Toniolo and Linder, Equation (2)

Percentage Accurate: 84.1% → 98.4%

Time: 17.5s

Alternatives: 12

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.1% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.5%

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

   1. sqrt-div 86.5%

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

   2. div-inv 86.5%

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

   3. add-sqr-sqrt 86.5%

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

   4. hypot-1-def 86.5%

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

   5. *-commutative 86.5%

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

   6. sqrt-prod 86.4%

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

   7. unpow2 86.4%

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

   8. sqrt-prod 51.1%

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

   9. add-sqr-sqrt 98.7%

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

4. Applied egg-rr 98.7%

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

   1. associate-*r/ 98.7%

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

   2. *-rgt-identity 98.7%

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

   3. associate-*l/ 98.7%

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

   4. associate-/l* 98.7%

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

6. Simplified 98.7%

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

   1. unpow2 98.7%

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

   2. clear-num 98.7%

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

   3. un-div-inv 98.7%

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

8. Applied egg-rr 98.7%

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

9. Final simplification 98.7%

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

Alternative 2: 97.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

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.pow(2.0, -0.5)))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, Om_, Omc_] := N[ArcSin[N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[(t / l), $MachinePrecision] / N[Power[2.0, -0.5], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 86.5%

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

3. Taylor expanded in Om around 0 70.3%

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

   1. sqrt-div 70.3%

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

   2. metadata-eval 70.3%

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

   3. add-sqr-sqrt 70.3%

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

   4. hypot-1-def 70.3%

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

   5. sqrt-prod 70.2%

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

   6. sqrt-div 72.1%

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

   7. unpow2 72.1%

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

   8. sqrt-prod 44.4%

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

   9. add-sqr-sqrt 83.2%

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

   10. unpow2 83.2%

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

   11. sqrt-prod 43.4%

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

   12. add-sqr-sqrt 97.9%

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

5. Applied egg-rr 97.9%

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

   1. *-commutative 97.9%

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

   2. associate-*l/ 97.9%

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

   3. associate-*r/ 97.9%

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

7. Simplified 97.9%

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

   1. clear-num 97.9%

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

   2. div-inv 97.9%

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

   3. div-inv 97.9%

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

   4. associate-/r* 97.8%

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

   5. pow1/2 97.8%

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

   6. pow-flip 97.9%

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

   7. metadata-eval 97.9%

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

9. Applied egg-rr 97.9%

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

10. Final simplification 97.9%

    \[\leadsto \sin^{-1} \left(\frac{1}{\mathsf{hypot}\left(1, \frac{\frac{t}{\ell}}{{2}^{-0.5}}\right)}\right) \]
11. Add Preprocessing

Alternative 3: 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 86.5%

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

3. Taylor expanded in Om around 0 70.3%

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

   1. sqrt-div 70.3%

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

   2. metadata-eval 70.3%

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

   3. add-sqr-sqrt 70.3%

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

   4. hypot-1-def 70.3%

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

   5. sqrt-prod 70.2%

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

   6. sqrt-div 72.1%

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

   7. unpow2 72.1%

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

   8. sqrt-prod 44.4%

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

   9. add-sqr-sqrt 83.2%

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

   10. unpow2 83.2%

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

   11. sqrt-prod 43.4%

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

   12. add-sqr-sqrt 97.9%

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

5. Applied egg-rr 97.9%

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

   1. *-commutative 97.9%

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

   2. associate-*l/ 97.9%

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

   3. associate-*r/ 97.9%

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

7. Simplified 97.9%

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

8. Final simplification 97.9%

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

Alternative 4: 97.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -6 \cdot 10^{+141}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \left(-\frac{\sqrt{0.5}}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+43}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -6e+141)
   (asin (* l (- (/ (sqrt 0.5) t))))
   (if (<= (/ t l) 1e+43)
     (asin (sqrt (/ 1.0 (+ 1.0 (* 2.0 (* (/ t l) (/ t l)))))))
     (asin (/ (sqrt 0.5) (/ t l))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -6e+141) {
		tmp = asin((l * -(sqrt(0.5) / t)));
	} else if ((t / l) <= 1e+43) {
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} else {
		tmp = asin((sqrt(0.5) / (t / l)));
	}
	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) <= (-6d+141)) then
        tmp = asin((l * -(sqrt(0.5d0) / t)))
    else if ((t / l) <= 1d+43) then
        tmp = asin(sqrt((1.0d0 / (1.0d0 + (2.0d0 * ((t / l) * (t / l)))))))
    else
        tmp = asin((sqrt(0.5d0) / (t / l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -6e+141) {
		tmp = Math.asin((l * -(Math.sqrt(0.5) / t)));
	} else if ((t / l) <= 1e+43) {
		tmp = Math.asin(Math.sqrt((1.0 / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} else {
		tmp = Math.asin((Math.sqrt(0.5) / (t / l)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -6e+141:
		tmp = math.asin((l * -(math.sqrt(0.5) / t)))
	elif (t / l) <= 1e+43:
		tmp = math.asin(math.sqrt((1.0 / (1.0 + (2.0 * ((t / l) * (t / l)))))))
	else:
		tmp = math.asin((math.sqrt(0.5) / (t / l)))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -6e+141)
		tmp = asin(Float64(l * Float64(-Float64(sqrt(0.5) / t))));
	elseif (Float64(t / l) <= 1e+43)
		tmp = asin(sqrt(Float64(1.0 / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) * Float64(t / l)))))));
	else
		tmp = asin(Float64(sqrt(0.5) / Float64(t / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -6e+141)
		tmp = asin((l * -(sqrt(0.5) / t)));
	elseif ((t / l) <= 1e+43)
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	else
		tmp = asin((sqrt(0.5) / (t / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -6e+141], N[ArcSin[N[(l * (-N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 1e+43], N[ArcSin[N[Sqrt[N[(1.0 / N[(1.0 + N[(2.0 * N[(N[(t / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -5.9999999999999998e141

   1. Initial program 53.5%

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

   3. Taylor expanded in Om around 0 51.0%

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

   4. Taylor expanded in t around -inf 99.5%

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

      1. mul-1-neg 99.5%

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

      2. associate-/l* 99.7%

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

      3. distribute-neg-frac 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in l around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. associate-*r/ 99.9%

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

      3. *-commutative 99.9%

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

      4. distribute-rgt-neg-in 99.9%

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

   9. Simplified 99.9%

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

   if -5.9999999999999998e141 < (/.f64 t l) < 1.00000000000000001e43

   1. Initial program 98.6%

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

   3. Taylor expanded in Om around 0 79.5%

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

      1. unpow2 79.5%

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

      2. unpow2 79.5%

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

      3. frac-times 97.9%

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

   5. Applied egg-rr 97.9%

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

   if 1.00000000000000001e43 < (/.f64 t l)

   1. Initial program 70.3%

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

   3. Taylor expanded in Om around 0 53.4%

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

   4. Taylor expanded in t around inf 97.6%

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

      1. associate-*l/ 97.6%

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

   6. Simplified 97.6%

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

      1. *-commutative 97.6%

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

      2. clear-num 96.7%

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

      3. un-div-inv 96.9%

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

   8. Applied egg-rr 96.9%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -6 \cdot 10^{+141}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \left(-\frac{\sqrt{0.5}}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+43}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50000000000:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.2:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -50000000000.0)
   (asin (/ (- l) (* t (sqrt 2.0))))
   (if (<= (/ t l) 0.2)
     (asin (sqrt (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))
     (asin (/ (sqrt 0.5) (/ t l))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -50000000000.0) {
		tmp = asin((-l / (t * sqrt(2.0))));
	} else if ((t / l) <= 0.2) {
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} else {
		tmp = asin((sqrt(0.5) / (t / l)));
	}
	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) <= (-50000000000.0d0)) then
        tmp = asin((-l / (t * sqrt(2.0d0))))
    else if ((t / l) <= 0.2d0) then
        tmp = asin(sqrt((1.0d0 - ((om / omc) / (omc / om)))))
    else
        tmp = asin((sqrt(0.5d0) / (t / l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -50000000000.0) {
		tmp = Math.asin((-l / (t * Math.sqrt(2.0))));
	} else if ((t / l) <= 0.2) {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} else {
		tmp = Math.asin((Math.sqrt(0.5) / (t / l)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -50000000000.0:
		tmp = math.asin((-l / (t * math.sqrt(2.0))))
	elif (t / l) <= 0.2:
		tmp = math.asin(math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))))
	else:
		tmp = math.asin((math.sqrt(0.5) / (t / l)))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -50000000000.0)
		tmp = asin(Float64(Float64(-l) / Float64(t * sqrt(2.0))));
	elseif (Float64(t / l) <= 0.2)
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))));
	else
		tmp = asin(Float64(sqrt(0.5) / Float64(t / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -50000000000.0)
		tmp = asin((-l / (t * sqrt(2.0))));
	elseif ((t / l) <= 0.2)
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	else
		tmp = asin((sqrt(0.5) / (t / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -50000000000.0], N[ArcSin[N[((-l) / N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 0.2], N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -5e10

   1. Initial program 72.6%

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

   3. Taylor expanded in Om around 0 47.2%

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

      1. sqrt-div 47.2%

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

      2. metadata-eval 47.2%

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

      3. add-sqr-sqrt 47.2%

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

      4. hypot-1-def 47.2%

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

      5. sqrt-prod 47.1%

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

      6. sqrt-div 50.1%

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

      7. unpow2 50.1%

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

      8. sqrt-prod 41.1%

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

      9. add-sqr-sqrt 66.5%

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

      10. unpow2 66.5%

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

      11. sqrt-prod 35.1%

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

      12. add-sqr-sqrt 98.6%

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

   5. Applied egg-rr 98.6%

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

      1. *-commutative 98.6%

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

      2. associate-*l/ 98.6%

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

      3. associate-*r/ 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in t around -inf 98.8%

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

      1. neg-mul-1 98.8%

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

      2. distribute-neg-frac 98.8%

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

   10. Simplified 98.8%

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

   if -5e10 < (/.f64 t l) < 0.20000000000000001

   1. Initial program 98.4%

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

   3. Taylor expanded in t around 0 90.1%

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

      1. unpow2 90.1%

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

      2. unpow2 90.1%

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

      3. times-frac 97.8%

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

      4. unpow2 97.8%

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

   5. Simplified 97.8%

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

      1. unpow2 98.4%

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

      2. clear-num 98.4%

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

      3. un-div-inv 98.4%

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

   7. Applied egg-rr 97.8%

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

   if 0.20000000000000001 < (/.f64 t l)

   1. Initial program 74.8%

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

   3. Taylor expanded in Om around 0 50.2%

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

   4. Taylor expanded in t around inf 97.2%

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

      1. associate-*l/ 97.1%

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

   6. Simplified 97.1%

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

      1. *-commutative 97.1%

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

      2. clear-num 96.4%

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

      3. un-div-inv 96.5%

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

   8. Applied egg-rr 96.5%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50000000000:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.2:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50000000000:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.2:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{1 + {\left(\frac{t}{\ell}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -50000000000.0)
   (asin (/ (- l) (* t (sqrt 2.0))))
   (if (<= (/ t l) 0.2)
     (asin (/ 1.0 (+ 1.0 (pow (/ t l) 2.0))))
     (asin (/ (sqrt 0.5) (/ t l))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -50000000000.0) {
		tmp = asin((-l / (t * sqrt(2.0))));
	} else if ((t / l) <= 0.2) {
		tmp = asin((1.0 / (1.0 + pow((t / l), 2.0))));
	} else {
		tmp = asin((sqrt(0.5) / (t / l)));
	}
	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) <= (-50000000000.0d0)) then
        tmp = asin((-l / (t * sqrt(2.0d0))))
    else if ((t / l) <= 0.2d0) then
        tmp = asin((1.0d0 / (1.0d0 + ((t / l) ** 2.0d0))))
    else
        tmp = asin((sqrt(0.5d0) / (t / l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -50000000000.0) {
		tmp = Math.asin((-l / (t * Math.sqrt(2.0))));
	} else if ((t / l) <= 0.2) {
		tmp = Math.asin((1.0 / (1.0 + Math.pow((t / l), 2.0))));
	} else {
		tmp = Math.asin((Math.sqrt(0.5) / (t / l)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -50000000000.0:
		tmp = math.asin((-l / (t * math.sqrt(2.0))))
	elif (t / l) <= 0.2:
		tmp = math.asin((1.0 / (1.0 + math.pow((t / l), 2.0))))
	else:
		tmp = math.asin((math.sqrt(0.5) / (t / l)))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -50000000000.0)
		tmp = asin(Float64(Float64(-l) / Float64(t * sqrt(2.0))));
	elseif (Float64(t / l) <= 0.2)
		tmp = asin(Float64(1.0 / Float64(1.0 + (Float64(t / l) ^ 2.0))));
	else
		tmp = asin(Float64(sqrt(0.5) / Float64(t / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -50000000000.0)
		tmp = asin((-l / (t * sqrt(2.0))));
	elseif ((t / l) <= 0.2)
		tmp = asin((1.0 / (1.0 + ((t / l) ^ 2.0))));
	else
		tmp = asin((sqrt(0.5) / (t / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -50000000000.0], N[ArcSin[N[((-l) / N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 0.2], N[ArcSin[N[(1.0 / N[(1.0 + N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -5e10

   1. Initial program 72.6%

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

   3. Taylor expanded in Om around 0 47.2%

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

      1. sqrt-div 47.2%

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

      2. metadata-eval 47.2%

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

      3. add-sqr-sqrt 47.2%

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

      4. hypot-1-def 47.2%

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

      5. sqrt-prod 47.1%

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

      6. sqrt-div 50.1%

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

      7. unpow2 50.1%

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

      8. sqrt-prod 41.1%

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

      9. add-sqr-sqrt 66.5%

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

      10. unpow2 66.5%

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

      11. sqrt-prod 35.1%

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

      12. add-sqr-sqrt 98.6%

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

   5. Applied egg-rr 98.6%

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

      1. *-commutative 98.6%

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

      2. associate-*l/ 98.6%

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

      3. associate-*r/ 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in t around -inf 98.8%

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

      1. neg-mul-1 98.8%

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

      2. distribute-neg-frac 98.8%

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

   10. Simplified 98.8%

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

   if -5e10 < (/.f64 t l) < 0.20000000000000001

   1. Initial program 98.4%

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

   3. Taylor expanded in Om around 0 90.3%

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

      1. sqrt-div 90.3%

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

      2. metadata-eval 90.3%

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

      3. add-sqr-sqrt 90.3%

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

      4. hypot-1-def 90.3%

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

      5. sqrt-prod 90.3%

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

      6. sqrt-div 90.3%

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

      7. unpow2 90.3%

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

      8. sqrt-prod 47.3%

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

      9. add-sqr-sqrt 95.8%

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

      10. unpow2 95.8%

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

      11. sqrt-prod 42.8%

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

      12. add-sqr-sqrt 97.9%

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

   5. Applied egg-rr 97.9%

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

      1. *-commutative 97.9%

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

      2. associate-*l/ 97.9%

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

      3. associate-*r/ 97.9%

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

   7. Simplified 97.9%

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

      1. clear-num 97.9%

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

      2. div-inv 97.9%

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

      3. div-inv 97.9%

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

      4. associate-/r* 97.9%

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

      5. pow1/2 97.9%

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

      6. pow-flip 97.9%

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

      7. metadata-eval 97.9%

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

   9. Applied egg-rr 97.9%

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

   10. Taylor expanded in t around 0 89.8%

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

       1. associate-*r/ 89.8%

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

       2. unpow2 89.8%

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

       3. *-commutative 89.8%

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

       4. unpow2 89.8%

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

       5. rem-square-sqrt 89.8%

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

       6. associate-*r* 89.8%

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

       7. metadata-eval 89.8%

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

       8. unpow2 89.8%

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

       9. associate-*l* 89.8%

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

       10. *-lft-identity 89.8%

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

       11. times-frac 97.4%

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

       12. unpow2 97.4%

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

   12. Simplified 97.4%

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

   if 0.20000000000000001 < (/.f64 t l)

   1. Initial program 74.8%

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

   3. Taylor expanded in Om around 0 50.2%

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

   4. Taylor expanded in t around inf 97.2%

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

      1. associate-*l/ 97.1%

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

   6. Simplified 97.1%

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

      1. *-commutative 97.1%

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

      2. clear-num 96.4%

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

      3. un-div-inv 96.5%

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

   8. Applied egg-rr 96.5%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50000000000:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.2:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{1 + {\left(\frac{t}{\ell}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50000000000:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \left(-\frac{\sqrt{0.5}}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.2:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -50000000000.0)
   (asin (* l (- (/ (sqrt 0.5) t))))
   (if (<= (/ t l) 0.2)
     (asin (- 1.0 (pow (/ t l) 2.0)))
     (asin (/ (sqrt 0.5) (/ t l))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -50000000000.0) {
		tmp = asin((l * -(sqrt(0.5) / t)));
	} else if ((t / l) <= 0.2) {
		tmp = asin((1.0 - pow((t / l), 2.0)));
	} else {
		tmp = asin((sqrt(0.5) / (t / l)));
	}
	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) <= (-50000000000.0d0)) then
        tmp = asin((l * -(sqrt(0.5d0) / t)))
    else if ((t / l) <= 0.2d0) then
        tmp = asin((1.0d0 - ((t / l) ** 2.0d0)))
    else
        tmp = asin((sqrt(0.5d0) / (t / l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -50000000000.0) {
		tmp = Math.asin((l * -(Math.sqrt(0.5) / t)));
	} else if ((t / l) <= 0.2) {
		tmp = Math.asin((1.0 - Math.pow((t / l), 2.0)));
	} else {
		tmp = Math.asin((Math.sqrt(0.5) / (t / l)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -50000000000.0:
		tmp = math.asin((l * -(math.sqrt(0.5) / t)))
	elif (t / l) <= 0.2:
		tmp = math.asin((1.0 - math.pow((t / l), 2.0)))
	else:
		tmp = math.asin((math.sqrt(0.5) / (t / l)))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -50000000000.0)
		tmp = asin(Float64(l * Float64(-Float64(sqrt(0.5) / t))));
	elseif (Float64(t / l) <= 0.2)
		tmp = asin(Float64(1.0 - (Float64(t / l) ^ 2.0)));
	else
		tmp = asin(Float64(sqrt(0.5) / Float64(t / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -50000000000.0)
		tmp = asin((l * -(sqrt(0.5) / t)));
	elseif ((t / l) <= 0.2)
		tmp = asin((1.0 - ((t / l) ^ 2.0)));
	else
		tmp = asin((sqrt(0.5) / (t / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -50000000000.0], N[ArcSin[N[(l * (-N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 0.2], N[ArcSin[N[(1.0 - N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -5e10

   1. Initial program 72.6%

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

   3. Taylor expanded in Om around 0 47.2%

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

   4. Taylor expanded in t around -inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. associate-/l* 98.8%

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

      3. distribute-neg-frac 98.8%

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

   6. Simplified 98.8%

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

   7. Taylor expanded in l around 0 98.7%

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

      1. mul-1-neg 98.7%

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

      2. associate-*r/ 98.8%

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

      3. *-commutative 98.8%

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

      4. distribute-rgt-neg-in 98.8%

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

   9. Simplified 98.8%

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

   if -5e10 < (/.f64 t l) < 0.20000000000000001

   1. Initial program 98.4%

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

   3. Taylor expanded in Om around 0 90.3%

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

   4. Taylor expanded in t around 0 89.8%

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

      1. mul-1-neg 89.8%

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

      2. unsub-neg 89.8%

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

      3. unpow2 89.8%

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

      4. unpow2 89.8%

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

      5. times-frac 97.4%

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

      6. unpow2 97.4%

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

   6. Simplified 97.4%

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

   if 0.20000000000000001 < (/.f64 t l)

   1. Initial program 74.8%

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

   3. Taylor expanded in Om around 0 50.2%

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

   4. Taylor expanded in t around inf 97.2%

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

      1. associate-*l/ 97.1%

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

   6. Simplified 97.1%

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

      1. *-commutative 97.1%

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

      2. clear-num 96.4%

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

      3. un-div-inv 96.5%

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

   8. Applied egg-rr 96.5%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50000000000:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \left(-\frac{\sqrt{0.5}}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.2:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50000000000:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.2:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -50000000000.0)
   (asin (/ (- l) (* t (sqrt 2.0))))
   (if (<= (/ t l) 0.2)
     (asin (- 1.0 (pow (/ t l) 2.0)))
     (asin (/ (sqrt 0.5) (/ t l))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -50000000000.0) {
		tmp = asin((-l / (t * sqrt(2.0))));
	} else if ((t / l) <= 0.2) {
		tmp = asin((1.0 - pow((t / l), 2.0)));
	} else {
		tmp = asin((sqrt(0.5) / (t / l)));
	}
	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) <= (-50000000000.0d0)) then
        tmp = asin((-l / (t * sqrt(2.0d0))))
    else if ((t / l) <= 0.2d0) then
        tmp = asin((1.0d0 - ((t / l) ** 2.0d0)))
    else
        tmp = asin((sqrt(0.5d0) / (t / l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -50000000000.0) {
		tmp = Math.asin((-l / (t * Math.sqrt(2.0))));
	} else if ((t / l) <= 0.2) {
		tmp = Math.asin((1.0 - Math.pow((t / l), 2.0)));
	} else {
		tmp = Math.asin((Math.sqrt(0.5) / (t / l)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -50000000000.0:
		tmp = math.asin((-l / (t * math.sqrt(2.0))))
	elif (t / l) <= 0.2:
		tmp = math.asin((1.0 - math.pow((t / l), 2.0)))
	else:
		tmp = math.asin((math.sqrt(0.5) / (t / l)))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -50000000000.0)
		tmp = asin(Float64(Float64(-l) / Float64(t * sqrt(2.0))));
	elseif (Float64(t / l) <= 0.2)
		tmp = asin(Float64(1.0 - (Float64(t / l) ^ 2.0)));
	else
		tmp = asin(Float64(sqrt(0.5) / Float64(t / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -50000000000.0)
		tmp = asin((-l / (t * sqrt(2.0))));
	elseif ((t / l) <= 0.2)
		tmp = asin((1.0 - ((t / l) ^ 2.0)));
	else
		tmp = asin((sqrt(0.5) / (t / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -50000000000.0], N[ArcSin[N[((-l) / N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 0.2], N[ArcSin[N[(1.0 - N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -5e10

   1. Initial program 72.6%

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

   3. Taylor expanded in Om around 0 47.2%

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

      1. sqrt-div 47.2%

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

      2. metadata-eval 47.2%

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

      3. add-sqr-sqrt 47.2%

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

      4. hypot-1-def 47.2%

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

      5. sqrt-prod 47.1%

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

      6. sqrt-div 50.1%

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

      7. unpow2 50.1%

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

      8. sqrt-prod 41.1%

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

      9. add-sqr-sqrt 66.5%

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

      10. unpow2 66.5%

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

      11. sqrt-prod 35.1%

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

      12. add-sqr-sqrt 98.6%

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

   5. Applied egg-rr 98.6%

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

      1. *-commutative 98.6%

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

      2. associate-*l/ 98.6%

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

      3. associate-*r/ 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in t around -inf 98.8%

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

      1. neg-mul-1 98.8%

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

      2. distribute-neg-frac 98.8%

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

   10. Simplified 98.8%

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

   if -5e10 < (/.f64 t l) < 0.20000000000000001

   1. Initial program 98.4%

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

   3. Taylor expanded in Om around 0 90.3%

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

   4. Taylor expanded in t around 0 89.8%

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

      1. mul-1-neg 89.8%

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

      2. unsub-neg 89.8%

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

      3. unpow2 89.8%

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

      4. unpow2 89.8%

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

      5. times-frac 97.4%

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

      6. unpow2 97.4%

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

   6. Simplified 97.4%

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

   if 0.20000000000000001 < (/.f64 t l)

   1. Initial program 74.8%

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

   3. Taylor expanded in Om around 0 50.2%

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

   4. Taylor expanded in t around inf 97.2%

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

      1. associate-*l/ 97.1%

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

   6. Simplified 97.1%

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

      1. *-commutative 97.1%

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

      2. clear-num 96.4%

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

      3. un-div-inv 96.5%

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

   8. Applied egg-rr 96.5%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50000000000:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.2:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+211} \lor \neg \left(\frac{t}{\ell} \leq 0.2\right):\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (or (<= (/ t l) -5e+211) (not (<= (/ t l) 0.2)))
   (asin (/ (sqrt 0.5) (/ t l)))
   (asin 1.0)))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (((t / l) <= -5e+211) || !((t / l) <= 0.2)) {
		tmp = asin((sqrt(0.5) / (t / l)));
	} 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 (((t / l) <= (-5d+211)) .or. (.not. ((t / l) <= 0.2d0))) then
        tmp = asin((sqrt(0.5d0) / (t / l)))
    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 (((t / l) <= -5e+211) || !((t / l) <= 0.2)) {
		tmp = Math.asin((Math.sqrt(0.5) / (t / l)));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if ((t / l) <= -5e+211) or not ((t / l) <= 0.2):
		tmp = math.asin((math.sqrt(0.5) / (t / l)))
	else:
		tmp = math.asin(1.0)
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if ((Float64(t / l) <= -5e+211) || !(Float64(t / l) <= 0.2))
		tmp = asin(Float64(sqrt(0.5) / Float64(t / l)));
	else
		tmp = asin(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (((t / l) <= -5e+211) || ~(((t / l) <= 0.2)))
		tmp = asin((sqrt(0.5) / (t / l)));
	else
		tmp = asin(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[Or[LessEqual[N[(t / l), $MachinePrecision], -5e+211], N[Not[LessEqual[N[(t / l), $MachinePrecision], 0.2]], $MachinePrecision]], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < -4.9999999999999995e211 or 0.20000000000000001 < (/.f64 t l)

   1. Initial program 69.8%

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

   3. Taylor expanded in Om around 0 53.6%

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

   4. Taylor expanded in t around inf 84.3%

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

      1. associate-*l/ 84.2%

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

   6. Simplified 84.2%

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

      1. *-commutative 84.2%

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

      2. clear-num 83.8%

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

      3. un-div-inv 83.9%

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

   8. Applied egg-rr 83.9%

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

   if -4.9999999999999995e211 < (/.f64 t l) < 0.20000000000000001

   1. Initial program 95.3%

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

   3. Taylor expanded in t around 0 72.6%

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

      1. unpow2 72.6%

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

      2. unpow2 72.6%

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

      3. times-frac 78.8%

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

      4. unpow2 78.8%

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

   5. Simplified 78.8%

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

   6. Taylor expanded in Om around 0 78.4%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+211} \lor \neg \left(\frac{t}{\ell} \leq 0.2\right):\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50000000000:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \left(-\frac{\sqrt{0.5}}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.2:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -50000000000.0)
   (asin (* l (- (/ (sqrt 0.5) t))))
   (if (<= (/ t l) 0.2) (asin 1.0) (asin (/ (sqrt 0.5) (/ t l))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -50000000000.0) {
		tmp = asin((l * -(sqrt(0.5) / t)));
	} else if ((t / l) <= 0.2) {
		tmp = asin(1.0);
	} else {
		tmp = asin((sqrt(0.5) / (t / l)));
	}
	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) <= (-50000000000.0d0)) then
        tmp = asin((l * -(sqrt(0.5d0) / t)))
    else if ((t / l) <= 0.2d0) then
        tmp = asin(1.0d0)
    else
        tmp = asin((sqrt(0.5d0) / (t / l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -50000000000.0) {
		tmp = Math.asin((l * -(Math.sqrt(0.5) / t)));
	} else if ((t / l) <= 0.2) {
		tmp = Math.asin(1.0);
	} else {
		tmp = Math.asin((Math.sqrt(0.5) / (t / l)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -50000000000.0:
		tmp = math.asin((l * -(math.sqrt(0.5) / t)))
	elif (t / l) <= 0.2:
		tmp = math.asin(1.0)
	else:
		tmp = math.asin((math.sqrt(0.5) / (t / l)))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -50000000000.0)
		tmp = asin(Float64(l * Float64(-Float64(sqrt(0.5) / t))));
	elseif (Float64(t / l) <= 0.2)
		tmp = asin(1.0);
	else
		tmp = asin(Float64(sqrt(0.5) / Float64(t / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -50000000000.0)
		tmp = asin((l * -(sqrt(0.5) / t)));
	elseif ((t / l) <= 0.2)
		tmp = asin(1.0);
	else
		tmp = asin((sqrt(0.5) / (t / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -50000000000.0], N[ArcSin[N[(l * (-N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 0.2], N[ArcSin[1.0], $MachinePrecision], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -5e10

   1. Initial program 72.6%

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

   3. Taylor expanded in Om around 0 47.2%

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

   4. Taylor expanded in t around -inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. associate-/l* 98.8%

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

      3. distribute-neg-frac 98.8%

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

   6. Simplified 98.8%

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

   7. Taylor expanded in l around 0 98.7%

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

      1. mul-1-neg 98.7%

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

      2. associate-*r/ 98.8%

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

      3. *-commutative 98.8%

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

      4. distribute-rgt-neg-in 98.8%

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

   9. Simplified 98.8%

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

   if -5e10 < (/.f64 t l) < 0.20000000000000001

   1. Initial program 98.4%

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

   3. Taylor expanded in t around 0 90.1%

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

      1. unpow2 90.1%

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

      2. unpow2 90.1%

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

      3. times-frac 97.8%

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

      4. unpow2 97.8%

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

   5. Simplified 97.8%

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

   6. Taylor expanded in Om around 0 97.3%

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

   if 0.20000000000000001 < (/.f64 t l)

   1. Initial program 74.8%

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

   3. Taylor expanded in Om around 0 50.2%

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

   4. Taylor expanded in t around inf 97.2%

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

      1. associate-*l/ 97.1%

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

   6. Simplified 97.1%

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

      1. *-commutative 97.1%

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

      2. clear-num 96.4%

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

      3. un-div-inv 96.5%

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

   8. Applied egg-rr 96.5%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50000000000:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \left(-\frac{\sqrt{0.5}}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.2:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.62 \cdot 10^{+106}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= t 1.62e+106) (asin 1.0) (asin (* (sqrt 0.5) (/ l t)))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (t <= 1.62e+106) {
		tmp = asin(1.0);
	} else {
		tmp = asin((sqrt(0.5) * (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 (t <= 1.62d+106) then
        tmp = asin(1.0d0)
    else
        tmp = asin((sqrt(0.5d0) * (l / t)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (t <= 1.62e+106) {
		tmp = Math.asin(1.0);
	} else {
		tmp = Math.asin((Math.sqrt(0.5) * (l / t)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if t <= 1.62e+106:
		tmp = math.asin(1.0)
	else:
		tmp = math.asin((math.sqrt(0.5) * (l / t)))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (t <= 1.62e+106)
		tmp = asin(1.0);
	else
		tmp = asin(Float64(sqrt(0.5) * Float64(l / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (t <= 1.62e+106)
		tmp = asin(1.0);
	else
		tmp = asin((sqrt(0.5) * (l / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[t, 1.62e+106], N[ArcSin[1.0], $MachinePrecision], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.62e106

   1. Initial program 87.4%

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

   3. Taylor expanded in t around 0 57.1%

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

      1. unpow2 57.1%

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

      2. unpow2 57.1%

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

      3. times-frac 61.8%

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

      4. unpow2 61.8%

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

   5. Simplified 61.8%

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

   6. Taylor expanded in Om around 0 61.5%

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

   if 1.62e106 < t

   1. Initial program 82.3%

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

   3. Taylor expanded in Om around 0 52.6%

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

   4. Taylor expanded in t around inf 67.4%

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

      1. associate-*l/ 67.4%

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

   6. Simplified 67.4%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.62 \cdot 10^{+106}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 50.7% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} 1 \end{array} \]

FPCore

(FPCore (t l Om Omc) :precision binary64 (asin 1.0))

C

double code(double t, double l, double Om, double Omc) {
	return asin(1.0);
}

Fortran

real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    code = asin(1.0d0)
end function

Java

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

Python

def code(t, l, Om, Omc):
	return math.asin(1.0)

Julia

function code(t, l, Om, Omc)
	return asin(1.0)
end

MATLAB

function tmp = code(t, l, Om, Omc)
	tmp = asin(1.0);
end

Wolfram

code[t_, l_, Om_, Omc_] := N[ArcSin[1.0], $MachinePrecision]

Derivation

1. Initial program 86.5%

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

3. Taylor expanded in t around 0 48.9%

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

   1. unpow2 48.9%

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

   2. unpow2 48.9%

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

   3. times-frac 53.3%

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

   4. unpow2 53.3%

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

5. Simplified 53.3%

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

6. Taylor expanded in Om around 0 53.0%

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

7. Final simplification 53.0%

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

Reproduce

herbie shell --seed 2024024 
(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)))))))