Toniolo and Linder, Equation (2)

Percentage Accurate: 83.4% → 98.4%

Time: 19.1s

Alternatives: 10

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.4% 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

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

FPCore

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.9%

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

   1. sqrt-div 80.9%

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

   2. clear-num 80.9%

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

   3. add-sqr-sqrt 80.9%

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

   4. hypot-1-def 80.9%

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

   5. *-commutative 80.9%

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

   6. sqrt-prod 80.8%

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

   7. sqrt-pow1 98.9%

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

   8. metadata-eval 98.9%

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

   9. pow1 98.9%

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

4. Applied egg-rr 98.9%

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

5. Final simplification 98.9%

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

Alternative 2: 98.4% accurate, 0.8× speedup

Math

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

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 80.9%

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

   1. sqrt-div 80.9%

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

   2. div-inv 80.9%

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

   3. add-sqr-sqrt 80.9%

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

   4. hypot-1-def 80.9%

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

   5. *-commutative 80.9%

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

   6. sqrt-prod 80.8%

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

   7. sqrt-pow1 98.9%

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

   8. metadata-eval 98.9%

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

   9. pow1 98.9%

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

4. Applied egg-rr 98.9%

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

   1. associate-*r/ 98.9%

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

   2. *-rgt-identity 98.9%

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

6. Simplified 98.9%

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

7. Final simplification 98.9%

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

Alternative 3: 97.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.9%

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

   1. sqrt-div 80.9%

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

   2. clear-num 80.9%

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

   3. add-sqr-sqrt 80.9%

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

   4. hypot-1-def 80.9%

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

   5. *-commutative 80.9%

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

   6. sqrt-prod 80.8%

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

   7. sqrt-pow1 98.9%

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

   8. metadata-eval 98.9%

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

   9. pow1 98.9%

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

4. Applied egg-rr 98.9%

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

5. Taylor expanded in Om around 0 65.4%

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

   1. associate-/l* 64.1%

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

   2. unpow2 64.1%

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

   3. unpow2 64.1%

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

   4. unpow2 64.1%

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

   5. times-frac 64.2%

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

   6. swap-sqr 80.1%

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

   7. associate-/l* 80.1%

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

   8. associate-/l* 80.1%

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

   9. metadata-eval 80.1%

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

   10. hypot-undefine 97.5%

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

   11. associate-/l* 97.9%

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

7. Simplified 97.9%

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

8. Final simplification 97.9%

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

Alternative 4: 97.7% 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 80.9%

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

   1. sqrt-div 80.9%

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

   2. clear-num 80.9%

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

   3. add-sqr-sqrt 80.9%

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

   4. hypot-1-def 80.9%

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

   5. *-commutative 80.9%

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

   6. sqrt-prod 80.8%

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

   7. sqrt-pow1 98.9%

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

   8. metadata-eval 98.9%

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

   9. pow1 98.9%

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

4. Applied egg-rr 98.9%

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

5. Taylor expanded in Om around 0 98.0%

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

   1. *-un-lft-identity 98.0%

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

   2. /-rgt-identity 98.0%

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

7. Applied egg-rr 98.0%

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

   1. *-lft-identity 98.0%

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

   2. *-commutative 98.0%

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

9. Simplified 98.0%

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

10. Final simplification 98.0%

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

Alternative 5: 78.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}{1 + 2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}}}\right)\\ \mathbf{if}\;t \leq 3.6 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+30}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{+50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+116}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{+178}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1
         (asin
          (sqrt
           (/
            (- 1.0 (/ Om (* Omc (/ Omc Om))))
            (+ 1.0 (* 2.0 (/ (* t (/ t l)) l))))))))
   (if (<= t 3.6e+21)
     t_1
     (if (<= t 1.9e+30)
       (asin (/ l (* t (sqrt 2.0))))
       (if (<= t 7.8e+50)
         t_1
         (if (<= t 1.9e+116)
           (asin (/ l (/ t (sqrt 0.5))))
           (if (<= t 1.55e+178) t_1 (asin (/ (/ l t) (sqrt 2.0))))))))))

C

double code(double t, double l, double Om, double Omc) {
	double t_1 = asin(sqrt(((1.0 - (Om / (Omc * (Omc / Om)))) / (1.0 + (2.0 * ((t * (t / l)) / l))))));
	double tmp;
	if (t <= 3.6e+21) {
		tmp = t_1;
	} else if (t <= 1.9e+30) {
		tmp = asin((l / (t * sqrt(2.0))));
	} else if (t <= 7.8e+50) {
		tmp = t_1;
	} else if (t <= 1.9e+116) {
		tmp = asin((l / (t / sqrt(0.5))));
	} else if (t <= 1.55e+178) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = asin(sqrt(((1.0d0 - (om / (omc * (omc / om)))) / (1.0d0 + (2.0d0 * ((t * (t / l)) / l))))))
    if (t <= 3.6d+21) then
        tmp = t_1
    else if (t <= 1.9d+30) then
        tmp = asin((l / (t * sqrt(2.0d0))))
    else if (t <= 7.8d+50) then
        tmp = t_1
    else if (t <= 1.9d+116) then
        tmp = asin((l / (t / sqrt(0.5d0))))
    else if (t <= 1.55d+178) then
        tmp = t_1
    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 t_1 = Math.asin(Math.sqrt(((1.0 - (Om / (Omc * (Omc / Om)))) / (1.0 + (2.0 * ((t * (t / l)) / l))))));
	double tmp;
	if (t <= 3.6e+21) {
		tmp = t_1;
	} else if (t <= 1.9e+30) {
		tmp = Math.asin((l / (t * Math.sqrt(2.0))));
	} else if (t <= 7.8e+50) {
		tmp = t_1;
	} else if (t <= 1.9e+116) {
		tmp = Math.asin((l / (t / Math.sqrt(0.5))));
	} else if (t <= 1.55e+178) {
		tmp = t_1;
	} else {
		tmp = Math.asin(((l / t) / Math.sqrt(2.0)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	t_1 = math.asin(math.sqrt(((1.0 - (Om / (Omc * (Omc / Om)))) / (1.0 + (2.0 * ((t * (t / l)) / l))))))
	tmp = 0
	if t <= 3.6e+21:
		tmp = t_1
	elif t <= 1.9e+30:
		tmp = math.asin((l / (t * math.sqrt(2.0))))
	elif t <= 7.8e+50:
		tmp = t_1
	elif t <= 1.9e+116:
		tmp = math.asin((l / (t / math.sqrt(0.5))))
	elif t <= 1.55e+178:
		tmp = t_1
	else:
		tmp = math.asin(((l / t) / math.sqrt(2.0)))
	return tmp

Julia

function code(t, l, Om, Omc)
	t_1 = asin(sqrt(Float64(Float64(1.0 - Float64(Om / Float64(Omc * Float64(Omc / Om)))) / Float64(1.0 + Float64(2.0 * Float64(Float64(t * Float64(t / l)) / l))))))
	tmp = 0.0
	if (t <= 3.6e+21)
		tmp = t_1;
	elseif (t <= 1.9e+30)
		tmp = asin(Float64(l / Float64(t * sqrt(2.0))));
	elseif (t <= 7.8e+50)
		tmp = t_1;
	elseif (t <= 1.9e+116)
		tmp = asin(Float64(l / Float64(t / sqrt(0.5))));
	elseif (t <= 1.55e+178)
		tmp = t_1;
	else
		tmp = asin(Float64(Float64(l / t) / sqrt(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	t_1 = asin(sqrt(((1.0 - (Om / (Omc * (Omc / Om)))) / (1.0 + (2.0 * ((t * (t / l)) / l))))));
	tmp = 0.0;
	if (t <= 3.6e+21)
		tmp = t_1;
	elseif (t <= 1.9e+30)
		tmp = asin((l / (t * sqrt(2.0))));
	elseif (t <= 7.8e+50)
		tmp = t_1;
	elseif (t <= 1.9e+116)
		tmp = asin((l / (t / sqrt(0.5))));
	elseif (t <= 1.55e+178)
		tmp = t_1;
	else
		tmp = asin(((l / t) / sqrt(2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(Om / N[(Omc * N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t * N[(t / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 3.6e+21], t$95$1, If[LessEqual[t, 1.9e+30], N[ArcSin[N[(l / N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 7.8e+50], t$95$1, If[LessEqual[t, 1.9e+116], N[ArcSin[N[(l / N[(t / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 1.55e+178], t$95$1, N[ArcSin[N[(N[(l / t), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 3.6e21 or 1.9000000000000001e30 < t < 7.79999999999999935e50 or 1.8999999999999999e116 < t < 1.54999999999999996e178

   1. Initial program 85.0%

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

      1. unpow2 85.0%

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

      2. associate-*r/ 82.8%

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

   4. Applied egg-rr 82.8%

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

      1. unpow2 57.7%

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

      2. clear-num 57.7%

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

      3. frac-times 57.7%

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

      4. *-un-lft-identity 57.7%

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

   6. Applied egg-rr 82.8%

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

   if 3.6e21 < t < 1.9000000000000001e30

   1. Initial program 8.6%

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

      1. sqrt-div 8.6%

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

      2. clear-num 8.6%

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

      3. add-sqr-sqrt 8.6%

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

      4. hypot-1-def 8.6%

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

      5. *-commutative 8.6%

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

      6. sqrt-prod 8.6%

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

      7. sqrt-pow1 99.2%

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

      8. metadata-eval 99.2%

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

      9. pow1 99.2%

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in Om around 0 99.2%

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

   6. Taylor expanded in t around inf 54.2%

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

   if 7.79999999999999935e50 < t < 1.8999999999999999e116

   1. Initial program 79.3%

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

   3. Taylor expanded in t around inf 19.0%

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

   4. Taylor expanded in Om around 0 54.4%

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

      1. associate-*r/ 54.3%

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

   6. Simplified 54.3%

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

      1. clear-num 54.3%

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

      2. un-div-inv 54.3%

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

   8. Applied egg-rr 54.3%

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

   if 1.54999999999999996e178 < t

   1. Initial program 61.4%

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

      1. sqrt-div 61.3%

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

      2. clear-num 61.3%

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

      3. add-sqr-sqrt 61.3%

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

      4. hypot-1-def 61.3%

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

      5. *-commutative 61.3%

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

      6. sqrt-prod 61.2%

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

      7. sqrt-pow1 99.4%

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

      8. metadata-eval 99.4%

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

      9. pow1 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in Om around 0 97.1%

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

      1. *-un-lft-identity 97.1%

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

      2. /-rgt-identity 97.1%

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

   7. Applied egg-rr 97.1%

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

      1. *-lft-identity 97.1%

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

      2. *-commutative 97.1%

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

   9. Simplified 97.1%

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

   10. Taylor expanded in t around inf 64.3%

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

       1. associate-/r* 64.3%

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

   12. Simplified 64.3%

       \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\frac{\ell}{t}}{\sqrt{2}}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.6 \cdot 10^{+21}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}{1 + 2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}}}\right)\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+30}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{+50}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}{1 + 2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}}}\right)\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+116}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{+178}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}{1 + 2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{+20}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= t 4.5e+20)
   (asin (sqrt (- 1.0 (/ Om (* Omc (/ Omc Om))))))
   (asin (/ (* l (sqrt 0.5)) t))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (t <= 4.5e+20) {
		tmp = asin(sqrt((1.0 - (Om / (Omc * (Omc / Om))))));
	} 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 <= 4.5d+20) then
        tmp = asin(sqrt((1.0d0 - (om / (omc * (omc / om))))))
    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 <= 4.5e+20) {
		tmp = Math.asin(Math.sqrt((1.0 - (Om / (Omc * (Omc / Om))))));
	} else {
		tmp = Math.asin(((l * Math.sqrt(0.5)) / t));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if t <= 4.5e+20:
		tmp = math.asin(math.sqrt((1.0 - (Om / (Omc * (Omc / Om))))))
	else:
		tmp = math.asin(((l * math.sqrt(0.5)) / t))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (t <= 4.5e+20)
		tmp = asin(sqrt(Float64(1.0 - Float64(Om / Float64(Omc * Float64(Omc / Om))))));
	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 <= 4.5e+20)
		tmp = asin(sqrt((1.0 - (Om / (Omc * (Omc / Om))))));
	else
		tmp = asin(((l * sqrt(0.5)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[t, 4.5e+20], N[ArcSin[N[Sqrt[N[(1.0 - N[(Om / N[(Omc * N[(Omc / Om), $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 2 regimes

2. if t < 4.5e20

   1. Initial program 86.0%

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

   3. Taylor expanded in t around 0 54.3%

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

      1. unpow2 54.3%

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

      2. unpow2 54.3%

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

      3. times-frac 60.0%

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

      4. unpow2 60.0%

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

   5. Simplified 60.0%

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

      1. unpow2 59.7%

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

      2. clear-num 59.7%

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

      3. frac-times 59.7%

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

      4. *-un-lft-identity 59.7%

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

   7. Applied egg-rr 60.0%

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

   if 4.5e20 < t

   1. Initial program 63.3%

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

   3. Taylor expanded in t around inf 44.8%

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

   4. Taylor expanded in Om around 0 55.8%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{+20}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{+20}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om}{Omc \cdot \frac{Omc}{Om}} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= t 9e+20)
   (asin (+ 1.0 (* (/ Om (* Omc (/ Omc Om))) -0.5)))
   (asin (* l (/ (sqrt 0.5) t)))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (t <= 9e+20) {
		tmp = asin((1.0 + ((Om / (Omc * (Omc / Om))) * -0.5)));
	} 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 <= 9d+20) then
        tmp = asin((1.0d0 + ((om / (omc * (omc / om))) * (-0.5d0))))
    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 <= 9e+20) {
		tmp = Math.asin((1.0 + ((Om / (Omc * (Omc / Om))) * -0.5)));
	} else {
		tmp = Math.asin((l * (Math.sqrt(0.5) / t)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if t <= 9e+20:
		tmp = math.asin((1.0 + ((Om / (Omc * (Omc / Om))) * -0.5)))
	else:
		tmp = math.asin((l * (math.sqrt(0.5) / t)))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (t <= 9e+20)
		tmp = asin(Float64(1.0 + Float64(Float64(Om / Float64(Omc * Float64(Omc / Om))) * -0.5)));
	else
		tmp = asin(Float64(l * Float64(sqrt(0.5) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (t <= 9e+20)
		tmp = asin((1.0 + ((Om / (Omc * (Omc / Om))) * -0.5)));
	else
		tmp = asin((l * (sqrt(0.5) / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[t, 9e+20], N[ArcSin[N[(1.0 + N[(N[(Om / N[(Omc * N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l * N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9e20

   1. Initial program 86.0%

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

   3. Taylor expanded in t around 0 54.3%

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

      1. unpow2 54.3%

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

      2. unpow2 54.3%

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

      3. times-frac 60.0%

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

      4. unpow2 60.0%

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

   5. Simplified 60.0%

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

   6. Taylor expanded in Om around 0 54.0%

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

      1. *-commutative 54.0%

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

      2. unpow2 54.0%

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

      3. unpow2 54.0%

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

      4. times-frac 59.7%

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

      5. unpow2 59.7%

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

   8. Simplified 59.7%

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

      1. unpow2 59.7%

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

      2. clear-num 59.7%

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

      3. frac-times 59.7%

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

      4. *-un-lft-identity 59.7%

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

   10. Applied egg-rr 59.7%

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

   if 9e20 < t

   1. Initial program 63.3%

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

   3. Taylor expanded in t around inf 44.8%

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

   4. Taylor expanded in Om around 0 55.8%

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

      1. associate-*r/ 55.8%

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

   6. Simplified 55.8%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{+20}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om}{Omc \cdot \frac{Omc}{Om}} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.35 \cdot 10^{+20}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om}{Omc \cdot \frac{Omc}{Om}} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= t 1.35e+20)
   (asin (+ 1.0 (* (/ Om (* Omc (/ Omc Om))) -0.5)))
   (asin (/ (* l (sqrt 0.5)) t))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (t <= 1.35e+20) {
		tmp = asin((1.0 + ((Om / (Omc * (Omc / Om))) * -0.5)));
	} 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 <= 1.35d+20) then
        tmp = asin((1.0d0 + ((om / (omc * (omc / om))) * (-0.5d0))))
    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 <= 1.35e+20) {
		tmp = Math.asin((1.0 + ((Om / (Omc * (Omc / Om))) * -0.5)));
	} else {
		tmp = Math.asin(((l * Math.sqrt(0.5)) / t));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if t <= 1.35e+20:
		tmp = math.asin((1.0 + ((Om / (Omc * (Omc / Om))) * -0.5)))
	else:
		tmp = math.asin(((l * math.sqrt(0.5)) / t))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (t <= 1.35e+20)
		tmp = asin(Float64(1.0 + Float64(Float64(Om / Float64(Omc * Float64(Omc / Om))) * -0.5)));
	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 <= 1.35e+20)
		tmp = asin((1.0 + ((Om / (Omc * (Omc / Om))) * -0.5)));
	else
		tmp = asin(((l * sqrt(0.5)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[t, 1.35e+20], N[ArcSin[N[(1.0 + N[(N[(Om / N[(Omc * N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.35e20

   1. Initial program 86.0%

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

   3. Taylor expanded in t around 0 54.3%

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

      1. unpow2 54.3%

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

      2. unpow2 54.3%

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

      3. times-frac 60.0%

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

      4. unpow2 60.0%

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

   5. Simplified 60.0%

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

   6. Taylor expanded in Om around 0 54.0%

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

      1. *-commutative 54.0%

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

      2. unpow2 54.0%

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

      3. unpow2 54.0%

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

      4. times-frac 59.7%

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

      5. unpow2 59.7%

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

   8. Simplified 59.7%

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

      1. unpow2 59.7%

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

      2. clear-num 59.7%

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

      3. frac-times 59.7%

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

      4. *-un-lft-identity 59.7%

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

   10. Applied egg-rr 59.7%

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

   if 1.35e20 < t

   1. Initial program 63.3%

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

   3. Taylor expanded in t around inf 44.8%

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

   4. Taylor expanded in Om around 0 55.8%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.35 \cdot 10^{+20}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om}{Omc \cdot \frac{Omc}{Om}} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} \left(1 + \frac{Om}{Omc \cdot \frac{Omc}{Om}} \cdot -0.5\right) \end{array} \]

FPCore

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

C

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

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 + ((om / (omc * (omc / om))) * (-0.5d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.9%

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

3. Taylor expanded in t around 0 45.0%

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

   1. unpow2 45.0%

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

   2. unpow2 45.0%

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

   3. times-frac 49.8%

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

   4. unpow2 49.8%

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

5. Simplified 49.8%

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

6. Taylor expanded in Om around 0 44.7%

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

   1. *-commutative 44.7%

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

   2. unpow2 44.7%

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

   3. unpow2 44.7%

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

   4. times-frac 49.5%

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

   5. unpow2 49.5%

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

8. Simplified 49.5%

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

   1. unpow2 49.5%

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

   2. clear-num 49.5%

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

   3. frac-times 49.5%

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

   4. *-un-lft-identity 49.5%

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

10. Applied egg-rr 49.5%

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

11. Final simplification 49.5%

    \[\leadsto \sin^{-1} \left(1 + \frac{Om}{Omc \cdot \frac{Omc}{Om}} \cdot -0.5\right) \]
12. Add Preprocessing

Alternative 10: 50.0% 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 80.9%

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

3. Taylor expanded in t around 0 45.0%

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

   1. unpow2 45.0%

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

   2. unpow2 45.0%

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

   3. times-frac 49.8%

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

   4. unpow2 49.8%

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

5. Simplified 49.8%

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

6. Taylor expanded in Om around 0 49.4%

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

7. Final simplification 49.4%

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

Reproduce

herbie shell --seed 2024112 
(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)))))))