Toniolo and Linder, Equation (2)

Percentage Accurate: 83.6% → 83.6%

Time: 16.7s

Alternatives: 11

Speedup: 1.0×

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: 83.6% 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: 83.6% 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]

Derivation

1. Initial program 81.7%

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

Alternative 2: 62.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \leq 2 \cdot 10^{-159}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{2}{\ell \cdot \ell}, t, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{Omc - Om}{Omc} \cdot \frac{Om + Omc}{Omc}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<=
      (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t l) 2.0))))
      2e-159)
   (asin (sqrt (/ 1.0 (fma (/ 2.0 (* l l)) t 1.0))))
   (asin (sqrt (* (/ (- Omc Om) Omc) (/ (+ Om Omc) Omc))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t / l), 2.0)))) <= 2e-159) {
		tmp = asin(sqrt((1.0 / fma((2.0 / (l * l)), t, 1.0))));
	} else {
		tmp = asin(sqrt((((Omc - Om) / Omc) * ((Om + Omc) / Omc))));
	}
	return tmp;
}

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t / l) ^ 2.0)))) <= 2e-159)
		tmp = asin(sqrt(Float64(1.0 / fma(Float64(2.0 / Float64(l * l)), t, 1.0))));
	else
		tmp = asin(sqrt(Float64(Float64(Float64(Omc - Om) / Omc) * Float64(Float64(Om + Omc) / Omc))));
	end
	return tmp
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[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], 2e-159], N[ArcSin[N[Sqrt[N[(1.0 / N[(N[(2.0 / N[(l * l), $MachinePrecision]), $MachinePrecision] * t + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(N[(Omc - Om), $MachinePrecision] / Omc), $MachinePrecision] * N[(N[(Om + Omc), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))) < 1.99999999999999998e-159

   1. Initial program 58.0%

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

   3. Taylor expanded in Om around 0

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

   5. Simplified 58.0%

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

      1. lift-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

   7. Applied egg-rr 53.4%

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

   8. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 34.2

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

   10. Simplified 34.2%

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

   if 1.99999999999999998e-159 < (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))

   1. Initial program 97.5%

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

   4. Applied egg-rr 73.6%

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

   5. Taylor expanded in Omc around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 37.5

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

   7. Simplified 37.5%

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

   8. Taylor expanded in t around 0

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

   10. Simplified 37.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift--.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-/l/ N/A

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

       6. lift--.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. difference-of-squares N/A

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

       10. *-commutative N/A

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

       11. neg-mul-1 N/A

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

       12. lift-*.f64 N/A

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

       13. distribute-lft-neg-in N/A

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

       14. times-frac N/A

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

       15. lower-*.f64 N/A

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

       16. lower-/.f64 N/A

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

       17. lower--.f64 N/A

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

       18. lower-neg.f64 N/A

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

       19. lower-/.f64 N/A

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

       20. lower-+.f64 78.7

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

   12. Applied egg-rr 78.7%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \leq 2 \cdot 10^{-159}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{2}{\ell \cdot \ell}, t, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{Omc - Om}{Omc} \cdot \frac{Om + Omc}{Omc}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \leq 0:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{\mathsf{fma}\left(t \cdot t, \frac{t}{\ell \cdot \ell}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<=
      (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t l) 2.0))))
      0.0)
   (asin (/ 1.0 (fma (* t t) (/ t (* l l)) 1.0)))
   (asin 1.0)))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t / l), 2.0)))) <= 0.0) {
		tmp = asin((1.0 / fma((t * t), (t / (l * l)), 1.0)));
	} else {
		tmp = asin(1.0);
	}
	return tmp;
}

Julia

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

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[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], 0.0], N[ArcSin[N[(1.0 / N[(N[(t * t), $MachinePrecision] * N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))) < 0.0

   1. Initial program 44.6%

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

   3. Taylor expanded in Om around 0

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

   5. Simplified 44.6%

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

   7. Applied egg-rr 44.6%

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

   8. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 44.6

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

   10. Simplified 44.6%

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

   11. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 44.6

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

   13. Simplified 44.6%

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

   if 0.0 < (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))

   1. Initial program 97.7%

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

   3. Taylor expanded in Om around 0

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

   5. Simplified 96.7%

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

   6. Taylor expanded in t around 0

      \[\leadsto \sin^{-1} \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 67.4%

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

Alternative 4: 83.6% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (asin
  (sqrt
   (/
    (- 1.0 (pow (/ Om Omc) 2.0))
    (+ 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 - pow((Om / Omc), 2.0)) / (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) ** 2.0d0)) / (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 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * (1.0 / ((l / t) * (l / t))))))));
}

Python

def code(t, l, Om, Omc):
	return math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (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(Om / Omc) ^ 2.0)) / 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) ^ 2.0)) / (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[Power[N[(Om / Omc), $MachinePrecision], 2.0], $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 81.7%

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

   1. lift-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. clear-num N/A

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

   6. clear-num N/A

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

   7. frac-times N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 81.7

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

4. Applied egg-rr 81.7%

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

Alternative 5: 62.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 7.6 \cdot 10^{-125}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{Omc - Om}{Omc} \cdot \frac{Om + Omc}{Omc}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2 \cdot t, \frac{t}{\ell \cdot \ell}, 1\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= t 7.6e-125)
   (asin (sqrt (* (/ (- Omc Om) Omc) (/ (+ Om Omc) Omc))))
   (asin (sqrt (/ 1.0 (fma (* 2.0 t) (/ t (* l l)) 1.0))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (t <= 7.6e-125) {
		tmp = asin(sqrt((((Omc - Om) / Omc) * ((Om + Omc) / Omc))));
	} else {
		tmp = asin(sqrt((1.0 / fma((2.0 * t), (t / (l * l)), 1.0))));
	}
	return tmp;
}

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (t <= 7.6e-125)
		tmp = asin(sqrt(Float64(Float64(Float64(Omc - Om) / Omc) * Float64(Float64(Om + Omc) / Omc))));
	else
		tmp = asin(sqrt(Float64(1.0 / fma(Float64(2.0 * t), Float64(t / Float64(l * l)), 1.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[t, 7.6e-125], N[ArcSin[N[Sqrt[N[(N[(N[(Omc - Om), $MachinePrecision] / Omc), $MachinePrecision] * N[(N[(Om + Omc), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 / N[(N[(2.0 * t), $MachinePrecision] * N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 7.6000000000000002e-125

   1. Initial program 85.4%

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

   4. Applied egg-rr 64.8%

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

   5. Taylor expanded in Omc around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 34.4

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

   7. Simplified 34.4%

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

   8. Taylor expanded in t around 0

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

   10. Simplified 28.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift--.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-/l/ N/A

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

       6. lift--.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. difference-of-squares N/A

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

       10. *-commutative N/A

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

       11. neg-mul-1 N/A

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

       12. lift-*.f64 N/A

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

       13. distribute-lft-neg-in N/A

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

       14. times-frac N/A

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

       15. lower-*.f64 N/A

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

       16. lower-/.f64 N/A

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

       17. lower--.f64 N/A

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

       18. lower-neg.f64 N/A

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

       19. lower-/.f64 N/A

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

       20. lower-+.f64 57.9

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

   12. Applied egg-rr 57.9%

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

   if 7.6000000000000002e-125 < t

   1. Initial program 76.4%

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

   3. Taylor expanded in Om around 0

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

   5. Simplified 75.8%

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

      1. lift-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-/r* N/A

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

      18. lift-*.f64 N/A

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

      19. lower-/.f64 69.5

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

   7. Applied egg-rr 69.5%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.6 \cdot 10^{-125}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{Omc - Om}{Omc} \cdot \frac{Om + Omc}{Omc}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2 \cdot t, \frac{t}{\ell \cdot \ell}, 1\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

3. Taylor expanded in Om around 0

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

5. Simplified 81.0%

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

   1. lift-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. +-commutative N/A

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

7. Applied egg-rr 81.0%

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

8. Final simplification 81.0%

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

Alternative 7: 79.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

3. Taylor expanded in Om around 0

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

5. Simplified 81.0%

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

   1. lift-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. +-commutative N/A

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

7. Applied egg-rr 78.8%

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

Alternative 8: 79.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

3. Taylor expanded in Om around 0

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

5. Simplified 81.0%

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

   1. lift-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. +-commutative N/A

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

7. Applied egg-rr 78.8%

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

   1. lift-/.f64 N/A

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

   2. associate-/r* N/A

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

   3. div-inv N/A

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

   4. lift-/.f64 N/A

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

   5. clear-num N/A

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

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. times-frac N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 78.8

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

9. Applied egg-rr 78.8%

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

Alternative 9: 55.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{-43}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{2}{\ell \cdot \ell}, t, 1\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= t 1.3e-43)
   (asin 1.0)
   (asin (sqrt (/ 1.0 (fma (/ 2.0 (* l l)) t 1.0))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (t <= 1.3e-43) {
		tmp = asin(1.0);
	} else {
		tmp = asin(sqrt((1.0 / fma((2.0 / (l * l)), t, 1.0))));
	}
	return tmp;
}

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (t <= 1.3e-43)
		tmp = asin(1.0);
	else
		tmp = asin(sqrt(Float64(1.0 / fma(Float64(2.0 / Float64(l * l)), t, 1.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[t, 1.3e-43], N[ArcSin[1.0], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 / N[(N[(2.0 / N[(l * l), $MachinePrecision]), $MachinePrecision] * t + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.3e-43

   1. Initial program 85.6%

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

   3. Taylor expanded in Om around 0

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

   5. Simplified 84.9%

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

   6. Taylor expanded in t around 0

      \[\leadsto \sin^{-1} \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 57.6%

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

   if 1.3e-43 < t

   1. Initial program 72.9%

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

   3. Taylor expanded in Om around 0

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

   5. Simplified 72.1%

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

      1. lift-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

   7. Applied egg-rr 72.1%

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

   8. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 51.1

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

   10. Simplified 51.1%

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

Alternative 10: 55.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{+166}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(t \cdot t\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) 5e+166) (asin 1.0) (asin (/ (* l (* l l)) (* t (* t t))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= 5e+166) {
		tmp = asin(1.0);
	} else {
		tmp = asin(((l * (l * l)) / (t * (t * 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) <= 5d+166) then
        tmp = asin(1.0d0)
    else
        tmp = asin(((l * (l * l)) / (t * (t * 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) <= 5e+166) {
		tmp = Math.asin(1.0);
	} else {
		tmp = Math.asin(((l * (l * l)) / (t * (t * t))));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= 5e+166:
		tmp = math.asin(1.0)
	else:
		tmp = math.asin(((l * (l * l)) / (t * (t * t))))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= 5e+166)
		tmp = asin(1.0);
	else
		tmp = asin(Float64(Float64(l * Float64(l * l)) / Float64(t * Float64(t * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= 5e+166)
		tmp = asin(1.0);
	else
		tmp = asin(((l * (l * l)) / (t * (t * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], 5e+166], N[ArcSin[1.0], $MachinePrecision], N[ArcSin[N[(N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(t * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 5.0000000000000002e166

   1. Initial program 86.7%

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

   3. Taylor expanded in Om around 0

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

   5. Simplified 85.8%

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

   6. Taylor expanded in t around 0

      \[\leadsto \sin^{-1} \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 54.6%

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

   if 5.0000000000000002e166 < (/.f64 t l)

   1. Initial program 47.1%

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

   3. Taylor expanded in Om around 0

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

   5. Simplified 47.1%

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

   7. Applied egg-rr 47.1%

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

   8. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 47.1

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

   10. Simplified 47.1%

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

   11. Taylor expanded in t around inf

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

       1. lower-/.f64 N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. cube-mult N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 47.1

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

   13. Simplified 47.1%

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

Alternative 11: 49.7% accurate, 3.5× 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 81.7%

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

3. Taylor expanded in Om around 0

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

5. Simplified 81.0%

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

6. Taylor expanded in t around 0

   \[\leadsto \sin^{-1} \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 48.2%

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

Reproduce

herbie shell --seed 2024214 
(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)))))))