Toniolo and Linder, Equation (2)

Percentage Accurate: 83.7% → 98.3%

Time: 16.0s

Alternatives: 13

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: 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, 1.0× speedup

Math

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

FPCore

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

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(0.5))))));
}

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(0.5))))));
}

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(0.5))))))

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(0.5))))))
end

MATLAB

function tmp = code(t, l, Om, Omc)
	tmp = asin((sqrt((1.0 - ((Om / Omc) / (Omc / Om)))) / hypot(1.0, (t / (l * sqrt(0.5))))));
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[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 83.6%

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

   1. sqrt-div 83.6%

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

   2. div-inv 83.6%

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

   3. add-sqr-sqrt 83.6%

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

   4. hypot-1-def 83.6%

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

   5. *-commutative 83.6%

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

   6. sqrt-prod 83.9%

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

   7. unpow2 83.9%

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

   8. sqrt-prod 55.2%

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

   9. add-sqr-sqrt 98.0%

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

3. Applied egg-rr 98.0%

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

   1. associate-*r/ 98.0%

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

   2. *-rgt-identity 98.0%

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

   3. associate-*l/ 98.0%

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

   4. associate-/l* 98.0%

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

5. Simplified 98.0%

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

   1. *-un-lft-identity 98.0%

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

   2. div-inv 98.0%

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

   3. times-frac 98.0%

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

   4. pow1/2 98.0%

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

   5. pow-flip 98.0%

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

   6. metadata-eval 98.0%

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

7. Applied egg-rr 98.0%

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

   1. associate-*r/ 98.0%

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

   2. associate-*l/ 98.1%

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

   3. *-lft-identity 98.1%

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

9. Simplified 98.1%

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

    1. unpow2 98.1%

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

    2. clear-num 98.1%

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

    3. un-div-inv 98.1%

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

11. Applied egg-rr 98.1%

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

    1. add-sqr-sqrt 97.6%

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

    2. sqrt-unprod 98.1%

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

    3. pow-prod-up 98.1%

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

    4. metadata-eval 98.1%

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

    5. metadata-eval 98.1%

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

    6. expm1-log1p-u 73.5%

       \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}}{\mathsf{hypot}\left(1, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{t}{\ell}}{\sqrt{0.5}}\right)\right)}\right)}\right) \]

    7. associate-/r* 73.5%

       \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}}{\mathsf{hypot}\left(1, \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{t}{\ell \cdot \sqrt{0.5}}}\right)\right)\right)}\right) \]

    8. expm1-udef 73.5%

       \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}}{\mathsf{hypot}\left(1, \color{blue}{e^{\mathsf{log1p}\left(\frac{t}{\ell \cdot \sqrt{0.5}}\right)} - 1}\right)}\right) \]

13. Applied egg-rr 73.5%

    \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}}{\mathsf{hypot}\left(1, \color{blue}{e^{\mathsf{log1p}\left(\frac{t}{\ell \cdot \sqrt{0.5}}\right)} - 1}\right)}\right) \]
14. Step-by-step derivation

    1. expm1-def 73.5%

       \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}}{\mathsf{hypot}\left(1, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t}{\ell \cdot \sqrt{0.5}}\right)\right)}\right)}\right) \]

    2. expm1-log1p 98.1%

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

15. Simplified 98.1%

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

16. Final simplification 98.1%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

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(Float64(t / 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[(N[(t / l), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 83.6%

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

   1. sqrt-div 83.6%

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

   2. add-sqr-sqrt 83.6%

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

   3. hypot-1-def 83.6%

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

   4. *-commutative 83.6%

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

   5. sqrt-prod 83.9%

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

   6. unpow2 83.9%

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

   7. sqrt-prod 55.2%

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

   8. add-sqr-sqrt 98.0%

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

3. Applied egg-rr 98.0%

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

   1. unpow2 98.1%

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

   2. clear-num 98.1%

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

   3. un-div-inv 98.1%

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

5. Applied egg-rr 98.0%

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

6. Final simplification 98.0%

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

Alternative 3: 79.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{t}{\ell}\right)}^{2}\\ \mathbf{if}\;t_1 \leq 0.1:\\ \;\;\;\;\sin^{-1} \left(1 - t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (pow (/ t l) 2.0)))
   (if (<= t_1 0.1) (asin (- 1.0 t_1)) (asin (* l (/ (sqrt 0.5) t))))))

C

double code(double t, double l, double Om, double Omc) {
	double t_1 = pow((t / l), 2.0);
	double tmp;
	if (t_1 <= 0.1) {
		tmp = asin((1.0 - t_1));
	} else {
		tmp = asin((l * (sqrt(0.5) / t)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t / l) ** 2.0d0
    if (t_1 <= 0.1d0) then
        tmp = asin((1.0d0 - t_1))
    else
        tmp = asin((l * (sqrt(0.5d0) / t)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double t_1 = Math.pow((t / l), 2.0);
	double tmp;
	if (t_1 <= 0.1) {
		tmp = Math.asin((1.0 - t_1));
	} else {
		tmp = Math.asin((l * (Math.sqrt(0.5) / t)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	t_1 = math.pow((t / l), 2.0)
	tmp = 0
	if t_1 <= 0.1:
		tmp = math.asin((1.0 - t_1))
	else:
		tmp = math.asin((l * (math.sqrt(0.5) / t)))
	return tmp

Julia

function code(t, l, Om, Omc)
	t_1 = Float64(t / l) ^ 2.0
	tmp = 0.0
	if (t_1 <= 0.1)
		tmp = asin(Float64(1.0 - t_1));
	else
		tmp = asin(Float64(l * Float64(sqrt(0.5) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	t_1 = (t / l) ^ 2.0;
	tmp = 0.0;
	if (t_1 <= 0.1)
		tmp = asin((1.0 - t_1));
	else
		tmp = asin((l * (sqrt(0.5) / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$1, 0.1], N[ArcSin[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l * N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 t l) 2) < 0.10000000000000001

   1. Initial program 97.8%

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

   2. Taylor expanded in Om around 0 82.4%

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

   3. Taylor expanded in t around 0 82.3%

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

      1. mul-1-neg 82.3%

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

      2. unpow2 82.3%

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

      3. unpow2 82.3%

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

      4. times-frac 94.3%

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

      5. unpow2 94.3%

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

      6. unsub-neg 94.3%

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

   5. Simplified 94.3%

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

   if 0.10000000000000001 < (pow.f64 (/.f64 t l) 2)

   1. Initial program 68.2%

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

   2. Taylor expanded in Om around 0 43.1%

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

   3. Taylor expanded in t around inf 62.8%

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

      1. *-commutative 62.8%

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

      2. associate-*l/ 62.8%

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

   5. Simplified 62.8%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{t}{\ell}\right)}^{2} \leq 0.1:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 4: 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 83.6%

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

2. Taylor expanded in Om around 0 63.5%

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

   1. sqrt-div 63.5%

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

   2. metadata-eval 63.5%

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

   3. add-sqr-sqrt 63.5%

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

   4. hypot-1-def 63.5%

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

   5. *-commutative 63.5%

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

   6. sqrt-prod 63.4%

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

   7. sqrt-div 65.4%

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

   8. unpow2 65.4%

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

   9. sqrt-prod 42.9%

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

   10. add-sqr-sqrt 79.2%

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

   11. unpow2 79.2%

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

   12. sqrt-prod 52.2%

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

   13. add-sqr-sqrt 96.3%

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

4. Applied egg-rr 96.3%

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

5. Final simplification 96.3%

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

Alternative 5: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+82}:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+105}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -4e+82)
   (- (asin (* l (/ (sqrt 0.5) t))))
   (if (<= (/ t l) 2e+105)
     (asin (sqrt (/ 1.0 (+ 1.0 (* 2.0 (* (/ t l) (/ t l)))))))
     (asin (* (sqrt 0.5) (/ l t))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -4e+82) {
		tmp = -asin((l * (sqrt(0.5) / t)));
	} else if ((t / l) <= 2e+105) {
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} 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 / l) <= (-4d+82)) then
        tmp = -asin((l * (sqrt(0.5d0) / t)))
    else if ((t / l) <= 2d+105) then
        tmp = asin(sqrt((1.0d0 / (1.0d0 + (2.0d0 * ((t / l) * (t / l)))))))
    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 / l) <= -4e+82) {
		tmp = -Math.asin((l * (Math.sqrt(0.5) / t)));
	} else if ((t / l) <= 2e+105) {
		tmp = Math.asin(Math.sqrt((1.0 / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} else {
		tmp = Math.asin((Math.sqrt(0.5) * (l / t)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -4e+82:
		tmp = -math.asin((l * (math.sqrt(0.5) / t)))
	elif (t / l) <= 2e+105:
		tmp = math.asin(math.sqrt((1.0 / (1.0 + (2.0 * ((t / l) * (t / l)))))))
	else:
		tmp = math.asin((math.sqrt(0.5) * (l / t)))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -4e+82)
		tmp = Float64(-asin(Float64(l * Float64(sqrt(0.5) / t))));
	elseif (Float64(t / l) <= 2e+105)
		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(l / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -4e+82)
		tmp = -asin((l * (sqrt(0.5) / t)));
	elseif ((t / l) <= 2e+105)
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	else
		tmp = asin((sqrt(0.5) * (l / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -4e+82], (-N[ArcSin[N[(l * N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[(t / l), $MachinePrecision], 2e+105], 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[(l / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -3.9999999999999999e82

   1. Initial program 59.4%

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

   2. Taylor expanded in Om around 0 44.3%

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

   3. Taylor expanded in t around -inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. asin-neg 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.5%

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

   7. Simplified 99.5%

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

   if -3.9999999999999999e82 < (/.f64 t l) < 1.9999999999999999e105

   1. Initial program 98.2%

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

   2. Taylor expanded in Om around 0 74.6%

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

      1. add-sqr-sqrt 74.6%

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

      2. pow2 74.6%

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

      3. sqrt-div 74.6%

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

      4. unpow2 74.6%

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

      5. sqrt-prod 43.7%

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

      6. add-sqr-sqrt 83.2%

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

      7. unpow2 83.2%

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

      8. sqrt-prod 49.5%

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

      9. add-sqr-sqrt 96.5%

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

      10. unpow2 96.5%

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

   4. Applied egg-rr 96.5%

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

   if 1.9999999999999999e105 < (/.f64 t l)

   1. Initial program 49.9%

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

   2. Taylor expanded in Om around 0 38.6%

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

   3. Taylor expanded in t around -inf 32.9%

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

      1. add-sqr-sqrt 32.9%

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

      2. sqrt-unprod 33.7%

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

      3. sqr-neg 33.7%

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

      4. mul-1-neg 33.7%

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

      5. mul-1-neg 33.7%

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

      6. sqrt-unprod 28.6%

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

      7. add-sqr-sqrt 95.8%

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

      8. *-commutative 95.8%

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

   5. Applied egg-rr 95.8%

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

      1. *-commutative 95.8%

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

      2. neg-mul-1 95.8%

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

      3. associate-*l/ 95.9%

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

      4. *-commutative 95.9%

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

      5. distribute-rgt-neg-in 95.9%

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

   7. Simplified 95.9%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+82}:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+105}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 6: 97.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -10000:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0002:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -10000.0)
   (- (asin (* l (/ (sqrt 0.5) t))))
   (if (<= (/ t l) 0.0002)
     (asin (sqrt (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))
     (asin (* (sqrt 0.5) (/ l t))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -10000.0) {
		tmp = -asin((l * (sqrt(0.5) / t)));
	} else if ((t / l) <= 0.0002) {
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} 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 / l) <= (-10000.0d0)) then
        tmp = -asin((l * (sqrt(0.5d0) / t)))
    else if ((t / l) <= 0.0002d0) then
        tmp = asin(sqrt((1.0d0 - ((om / omc) / (omc / om)))))
    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 / l) <= -10000.0) {
		tmp = -Math.asin((l * (Math.sqrt(0.5) / t)));
	} else if ((t / l) <= 0.0002) {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} else {
		tmp = Math.asin((Math.sqrt(0.5) * (l / t)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -10000.0:
		tmp = -math.asin((l * (math.sqrt(0.5) / t)))
	elif (t / l) <= 0.0002:
		tmp = math.asin(math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))))
	else:
		tmp = math.asin((math.sqrt(0.5) * (l / t)))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -10000.0)
		tmp = Float64(-asin(Float64(l * Float64(sqrt(0.5) / t))));
	elseif (Float64(t / l) <= 0.0002)
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))));
	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 / l) <= -10000.0)
		tmp = -asin((l * (sqrt(0.5) / t)));
	elseif ((t / l) <= 0.0002)
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	else
		tmp = asin((sqrt(0.5) * (l / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -10000.0], (-N[ArcSin[N[(l * N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[(t / l), $MachinePrecision], 0.0002], 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[(l / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -1e4

   1. Initial program 69.6%

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

   2. Taylor expanded in Om around 0 46.7%

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

   3. Taylor expanded in t around -inf 98.7%

      \[\leadsto \sin^{-1} \color{blue}{\left(-1 \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)} \]
   4. 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. asin-neg 98.7%

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

   5. Applied egg-rr 98.7%

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

      1. associate-*r/ 98.7%

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

   7. Simplified 98.7%

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

   if -1e4 < (/.f64 t l) < 2.0000000000000001e-4

   1. Initial program 97.8%

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

   2. Taylor expanded in t around 0 82.6%

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

      1. unpow2 82.6%

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

      2. unpow2 82.6%

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

      3. times-frac 95.8%

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

      4. unpow2 95.8%

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

   4. Simplified 95.8%

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

      1. unpow2 97.8%

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

      2. clear-num 97.8%

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

      3. un-div-inv 97.8%

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

   6. Applied egg-rr 95.8%

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

   if 2.0000000000000001e-4 < (/.f64 t l)

   1. Initial program 66.9%

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

   2. Taylor expanded in Om around 0 39.8%

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

   3. Taylor expanded in t around -inf 22.3%

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

      1. add-sqr-sqrt 22.3%

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

      2. sqrt-unprod 22.8%

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

      3. sqr-neg 22.8%

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

      4. mul-1-neg 22.8%

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

      5. mul-1-neg 22.8%

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

      6. sqrt-unprod 18.8%

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

      7. add-sqr-sqrt 94.9%

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

      8. *-commutative 94.9%

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

   5. Applied egg-rr 94.9%

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

      1. *-commutative 94.9%

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

      2. neg-mul-1 94.9%

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

      3. associate-*l/ 94.9%

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

      4. *-commutative 94.9%

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

      5. distribute-rgt-neg-in 94.9%

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

   7. Simplified 94.9%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -10000:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0002:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 7: 96.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -10000:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0002:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -10000.0)
   (asin (/ (- l) (/ t (sqrt 0.5))))
   (if (<= (/ t l) 0.0002)
     (asin (- 1.0 (pow (/ t l) 2.0)))
     (asin (* l (/ (sqrt 0.5) t))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -10000.0) {
		tmp = asin((-l / (t / sqrt(0.5))));
	} else if ((t / l) <= 0.0002) {
		tmp = asin((1.0 - pow((t / l), 2.0)));
	} else {
		tmp = asin((l * (sqrt(0.5) / t)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t / l) <= (-10000.0d0)) then
        tmp = asin((-l / (t / sqrt(0.5d0))))
    else if ((t / l) <= 0.0002d0) then
        tmp = asin((1.0d0 - ((t / l) ** 2.0d0)))
    else
        tmp = asin((l * (sqrt(0.5d0) / t)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -10000.0) {
		tmp = Math.asin((-l / (t / Math.sqrt(0.5))));
	} else if ((t / l) <= 0.0002) {
		tmp = Math.asin((1.0 - Math.pow((t / l), 2.0)));
	} else {
		tmp = Math.asin((l * (Math.sqrt(0.5) / t)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -10000.0:
		tmp = math.asin((-l / (t / math.sqrt(0.5))))
	elif (t / l) <= 0.0002:
		tmp = math.asin((1.0 - math.pow((t / l), 2.0)))
	else:
		tmp = math.asin((l * (math.sqrt(0.5) / t)))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -10000.0)
		tmp = asin(Float64(Float64(-l) / Float64(t / sqrt(0.5))));
	elseif (Float64(t / l) <= 0.0002)
		tmp = asin(Float64(1.0 - (Float64(t / l) ^ 2.0)));
	else
		tmp = asin(Float64(l * Float64(sqrt(0.5) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -10000.0)
		tmp = asin((-l / (t / sqrt(0.5))));
	elseif ((t / l) <= 0.0002)
		tmp = asin((1.0 - ((t / l) ^ 2.0)));
	else
		tmp = asin((l * (sqrt(0.5) / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -10000.0], N[ArcSin[N[((-l) / N[(t / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 0.0002], N[ArcSin[N[(1.0 - N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l * N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -1e4

   1. Initial program 69.6%

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

   2. Taylor expanded in Om around 0 46.7%

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

   3. Taylor expanded in t around -inf 98.7%

      \[\leadsto \sin^{-1} \color{blue}{\left(-1 \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)} \]
   4. 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* 98.7%

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

      3. distribute-neg-frac 98.7%

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

   5. Simplified 98.7%

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

   if -1e4 < (/.f64 t l) < 2.0000000000000001e-4

   1. Initial program 97.8%

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

   2. Taylor expanded in Om around 0 82.4%

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

   3. Taylor expanded in t around 0 82.3%

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

      1. mul-1-neg 82.3%

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

      2. unpow2 82.3%

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

      3. unpow2 82.3%

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

      4. times-frac 94.3%

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

      5. unpow2 94.3%

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

      6. unsub-neg 94.3%

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

   5. Simplified 94.3%

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

   if 2.0000000000000001e-4 < (/.f64 t l)

   1. Initial program 66.9%

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

   2. Taylor expanded in Om around 0 39.8%

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

   3. Taylor expanded in t around inf 94.9%

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

      1. *-commutative 94.9%

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

      2. associate-*l/ 95.0%

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

   5. Simplified 95.0%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -10000:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0002:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 8: 96.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{if}\;\frac{t}{\ell} \leq -10000:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0002:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (asin (* l (/ (sqrt 0.5) t)))))
   (if (<= (/ t l) -10000.0)
     (- t_1)
     (if (<= (/ t l) 0.0002) (asin (- 1.0 (pow (/ t l) 2.0))) t_1))))

C

double code(double t, double l, double Om, double Omc) {
	double t_1 = asin((l * (sqrt(0.5) / t)));
	double tmp;
	if ((t / l) <= -10000.0) {
		tmp = -t_1;
	} else if ((t / l) <= 0.0002) {
		tmp = asin((1.0 - pow((t / l), 2.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: t_1
    real(8) :: tmp
    t_1 = asin((l * (sqrt(0.5d0) / t)))
    if ((t / l) <= (-10000.0d0)) then
        tmp = -t_1
    else if ((t / l) <= 0.0002d0) then
        tmp = asin((1.0d0 - ((t / l) ** 2.0d0)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double t_1 = Math.asin((l * (Math.sqrt(0.5) / t)));
	double tmp;
	if ((t / l) <= -10000.0) {
		tmp = -t_1;
	} else if ((t / l) <= 0.0002) {
		tmp = Math.asin((1.0 - Math.pow((t / l), 2.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	t_1 = math.asin((l * (math.sqrt(0.5) / t)))
	tmp = 0
	if (t / l) <= -10000.0:
		tmp = -t_1
	elif (t / l) <= 0.0002:
		tmp = math.asin((1.0 - math.pow((t / l), 2.0)))
	else:
		tmp = t_1
	return tmp

Julia

function code(t, l, Om, Omc)
	t_1 = asin(Float64(l * Float64(sqrt(0.5) / t)))
	tmp = 0.0
	if (Float64(t / l) <= -10000.0)
		tmp = Float64(-t_1);
	elseif (Float64(t / l) <= 0.0002)
		tmp = asin(Float64(1.0 - (Float64(t / l) ^ 2.0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	t_1 = asin((l * (sqrt(0.5) / t)));
	tmp = 0.0;
	if ((t / l) <= -10000.0)
		tmp = -t_1;
	elseif ((t / l) <= 0.0002)
		tmp = asin((1.0 - ((t / l) ^ 2.0)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[ArcSin[N[(l * N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -10000.0], (-t$95$1), If[LessEqual[N[(t / l), $MachinePrecision], 0.0002], N[ArcSin[N[(1.0 - N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -1e4

   1. Initial program 69.6%

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

   2. Taylor expanded in Om around 0 46.7%

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

   3. Taylor expanded in t around -inf 98.7%

      \[\leadsto \sin^{-1} \color{blue}{\left(-1 \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)} \]
   4. 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. asin-neg 98.7%

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

   5. Applied egg-rr 98.7%

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

      1. associate-*r/ 98.7%

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

   7. Simplified 98.7%

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

   if -1e4 < (/.f64 t l) < 2.0000000000000001e-4

   1. Initial program 97.8%

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

   2. Taylor expanded in Om around 0 82.4%

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

   3. Taylor expanded in t around 0 82.3%

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

      1. mul-1-neg 82.3%

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

      2. unpow2 82.3%

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

      3. unpow2 82.3%

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

      4. times-frac 94.3%

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

      5. unpow2 94.3%

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

      6. unsub-neg 94.3%

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

   5. Simplified 94.3%

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

   if 2.0000000000000001e-4 < (/.f64 t l)

   1. Initial program 66.9%

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

   2. Taylor expanded in Om around 0 39.8%

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

   3. Taylor expanded in t around inf 94.9%

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

      1. *-commutative 94.9%

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

      2. associate-*l/ 95.0%

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

   5. Simplified 95.0%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -10000:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0002:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 9: 96.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -10000:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0002:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -10000.0)
   (- (asin (* l (/ (sqrt 0.5) t))))
   (if (<= (/ t l) 0.0002)
     (asin (- 1.0 (pow (/ t l) 2.0)))
     (asin (* (sqrt 0.5) (/ l t))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -10000.0) {
		tmp = -asin((l * (sqrt(0.5) / t)));
	} else if ((t / l) <= 0.0002) {
		tmp = asin((1.0 - pow((t / l), 2.0)));
	} 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 / l) <= (-10000.0d0)) then
        tmp = -asin((l * (sqrt(0.5d0) / t)))
    else if ((t / l) <= 0.0002d0) then
        tmp = asin((1.0d0 - ((t / l) ** 2.0d0)))
    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 / l) <= -10000.0) {
		tmp = -Math.asin((l * (Math.sqrt(0.5) / t)));
	} else if ((t / l) <= 0.0002) {
		tmp = Math.asin((1.0 - Math.pow((t / l), 2.0)));
	} else {
		tmp = Math.asin((Math.sqrt(0.5) * (l / t)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -10000.0:
		tmp = -math.asin((l * (math.sqrt(0.5) / t)))
	elif (t / l) <= 0.0002:
		tmp = math.asin((1.0 - math.pow((t / l), 2.0)))
	else:
		tmp = math.asin((math.sqrt(0.5) * (l / t)))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -10000.0)
		tmp = Float64(-asin(Float64(l * Float64(sqrt(0.5) / t))));
	elseif (Float64(t / l) <= 0.0002)
		tmp = asin(Float64(1.0 - (Float64(t / l) ^ 2.0)));
	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 / l) <= -10000.0)
		tmp = -asin((l * (sqrt(0.5) / t)));
	elseif ((t / l) <= 0.0002)
		tmp = asin((1.0 - ((t / l) ^ 2.0)));
	else
		tmp = asin((sqrt(0.5) * (l / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -10000.0], (-N[ArcSin[N[(l * N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[(t / l), $MachinePrecision], 0.0002], 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[(l / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -1e4

   1. Initial program 69.6%

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

   2. Taylor expanded in Om around 0 46.7%

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

   3. Taylor expanded in t around -inf 98.7%

      \[\leadsto \sin^{-1} \color{blue}{\left(-1 \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)} \]
   4. 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. asin-neg 98.7%

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

   5. Applied egg-rr 98.7%

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

      1. associate-*r/ 98.7%

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

   7. Simplified 98.7%

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

   if -1e4 < (/.f64 t l) < 2.0000000000000001e-4

   1. Initial program 97.8%

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

   2. Taylor expanded in Om around 0 82.4%

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

   3. Taylor expanded in t around 0 82.3%

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

      1. mul-1-neg 82.3%

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

      2. unpow2 82.3%

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

      3. unpow2 82.3%

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

      4. times-frac 94.3%

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

      5. unpow2 94.3%

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

      6. unsub-neg 94.3%

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

   5. Simplified 94.3%

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

   if 2.0000000000000001e-4 < (/.f64 t l)

   1. Initial program 66.9%

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

   2. Taylor expanded in Om around 0 39.8%

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

   3. Taylor expanded in t around -inf 22.3%

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

      1. add-sqr-sqrt 22.3%

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

      2. sqrt-unprod 22.8%

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

      3. sqr-neg 22.8%

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

      4. mul-1-neg 22.8%

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

      5. mul-1-neg 22.8%

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

      6. sqrt-unprod 18.8%

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

      7. add-sqr-sqrt 94.9%

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

      8. *-commutative 94.9%

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

   5. Applied egg-rr 94.9%

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

      1. *-commutative 94.9%

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

      2. neg-mul-1 94.9%

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

      3. associate-*l/ 94.9%

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

      4. *-commutative 94.9%

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

      5. distribute-rgt-neg-in 94.9%

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

   7. Simplified 94.9%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -10000:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0002:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 10: 62.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.1 \cdot 10^{-164}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 2.45 \cdot 10^{-58} \lor \neg \left(\ell \leq 3.4 \cdot 10^{+101}\right) \land \ell \leq 5.6 \cdot 10^{+110}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= l -1.1e-164)
   (asin 1.0)
   (if (or (<= l 2.45e-58) (and (not (<= l 3.4e+101)) (<= l 5.6e+110)))
     (asin (* l (/ (sqrt 0.5) t)))
     (asin 1.0))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= -1.1e-164) {
		tmp = asin(1.0);
	} else if ((l <= 2.45e-58) || (!(l <= 3.4e+101) && (l <= 5.6e+110))) {
		tmp = asin((l * (sqrt(0.5) / t)));
	} 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 <= (-1.1d-164)) then
        tmp = asin(1.0d0)
    else if ((l <= 2.45d-58) .or. (.not. (l <= 3.4d+101)) .and. (l <= 5.6d+110)) then
        tmp = asin((l * (sqrt(0.5d0) / t)))
    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 <= -1.1e-164) {
		tmp = Math.asin(1.0);
	} else if ((l <= 2.45e-58) || (!(l <= 3.4e+101) && (l <= 5.6e+110))) {
		tmp = Math.asin((l * (Math.sqrt(0.5) / t)));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if l <= -1.1e-164:
		tmp = math.asin(1.0)
	elif (l <= 2.45e-58) or (not (l <= 3.4e+101) and (l <= 5.6e+110)):
		tmp = math.asin((l * (math.sqrt(0.5) / t)))
	else:
		tmp = math.asin(1.0)
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (l <= -1.1e-164)
		tmp = asin(1.0);
	elseif ((l <= 2.45e-58) || (!(l <= 3.4e+101) && (l <= 5.6e+110)))
		tmp = asin(Float64(l * Float64(sqrt(0.5) / t)));
	else
		tmp = asin(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (l <= -1.1e-164)
		tmp = asin(1.0);
	elseif ((l <= 2.45e-58) || (~((l <= 3.4e+101)) && (l <= 5.6e+110)))
		tmp = asin((l * (sqrt(0.5) / t)));
	else
		tmp = asin(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[l, -1.1e-164], N[ArcSin[1.0], $MachinePrecision], If[Or[LessEqual[l, 2.45e-58], And[N[Not[LessEqual[l, 3.4e+101]], $MachinePrecision], LessEqual[l, 5.6e+110]]], N[ArcSin[N[(l * N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -1.09999999999999994e-164 or 2.45000000000000015e-58 < l < 3.40000000000000017e101 or 5.59999999999999973e110 < l

   1. Initial program 88.4%

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

   2. Taylor expanded in Om around 0 74.6%

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

   3. Taylor expanded in t around 0 70.3%

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

   if -1.09999999999999994e-164 < l < 2.45000000000000015e-58 or 3.40000000000000017e101 < l < 5.59999999999999973e110

   1. Initial program 74.0%

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

   2. Taylor expanded in Om around 0 41.5%

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

   3. Taylor expanded in t around inf 60.7%

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

      1. *-commutative 60.7%

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

      2. associate-*l/ 60.7%

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

   5. Simplified 60.7%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.1 \cdot 10^{-164}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 2.45 \cdot 10^{-58} \lor \neg \left(\ell \leq 3.4 \cdot 10^{+101}\right) \land \ell \leq 5.6 \cdot 10^{+110}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 11: 62.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{-164}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 2.45 \cdot 10^{-58}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{+101}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{+110}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= l -6.5e-164)
   (asin 1.0)
   (if (<= l 2.45e-58)
     (asin (* l (/ (sqrt 0.5) t)))
     (if (<= l 3.4e+101)
       (asin 1.0)
       (if (<= l 5.6e+110) (asin (/ l (/ t (sqrt 0.5)))) (asin 1.0))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= -6.5e-164) {
		tmp = asin(1.0);
	} else if (l <= 2.45e-58) {
		tmp = asin((l * (sqrt(0.5) / t)));
	} else if (l <= 3.4e+101) {
		tmp = asin(1.0);
	} else if (l <= 5.6e+110) {
		tmp = asin((l / (t / sqrt(0.5))));
	} else {
		tmp = asin(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (l <= (-6.5d-164)) then
        tmp = asin(1.0d0)
    else if (l <= 2.45d-58) then
        tmp = asin((l * (sqrt(0.5d0) / t)))
    else if (l <= 3.4d+101) then
        tmp = asin(1.0d0)
    else if (l <= 5.6d+110) then
        tmp = asin((l / (t / sqrt(0.5d0))))
    else
        tmp = asin(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= -6.5e-164) {
		tmp = Math.asin(1.0);
	} else if (l <= 2.45e-58) {
		tmp = Math.asin((l * (Math.sqrt(0.5) / t)));
	} else if (l <= 3.4e+101) {
		tmp = Math.asin(1.0);
	} else if (l <= 5.6e+110) {
		tmp = Math.asin((l / (t / Math.sqrt(0.5))));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if l <= -6.5e-164:
		tmp = math.asin(1.0)
	elif l <= 2.45e-58:
		tmp = math.asin((l * (math.sqrt(0.5) / t)))
	elif l <= 3.4e+101:
		tmp = math.asin(1.0)
	elif l <= 5.6e+110:
		tmp = math.asin((l / (t / math.sqrt(0.5))))
	else:
		tmp = math.asin(1.0)
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (l <= -6.5e-164)
		tmp = asin(1.0);
	elseif (l <= 2.45e-58)
		tmp = asin(Float64(l * Float64(sqrt(0.5) / t)));
	elseif (l <= 3.4e+101)
		tmp = asin(1.0);
	elseif (l <= 5.6e+110)
		tmp = asin(Float64(l / Float64(t / sqrt(0.5))));
	else
		tmp = asin(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (l <= -6.5e-164)
		tmp = asin(1.0);
	elseif (l <= 2.45e-58)
		tmp = asin((l * (sqrt(0.5) / t)));
	elseif (l <= 3.4e+101)
		tmp = asin(1.0);
	elseif (l <= 5.6e+110)
		tmp = asin((l / (t / sqrt(0.5))));
	else
		tmp = asin(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[l, -6.5e-164], N[ArcSin[1.0], $MachinePrecision], If[LessEqual[l, 2.45e-58], N[ArcSin[N[(l * N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3.4e+101], N[ArcSin[1.0], $MachinePrecision], If[LessEqual[l, 5.6e+110], N[ArcSin[N[(l / N[(t / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.50000000000000004e-164 or 2.45000000000000015e-58 < l < 3.40000000000000017e101 or 5.59999999999999973e110 < l

   1. Initial program 88.4%

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

   2. Taylor expanded in Om around 0 74.6%

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

   3. Taylor expanded in t around 0 70.3%

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

   if -6.50000000000000004e-164 < l < 2.45000000000000015e-58

   1. Initial program 72.1%

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

   2. Taylor expanded in Om around 0 44.1%

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

   3. Taylor expanded in t around inf 60.3%

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

      1. *-commutative 60.3%

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

      2. associate-*l/ 60.3%

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

   5. Simplified 60.3%

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

   if 3.40000000000000017e101 < l < 5.59999999999999973e110

   1. Initial program 99.2%

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

   2. Taylor expanded in Om around 0 7.8%

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

   3. Taylor expanded in t around inf 66.6%

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

      1. associate-/l* 66.6%

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

   5. Simplified 66.6%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{-164}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 2.45 \cdot 10^{-58}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{+101}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{+110}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 12: 62.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{-165}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{-59}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{+101}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{+110}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= l -1.4e-165)
   (asin 1.0)
   (if (<= l 5.8e-59)
     (asin (* l (/ (sqrt 0.5) t)))
     (if (<= l 2.7e+101)
       (asin 1.0)
       (if (<= l 5.6e+110) (asin (/ (* l (sqrt 0.5)) t)) (asin 1.0))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= -1.4e-165) {
		tmp = asin(1.0);
	} else if (l <= 5.8e-59) {
		tmp = asin((l * (sqrt(0.5) / t)));
	} else if (l <= 2.7e+101) {
		tmp = asin(1.0);
	} else if (l <= 5.6e+110) {
		tmp = asin(((l * sqrt(0.5)) / t));
	} 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 <= (-1.4d-165)) then
        tmp = asin(1.0d0)
    else if (l <= 5.8d-59) then
        tmp = asin((l * (sqrt(0.5d0) / t)))
    else if (l <= 2.7d+101) then
        tmp = asin(1.0d0)
    else if (l <= 5.6d+110) then
        tmp = asin(((l * sqrt(0.5d0)) / t))
    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 <= -1.4e-165) {
		tmp = Math.asin(1.0);
	} else if (l <= 5.8e-59) {
		tmp = Math.asin((l * (Math.sqrt(0.5) / t)));
	} else if (l <= 2.7e+101) {
		tmp = Math.asin(1.0);
	} else if (l <= 5.6e+110) {
		tmp = Math.asin(((l * Math.sqrt(0.5)) / t));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if l <= -1.4e-165:
		tmp = math.asin(1.0)
	elif l <= 5.8e-59:
		tmp = math.asin((l * (math.sqrt(0.5) / t)))
	elif l <= 2.7e+101:
		tmp = math.asin(1.0)
	elif l <= 5.6e+110:
		tmp = math.asin(((l * math.sqrt(0.5)) / t))
	else:
		tmp = math.asin(1.0)
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (l <= -1.4e-165)
		tmp = asin(1.0);
	elseif (l <= 5.8e-59)
		tmp = asin(Float64(l * Float64(sqrt(0.5) / t)));
	elseif (l <= 2.7e+101)
		tmp = asin(1.0);
	elseif (l <= 5.6e+110)
		tmp = asin(Float64(Float64(l * sqrt(0.5)) / t));
	else
		tmp = asin(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (l <= -1.4e-165)
		tmp = asin(1.0);
	elseif (l <= 5.8e-59)
		tmp = asin((l * (sqrt(0.5) / t)));
	elseif (l <= 2.7e+101)
		tmp = asin(1.0);
	elseif (l <= 5.6e+110)
		tmp = asin(((l * sqrt(0.5)) / t));
	else
		tmp = asin(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[l, -1.4e-165], N[ArcSin[1.0], $MachinePrecision], If[LessEqual[l, 5.8e-59], N[ArcSin[N[(l * N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.7e+101], N[ArcSin[1.0], $MachinePrecision], If[LessEqual[l, 5.6e+110], N[ArcSin[N[(N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.4e-165 or 5.80000000000000033e-59 < l < 2.70000000000000006e101 or 5.59999999999999973e110 < l

   1. Initial program 88.4%

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

   2. Taylor expanded in Om around 0 74.6%

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

   3. Taylor expanded in t around 0 70.3%

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

   if -1.4e-165 < l < 5.80000000000000033e-59

   1. Initial program 72.1%

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

   2. Taylor expanded in Om around 0 44.1%

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

   3. Taylor expanded in t around inf 60.3%

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

      1. *-commutative 60.3%

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

      2. associate-*l/ 60.3%

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

   5. Simplified 60.3%

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

   if 2.70000000000000006e101 < l < 5.59999999999999973e110

   1. Initial program 99.2%

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

   2. Taylor expanded in Om around 0 7.8%

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

   3. Taylor expanded in t around inf 66.6%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{-165}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{-59}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{+101}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{+110}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 13: 49.7% 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 83.6%

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

2. Taylor expanded in Om around 0 63.5%

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

3. Taylor expanded in t around 0 51.1%

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

4. Final simplification 51.1%

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

Reproduce

herbie shell --seed 2023322 
(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)))))))