Toniolo and Linder, Equation (2)

Percentage Accurate: 83.4% → 98.5%

Time: 19.3s

Alternatives: 6

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.4% 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: 98.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

   1. sqrt-div 83.7%

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

   2. add-sqr-sqrt 83.7%

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

   3. hypot-1-def 83.7%

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

   4. *-commutative 83.7%

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

   5. sqrt-prod 83.7%

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

   6. sqrt-pow1 99.2%

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

   7. metadata-eval 99.2%

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

   8. pow1 99.2%

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

4. Applied egg-rr 99.2%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 * Math.sqrt(2.0)) / l))));
}

Python

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

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 * sqrt(2.0)) / l))))
end

MATLAB

function tmp = code(t, l, Om, Omc)
	tmp = asin((sqrt((1.0 - ((Om / Omc) / (Omc / Om)))) / hypot(1.0, ((t * sqrt(2.0)) / l))));
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 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 83.7%

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

   1. sqrt-div 83.7%

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

   2. div-inv 83.7%

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

   3. add-sqr-sqrt 83.7%

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

   4. hypot-1-def 83.7%

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

   5. *-commutative 83.7%

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

   6. sqrt-prod 83.7%

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

   7. sqrt-pow1 99.2%

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

   8. metadata-eval 99.2%

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

   9. pow1 99.2%

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

4. Applied egg-rr 99.2%

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

   1. associate-*r/ 99.2%

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

   2. *-rgt-identity 99.2%

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

   3. *-commutative 99.2%

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

   4. associate-*r/ 98.8%

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

6. Simplified 98.8%

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

   1. unpow2 98.8%

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

   2. clear-num 98.8%

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

   3. un-div-inv 98.8%

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

8. Applied egg-rr 98.8%

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

9. Final simplification 98.8%

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

Alternative 3: 97.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

   1. add-exp-log 82.4%

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

   2. +-commutative 82.4%

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

   3. fma-define 82.4%

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

4. Applied egg-rr 82.4%

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

5. Taylor expanded in Om around 0 66.5%

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

   1. +-commutative 66.5%

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

   2. fma-define 66.5%

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

   3. unpow2 66.5%

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

   4. unpow2 66.5%

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

   5. times-frac 82.0%

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

   6. unpow2 82.0%

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

7. Simplified 82.0%

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

   1. sqrt-div 82.0%

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

   2. metadata-eval 82.0%

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

9. Applied egg-rr 82.0%

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

    1. unpow2 82.0%

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

    2. times-frac 66.5%

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

    3. unpow2 66.5%

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

    4. unpow2 66.5%

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

    5. fma-define 66.5%

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

    6. +-commutative 66.5%

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

    7. metadata-eval 66.5%

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

    8. rem-square-sqrt 66.5%

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

    9. unpow2 66.5%

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

    10. unpow2 66.5%

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

    11. times-frac 82.0%

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

    12. swap-sqr 82.0%

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

    13. hypot-undefine 95.9%

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

11. Simplified 95.9%

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

    1. *-un-lft-identity 95.9%

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

    2. rem-exp-log 98.6%

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

    3. associate-*r/ 98.3%

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

13. Applied egg-rr 98.3%

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

    1. *-lft-identity 98.3%

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

    2. *-commutative 98.3%

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

    3. associate-/l* 98.6%

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

15. Simplified 98.6%

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

Alternative 4: 55.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{if}\;t \leq 3.8 \cdot 10^{-55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-22}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+83}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{t} \cdot \left(\ell \cdot \sqrt{0.5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (asin (sqrt (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))))
   (if (<= t 3.8e-55)
     t_1
     (if (<= t 2.9e-22)
       (asin (/ l (/ t (sqrt 0.5))))
       (if (<= t 7e+83) t_1 (asin (* (/ 1.0 t) (* l (sqrt 0.5)))))))))

C

double code(double t, double l, double Om, double Omc) {
	double t_1 = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	double tmp;
	if (t <= 3.8e-55) {
		tmp = t_1;
	} else if (t <= 2.9e-22) {
		tmp = asin((l / (t / sqrt(0.5))));
	} else if (t <= 7e+83) {
		tmp = t_1;
	} else {
		tmp = asin(((1.0 / t) * (l * sqrt(0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: t_1
    real(8) :: tmp
    t_1 = asin(sqrt((1.0d0 - ((om / omc) / (omc / om)))))
    if (t <= 3.8d-55) then
        tmp = t_1
    else if (t <= 2.9d-22) then
        tmp = asin((l / (t / sqrt(0.5d0))))
    else if (t <= 7d+83) then
        tmp = t_1
    else
        tmp = asin(((1.0d0 / t) * (l * sqrt(0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double t_1 = Math.asin(Math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	double tmp;
	if (t <= 3.8e-55) {
		tmp = t_1;
	} else if (t <= 2.9e-22) {
		tmp = Math.asin((l / (t / Math.sqrt(0.5))));
	} else if (t <= 7e+83) {
		tmp = t_1;
	} else {
		tmp = Math.asin(((1.0 / t) * (l * Math.sqrt(0.5))));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	t_1 = math.asin(math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))))
	tmp = 0
	if t <= 3.8e-55:
		tmp = t_1
	elif t <= 2.9e-22:
		tmp = math.asin((l / (t / math.sqrt(0.5))))
	elif t <= 7e+83:
		tmp = t_1
	else:
		tmp = math.asin(((1.0 / t) * (l * math.sqrt(0.5))))
	return tmp

Julia

function code(t, l, Om, Omc)
	t_1 = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))))
	tmp = 0.0
	if (t <= 3.8e-55)
		tmp = t_1;
	elseif (t <= 2.9e-22)
		tmp = asin(Float64(l / Float64(t / sqrt(0.5))));
	elseif (t <= 7e+83)
		tmp = t_1;
	else
		tmp = asin(Float64(Float64(1.0 / t) * Float64(l * sqrt(0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	t_1 = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	tmp = 0.0;
	if (t <= 3.8e-55)
		tmp = t_1;
	elseif (t <= 2.9e-22)
		tmp = asin((l / (t / sqrt(0.5))));
	elseif (t <= 7e+83)
		tmp = t_1;
	else
		tmp = asin(((1.0 / t) * (l * sqrt(0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 3.8e-55], t$95$1, If[LessEqual[t, 2.9e-22], N[ArcSin[N[(l / N[(t / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 7e+83], t$95$1, N[ArcSin[N[(N[(1.0 / t), $MachinePrecision] * N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 3.7999999999999997e-55 or 2.9000000000000002e-22 < t < 6.99999999999999954e83

   1. Initial program 87.7%

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

   3. Taylor expanded in t around 0 55.7%

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

      1. unpow2 55.7%

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

      2. unpow2 55.7%

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

      3. times-frac 61.7%

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

      4. unpow2 61.7%

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

   5. Simplified 61.7%

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

      1. unpow2 99.1%

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

      2. clear-num 99.1%

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

      3. un-div-inv 99.1%

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

   7. Applied egg-rr 61.7%

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

   if 3.7999999999999997e-55 < t < 2.9000000000000002e-22

   1. Initial program 52.8%

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

   3. Taylor expanded in t around inf 50.5%

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

      1. *-commutative 50.5%

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

      2. unpow2 50.5%

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

      3. unpow2 50.5%

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

      4. times-frac 50.5%

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

      5. unpow2 50.5%

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

      6. associate-/l* 50.2%

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

   5. Simplified 50.2%

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

      1. unpow2 99.3%

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

      2. clear-num 99.3%

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

      3. un-div-inv 99.3%

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

   7. Applied egg-rr 50.2%

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

   8. Taylor expanded in Om around 0 50.5%

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

      1. associate-/l* 50.2%

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

   10. Simplified 50.2%

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

       1. clear-num 50.2%

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

       2. un-div-inv 50.5%

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

   12. Applied egg-rr 50.5%

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

   if 6.99999999999999954e83 < t

   1. Initial program 72.4%

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

   3. Taylor expanded in t around inf 58.5%

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

      1. *-commutative 58.5%

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

      2. unpow2 58.5%

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

      3. unpow2 58.5%

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

      4. times-frac 60.8%

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

      5. unpow2 60.8%

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

      6. associate-/l* 60.9%

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

   5. Simplified 60.9%

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

      1. unpow2 97.3%

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

      2. clear-num 97.3%

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

      3. un-div-inv 97.3%

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

   7. Applied egg-rr 60.9%

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

   8. Taylor expanded in Om around 0 60.8%

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

      1. associate-/l* 60.9%

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

   10. Simplified 60.9%

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

       1. associate-*r/ 60.8%

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

       2. clear-num 60.7%

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

   12. Applied egg-rr 60.7%

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

       1. associate-/r/ 60.9%

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

   14. Simplified 60.9%

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

Alternative 5: 31.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double t, double l, double Om, double Omc) {
	return asin(((1.0 / t) * (l * sqrt(0.5))));
}

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 / t) * (l * sqrt(0.5d0))))
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	return Math.asin(((1.0 / t) * (l * Math.sqrt(0.5))));
}

Python

def code(t, l, Om, Omc):
	return math.asin(((1.0 / t) * (l * math.sqrt(0.5))))

Julia

function code(t, l, Om, Omc)
	return asin(Float64(Float64(1.0 / t) * Float64(l * sqrt(0.5))))
end

MATLAB

function tmp = code(t, l, Om, Omc)
	tmp = asin(((1.0 / t) * (l * sqrt(0.5))));
end

Wolfram

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

Derivation

1. Initial program 83.7%

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

3. Taylor expanded in t around inf 32.3%

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

   1. *-commutative 32.3%

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

   2. unpow2 32.3%

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

   3. unpow2 32.3%

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

   4. times-frac 33.9%

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

   5. unpow2 33.9%

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

   6. associate-/l* 33.9%

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

5. Simplified 33.9%

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

   1. unpow2 98.8%

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

   2. clear-num 98.8%

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

   3. un-div-inv 98.8%

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

7. Applied egg-rr 33.9%

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

8. Taylor expanded in Om around 0 33.9%

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

   1. associate-/l* 33.9%

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

10. Simplified 33.9%

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

    1. associate-*r/ 33.9%

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

    2. clear-num 33.8%

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

12. Applied egg-rr 33.8%

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

    1. associate-/r/ 33.9%

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

14. Simplified 33.9%

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

Alternative 6: 31.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double t, double l, double Om, double Omc) {
	return asin((l * (sqrt(0.5) / 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((l * (sqrt(0.5d0) / t)))
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	return Math.asin((l * (Math.sqrt(0.5) / t)));
}

Python

def code(t, l, Om, Omc):
	return math.asin((l * (math.sqrt(0.5) / t)))

Julia

function code(t, l, Om, Omc)
	return asin(Float64(l * Float64(sqrt(0.5) / t)))
end

MATLAB

function tmp = code(t, l, Om, Omc)
	tmp = asin((l * (sqrt(0.5) / t)));
end

Wolfram

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

Derivation

1. Initial program 83.7%

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

3. Taylor expanded in t around inf 32.3%

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

   1. *-commutative 32.3%

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

   2. unpow2 32.3%

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

   3. unpow2 32.3%

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

   4. times-frac 33.9%

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

   5. unpow2 33.9%

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

   6. associate-/l* 33.9%

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

5. Simplified 33.9%

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

   1. unpow2 98.8%

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

   2. clear-num 98.8%

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

   3. un-div-inv 98.8%

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

7. Applied egg-rr 33.9%

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

8. Taylor expanded in Om around 0 33.9%

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

   1. associate-/l* 33.9%

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

10. Simplified 33.9%

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

Reproduce

herbie shell --seed 2024088 
(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)))))))