Toniolo and Linder, Equation (2)

Percentage Accurate: 83.4% → 98.4%

Time: 20.0s

Alternatives: 10

Speedup: 1.8×

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

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

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. add-sqr-sqrt 80.9%

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

   3. hypot-1-def 80.9%

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

   4. *-commutative 80.9%

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

   5. sqrt-prod 80.8%

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

   6. sqrt-pow1 98.9%

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

   7. metadata-eval 98.9%

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

   8. pow1 98.9%

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

4. Applied egg-rr 98.9%

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

Alternative 2: 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. add-sqr-sqrt 80.9%

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

   3. hypot-1-def 80.9%

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

   4. *-commutative 80.9%

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

   5. sqrt-prod 80.8%

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

   6. sqrt-pow1 98.9%

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

   7. metadata-eval 98.9%

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

   8. pow1 98.9%

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

4. Applied egg-rr 98.9%

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

5. Taylor expanded in Om around 0 98.0%

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

Alternative 3: 91.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 4 \cdot 10^{+139}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) 4e+139)
   (asin
    (sqrt
     (/
      (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
      (+ 1.0 (* 2.0 (* (/ t l) (/ t l)))))))
   (asin (/ (* l (sqrt 0.5)) t))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= 4e+139) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} else {
		tmp = asin(((l * sqrt(0.5)) / t));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t / l) <= 4d+139) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * ((t / l) * (t / l)))))))
    else
        tmp = asin(((l * sqrt(0.5d0)) / t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= 4e+139) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} else {
		tmp = Math.asin(((l * Math.sqrt(0.5)) / t));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= 4e+139:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) * (t / l)))))))
	else:
		tmp = math.asin(((l * math.sqrt(0.5)) / t))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= 4e+139)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om))) / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) * Float64(t / l)))))));
	else
		tmp = asin(Float64(Float64(l * sqrt(0.5)) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= 4e+139)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	else
		tmp = asin(((l * sqrt(0.5)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], 4e+139], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 4.00000000000000013e139

   1. Initial program 89.4%

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

      1. unpow2 89.4%

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

      2. associate-*r/ 85.0%

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

   4. Applied egg-rr 85.0%

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

      1. unpow2 85.0%

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

      2. clear-num 85.0%

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

      3. un-div-inv 85.0%

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

   6. Applied egg-rr 85.0%

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

      1. associate-/l* 89.4%

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

   8. Applied egg-rr 89.4%

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

   if 4.00000000000000013e139 < (/.f64 t l)

   1. Initial program 37.5%

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

      1. add-cube-cbrt 37.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + \color{blue}{\left(\sqrt[3]{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt[3]{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right) \cdot \sqrt[3]{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]

      2. pow3 37.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + \color{blue}{{\left(\sqrt[3]{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}^{3}}}}\right) \]

   4. Applied egg-rr 37.5%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + \color{blue}{{\left(\sqrt[3]{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}^{3}}}}\right) \]

   5. Taylor expanded in Om around 0 37.5%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{1}}{1 + {\left(\sqrt[3]{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}^{3}}}\right) \]

   6. Taylor expanded in t around inf 97.9%

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

      1. associate-*l/ 98.0%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\ell \cdot \sqrt{\frac{1}{{\left(\sqrt[3]{2}\right)}^{3}}}}{t}\right)} \]

      2. rem-cube-cbrt 99.5%

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

      3. metadata-eval 99.5%

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

      4. associate-/l* 99.3%

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

   8. Simplified 99.3%

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

   9. Taylor expanded in l around 0 99.5%

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

Alternative 4: 80.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.08 \cdot 10^{+179}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{\sqrt{2}}}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= t 1.08e+179)
   (asin (sqrt (/ 1.0 (+ 1.0 (* 2.0 (/ (* t (/ t l)) l))))))
   (asin (/ (/ l (sqrt 2.0)) t))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (t <= 1.08e+179) {
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 * ((t * (t / l)) / l))))));
	} else {
		tmp = asin(((l / sqrt(2.0)) / 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 <= 1.08d+179) then
        tmp = asin(sqrt((1.0d0 / (1.0d0 + (2.0d0 * ((t * (t / l)) / l))))))
    else
        tmp = asin(((l / sqrt(2.0d0)) / t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (t <= 1.08e+179) {
		tmp = Math.asin(Math.sqrt((1.0 / (1.0 + (2.0 * ((t * (t / l)) / l))))));
	} else {
		tmp = Math.asin(((l / Math.sqrt(2.0)) / t));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if t <= 1.08e+179:
		tmp = math.asin(math.sqrt((1.0 / (1.0 + (2.0 * ((t * (t / l)) / l))))))
	else:
		tmp = math.asin(((l / math.sqrt(2.0)) / t))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (t <= 1.08e+179)
		tmp = asin(sqrt(Float64(1.0 / Float64(1.0 + Float64(2.0 * Float64(Float64(t * Float64(t / l)) / l))))));
	else
		tmp = asin(Float64(Float64(l / sqrt(2.0)) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (t <= 1.08e+179)
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 * ((t * (t / l)) / l))))));
	else
		tmp = asin(((l / sqrt(2.0)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[t, 1.08e+179], N[ArcSin[N[Sqrt[N[(1.0 / N[(1.0 + N[(2.0 * N[(N[(t * N[(t / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.08000000000000007e179

   1. Initial program 83.4%

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

      1. unpow2 83.4%

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

      2. associate-*r/ 81.3%

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

   4. Applied egg-rr 81.3%

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

   5. Taylor expanded in Om around 0 80.8%

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

   if 1.08000000000000007e179 < 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. add-sqr-sqrt 61.3%

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

      3. hypot-1-def 61.3%

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

      4. *-commutative 61.3%

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

      5. sqrt-prod 61.2%

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

      6. sqrt-pow1 99.4%

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

      7. metadata-eval 99.4%

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

      8. pow1 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in Om around 0 97.1%

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

      1. associate-*l/ 97.0%

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

      2. clear-num 97.0%

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

   7. Applied egg-rr 97.0%

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

      1. associate-/r/ 97.0%

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

   9. Simplified 97.0%

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

   10. Taylor expanded in l around 0 64.3%

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

       1. *-commutative 64.3%

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

       2. associate-/r* 64.4%

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

   12. Simplified 64.4%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.08 \cdot 10^{+179}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{\sqrt{2}}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 6.6 \cdot 10^{+179}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t}{\ell \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{\sqrt{2}}}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= t 6.6e+179)
   (asin (sqrt (/ 1.0 (+ 1.0 (* 2.0 (/ t (* l (/ l t))))))))
   (asin (/ (/ l (sqrt 2.0)) t))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (t <= 6.6e+179) {
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 * (t / (l * (l / t))))))));
	} else {
		tmp = asin(((l / sqrt(2.0)) / 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 <= 6.6d+179) then
        tmp = asin(sqrt((1.0d0 / (1.0d0 + (2.0d0 * (t / (l * (l / t))))))))
    else
        tmp = asin(((l / sqrt(2.0d0)) / t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (t <= 6.6e+179) {
		tmp = Math.asin(Math.sqrt((1.0 / (1.0 + (2.0 * (t / (l * (l / t))))))));
	} else {
		tmp = Math.asin(((l / Math.sqrt(2.0)) / t));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if t <= 6.6e+179:
		tmp = math.asin(math.sqrt((1.0 / (1.0 + (2.0 * (t / (l * (l / t))))))))
	else:
		tmp = math.asin(((l / math.sqrt(2.0)) / t))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (t <= 6.6e+179)
		tmp = asin(sqrt(Float64(1.0 / Float64(1.0 + Float64(2.0 * Float64(t / Float64(l * Float64(l / t))))))));
	else
		tmp = asin(Float64(Float64(l / sqrt(2.0)) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (t <= 6.6e+179)
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 * (t / (l * (l / t))))))));
	else
		tmp = asin(((l / sqrt(2.0)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[t, 6.6e+179], N[ArcSin[N[Sqrt[N[(1.0 / N[(1.0 + N[(2.0 * N[(t / N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 6.59999999999999955e179

   1. Initial program 83.4%

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

      1. unpow2 83.4%

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

      2. clear-num 83.5%

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

      3. frac-times 82.9%

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

      4. *-un-lft-identity 82.9%

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

   4. Applied egg-rr 82.9%

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

   5. Taylor expanded in Om around 0 82.4%

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

   if 6.59999999999999955e179 < 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. add-sqr-sqrt 61.3%

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

      3. hypot-1-def 61.3%

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

      4. *-commutative 61.3%

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

      5. sqrt-prod 61.2%

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

      6. sqrt-pow1 99.4%

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

      7. metadata-eval 99.4%

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

      8. pow1 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in Om around 0 97.1%

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

      1. associate-*l/ 97.0%

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

      2. clear-num 97.0%

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

   7. Applied egg-rr 97.0%

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

      1. associate-/r/ 97.0%

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

   9. Simplified 97.0%

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

   10. Taylor expanded in l around 0 64.3%

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

       1. *-commutative 64.3%

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

       2. associate-/r* 64.4%

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

   12. Simplified 64.4%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.6 \cdot 10^{+179}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t}{\ell \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{\sqrt{2}}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.45 \cdot 10^{+20}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= t 1.45e+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 <= 1.45e+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 <= 1.45d+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 <= 1.45e+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 <= 1.45e+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 <= 1.45e+20)
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / 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 <= 1.45e+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, 1.45e+20], 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[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.45e20

   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 83.6%

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

      2. clear-num 83.6%

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

      3. un-div-inv 83.6%

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

   7. Applied egg-rr 60.0%

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

   if 1.45e20 < 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. Step-by-step derivation

      1. add-cube-cbrt 63.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + \color{blue}{\left(\sqrt[3]{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt[3]{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right) \cdot \sqrt[3]{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]

      2. pow3 63.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + \color{blue}{{\left(\sqrt[3]{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}^{3}}}}\right) \]

   4. Applied egg-rr 63.1%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + \color{blue}{{\left(\sqrt[3]{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}^{3}}}}\right) \]

   5. Taylor expanded in Om around 0 62.0%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{1}}{1 + {\left(\sqrt[3]{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}^{3}}}\right) \]

   6. Taylor expanded in t around inf 55.2%

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

      1. associate-*l/ 55.2%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\ell \cdot \sqrt{\frac{1}{{\left(\sqrt[3]{2}\right)}^{3}}}}{t}\right)} \]

      2. rem-cube-cbrt 55.8%

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

      3. metadata-eval 55.8%

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

      4. associate-/l* 55.8%

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

   8. Simplified 55.8%

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

   9. Taylor expanded in l 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. Add Preprocessing

Alternative 7: 56.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.6 \cdot 10^{+20}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= t 3.6e+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 <= 3.6e+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 <= 3.6d+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 <= 3.6e+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 <= 3.6e+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 <= 3.6e+20)
		tmp = asin(Float64(1.0 + Float64(Float64(Float64(Om / 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 <= 3.6e+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, 3.6e+20], N[ArcSin[N[(1.0 + N[(N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $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 < 3.6e20

   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 83.6%

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

      2. clear-num 83.6%

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

      3. un-div-inv 83.6%

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

   10. Applied egg-rr 59.7%

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

   if 3.6e20 < 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. Step-by-step derivation

      1. add-cube-cbrt 63.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + \color{blue}{\left(\sqrt[3]{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt[3]{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right) \cdot \sqrt[3]{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]

      2. pow3 63.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + \color{blue}{{\left(\sqrt[3]{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}^{3}}}}\right) \]

   4. Applied egg-rr 63.1%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + \color{blue}{{\left(\sqrt[3]{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}^{3}}}}\right) \]

   5. Taylor expanded in Om around 0 62.0%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{1}}{1 + {\left(\sqrt[3]{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}^{3}}}\right) \]

   6. Taylor expanded in t around inf 55.2%

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

      1. associate-*l/ 55.2%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\ell \cdot \sqrt{\frac{1}{{\left(\sqrt[3]{2}\right)}^{3}}}}{t}\right)} \]

      2. rem-cube-cbrt 55.8%

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

      3. metadata-eval 55.8%

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

      4. associate-/l* 55.8%

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

   8. Simplified 55.8%

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

   9. Taylor expanded in l 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. Add Preprocessing

Alternative 8: 56.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.12 \cdot 10^{+20}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= t 1.12e+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 <= 1.12e+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 <= 1.12d+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 <= 1.12e+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 <= 1.12e+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 <= 1.12e+20)
		tmp = asin(Float64(1.0 + Float64(Float64(Float64(Om / 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 <= 1.12e+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, 1.12e+20], N[ArcSin[N[(1.0 + N[(N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $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 < 1.12e20

   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 83.6%

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

      2. clear-num 83.6%

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

      3. un-div-inv 83.6%

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

   10. Applied egg-rr 59.7%

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

   if 1.12e20 < 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. Step-by-step derivation

      1. add-cube-cbrt 63.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + \color{blue}{\left(\sqrt[3]{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt[3]{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right) \cdot \sqrt[3]{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]

      2. pow3 63.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + \color{blue}{{\left(\sqrt[3]{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}^{3}}}}\right) \]

   4. Applied egg-rr 63.1%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + \color{blue}{{\left(\sqrt[3]{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}^{3}}}}\right) \]

   5. Taylor expanded in Om around 0 62.0%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{1}}{1 + {\left(\sqrt[3]{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}^{3}}}\right) \]

   6. Taylor expanded in t around inf 55.2%

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

      1. associate-*l/ 55.2%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\ell \cdot \sqrt{\frac{1}{{\left(\sqrt[3]{2}\right)}^{3}}}}{t}\right)} \]

      2. rem-cube-cbrt 55.8%

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

      3. metadata-eval 55.8%

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

      4. associate-/l* 55.8%

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

   8. 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. Add Preprocessing

Alternative 9: 50.2% accurate, 3.7× speedup

Math

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

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(Float64(Om / 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[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $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 76.8%

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

   2. clear-num 76.8%

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

   3. un-div-inv 76.8%

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

10. Applied egg-rr 49.5%

    \[\leadsto \sin^{-1} \left(1 + \color{blue}{\frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}} \cdot -0.5\right) \]
11. 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. 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)))))))