Toniolo and Linder, Equation (2)

Percentage Accurate: 84.7% → 98.4%

Time: 20.9s

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: 84.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.7%

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

   1. sqrt-div 84.7%

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

   2. add-sqr-sqrt 84.7%

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

   3. hypot-1-def 84.7%

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

   4. *-commutative 84.7%

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

   5. sqrt-prod 84.7%

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

   6. unpow2 84.7%

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

   7. sqrt-prod 58.5%

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

   8. add-sqr-sqrt 99.1%

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

4. Applied egg-rr 99.1%

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

5. Final simplification 99.1%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

def code(t, l, Om, Omc):
	return math.asin((math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))) / 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(Float64(Om / Omc) / Float64(Omc / Om)))) / 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) / (Omc / Om)))) / hypot(1.0, (t / (l / sqrt(2.0))))));
end

Wolfram

code[t_, l_, Om_, Omc_] := N[ArcSin[N[(N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $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 84.7%

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

   1. sqrt-div 84.7%

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

   2. div-inv 84.7%

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

   3. add-sqr-sqrt 84.7%

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

   4. hypot-1-def 84.7%

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

   5. *-commutative 84.7%

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

   6. sqrt-prod 84.7%

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

   7. unpow2 84.7%

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

   8. sqrt-prod 58.5%

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

   9. add-sqr-sqrt 99.1%

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

4. Applied egg-rr 99.1%

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

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

   3. associate-*l/ 99.1%

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

   4. associate-/l* 99.1%

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

6. Simplified 99.1%

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

   1. unpow2 52.1%

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

   2. clear-num 52.1%

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

   3. un-div-inv 52.1%

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

8. Applied egg-rr 99.1%

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

9. Final simplification 99.1%

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

Alternative 3: 97.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.7%

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

3. Taylor expanded in Om around 0 70.9%

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

   1. sqrt-div 70.9%

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

   2. metadata-eval 70.9%

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

   3. add-sqr-sqrt 70.9%

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

   4. hypot-1-def 70.9%

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

   5. sqrt-prod 70.8%

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

   6. sqrt-div 75.6%

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

   7. unpow2 75.6%

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

   8. sqrt-prod 41.6%

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

   9. add-sqr-sqrt 85.3%

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

   10. unpow2 85.3%

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

   11. sqrt-prod 46.3%

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

   12. add-sqr-sqrt 98.5%

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

5. Applied egg-rr 98.5%

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

6. Final simplification 98.5%

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

Alternative 4: 98.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \left(-\sqrt{0.5}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+50}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{{0.5}^{-0.5}}}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -2e+154)
   (asin (* (/ l t) (- (sqrt 0.5))))
   (if (<= (/ t l) 2e+50)
     (asin (sqrt (/ 1.0 (+ 1.0 (* 2.0 (* (/ t l) (/ t l)))))))
     (asin (/ (/ l (pow 0.5 -0.5)) t)))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -2e+154) {
		tmp = asin(((l / t) * -sqrt(0.5)));
	} else if ((t / l) <= 2e+50) {
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} else {
		tmp = asin(((l / pow(0.5, -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) <= (-2d+154)) then
        tmp = asin(((l / t) * -sqrt(0.5d0)))
    else if ((t / l) <= 2d+50) then
        tmp = asin(sqrt((1.0d0 / (1.0d0 + (2.0d0 * ((t / l) * (t / l)))))))
    else
        tmp = asin(((l / (0.5d0 ** (-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) <= -2e+154) {
		tmp = Math.asin(((l / t) * -Math.sqrt(0.5)));
	} else if ((t / l) <= 2e+50) {
		tmp = Math.asin(Math.sqrt((1.0 / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} else {
		tmp = Math.asin(((l / Math.pow(0.5, -0.5)) / t));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -2e+154:
		tmp = math.asin(((l / t) * -math.sqrt(0.5)))
	elif (t / l) <= 2e+50:
		tmp = math.asin(math.sqrt((1.0 / (1.0 + (2.0 * ((t / l) * (t / l)))))))
	else:
		tmp = math.asin(((l / math.pow(0.5, -0.5)) / t))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -2e+154)
		tmp = asin(Float64(Float64(l / t) * Float64(-sqrt(0.5))));
	elseif (Float64(t / l) <= 2e+50)
		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 / (0.5 ^ -0.5)) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -2e+154)
		tmp = asin(((l / t) * -sqrt(0.5)));
	elseif ((t / l) <= 2e+50)
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	else
		tmp = asin(((l / (0.5 ^ -0.5)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -2e+154], N[ArcSin[N[(N[(l / t), $MachinePrecision] * (-N[Sqrt[0.5], $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 2e+50], 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[Power[0.5, -0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -2.00000000000000007e154

   1. Initial program 44.2%

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

   3. Taylor expanded in Om around 0 44.2%

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

   4. Taylor expanded in t around -inf 99.5%

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

      1. mul-1-neg 99.5%

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

      2. associate-/l* 99.4%

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

      3. distribute-neg-frac 99.4%

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

   6. Simplified 99.4%

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

      1. frac-2neg 99.4%

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

      2. associate-/r/ 99.7%

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

      3. frac-2neg 99.7%

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

   8. Applied egg-rr 99.7%

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

   if -2.00000000000000007e154 < (/.f64 t l) < 2.0000000000000002e50

   1. Initial program 99.3%

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

   3. Taylor expanded in Om around 0 82.9%

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

      1. add-sqr-sqrt 82.8%

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

      2. pow2 82.8%

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

      3. sqrt-div 82.9%

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

      4. unpow2 82.9%

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

      5. sqrt-prod 43.9%

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

      6. add-sqr-sqrt 89.9%

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

      7. unpow2 89.9%

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

      8. sqrt-prod 46.4%

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

      9. add-sqr-sqrt 98.6%

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

      10. unpow2 98.6%

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

   5. Applied egg-rr 98.6%

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

   if 2.0000000000000002e50 < (/.f64 t l)

   1. Initial program 64.5%

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

   3. Taylor expanded in Om around 0 49.9%

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

   4. Taylor expanded in t around -inf 34.6%

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

      1. mul-1-neg 34.6%

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

      2. associate-/l* 34.6%

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

      3. distribute-neg-frac 34.6%

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

   6. Simplified 34.6%

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

      1. *-un-lft-identity 34.6%

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

      2. div-inv 34.6%

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

      3. times-frac 34.6%

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

      4. add-sqr-sqrt 20.0%

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

      5. sqrt-unprod 54.6%

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

      6. sqr-neg 54.6%

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

      7. sqrt-unprod 45.0%

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

      8. add-sqr-sqrt 98.3%

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

      9. pow1/2 98.3%

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

      10. pow-flip 98.3%

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

      11. metadata-eval 98.3%

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

   8. Applied egg-rr 98.3%

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

      1. associate-*l/ 98.3%

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

      2. *-lft-identity 98.3%

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

   10. Simplified 98.3%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \left(-\sqrt{0.5}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+50}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{{0.5}^{-0.5}}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -100:\\ \;\;\;\;-\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0005:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{{0.5}^{-0.5}}}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -100.0)
   (- (asin (/ (* l (sqrt 0.5)) t)))
   (if (<= (/ t l) 0.0005)
     (asin (sqrt (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))
     (asin (/ (/ l (pow 0.5 -0.5)) t)))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -100.0) {
		tmp = -asin(((l * sqrt(0.5)) / t));
	} else if ((t / l) <= 0.0005) {
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} else {
		tmp = asin(((l / pow(0.5, -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) <= (-100.0d0)) then
        tmp = -asin(((l * sqrt(0.5d0)) / t))
    else if ((t / l) <= 0.0005d0) then
        tmp = asin(sqrt((1.0d0 - ((om / omc) / (omc / om)))))
    else
        tmp = asin(((l / (0.5d0 ** (-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) <= -100.0) {
		tmp = -Math.asin(((l * Math.sqrt(0.5)) / t));
	} else if ((t / l) <= 0.0005) {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} else {
		tmp = Math.asin(((l / Math.pow(0.5, -0.5)) / t));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -100.0:
		tmp = -math.asin(((l * math.sqrt(0.5)) / t))
	elif (t / l) <= 0.0005:
		tmp = math.asin(math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))))
	else:
		tmp = math.asin(((l / math.pow(0.5, -0.5)) / t))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -100.0)
		tmp = Float64(-asin(Float64(Float64(l * sqrt(0.5)) / t)));
	elseif (Float64(t / l) <= 0.0005)
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))));
	else
		tmp = asin(Float64(Float64(l / (0.5 ^ -0.5)) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -100.0)
		tmp = -asin(((l * sqrt(0.5)) / t));
	elseif ((t / l) <= 0.0005)
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	else
		tmp = asin(((l / (0.5 ^ -0.5)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -100.0], (-N[ArcSin[N[(N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[(t / l), $MachinePrecision], 0.0005], N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l / N[Power[0.5, -0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -100

   1. Initial program 70.1%

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

   3. Taylor expanded in Om around 0 52.0%

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

   4. Taylor expanded in t around -inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. associate-/l* 98.6%

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

      3. distribute-neg-frac 98.6%

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

   6. Simplified 98.6%

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

      1. distribute-frac-neg 98.6%

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

      2. asin-neg 98.6%

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

   8. Applied egg-rr 98.6%

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

      1. associate-/l* 98.7%

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

   10. Simplified 98.7%

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

   if -100 < (/.f64 t l) < 5.0000000000000001e-4

   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 86.9%

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

      1. unpow2 86.9%

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

      2. unpow2 86.9%

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

      3. times-frac 98.4%

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

      4. unpow2 98.4%

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

   5. Simplified 98.4%

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

      1. unpow2 98.4%

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

      2. clear-num 98.4%

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

      3. un-div-inv 98.4%

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

   7. Applied egg-rr 98.4%

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

   if 5.0000000000000001e-4 < (/.f64 t l)

   1. Initial program 70.0%

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

   3. Taylor expanded in Om around 0 51.5%

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

   4. Taylor expanded in t around -inf 29.4%

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

      1. mul-1-neg 29.4%

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

      2. associate-/l* 29.4%

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

      3. distribute-neg-frac 29.4%

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

   6. Simplified 29.4%

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

      1. *-un-lft-identity 29.4%

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

      2. div-inv 29.4%

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

      3. times-frac 29.4%

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

      4. add-sqr-sqrt 17.0%

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

      5. sqrt-unprod 50.8%

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

      6. sqr-neg 50.8%

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

      7. sqrt-unprod 43.4%

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

      8. add-sqr-sqrt 96.9%

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

      9. pow1/2 96.9%

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

      10. pow-flip 96.7%

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

      11. metadata-eval 96.7%

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

   8. Applied egg-rr 96.7%

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

      1. associate-*l/ 96.8%

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

      2. *-lft-identity 96.8%

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

   10. Simplified 96.8%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -100:\\ \;\;\;\;-\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0005:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{{0.5}^{-0.5}}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.08 \cdot 10^{+76}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{-\ell}{t}}{\sqrt{2}}\right)\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-17}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= l -1.08e+76)
   (asin 1.0)
   (if (<= l -5e-311)
     (asin (/ (/ (- l) t) (sqrt 2.0)))
     (if (<= l 2.5e-17) (asin (* (/ l t) (sqrt 0.5))) (asin 1.0)))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= -1.08e+76) {
		tmp = asin(1.0);
	} else if (l <= -5e-311) {
		tmp = asin(((-l / t) / sqrt(2.0)));
	} else if (l <= 2.5e-17) {
		tmp = asin(((l / t) * sqrt(0.5)));
	} else {
		tmp = asin(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (l <= (-1.08d+76)) then
        tmp = asin(1.0d0)
    else if (l <= (-5d-311)) then
        tmp = asin(((-l / t) / sqrt(2.0d0)))
    else if (l <= 2.5d-17) then
        tmp = asin(((l / t) * sqrt(0.5d0)))
    else
        tmp = asin(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= -1.08e+76) {
		tmp = Math.asin(1.0);
	} else if (l <= -5e-311) {
		tmp = Math.asin(((-l / t) / Math.sqrt(2.0)));
	} else if (l <= 2.5e-17) {
		tmp = Math.asin(((l / t) * Math.sqrt(0.5)));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if l <= -1.08e+76:
		tmp = math.asin(1.0)
	elif l <= -5e-311:
		tmp = math.asin(((-l / t) / math.sqrt(2.0)))
	elif l <= 2.5e-17:
		tmp = math.asin(((l / t) * math.sqrt(0.5)))
	else:
		tmp = math.asin(1.0)
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (l <= -1.08e+76)
		tmp = asin(1.0);
	elseif (l <= -5e-311)
		tmp = asin(Float64(Float64(Float64(-l) / t) / sqrt(2.0)));
	elseif (l <= 2.5e-17)
		tmp = asin(Float64(Float64(l / t) * sqrt(0.5)));
	else
		tmp = asin(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (l <= -1.08e+76)
		tmp = asin(1.0);
	elseif (l <= -5e-311)
		tmp = asin(((-l / t) / sqrt(2.0)));
	elseif (l <= 2.5e-17)
		tmp = asin(((l / t) * sqrt(0.5)));
	else
		tmp = asin(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[l, -1.08e+76], N[ArcSin[1.0], $MachinePrecision], If[LessEqual[l, -5e-311], N[ArcSin[N[(N[((-l) / t), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.5e-17], N[ArcSin[N[(N[(l / t), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.07999999999999999e76 or 2.4999999999999999e-17 < l

   1. Initial program 97.0%

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

   3. Taylor expanded in Om around 0 80.7%

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

   4. Taylor expanded in t around 0 85.5%

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

   if -1.07999999999999999e76 < l < -5.00000000000023e-311

   1. Initial program 80.6%

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

   3. Taylor expanded in Om around 0 68.9%

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

      1. sqrt-div 68.9%

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

      2. metadata-eval 68.9%

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

      3. add-sqr-sqrt 68.9%

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

      4. hypot-1-def 68.9%

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

      5. sqrt-prod 68.8%

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

      6. sqrt-div 75.1%

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

      7. unpow2 75.1%

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

      8. sqrt-prod 39.8%

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

      9. add-sqr-sqrt 82.6%

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

      10. unpow2 82.6%

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

      11. sqrt-prod 0.0%

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

      12. add-sqr-sqrt 98.0%

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

   5. Applied egg-rr 98.0%

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

   6. Taylor expanded in t around -inf 48.2%

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

      1. mul-1-neg 48.2%

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

      2. associate-/r* 48.2%

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

   8. Simplified 48.2%

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

   if -5.00000000000023e-311 < l < 2.4999999999999999e-17

   1. Initial program 68.8%

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

   3. Taylor expanded in Om around 0 56.1%

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

   4. Taylor expanded in t around -inf 55.8%

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

      1. mul-1-neg 55.8%

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

      2. associate-/l* 55.9%

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

      3. distribute-neg-frac 55.9%

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

   6. Simplified 55.9%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 44.5%

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

      3. sqr-neg 44.5%

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

      4. sqrt-unprod 53.4%

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

      5. add-sqr-sqrt 53.5%

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

      6. associate-/r/ 53.5%

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

   8. Applied egg-rr 53.5%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.08 \cdot 10^{+76}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{-\ell}{t}}{\sqrt{2}}\right)\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-17}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -100:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{-\ell}{t}}{\sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0005:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -100.0)
   (asin (/ (/ (- l) t) (sqrt 2.0)))
   (if (<= (/ t l) 0.0005)
     (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) <= -100.0) {
		tmp = asin(((-l / t) / sqrt(2.0)));
	} else if ((t / l) <= 0.0005) {
		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) <= (-100.0d0)) then
        tmp = asin(((-l / t) / sqrt(2.0d0)))
    else if ((t / l) <= 0.0005d0) 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) <= -100.0) {
		tmp = Math.asin(((-l / t) / Math.sqrt(2.0)));
	} else if ((t / l) <= 0.0005) {
		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) <= -100.0:
		tmp = math.asin(((-l / t) / math.sqrt(2.0)))
	elif (t / l) <= 0.0005:
		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) <= -100.0)
		tmp = asin(Float64(Float64(Float64(-l) / t) / sqrt(2.0)));
	elseif (Float64(t / l) <= 0.0005)
		tmp = asin(Float64(1.0 - (Float64(t / l) ^ 2.0)));
	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 / l) <= -100.0)
		tmp = asin(((-l / t) / sqrt(2.0)));
	elseif ((t / l) <= 0.0005)
		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], -100.0], N[ArcSin[N[(N[((-l) / t), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 0.0005], N[ArcSin[N[(1.0 - N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l / t), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -100

   1. Initial program 70.1%

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

   3. Taylor expanded in Om around 0 52.0%

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

      1. sqrt-div 52.0%

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

      2. metadata-eval 52.0%

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

      3. add-sqr-sqrt 52.0%

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

      4. hypot-1-def 52.0%

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

      5. sqrt-prod 51.8%

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

      6. sqrt-div 63.5%

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

      7. unpow2 63.5%

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

      8. sqrt-prod 44.0%

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

      9. add-sqr-sqrt 75.6%

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

      10. unpow2 75.6%

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

      11. sqrt-prod 44.3%

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

      12. add-sqr-sqrt 98.7%

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

   5. Applied egg-rr 98.7%

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

   6. Taylor expanded in t around -inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. associate-/r* 98.8%

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

   8. Simplified 98.8%

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

   if -100 < (/.f64 t l) < 5.0000000000000001e-4

   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 89.7%

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

   4. Taylor expanded in t around 0 89.6%

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

      1. mul-1-neg 89.6%

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

      2. unsub-neg 89.6%

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

      3. unpow2 89.6%

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

      4. unpow2 89.6%

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

      5. times-frac 98.1%

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

      6. unpow2 98.1%

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

   6. Simplified 98.1%

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

   if 5.0000000000000001e-4 < (/.f64 t l)

   1. Initial program 70.0%

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

   3. Taylor expanded in Om around 0 51.5%

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

   4. Taylor expanded in t around -inf 29.4%

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

      1. mul-1-neg 29.4%

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

      2. associate-/l* 29.4%

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

      3. distribute-neg-frac 29.4%

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

   6. Simplified 29.4%

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

      1. add-sqr-sqrt 17.0%

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

      2. sqrt-unprod 50.9%

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

      3. sqr-neg 50.9%

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

      4. sqrt-unprod 43.4%

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

      5. add-sqr-sqrt 96.9%

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

      6. associate-/r/ 96.9%

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

   8. Applied egg-rr 96.9%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -100:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{-\ell}{t}}{\sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0005:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -100:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{-\ell}{t}}{\sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0005:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{{0.5}^{-0.5}}}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -100.0)
   (asin (/ (/ (- l) t) (sqrt 2.0)))
   (if (<= (/ t l) 0.0005)
     (asin (- 1.0 (pow (/ t l) 2.0)))
     (asin (/ (/ l (pow 0.5 -0.5)) t)))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -100.0) {
		tmp = asin(((-l / t) / sqrt(2.0)));
	} else if ((t / l) <= 0.0005) {
		tmp = asin((1.0 - pow((t / l), 2.0)));
	} else {
		tmp = asin(((l / pow(0.5, -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) <= (-100.0d0)) then
        tmp = asin(((-l / t) / sqrt(2.0d0)))
    else if ((t / l) <= 0.0005d0) then
        tmp = asin((1.0d0 - ((t / l) ** 2.0d0)))
    else
        tmp = asin(((l / (0.5d0 ** (-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) <= -100.0) {
		tmp = Math.asin(((-l / t) / Math.sqrt(2.0)));
	} else if ((t / l) <= 0.0005) {
		tmp = Math.asin((1.0 - Math.pow((t / l), 2.0)));
	} else {
		tmp = Math.asin(((l / Math.pow(0.5, -0.5)) / t));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -100.0:
		tmp = math.asin(((-l / t) / math.sqrt(2.0)))
	elif (t / l) <= 0.0005:
		tmp = math.asin((1.0 - math.pow((t / l), 2.0)))
	else:
		tmp = math.asin(((l / math.pow(0.5, -0.5)) / t))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -100.0)
		tmp = asin(Float64(Float64(Float64(-l) / t) / sqrt(2.0)));
	elseif (Float64(t / l) <= 0.0005)
		tmp = asin(Float64(1.0 - (Float64(t / l) ^ 2.0)));
	else
		tmp = asin(Float64(Float64(l / (0.5 ^ -0.5)) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -100.0)
		tmp = asin(((-l / t) / sqrt(2.0)));
	elseif ((t / l) <= 0.0005)
		tmp = asin((1.0 - ((t / l) ^ 2.0)));
	else
		tmp = asin(((l / (0.5 ^ -0.5)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -100.0], N[ArcSin[N[(N[((-l) / t), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 0.0005], N[ArcSin[N[(1.0 - N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l / N[Power[0.5, -0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -100

   1. Initial program 70.1%

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

   3. Taylor expanded in Om around 0 52.0%

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

      1. sqrt-div 52.0%

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

      2. metadata-eval 52.0%

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

      3. add-sqr-sqrt 52.0%

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

      4. hypot-1-def 52.0%

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

      5. sqrt-prod 51.8%

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

      6. sqrt-div 63.5%

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

      7. unpow2 63.5%

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

      8. sqrt-prod 44.0%

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

      9. add-sqr-sqrt 75.6%

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

      10. unpow2 75.6%

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

      11. sqrt-prod 44.3%

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

      12. add-sqr-sqrt 98.7%

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

   5. Applied egg-rr 98.7%

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

   6. Taylor expanded in t around -inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. associate-/r* 98.8%

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

   8. Simplified 98.8%

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

   if -100 < (/.f64 t l) < 5.0000000000000001e-4

   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 89.7%

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

   4. Taylor expanded in t around 0 89.6%

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

      1. mul-1-neg 89.6%

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

      2. unsub-neg 89.6%

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

      3. unpow2 89.6%

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

      4. unpow2 89.6%

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

      5. times-frac 98.1%

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

      6. unpow2 98.1%

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

   6. Simplified 98.1%

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

   if 5.0000000000000001e-4 < (/.f64 t l)

   1. Initial program 70.0%

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

   3. Taylor expanded in Om around 0 51.5%

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

   4. Taylor expanded in t around -inf 29.4%

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

      1. mul-1-neg 29.4%

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

      2. associate-/l* 29.4%

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

      3. distribute-neg-frac 29.4%

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

   6. Simplified 29.4%

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

      1. *-un-lft-identity 29.4%

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

      2. div-inv 29.4%

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

      3. times-frac 29.4%

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

      4. add-sqr-sqrt 17.0%

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

      5. sqrt-unprod 50.8%

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

      6. sqr-neg 50.8%

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

      7. sqrt-unprod 43.4%

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

      8. add-sqr-sqrt 96.9%

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

      9. pow1/2 96.9%

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

      10. pow-flip 96.7%

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

      11. metadata-eval 96.7%

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

   8. Applied egg-rr 96.7%

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

      1. associate-*l/ 96.8%

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

      2. *-lft-identity 96.8%

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

   10. Simplified 96.8%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -100:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{-\ell}{t}}{\sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0005:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{{0.5}^{-0.5}}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -100:\\ \;\;\;\;-\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0005:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{{0.5}^{-0.5}}}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -100.0)
   (- (asin (/ (* l (sqrt 0.5)) t)))
   (if (<= (/ t l) 0.0005)
     (asin (- 1.0 (pow (/ t l) 2.0)))
     (asin (/ (/ l (pow 0.5 -0.5)) t)))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -100.0) {
		tmp = -asin(((l * sqrt(0.5)) / t));
	} else if ((t / l) <= 0.0005) {
		tmp = asin((1.0 - pow((t / l), 2.0)));
	} else {
		tmp = asin(((l / pow(0.5, -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) <= (-100.0d0)) then
        tmp = -asin(((l * sqrt(0.5d0)) / t))
    else if ((t / l) <= 0.0005d0) then
        tmp = asin((1.0d0 - ((t / l) ** 2.0d0)))
    else
        tmp = asin(((l / (0.5d0 ** (-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) <= -100.0) {
		tmp = -Math.asin(((l * Math.sqrt(0.5)) / t));
	} else if ((t / l) <= 0.0005) {
		tmp = Math.asin((1.0 - Math.pow((t / l), 2.0)));
	} else {
		tmp = Math.asin(((l / Math.pow(0.5, -0.5)) / t));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -100.0:
		tmp = -math.asin(((l * math.sqrt(0.5)) / t))
	elif (t / l) <= 0.0005:
		tmp = math.asin((1.0 - math.pow((t / l), 2.0)))
	else:
		tmp = math.asin(((l / math.pow(0.5, -0.5)) / t))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -100.0)
		tmp = Float64(-asin(Float64(Float64(l * sqrt(0.5)) / t)));
	elseif (Float64(t / l) <= 0.0005)
		tmp = asin(Float64(1.0 - (Float64(t / l) ^ 2.0)));
	else
		tmp = asin(Float64(Float64(l / (0.5 ^ -0.5)) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -100.0)
		tmp = -asin(((l * sqrt(0.5)) / t));
	elseif ((t / l) <= 0.0005)
		tmp = asin((1.0 - ((t / l) ^ 2.0)));
	else
		tmp = asin(((l / (0.5 ^ -0.5)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -100.0], (-N[ArcSin[N[(N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[(t / l), $MachinePrecision], 0.0005], N[ArcSin[N[(1.0 - N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l / N[Power[0.5, -0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -100

   1. Initial program 70.1%

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

   3. Taylor expanded in Om around 0 52.0%

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

   4. Taylor expanded in t around -inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. associate-/l* 98.6%

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

      3. distribute-neg-frac 98.6%

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

   6. Simplified 98.6%

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

      1. distribute-frac-neg 98.6%

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

      2. asin-neg 98.6%

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

   8. Applied egg-rr 98.6%

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

      1. associate-/l* 98.7%

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

   10. Simplified 98.7%

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

   if -100 < (/.f64 t l) < 5.0000000000000001e-4

   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 89.7%

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

   4. Taylor expanded in t around 0 89.6%

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

      1. mul-1-neg 89.6%

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

      2. unsub-neg 89.6%

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

      3. unpow2 89.6%

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

      4. unpow2 89.6%

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

      5. times-frac 98.1%

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

      6. unpow2 98.1%

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

   6. Simplified 98.1%

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

   if 5.0000000000000001e-4 < (/.f64 t l)

   1. Initial program 70.0%

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

   3. Taylor expanded in Om around 0 51.5%

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

   4. Taylor expanded in t around -inf 29.4%

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

      1. mul-1-neg 29.4%

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

      2. associate-/l* 29.4%

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

      3. distribute-neg-frac 29.4%

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

   6. Simplified 29.4%

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

      1. *-un-lft-identity 29.4%

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

      2. div-inv 29.4%

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

      3. times-frac 29.4%

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

      4. add-sqr-sqrt 17.0%

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

      5. sqrt-unprod 50.8%

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

      6. sqr-neg 50.8%

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

      7. sqrt-unprod 43.4%

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

      8. add-sqr-sqrt 96.9%

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

      9. pow1/2 96.9%

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

      10. pow-flip 96.7%

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

      11. metadata-eval 96.7%

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

   8. Applied egg-rr 96.7%

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

      1. associate-*l/ 96.8%

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

      2. *-lft-identity 96.8%

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

   10. Simplified 96.8%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -100:\\ \;\;\;\;-\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0005:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{{0.5}^{-0.5}}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.4 \cdot 10^{-132}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 1.18 \cdot 10^{-15}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= l -4.4e-132)
   (asin 1.0)
   (if (<= l 1.18e-15) (asin (* (/ l t) (sqrt 0.5))) (asin 1.0))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= -4.4e-132) {
		tmp = asin(1.0);
	} else if (l <= 1.18e-15) {
		tmp = asin(((l / t) * sqrt(0.5)));
	} else {
		tmp = asin(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (l <= (-4.4d-132)) then
        tmp = asin(1.0d0)
    else if (l <= 1.18d-15) then
        tmp = asin(((l / t) * sqrt(0.5d0)))
    else
        tmp = asin(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= -4.4e-132) {
		tmp = Math.asin(1.0);
	} else if (l <= 1.18e-15) {
		tmp = Math.asin(((l / t) * Math.sqrt(0.5)));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if l <= -4.4e-132:
		tmp = math.asin(1.0)
	elif l <= 1.18e-15:
		tmp = math.asin(((l / t) * math.sqrt(0.5)))
	else:
		tmp = math.asin(1.0)
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (l <= -4.4e-132)
		tmp = asin(1.0);
	elseif (l <= 1.18e-15)
		tmp = asin(Float64(Float64(l / t) * sqrt(0.5)));
	else
		tmp = asin(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (l <= -4.4e-132)
		tmp = asin(1.0);
	elseif (l <= 1.18e-15)
		tmp = asin(((l / t) * sqrt(0.5)));
	else
		tmp = asin(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[l, -4.4e-132], N[ArcSin[1.0], $MachinePrecision], If[LessEqual[l, 1.18e-15], N[ArcSin[N[(N[(l / t), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -4.39999999999999981e-132 or 1.18000000000000004e-15 < l

   1. Initial program 92.2%

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

   3. Taylor expanded in Om around 0 80.4%

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

   4. Taylor expanded in t around 0 71.2%

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

   if -4.39999999999999981e-132 < l < 1.18000000000000004e-15

   1. Initial program 72.0%

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

   3. Taylor expanded in Om around 0 54.5%

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

   4. Taylor expanded in t around -inf 58.2%

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

      1. mul-1-neg 58.2%

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

      2. associate-/l* 58.2%

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

      3. distribute-neg-frac 58.2%

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

   6. Simplified 58.2%

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

      1. add-sqr-sqrt 22.6%

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

      2. sqrt-unprod 45.1%

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

      3. sqr-neg 45.1%

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

      4. sqrt-unprod 34.1%

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

      5. add-sqr-sqrt 56.2%

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

      6. associate-/r/ 56.2%

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

   8. Applied egg-rr 56.2%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.4 \cdot 10^{-132}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 1.18 \cdot 10^{-15}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.7% 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 84.7%

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

3. Taylor expanded in Om around 0 70.9%

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

4. Taylor expanded in t around 0 51.6%

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

5. Final simplification 51.6%

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

Reproduce

herbie shell --seed 2024026 
(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)))))))