Toniolo and Linder, Equation (2)

Percentage Accurate: 84.0% → 98.7%

Time: 22.2s

Alternatives: 5

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: 84.0% 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: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{t}{\ell}\right)}^{2} \leq 2 \cdot 10^{+298}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{1}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left|\frac{\ell}{t \cdot \sqrt{2}}\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (pow (/ t l) 2.0) 2e+298)
   (asin
    (sqrt
     (/
      (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
      (+ 1.0 (* 2.0 (/ 1.0 (* (/ l t) (/ l t))))))))
   (asin (fabs (/ l (* t (sqrt 2.0)))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (pow((t / l), 2.0) <= 2e+298) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * (1.0 / ((l / t) * (l / t))))))));
	} else {
		tmp = asin(fabs((l / (t * sqrt(2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (((t / l) ** 2.0d0) <= 2d+298) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * (1.0d0 / ((l / t) * (l / t))))))))
    else
        tmp = asin(abs((l / (t * sqrt(2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (Math.pow((t / l), 2.0) <= 2e+298) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * (1.0 / ((l / t) * (l / t))))))));
	} else {
		tmp = Math.asin(Math.abs((l / (t * Math.sqrt(2.0)))));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if math.pow((t / l), 2.0) <= 2e+298:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * (1.0 / ((l / t) * (l / t))))))))
	else:
		tmp = math.asin(math.fabs((l / (t * math.sqrt(2.0)))))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if ((Float64(t / l) ^ 2.0) <= 2e+298)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om))) / Float64(1.0 + Float64(2.0 * Float64(1.0 / Float64(Float64(l / t) * Float64(l / t))))))));
	else
		tmp = asin(abs(Float64(l / Float64(t * sqrt(2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (((t / l) ^ 2.0) <= 2e+298)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * (1.0 / ((l / t) * (l / t))))))));
	else
		tmp = asin(abs((l / (t * sqrt(2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision], 2e+298], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(1.0 / N[(N[(l / t), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Abs[N[(l / N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 t l) 2) < 1.9999999999999999e298

   1. Initial program 99.0%

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

      1. unpow2 99.0%

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

      2. clear-num 99.0%

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

      3. clear-num 99.0%

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

      4. frac-times 99.0%

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

      5. metadata-eval 99.0%

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

   4. Applied egg-rr 99.0%

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

      1. unpow2 99.0%

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

      2. clear-num 99.0%

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

      3. un-div-inv 99.0%

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

   6. Applied egg-rr 99.0%

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

   if 1.9999999999999999e298 < (pow.f64 (/.f64 t l) 2)

   1. Initial program 45.5%

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

      1. sqrt-div 45.5%

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

      2. div-inv 45.5%

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

      3. add-sqr-sqrt 45.5%

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

      4. hypot-1-def 45.5%

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

      5. *-commutative 45.5%

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

      6. sqrt-prod 45.5%

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

      7. unpow2 45.5%

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

      8. sqrt-prod 53.3%

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

      9. add-sqr-sqrt 99.0%

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

   4. Applied egg-rr 99.0%

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

      1. associate-*r/ 99.0%

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

      2. *-rgt-identity 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in t around inf 67.2%

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

      1. *-commutative 67.2%

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

      2. unpow2 67.2%

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

      3. unpow2 67.2%

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

      4. times-frac 74.3%

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

      5. unpow2 74.3%

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

   9. Simplified 74.3%

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

   10. Taylor expanded in Om around 0 74.3%

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

       1. add-sqr-sqrt 72.1%

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

       2. sqrt-unprod 46.4%

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

       3. pow2 46.4%

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

       4. associate-/r* 46.4%

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

   12. Applied egg-rr 46.4%

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

       1. unpow2 46.4%

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

       2. rem-sqrt-square 99.6%

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

       3. associate-/r* 99.6%

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

   14. Simplified 99.6%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{t}{\ell}\right)}^{2} \leq 2 \cdot 10^{+298}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{1}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left|\frac{\ell}{t \cdot \sqrt{2}}\right|\right)\\ \end{array} \]
5. Add Preprocessing

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 80.8%

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

   1. sqrt-div 80.7%

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

   2. div-inv 80.7%

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

   3. add-sqr-sqrt 80.7%

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

   4. hypot-1-def 80.7%

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

   5. *-commutative 80.7%

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

   6. sqrt-prod 80.7%

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

   7. unpow2 80.7%

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

   8. sqrt-prod 56.3%

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

   9. add-sqr-sqrt 98.9%

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

4. Applied egg-rr 98.9%

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

   1. associate-*r/ 98.9%

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

   2. *-rgt-identity 98.9%

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

6. Simplified 98.9%

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

   1. unpow2 80.8%

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

   2. clear-num 80.8%

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

   3. un-div-inv 80.8%

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

8. Applied egg-rr 98.9%

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

9. Final simplification 98.9%

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

Alternative 3: 84.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

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) / (omc / om))) / (1.0d0 + (2.0d0 * (1.0d0 / ((l / t) * (l / t))))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.8%

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

   1. unpow2 80.8%

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

   2. clear-num 80.8%

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

   3. clear-num 80.8%

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

   4. frac-times 80.8%

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

   5. metadata-eval 80.8%

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

4. Applied egg-rr 80.8%

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

   1. unpow2 80.8%

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

   2. clear-num 80.8%

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

   3. un-div-inv 80.8%

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

6. Applied egg-rr 80.8%

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

7. Final simplification 80.8%

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

Alternative 4: 64.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-149} \lor \neg \left(\ell \leq 2.9 \cdot 10^{-36}\right):\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{t \cdot \frac{\sqrt{2}}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (or (<= l -5e-149) (not (<= l 2.9e-36)))
   (asin (sqrt (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))
   (asin (/ 1.0 (* t (/ (sqrt 2.0) l))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((l <= -5e-149) || !(l <= 2.9e-36)) {
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} else {
		tmp = asin((1.0 / (t * (sqrt(2.0) / 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 ((l <= (-5d-149)) .or. (.not. (l <= 2.9d-36))) then
        tmp = asin(sqrt((1.0d0 - ((om / omc) / (omc / om)))))
    else
        tmp = asin((1.0d0 / (t * (sqrt(2.0d0) / l))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((l <= -5e-149) || !(l <= 2.9e-36)) {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} else {
		tmp = Math.asin((1.0 / (t * (Math.sqrt(2.0) / l))));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (l <= -5e-149) or not (l <= 2.9e-36):
		tmp = math.asin(math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))))
	else:
		tmp = math.asin((1.0 / (t * (math.sqrt(2.0) / l))))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if ((l <= -5e-149) || !(l <= 2.9e-36))
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))));
	else
		tmp = asin(Float64(1.0 / Float64(t * Float64(sqrt(2.0) / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((l <= -5e-149) || ~((l <= 2.9e-36)))
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	else
		tmp = asin((1.0 / (t * (sqrt(2.0) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[Or[LessEqual[l, -5e-149], N[Not[LessEqual[l, 2.9e-36]], $MachinePrecision]], N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(1.0 / N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.99999999999999968e-149 or 2.90000000000000013e-36 < l

   1. Initial program 86.2%

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

   3. Taylor expanded in t around 0 58.1%

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

      1. unpow2 58.1%

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

      2. unpow2 58.1%

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

      3. times-frac 68.2%

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

      4. unpow2 68.2%

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

   5. Simplified 68.2%

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

      1. unpow2 86.2%

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

      2. clear-num 86.2%

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

      3. un-div-inv 86.2%

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

   7. Applied egg-rr 68.2%

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

   if -4.99999999999999968e-149 < l < 2.90000000000000013e-36

   1. Initial program 72.7%

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

      1. sqrt-div 72.7%

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

      2. div-inv 72.7%

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

      3. add-sqr-sqrt 72.7%

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

      4. hypot-1-def 72.7%

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

      5. *-commutative 72.7%

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

      6. sqrt-prod 72.6%

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

      7. unpow2 72.6%

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

      8. sqrt-prod 48.4%

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

      9. add-sqr-sqrt 99.0%

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

   4. Applied egg-rr 99.0%

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

      1. associate-*r/ 99.0%

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

      2. *-rgt-identity 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in t around inf 50.9%

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

      1. *-commutative 50.9%

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

      2. unpow2 50.9%

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

      3. unpow2 50.9%

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

      4. times-frac 59.9%

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

      5. unpow2 59.9%

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

   9. Simplified 59.9%

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

   10. Taylor expanded in Om around 0 59.9%

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

       1. clear-num 59.5%

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

       2. associate-*l/ 59.6%

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

       3. inv-pow 59.6%

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

       4. *-commutative 59.6%

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

       5. clear-num 59.5%

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

       6. un-div-inv 59.6%

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

   12. Applied egg-rr 59.6%

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

       1. unpow-1 59.6%

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

       2. associate-/r/ 59.6%

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

   14. Simplified 59.6%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-149} \lor \neg \left(\ell \leq 2.9 \cdot 10^{-36}\right):\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{t \cdot \frac{\sqrt{2}}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 31.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double t, double l, double Om, double Omc) {
	return asin((l / (t * sqrt(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((l / (t * sqrt(2.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.8%

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

   1. sqrt-div 80.7%

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

   2. div-inv 80.7%

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

   3. add-sqr-sqrt 80.7%

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

   4. hypot-1-def 80.7%

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

   5. *-commutative 80.7%

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

   6. sqrt-prod 80.7%

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

   7. unpow2 80.7%

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

   8. sqrt-prod 56.3%

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

   9. add-sqr-sqrt 98.9%

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

4. Applied egg-rr 98.9%

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

   1. associate-*r/ 98.9%

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

   2. *-rgt-identity 98.9%

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

6. Simplified 98.9%

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

7. Taylor expanded in t around inf 30.4%

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

   1. *-commutative 30.4%

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

   2. unpow2 30.4%

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

   3. unpow2 30.4%

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

   4. times-frac 34.4%

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

   5. unpow2 34.4%

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

9. Simplified 34.4%

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

10. Taylor expanded in Om around 0 34.3%

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

11. Final simplification 34.3%

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

Reproduce

herbie shell --seed 2024017 
(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)))))))