5))));
	} else if (l <= 9.5e-13) {
		tmp = Math.asin(((l * Math.sqrt(0.5)) / t));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if l <= -5e+136:
		tmp = math.asin(1.0)
	elif l <= -2.8e-308:
		tmp = math.asin((-l / (t / math.sqrt(0.5))))
	elif l <= 9.5e-13:
		tmp = math.asin(((l * math.sqrt(0.5)) / t))
	else:
		tmp = math.asin(1.0)
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (l <= -5e+136)
		tmp = asin(1.0);
	elseif (l <= -2.8e-308)
		tmp = asin(Float64(Float64(-l) / Float64(t / sqrt(0.5))));
	elseif (l <= 9.5e-13)
		tmp = asin(Float64(Float64(l * sqrt(0.5)) / t));
	else
		tmp = asin(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (l <= -5e+136)
		tmp = asin(1.0);
	elseif (l <= -2.8e-308)
		tmp = asin((-l / (t / sqrt(0.5))));
	elseif (l <= 9.5e-13)
		tmp = asin(((l * sqrt(0.5)) / t));
	else
		tmp = asin(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[l, -5e+136], N[ArcSin[1.0], $MachinePrecision], If[LessEqual[l, -2.8e-308], N[ArcSin[N[((-l) / N[(t / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 9.5e-13], N[ArcSin[N[(N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.0000000000000002e136 or 9.49999999999999991e-13 < l

   1. Initial program 96.8%

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

   3. Taylor expanded in t around 0 76.2%

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

      1. unpow2 76.2%

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

      2. unpow2 76.2%

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

      3. times-frac 86.0%

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

      4. unpow2 86.0%

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

   5. Simplified 86.0%

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

   6. Taylor expanded in Om around 0 85.3%

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

   if -5.0000000000000002e136 < l < -2.79999999999999984e-308

   1. Initial program 74.4%

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

   3. Taylor expanded in Om around 0 50.8%

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

   4. Taylor expanded in t around -inf 51.2%

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

      1. associate-*r/ 51.2%

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

      2. neg-mul-1 51.2%

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

      3. distribute-rgt-neg-in 51.2%

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

   6. Simplified 51.2%

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

   7. Taylor expanded in l around 0 51.2%

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

      1. mul-1-neg 51.2%

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

      2. associate-/l* 51.1%

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

   9. Simplified 51.1%

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

   if -2.79999999999999984e-308 < l < 9.49999999999999991e-13

   1. Initial program 73.4%

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

   3. Taylor expanded in Om around 0 57.2%

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

   4. Taylor expanded in t around inf 50.4%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{+136}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq -2.8 \cdot 10^{-308}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{-13}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -20:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -20.0)
   (asin (/ (- l) (/ t (sqrt 0.5))))
   (if (<= (/ t l) 2e-8)
     (asin (- 1.0 (pow (/ t l) 2.0)))
     (asin (/ l (* t (sqrt 2.0)))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -20.0) {
		tmp = asin((-l / (t / sqrt(0.5))));
	} else if ((t / l) <= 2e-8) {
		tmp = asin((1.0 - pow((t / l), 2.0)));
	} 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 / l) <= (-20.0d0)) then
        tmp = asin((-l / (t / sqrt(0.5d0))))
    else if ((t / l) <= 2d-8) then
        tmp = asin((1.0d0 - ((t / l) ** 2.0d0)))
    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 / l) <= -20.0) {
		tmp = Math.asin((-l / (t / Math.sqrt(0.5))));
	} else if ((t / l) <= 2e-8) {
		tmp = Math.asin((1.0 - Math.pow((t / l), 2.0)));
	} else {
		tmp = Math.asin((l / (t * Math.sqrt(2.0))));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -20.0:
		tmp = math.asin((-l / (t / math.sqrt(0.5))))
	elif (t / l) <= 2e-8:
		tmp = math.asin((1.0 - math.pow((t / l), 2.0)))
	else:
		tmp = math.asin((l / (t * math.sqrt(2.0))))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -20.0)
		tmp = asin(Float64(Float64(-l) / Float64(t / sqrt(0.5))));
	elseif (Float64(t / l) <= 2e-8)
		tmp = asin(Float64(1.0 - (Float64(t / l) ^ 2.0)));
	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 / l) <= -20.0)
		tmp = asin((-l / (t / sqrt(0.5))));
	elseif ((t / l) <= 2e-8)
		tmp = asin((1.0 - ((t / l) ^ 2.0)));
	else
		tmp = asin((l / (t * sqrt(2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -20.0], N[ArcSin[N[((-l) / N[(t / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 2e-8], N[ArcSin[N[(1.0 - N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l / N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -20

   1. Initial program 68.9%

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

   3. Taylor expanded in Om around 0 40.2%

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

   4. Taylor expanded in t around -inf 97.7%

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

      1. associate-*r/ 97.7%

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

      2. neg-mul-1 97.7%

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

      3. distribute-rgt-neg-in 97.7%

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

   6. Simplified 97.7%

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

   7. Taylor expanded in l around 0 97.7%

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

      1. mul-1-neg 97.7%

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

      2. associate-/l* 97.6%

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

   9. Simplified 97.6%

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

   if -20 < (/.f64 t l) < 2e-8

   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. Add Preprocessing

   3. Taylor expanded in Om around 0 91.5%

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

   4. Taylor expanded in t around 0 90.0%

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

      1. mul-1-neg 90.0%

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

      2. unpow2 90.0%

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

      3. unpow2 90.0%

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

      4. times-frac 96.6%

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

      5. unpow2 96.6%

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

      6. unsub-neg 96.6%

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

   6. Simplified 96.6%

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

   if 2e-8 < (/.f64 t l)

   1. Initial program 67.7%

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

      1. sqrt-div 67.7%

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

      2. div-inv 67.7%

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

      3. add-sqr-sqrt 67.7%

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

      4. hypot-1-def 67.7%

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

      5. *-commutative 67.7%

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

      6. sqrt-prod 67.6%

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

      7. unpow2 67.6%

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

      8. sqrt-prod 98.6%

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

      9. add-sqr-sqrt 98.7%

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

   4. Applied egg-rr 98.7%

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

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

      3. associate-*l/ 98.7%

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

      4. associate-/l* 98.8%

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

   6. Simplified 98.8%

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

   7. Taylor expanded in Om around 0 98.8%

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

   8. Taylor expanded in t around inf 98.8%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -20:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -20:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \left(-\ell\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -20.0)
   (asin (/ (* (sqrt 0.5) (- l)) t))
   (if (<= (/ t l) 2e-8)
     (asin (- 1.0 (pow (/ t l) 2.0)))
     (asin (/ l (* t (sqrt 2.0)))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -20.0) {
		tmp = asin(((sqrt(0.5) * -l) / t));
	} else if ((t / l) <= 2e-8) {
		tmp = asin((1.0 - pow((t / l), 2.0)));
	} 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 / l) <= (-20.0d0)) then
        tmp = asin(((sqrt(0.5d0) * -l) / t))
    else if ((t / l) <= 2d-8) then
        tmp = asin((1.0d0 - ((t / l) ** 2.0d0)))
    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 / l) <= -20.0) {
		tmp = Math.asin(((Math.sqrt(0.5) * -l) / t));
	} else if ((t / l) <= 2e-8) {
		tmp = Math.asin((1.0 - Math.pow((t / l), 2.0)));
	} else {
		tmp = Math.asin((l / (t * Math.sqrt(2.0))));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -20.0:
		tmp = math.asin(((math.sqrt(0.5) * -l) / t))
	elif (t / l) <= 2e-8:
		tmp = math.asin((1.0 - math.pow((t / l), 2.0)))
	else:
		tmp = math.asin((l / (t * math.sqrt(2.0))))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -20.0)
		tmp = asin(Float64(Float64(sqrt(0.5) * Float64(-l)) / t));
	elseif (Float64(t / l) <= 2e-8)
		tmp = asin(Float64(1.0 - (Float64(t / l) ^ 2.0)));
	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 / l) <= -20.0)
		tmp = asin(((sqrt(0.5) * -l) / t));
	elseif ((t / l) <= 2e-8)
		tmp = asin((1.0 - ((t / l) ^ 2.0)));
	else
		tmp = asin((l / (t * sqrt(2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -20.0], N[ArcSin[N[(N[(N[Sqrt[0.5], $MachinePrecision] * (-l)), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 2e-8], N[ArcSin[N[(1.0 - N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l / N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -20

   1. Initial program 68.9%

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

   3. Taylor expanded in Om around 0 40.2%

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

   4. Taylor expanded in t around -inf 97.7%

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

      1. associate-*r/ 97.7%

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

      2. neg-mul-1 97.7%

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

      3. distribute-rgt-neg-in 97.7%

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

   6. Simplified 97.7%

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

   if -20 < (/.f64 t l) < 2e-8

   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. Add Preprocessing

   3. Taylor expanded in Om around 0 91.5%

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

   4. Taylor expanded in t around 0 90.0%

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

      1. mul-1-neg 90.0%

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

      2. unpow2 90.0%

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

      3. unpow2 90.0%

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

      4. times-frac 96.6%

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

      5. unpow2 96.6%

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

      6. unsub-neg 96.6%

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

   6. Simplified 96.6%

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

   if 2e-8 < (/.f64 t l)

   1. Initial program 67.7%

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

      1. sqrt-div 67.7%

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

      2. div-inv 67.7%

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

      3. add-sqr-sqrt 67.7%

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

      4. hypot-1-def 67.7%

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

      5. *-commutative 67.7%

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

      6. sqrt-prod 67.6%

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

      7. unpow2 67.6%

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

      8. sqrt-prod 98.6%

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

      9. add-sqr-sqrt 98.7%

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

   4. Applied egg-rr 98.7%

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

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

      3. associate-*l/ 98.7%

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

      4. associate-/l* 98.8%

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

   6. Simplified 98.8%

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

   7. Taylor expanded in Om around 0 98.8%

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

   8. Taylor expanded in t around inf 98.8%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -20:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \left(-\ell\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{-186}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{-9}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= l -2.4e-186)
   (asin 1.0)
   (if (<= l 3.5e-9) (asin (/ l (* t (sqrt 2.0)))) (asin 1.0))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= -2.4e-186) {
		tmp = asin(1.0);
	} else if (l <= 3.5e-9) {
		tmp = asin((l / (t * sqrt(2.0))));
	} else {
		tmp = asin(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (l <= (-2.4d-186)) then
        tmp = asin(1.0d0)
    else if (l <= 3.5d-9) then
        tmp = asin((l / (t * sqrt(2.0d0))))
    else
        tmp = asin(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= -2.4e-186) {
		tmp = Math.asin(1.0);
	} else if (l <= 3.5e-9) {
		tmp = Math.asin((l / (t * Math.sqrt(2.0))));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if l <= -2.4e-186:
		tmp = math.asin(1.0)
	elif l <= 3.5e-9:
		tmp = math.asin((l / (t * math.sqrt(2.0))))
	else:
		tmp = math.asin(1.0)
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (l <= -2.4e-186)
		tmp = asin(1.0);
	elseif (l <= 3.5e-9)
		tmp = asin(Float64(l / Float64(t * sqrt(2.0))));
	else
		tmp = asin(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (l <= -2.4e-186)
		tmp = asin(1.0);
	elseif (l <= 3.5e-9)
		tmp = asin((l / (t * sqrt(2.0))));
	else
		tmp = asin(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[l, -2.4e-186], N[ArcSin[1.0], $MachinePrecision], If[LessEqual[l, 3.5e-9], N[ArcSin[N[(l / N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -2.40000000000000003e-186 or 3.4999999999999999e-9 < l

   1. Initial program 87.4%

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

   3. Taylor expanded in t around 0 56.5%

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

      1. unpow2 56.5%

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

      2. unpow2 56.5%

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

      3. times-frac 64.9%

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

      4. unpow2 64.9%

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

   5. Simplified 64.9%

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

   6. Taylor expanded in Om around 0 64.1%

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

   if -2.40000000000000003e-186 < l < 3.4999999999999999e-9

   1. Initial program 73.9%

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

      1. sqrt-div 73.8%

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

      2. div-inv 73.8%

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

      3. add-sqr-sqrt 73.8%

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

      4. hypot-1-def 73.8%

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

      5. *-commutative 73.8%

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

      6. sqrt-prod 73.7%

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

      7. unpow2 73.7%

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

      8. sqrt-prod 37.2%

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

      9. add-sqr-sqrt 98.6%

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

   4. Applied egg-rr 98.6%

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

      1. associate-*r/ 98.6%

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

      2. *-rgt-identity 98.6%

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

      3. associate-*l/ 98.6%

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

      4. associate-/l* 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in Om around 0 98.3%

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

   8. Taylor expanded in t around inf 53.1%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{-186}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{-9}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.6 \cdot 10^{-187}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{-12}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= l -2.6e-187)
   (asin 1.0)
   (if (<= l 1.3e-12) (asin (/ (* l (sqrt 0.5)) t)) (asin 1.0))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= -2.6e-187) {
		tmp = asin(1.0);
	} else if (l <= 1.3e-12) {
		tmp = asin(((l * sqrt(0.5)) / t));
	} else {
		tmp = asin(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (l <= (-2.6d-187)) then
        tmp = asin(1.0d0)
    else if (l <= 1.3d-12) then
        tmp = asin(((l * sqrt(0.5d0)) / t))
    else
        tmp = asin(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= -2.6e-187) {
		tmp = Math.asin(1.0);
	} else if (l <= 1.3e-12) {
		tmp = Math.asin(((l * Math.sqrt(0.5)) / t));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if l <= -2.6e-187:
		tmp = math.asin(1.0)
	elif l <= 1.3e-12:
		tmp = math.asin(((l * math.sqrt(0.5)) / t))
	else:
		tmp = math.asin(1.0)
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (l <= -2.6e-187)
		tmp = asin(1.0);
	elseif (l <= 1.3e-12)
		tmp = asin(Float64(Float64(l * sqrt(0.5)) / t));
	else
		tmp = asin(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (l <= -2.6e-187)
		tmp = asin(1.0);
	elseif (l <= 1.3e-12)
		tmp = asin(((l * sqrt(0.5)) / t));
	else
		tmp = asin(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[l, -2.6e-187], N[ArcSin[1.0], $MachinePrecision], If[LessEqual[l, 1.3e-12], N[ArcSin[N[(N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -2.5999999999999999e-187 or 1.29999999999999991e-12 < l

   1. Initial program 87.4%

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

   3. Taylor expanded in t around 0 56.5%

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

      1. unpow2 56.5%

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

      2. unpow2 56.5%

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

      3. times-frac 64.9%

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

      4. unpow2 64.9%

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

   5. Simplified 64.9%

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

   6. Taylor expanded in Om around 0 64.1%

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

   if -2.5999999999999999e-187 < l < 1.29999999999999991e-12

   1. Initial program 73.9%

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

   3. Taylor expanded in Om around 0 51.9%

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

   4. Taylor expanded in t around inf 53.1%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.6 \cdot 10^{-187}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{-12}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 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 82.3%

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

3. Taylor expanded in t around 0 39.9%

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

   1. unpow2 39.9%

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

   2. unpow2 39.9%

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

   3. times-frac 46.5%

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

   4. unpow2 46.5%

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

5. Simplified 46.5%

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

6. Taylor expanded in Om around 0 46.0%

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

7. Final simplification 46.0%

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

Reproduce

herbie shell --seed 2024031 
(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)))))))