Toniolo and Linder, Equation (2)

Percentage Accurate: 83.7% → 98.3%

Time: 17.4s

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: 83.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% 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 79.8%

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

   1. sqrt-div 79.8%

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

   2. div-inv 79.8%

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

   3. add-sqr-sqrt 79.8%

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

   4. hypot-1-def 79.8%

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

   5. *-commutative 79.8%

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

   6. sqrt-prod 79.8%

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

   7. sqrt-pow1 98.3%

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

   8. metadata-eval 98.3%

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

   9. pow1 98.3%

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

4. Applied egg-rr 98.3%

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

   1. associate-*r/ 98.3%

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

   2. *-rgt-identity 98.3%

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

6. Simplified 98.3%

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

Alternative 2: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} \left(\frac{1 + {\left(\frac{Om}{Omc}\right)}^{2} \cdot -0.5}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (asin
  (/
   (+ 1.0 (* (pow (/ Om Omc) 2.0) -0.5))
   (hypot 1.0 (* (/ t l) (sqrt 2.0))))))

C

double code(double t, double l, double Om, double Omc) {
	return asin(((1.0 + (pow((Om / Omc), 2.0) * -0.5)) / 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.pow((Om / Omc), 2.0) * -0.5)) / Math.hypot(1.0, ((t / l) * Math.sqrt(2.0)))));
}

Python

def code(t, l, Om, Omc):
	return math.asin(((1.0 + (math.pow((Om / Omc), 2.0) * -0.5)) / math.hypot(1.0, ((t / l) * math.sqrt(2.0)))))

Julia

function code(t, l, Om, Omc)
	return asin(Float64(Float64(1.0 + Float64((Float64(Om / Omc) ^ 2.0) * -0.5)) / hypot(1.0, Float64(Float64(t / l) * sqrt(2.0)))))
end

MATLAB

function tmp = code(t, l, Om, Omc)
	tmp = asin(((1.0 + (((Om / Omc) ^ 2.0) * -0.5)) / hypot(1.0, ((t / l) * sqrt(2.0)))));
end

Wolfram

code[t_, l_, Om_, Omc_] := N[ArcSin[N[(N[(1.0 + N[(N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $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 79.8%

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

   1. sqrt-div 79.8%

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

   2. div-inv 79.8%

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

   3. add-sqr-sqrt 79.8%

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

   4. hypot-1-def 79.8%

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

   5. *-commutative 79.8%

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

   6. sqrt-prod 79.8%

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

   7. sqrt-pow1 98.3%

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

   8. metadata-eval 98.3%

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

   9. pow1 98.3%

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

4. Applied egg-rr 98.3%

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

   1. associate-*r/ 98.3%

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

   2. *-rgt-identity 98.3%

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

6. Simplified 98.3%

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

7. Taylor expanded in Om around 0 88.1%

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

   1. *-commutative 88.1%

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

   2. unpow2 88.1%

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

   3. unpow2 88.1%

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

   4. times-frac 98.1%

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

   5. unpow2 98.1%

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

9. Simplified 98.1%

   \[\leadsto \sin^{-1} \left(\frac{\color{blue}{1 + {\left(\frac{Om}{Omc}\right)}^{2} \cdot -0.5}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
10. Add Preprocessing

Alternative 3: 97.4% 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 79.8%

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

   1. sqrt-div 79.8%

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

   2. div-inv 79.8%

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

   3. add-sqr-sqrt 79.8%

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

   4. hypot-1-def 79.8%

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

   5. *-commutative 79.8%

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

   6. sqrt-prod 79.8%

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

   7. sqrt-pow1 98.3%

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

   8. metadata-eval 98.3%

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

   9. pow1 98.3%

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

4. Applied egg-rr 98.3%

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

   1. associate-*r/ 98.3%

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

   2. *-rgt-identity 98.3%

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

6. Simplified 98.3%

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

7. Taylor expanded in Om around 0 97.8%

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

Alternative 4: 97.4% 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 79.8%

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

   1. clear-num 79.8%

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

   2. sqrt-div 79.7%

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

   3. metadata-eval 79.7%

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

   4. +-commutative 79.7%

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

   5. fma-define 79.7%

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

4. Applied egg-rr 79.7%

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

5. Taylor expanded in Om around 0 62.9%

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

   1. metadata-eval 62.9%

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

   2. *-commutative 62.9%

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

   3. unpow2 62.9%

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

   4. unpow2 62.9%

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

   5. times-frac 79.3%

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

   6. rem-square-sqrt 79.3%

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

   7. swap-sqr 79.4%

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

   8. hypot-undefine 97.8%

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

   9. associate-*l/ 97.8%

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

   10. associate-/l* 97.8%

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

7. Simplified 97.8%

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

Alternative 5: 56.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.45 \cdot 10^{-162}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= l 1.45e-162)
   (asin (/ l (* t (sqrt 2.0))))
   (asin
    (sqrt
     (/
      (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
      (+ 1.0 (* 2.0 (/ (/ t l) (/ l t)))))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= 1.45e-162) {
		tmp = asin((l / (t * sqrt(2.0))));
	} else {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) / (l / 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 (l <= 1.45d-162) then
        tmp = asin((l / (t * sqrt(2.0d0))))
    else
        tmp = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * ((t / l) / (l / t)))))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= 1.45e-162) {
		tmp = Math.asin((l / (t * Math.sqrt(2.0))));
	} else {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if l <= 1.45e-162:
		tmp = math.asin((l / (t * math.sqrt(2.0))))
	else:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) / (l / t)))))))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (l <= 1.45e-162)
		tmp = asin(Float64(l / Float64(t * sqrt(2.0))));
	else
		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(l / t)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (l <= 1.45e-162)
		tmp = asin((l / (t * sqrt(2.0))));
	else
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[l, 1.45e-162], N[ArcSin[N[(l / N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 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[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.4500000000000001e-162

   1. Initial program 75.1%

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

      1. clear-num 75.1%

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

      2. sqrt-div 75.0%

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

      3. metadata-eval 75.0%

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

      4. +-commutative 75.0%

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

      5. fma-define 75.0%

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

   4. Applied egg-rr 75.0%

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

   5. Taylor expanded in Om around 0 55.6%

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

      1. metadata-eval 55.6%

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

      2. *-commutative 55.6%

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

      3. unpow2 55.6%

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

      4. unpow2 55.6%

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

      5. times-frac 75.0%

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

      6. rem-square-sqrt 75.0%

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

      7. swap-sqr 75.0%

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

      8. hypot-undefine 98.0%

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

      9. associate-*l/ 98.0%

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

      10. associate-/l* 98.0%

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

   7. Simplified 98.0%

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

   8. Taylor expanded in t around inf 47.6%

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

   if 1.4500000000000001e-162 < l

   1. Initial program 87.4%

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

      1. unpow2 87.4%

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

      2. clear-num 87.4%

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

      3. un-div-inv 87.4%

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

   4. Applied egg-rr 87.4%

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

      1. unpow2 87.4%

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

      2. clear-num 87.4%

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

      3. un-div-inv 87.4%

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

   6. Applied egg-rr 87.4%

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

Alternative 6: 56.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= l 1.6e-162)
   (asin (/ l (* t (sqrt 2.0))))
   (asin (sqrt (/ 1.0 (+ 1.0 (* 2.0 (/ (/ t l) (/ l t)))))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= 1.6e-162) {
		tmp = asin((l / (t * sqrt(2.0))));
	} else {
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 * ((t / l) / (l / 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 (l <= 1.6d-162) then
        tmp = asin((l / (t * sqrt(2.0d0))))
    else
        tmp = asin(sqrt((1.0d0 / (1.0d0 + (2.0d0 * ((t / l) / (l / t)))))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= 1.6e-162) {
		tmp = Math.asin((l / (t * Math.sqrt(2.0))));
	} else {
		tmp = Math.asin(Math.sqrt((1.0 / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if l <= 1.6e-162:
		tmp = math.asin((l / (t * math.sqrt(2.0))))
	else:
		tmp = math.asin(math.sqrt((1.0 / (1.0 + (2.0 * ((t / l) / (l / t)))))))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (l <= 1.6e-162)
		tmp = asin(Float64(l / Float64(t * sqrt(2.0))));
	else
		tmp = asin(sqrt(Float64(1.0 / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) / Float64(l / t)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (l <= 1.6e-162)
		tmp = asin((l / (t * sqrt(2.0))));
	else
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[l, 1.6e-162], N[ArcSin[N[(l / N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 / N[(1.0 + N[(2.0 * N[(N[(t / l), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.59999999999999988e-162

   1. Initial program 75.1%

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

      1. clear-num 75.1%

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

      2. sqrt-div 75.0%

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

      3. metadata-eval 75.0%

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

      4. +-commutative 75.0%

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

      5. fma-define 75.0%

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

   4. Applied egg-rr 75.0%

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

   5. Taylor expanded in Om around 0 55.6%

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

      1. metadata-eval 55.6%

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

      2. *-commutative 55.6%

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

      3. unpow2 55.6%

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

      4. unpow2 55.6%

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

      5. times-frac 75.0%

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

      6. rem-square-sqrt 75.0%

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

      7. swap-sqr 75.0%

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

      8. hypot-undefine 98.0%

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

      9. associate-*l/ 98.0%

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

      10. associate-/l* 98.0%

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

   7. Simplified 98.0%

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

   8. Taylor expanded in t around inf 47.6%

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

   if 1.59999999999999988e-162 < l

   1. Initial program 87.4%

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

      1. unpow2 87.4%

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

      2. clear-num 87.4%

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

      3. un-div-inv 87.4%

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

   4. Applied egg-rr 87.4%

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

   5. Taylor expanded in Om around 0 86.3%

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

Alternative 7: 44.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.5 \cdot 10^{+80}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= l 3.5e+80)
   (asin (/ l (* t (sqrt 2.0))))
   (asin (sqrt (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= 3.5e+80) {
		tmp = asin((l / (t * sqrt(2.0))));
	} else {
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (l <= 3.5d+80) then
        tmp = asin((l / (t * sqrt(2.0d0))))
    else
        tmp = asin(sqrt((1.0d0 - ((om / omc) / (omc / om)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= 3.5e+80) {
		tmp = Math.asin((l / (t * Math.sqrt(2.0))));
	} else {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if l <= 3.5e+80:
		tmp = math.asin((l / (t * math.sqrt(2.0))))
	else:
		tmp = math.asin(math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (l <= 3.5e+80)
		tmp = asin(Float64(l / Float64(t * sqrt(2.0))));
	else
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (l <= 3.5e+80)
		tmp = asin((l / (t * sqrt(2.0))));
	else
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[l, 3.5e+80], N[ArcSin[N[(l / N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.49999999999999994e80

   1. Initial program 75.4%

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

      1. clear-num 75.4%

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

      2. sqrt-div 75.4%

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

      3. metadata-eval 75.4%

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

      4. +-commutative 75.4%

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

      5. fma-define 75.4%

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

   4. Applied egg-rr 75.4%

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

   5. Taylor expanded in Om around 0 59.5%

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

      1. metadata-eval 59.5%

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

      2. *-commutative 59.5%

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

      3. unpow2 59.5%

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

      4. unpow2 59.5%

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

      5. times-frac 75.4%

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

      6. rem-square-sqrt 75.3%

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

      7. swap-sqr 75.4%

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

      8. hypot-undefine 98.2%

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

      9. associate-*l/ 98.2%

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

      10. associate-/l* 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in t around inf 45.7%

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

   if 3.49999999999999994e80 < l

   1. Initial program 98.3%

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

   3. Taylor expanded in t around 0 71.1%

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

      1. unpow2 71.1%

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

      2. unpow2 71.1%

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

      3. times-frac 87.2%

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

      4. unpow2 87.2%

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

   5. Simplified 87.2%

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

      1. unpow2 98.3%

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

      2. clear-num 98.3%

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

      3. un-div-inv 98.3%

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

   7. Applied egg-rr 87.2%

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

Alternative 8: 44.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.3 \cdot 10^{+80}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(1 + {\left(\frac{Om}{Omc}\right)}^{2} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= l 2.3e+80)
   (asin (/ l (* t (sqrt 2.0))))
   (asin (+ 1.0 (* (pow (/ Om Omc) 2.0) -0.5)))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= 2.3e+80) {
		tmp = asin((l / (t * sqrt(2.0))));
	} else {
		tmp = asin((1.0 + (pow((Om / Omc), 2.0) * -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (l <= 2.3d+80) then
        tmp = asin((l / (t * sqrt(2.0d0))))
    else
        tmp = asin((1.0d0 + (((om / omc) ** 2.0d0) * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= 2.3e+80) {
		tmp = Math.asin((l / (t * Math.sqrt(2.0))));
	} else {
		tmp = Math.asin((1.0 + (Math.pow((Om / Omc), 2.0) * -0.5)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if l <= 2.3e+80:
		tmp = math.asin((l / (t * math.sqrt(2.0))))
	else:
		tmp = math.asin((1.0 + (math.pow((Om / Omc), 2.0) * -0.5)))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (l <= 2.3e+80)
		tmp = asin(Float64(l / Float64(t * sqrt(2.0))));
	else
		tmp = asin(Float64(1.0 + Float64((Float64(Om / Omc) ^ 2.0) * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (l <= 2.3e+80)
		tmp = asin((l / (t * sqrt(2.0))));
	else
		tmp = asin((1.0 + (((Om / Omc) ^ 2.0) * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[l, 2.3e+80], N[ArcSin[N[(l / N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(1.0 + N[(N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.30000000000000004e80

   1. Initial program 75.4%

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

      1. clear-num 75.4%

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

      2. sqrt-div 75.4%

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

      3. metadata-eval 75.4%

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

      4. +-commutative 75.4%

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

      5. fma-define 75.4%

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

   4. Applied egg-rr 75.4%

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

   5. Taylor expanded in Om around 0 59.5%

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

      1. metadata-eval 59.5%

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

      2. *-commutative 59.5%

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

      3. unpow2 59.5%

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

      4. unpow2 59.5%

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

      5. times-frac 75.4%

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

      6. rem-square-sqrt 75.3%

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

      7. swap-sqr 75.4%

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

      8. hypot-undefine 98.2%

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

      9. associate-*l/ 98.2%

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

      10. associate-/l* 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in t around inf 45.7%

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

   if 2.30000000000000004e80 < l

   1. Initial program 98.3%

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

   3. Taylor expanded in t around 0 71.1%

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

      1. unpow2 71.1%

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

      2. unpow2 71.1%

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

      3. times-frac 87.2%

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

      4. unpow2 87.2%

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

   5. Simplified 87.2%

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

   6. Taylor expanded in Om around 0 70.3%

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

      1. *-commutative 77.7%

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

      2. unpow2 77.7%

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

      3. unpow2 77.7%

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

      4. times-frac 97.5%

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

      5. unpow2 97.5%

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

   8. Simplified 86.3%

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

Alternative 9: 44.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.9 \cdot 10^{+80}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= l 1.9e+80) (asin (/ l (* t (sqrt 2.0)))) (asin 1.0)))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= 1.9e+80) {
		tmp = asin((l / (t * sqrt(2.0))));
	} else {
		tmp = asin(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (l <= 1.9d+80) then
        tmp = asin((l / (t * sqrt(2.0d0))))
    else
        tmp = asin(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= 1.9e+80) {
		tmp = Math.asin((l / (t * Math.sqrt(2.0))));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if l <= 1.9e+80:
		tmp = math.asin((l / (t * math.sqrt(2.0))))
	else:
		tmp = math.asin(1.0)
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (l <= 1.9e+80)
		tmp = asin(Float64(l / Float64(t * sqrt(2.0))));
	else
		tmp = asin(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (l <= 1.9e+80)
		tmp = asin((l / (t * sqrt(2.0))));
	else
		tmp = asin(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[l, 1.9e+80], N[ArcSin[N[(l / N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.89999999999999999e80

   1. Initial program 75.4%

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

      1. clear-num 75.4%

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

      2. sqrt-div 75.4%

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

      3. metadata-eval 75.4%

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

      4. +-commutative 75.4%

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

      5. fma-define 75.4%

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

   4. Applied egg-rr 75.4%

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

   5. Taylor expanded in Om around 0 59.5%

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

      1. metadata-eval 59.5%

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

      2. *-commutative 59.5%

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

      3. unpow2 59.5%

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

      4. unpow2 59.5%

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

      5. times-frac 75.4%

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

      6. rem-square-sqrt 75.3%

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

      7. swap-sqr 75.4%

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

      8. hypot-undefine 98.2%

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

      9. associate-*l/ 98.2%

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

      10. associate-/l* 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in t around inf 45.7%

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

   if 1.89999999999999999e80 < l

   1. Initial program 98.3%

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

   3. Taylor expanded in t around 0 71.1%

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

      1. unpow2 71.1%

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

      2. unpow2 71.1%

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

      3. times-frac 87.2%

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

      4. unpow2 87.2%

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

   5. Simplified 87.2%

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

   6. Taylor expanded in Om around 0 85.8%

      \[\leadsto \sin^{-1} \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 49.8% 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 79.8%

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

3. Taylor expanded in t around 0 39.0%

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

   1. unpow2 39.0%

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

   2. unpow2 39.0%

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

   3. times-frac 46.4%

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

   4. unpow2 46.4%

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

5. Simplified 46.4%

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

6. Taylor expanded in Om around 0 46.2%

   \[\leadsto \sin^{-1} \color{blue}{1} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024147 
(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)))))))