Toniolo and Linder, Equation (2)

Percentage Accurate: 83.8% → 98.4%

Time: 18.6s

Alternatives: 11

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 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 83.6%

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

   1. sqrt-div 83.5%

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

   2. add-sqr-sqrt 83.5%

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

   3. hypot-1-def 83.5%

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

   4. *-commutative 83.5%

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

   5. sqrt-prod 83.5%

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

   6. unpow2 83.5%

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

   7. sqrt-prod 59.5%

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

   8. add-sqr-sqrt 98.3%

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

3. Applied egg-rr 98.3%

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

4. Final simplification 98.3%

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

Alternative 2: 97.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.6%

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

2. Taylor expanded in Om around 0 66.2%

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

   1. unpow2 66.2%

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

   2. unpow2 66.2%

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

4. Simplified 66.2%

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

   1. expm1-log1p-u 66.2%

      \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)\right)\right)} \]

   2. expm1-udef 58.3%

      \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} - 1\right)} \]

   3. sqrt-div 58.3%

      \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}}\right)} - 1\right) \]

   4. metadata-eval 58.3%

      \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} - 1\right) \]

   5. +-commutative 58.3%

      \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{2 \cdot \frac{t \cdot t}{\ell \cdot \ell} + 1}}}\right)} - 1\right) \]

   6. times-frac 65.4%

      \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2 \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} + 1}}\right)} - 1\right) \]

   7. unpow2 65.4%

      \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2 \cdot \color{blue}{{\left(\frac{t}{\ell}\right)}^{2}} + 1}}\right)} - 1\right) \]

   8. fma-udef 65.4%

      \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}}\right)} - 1\right) \]

6. Applied egg-rr 65.4%

   \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)} - 1\right)} \]
7. Step-by-step derivation

   1. expm1-def 82.9%

      \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)} \]

   2. expm1-log1p 82.9%

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

   3. fma-udef 82.9%

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

   4. rem-square-sqrt 82.8%

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

   5. unpow2 82.8%

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

   6. swap-sqr 82.9%

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

   7. *-commutative 82.9%

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

   8. *-commutative 82.9%

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

   9. +-commutative 82.9%

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

   10. hypot-1-def 97.7%

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

   11. *-commutative 97.7%

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

8. Simplified 97.7%

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

9. Final simplification 97.7%

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

Alternative 3: 97.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \sqrt{2}\\ \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+157}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t_1}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+73}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (* t (sqrt 2.0))))
   (if (<= (/ t l) -4e+157)
     (asin (/ (- l) t_1))
     (if (<= (/ t l) 2e+73)
       (asin (sqrt (/ 1.0 (+ 1.0 (* 2.0 (* (/ t l) (/ t l)))))))
       (asin (/ l t_1))))))

C

double code(double t, double l, double Om, double Omc) {
	double t_1 = t * sqrt(2.0);
	double tmp;
	if ((t / l) <= -4e+157) {
		tmp = asin((-l / t_1));
	} else if ((t / l) <= 2e+73) {
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} else {
		tmp = asin((l / t_1));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * sqrt(2.0d0)
    if ((t / l) <= (-4d+157)) then
        tmp = asin((-l / t_1))
    else if ((t / l) <= 2d+73) then
        tmp = asin(sqrt((1.0d0 / (1.0d0 + (2.0d0 * ((t / l) * (t / l)))))))
    else
        tmp = asin((l / t_1))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double t_1 = t * Math.sqrt(2.0);
	double tmp;
	if ((t / l) <= -4e+157) {
		tmp = Math.asin((-l / t_1));
	} else if ((t / l) <= 2e+73) {
		tmp = Math.asin(Math.sqrt((1.0 / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} else {
		tmp = Math.asin((l / t_1));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	t_1 = t * math.sqrt(2.0)
	tmp = 0
	if (t / l) <= -4e+157:
		tmp = math.asin((-l / t_1))
	elif (t / l) <= 2e+73:
		tmp = math.asin(math.sqrt((1.0 / (1.0 + (2.0 * ((t / l) * (t / l)))))))
	else:
		tmp = math.asin((l / t_1))
	return tmp

Julia

function code(t, l, Om, Omc)
	t_1 = Float64(t * sqrt(2.0))
	tmp = 0.0
	if (Float64(t / l) <= -4e+157)
		tmp = asin(Float64(Float64(-l) / t_1));
	elseif (Float64(t / l) <= 2e+73)
		tmp = asin(sqrt(Float64(1.0 / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) * Float64(t / l)))))));
	else
		tmp = asin(Float64(l / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	t_1 = t * sqrt(2.0);
	tmp = 0.0;
	if ((t / l) <= -4e+157)
		tmp = asin((-l / t_1));
	elseif ((t / l) <= 2e+73)
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	else
		tmp = asin((l / t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -4e+157], N[ArcSin[N[((-l) / t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 2e+73], N[ArcSin[N[Sqrt[N[(1.0 / N[(1.0 + N[(2.0 * N[(N[(t / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l / t$95$1), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -3.99999999999999993e157

   1. Initial program 51.2%

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

   2. Taylor expanded in Om around 0 51.2%

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

      1. unpow2 51.2%

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

      2. unpow2 51.2%

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

   4. Simplified 51.2%

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

      1. expm1-log1p-u 51.2%

         \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)\right)\right)} \]

      2. expm1-udef 51.2%

         \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} - 1\right)} \]

      3. sqrt-div 51.2%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}}\right)} - 1\right) \]

      4. metadata-eval 51.2%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} - 1\right) \]

      5. +-commutative 51.2%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{2 \cdot \frac{t \cdot t}{\ell \cdot \ell} + 1}}}\right)} - 1\right) \]

      6. times-frac 51.2%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2 \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} + 1}}\right)} - 1\right) \]

      7. unpow2 51.2%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2 \cdot \color{blue}{{\left(\frac{t}{\ell}\right)}^{2}} + 1}}\right)} - 1\right) \]

      8. fma-udef 51.2%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}}\right)} - 1\right) \]

   6. Applied egg-rr 51.2%

      \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 51.2%

         \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)} \]

      2. expm1-log1p 51.2%

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

      3. fma-udef 51.2%

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

      4. rem-square-sqrt 51.2%

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

      5. unpow2 51.2%

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

      6. swap-sqr 51.2%

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

      7. *-commutative 51.2%

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

      8. *-commutative 51.2%

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

      9. +-commutative 51.2%

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

      10. hypot-1-def 97.5%

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

      11. *-commutative 97.5%

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

   8. Simplified 97.5%

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

   9. Taylor expanded in t around -inf 99.7%

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

       1. associate-*r/ 99.7%

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

       2. mul-1-neg 99.7%

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

   11. Simplified 99.7%

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

   if -3.99999999999999993e157 < (/.f64 t l) < 1.99999999999999997e73

   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. Taylor expanded in Om around 0 76.6%

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

      1. unpow2 76.6%

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

      2. unpow2 76.6%

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

   4. Simplified 76.6%

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

      1. times-frac 96.9%

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

   6. Applied egg-rr 96.9%

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

   if 1.99999999999999997e73 < (/.f64 t l)

   1. Initial program 54.5%

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

   2. Taylor expanded in Om around 0 40.9%

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

      1. unpow2 40.9%

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

      2. unpow2 40.9%

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

   4. Simplified 40.9%

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

      1. expm1-log1p-u 40.9%

         \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)\right)\right)} \]

      2. expm1-udef 28.9%

         \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} - 1\right)} \]

      3. sqrt-div 28.9%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}}\right)} - 1\right) \]

      4. metadata-eval 28.9%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} - 1\right) \]

      5. +-commutative 28.9%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{2 \cdot \frac{t \cdot t}{\ell \cdot \ell} + 1}}}\right)} - 1\right) \]

      6. times-frac 28.9%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2 \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} + 1}}\right)} - 1\right) \]

      7. unpow2 28.9%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2 \cdot \color{blue}{{\left(\frac{t}{\ell}\right)}^{2}} + 1}}\right)} - 1\right) \]

      8. fma-udef 28.9%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}}\right)} - 1\right) \]

   6. Applied egg-rr 28.9%

      \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 54.4%

         \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)} \]

      2. expm1-log1p 54.4%

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

      3. fma-udef 54.4%

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

      4. rem-square-sqrt 54.3%

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

      5. unpow2 54.3%

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

      6. swap-sqr 54.4%

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

      7. *-commutative 54.4%

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

      8. *-commutative 54.4%

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

      9. +-commutative 54.4%

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

      10. hypot-1-def 98.9%

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

      11. *-commutative 98.9%

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

   8. Simplified 98.9%

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

   9. Taylor expanded in t around inf 99.6%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+157}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+73}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \end{array} \]

Alternative 4: 97.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \sqrt{2}\\ \mathbf{if}\;\frac{t}{\ell} \leq -10:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t_1}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0001:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (* t (sqrt 2.0))))
   (if (<= (/ t l) -10.0)
     (asin (/ (- l) t_1))
     (if (<= (/ t l) 0.0001)
       (asin (sqrt (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))
       (asin (/ l t_1))))))

C

double code(double t, double l, double Om, double Omc) {
	double t_1 = t * sqrt(2.0);
	double tmp;
	if ((t / l) <= -10.0) {
		tmp = asin((-l / t_1));
	} else if ((t / l) <= 0.0001) {
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} else {
		tmp = asin((l / t_1));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * sqrt(2.0d0)
    if ((t / l) <= (-10.0d0)) then
        tmp = asin((-l / t_1))
    else if ((t / l) <= 0.0001d0) then
        tmp = asin(sqrt((1.0d0 - ((om / omc) / (omc / om)))))
    else
        tmp = asin((l / t_1))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double t_1 = t * Math.sqrt(2.0);
	double tmp;
	if ((t / l) <= -10.0) {
		tmp = Math.asin((-l / t_1));
	} else if ((t / l) <= 0.0001) {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} else {
		tmp = Math.asin((l / t_1));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	t_1 = t * math.sqrt(2.0)
	tmp = 0
	if (t / l) <= -10.0:
		tmp = math.asin((-l / t_1))
	elif (t / l) <= 0.0001:
		tmp = math.asin(math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))))
	else:
		tmp = math.asin((l / t_1))
	return tmp

Julia

function code(t, l, Om, Omc)
	t_1 = Float64(t * sqrt(2.0))
	tmp = 0.0
	if (Float64(t / l) <= -10.0)
		tmp = asin(Float64(Float64(-l) / t_1));
	elseif (Float64(t / l) <= 0.0001)
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))));
	else
		tmp = asin(Float64(l / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	t_1 = t * sqrt(2.0);
	tmp = 0.0;
	if ((t / l) <= -10.0)
		tmp = asin((-l / t_1));
	elseif ((t / l) <= 0.0001)
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	else
		tmp = asin((l / t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -10.0], N[ArcSin[N[((-l) / t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 0.0001], N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l / t$95$1), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -10

   1. Initial program 71.3%

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

   2. Taylor expanded in Om around 0 51.4%

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

      1. unpow2 51.4%

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

      2. unpow2 51.4%

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

   4. Simplified 51.4%

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

      1. expm1-log1p-u 51.4%

         \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)\right)\right)} \]

      2. expm1-udef 35.3%

         \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} - 1\right)} \]

      3. sqrt-div 35.3%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}}\right)} - 1\right) \]

      4. metadata-eval 35.3%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} - 1\right) \]

      5. +-commutative 35.3%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{2 \cdot \frac{t \cdot t}{\ell \cdot \ell} + 1}}}\right)} - 1\right) \]

      6. times-frac 36.9%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2 \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} + 1}}\right)} - 1\right) \]

      7. unpow2 36.9%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2 \cdot \color{blue}{{\left(\frac{t}{\ell}\right)}^{2}} + 1}}\right)} - 1\right) \]

      8. fma-udef 36.9%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}}\right)} - 1\right) \]

   6. Applied egg-rr 36.9%

      \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 71.3%

         \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)} \]

      2. expm1-log1p 71.3%

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

      3. fma-udef 71.3%

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

      4. rem-square-sqrt 71.1%

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

      5. unpow2 71.1%

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

      6. swap-sqr 71.3%

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

      7. *-commutative 71.3%

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

      8. *-commutative 71.3%

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

      9. +-commutative 71.3%

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

      10. hypot-1-def 98.4%

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

      11. *-commutative 98.4%

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

   8. Simplified 98.4%

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

   9. Taylor expanded in t around -inf 95.4%

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

       1. associate-*r/ 95.4%

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

       2. mul-1-neg 95.4%

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

   11. Simplified 95.4%

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

   if -10 < (/.f64 t l) < 1.00000000000000005e-4

   1. Initial program 97.9%

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

   2. Taylor expanded in t around 0 87.5%

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

      1. unpow2 87.5%

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

      2. unpow2 87.5%

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

      3. times-frac 96.6%

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

      4. unpow2 96.6%

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

   4. Simplified 96.6%

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

      1. unpow2 96.6%

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

      2. clear-num 96.6%

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

      3. un-div-inv 96.6%

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

   6. Applied egg-rr 96.6%

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

   if 1.00000000000000005e-4 < (/.f64 t l)

   1. Initial program 64.6%

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

   2. Taylor expanded in Om around 0 40.9%

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

      1. unpow2 40.9%

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

      2. unpow2 40.9%

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

   4. Simplified 40.9%

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

      1. expm1-log1p-u 40.9%

         \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)\right)\right)} \]

      2. expm1-udef 23.4%

         \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} - 1\right)} \]

      3. sqrt-div 23.4%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}}\right)} - 1\right) \]

      4. metadata-eval 23.4%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} - 1\right) \]

      5. +-commutative 23.4%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{2 \cdot \frac{t \cdot t}{\ell \cdot \ell} + 1}}}\right)} - 1\right) \]

      6. times-frac 25.0%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2 \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} + 1}}\right)} - 1\right) \]

      7. unpow2 25.0%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2 \cdot \color{blue}{{\left(\frac{t}{\ell}\right)}^{2}} + 1}}\right)} - 1\right) \]

      8. fma-udef 25.0%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}}\right)} - 1\right) \]

   6. Applied egg-rr 25.0%

      \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 64.5%

         \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)} \]

      2. expm1-log1p 64.5%

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

      3. fma-udef 64.5%

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

      4. rem-square-sqrt 64.2%

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

      5. unpow2 64.2%

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

      6. swap-sqr 64.4%

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

      7. *-commutative 64.4%

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

      8. *-commutative 64.4%

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

      9. +-commutative 64.4%

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

      10. hypot-1-def 99.0%

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

      11. *-commutative 99.0%

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

   8. Simplified 99.0%

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

   9. Taylor expanded in t around inf 98.7%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -10:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0001:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \end{array} \]

Alternative 5: 97.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -10:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{-\ell}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0001:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -10.0)
   (asin (* (sqrt 0.5) (/ (- l) t)))
   (if (<= (/ t l) 0.0001)
     (asin (- 1.0 (pow (/ t l) 2.0)))
     (asin (/ l (* t (sqrt 2.0)))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -10.0) {
		tmp = asin((sqrt(0.5) * (-l / t)));
	} else if ((t / l) <= 0.0001) {
		tmp = asin((1.0 - pow((t / l), 2.0)));
	} else {
		tmp = asin((l / (t * sqrt(2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t / l) <= (-10.0d0)) then
        tmp = asin((sqrt(0.5d0) * (-l / t)))
    else if ((t / l) <= 0.0001d0) then
        tmp = asin((1.0d0 - ((t / l) ** 2.0d0)))
    else
        tmp = asin((l / (t * sqrt(2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -10.0) {
		tmp = Math.asin((Math.sqrt(0.5) * (-l / t)));
	} else if ((t / l) <= 0.0001) {
		tmp = Math.asin((1.0 - Math.pow((t / l), 2.0)));
	} else {
		tmp = Math.asin((l / (t * Math.sqrt(2.0))));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -10.0:
		tmp = math.asin((math.sqrt(0.5) * (-l / t)))
	elif (t / l) <= 0.0001:
		tmp = math.asin((1.0 - math.pow((t / l), 2.0)))
	else:
		tmp = math.asin((l / (t * math.sqrt(2.0))))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -10.0)
		tmp = asin(Float64(sqrt(0.5) * Float64(Float64(-l) / t)));
	elseif (Float64(t / l) <= 0.0001)
		tmp = asin(Float64(1.0 - (Float64(t / l) ^ 2.0)));
	else
		tmp = asin(Float64(l / Float64(t * sqrt(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -10.0)
		tmp = asin((sqrt(0.5) * (-l / t)));
	elseif ((t / l) <= 0.0001)
		tmp = asin((1.0 - ((t / l) ^ 2.0)));
	else
		tmp = asin((l / (t * sqrt(2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -10.0], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] * N[((-l) / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 0.0001], N[ArcSin[N[(1.0 - N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l / N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -10

   1. Initial program 71.3%

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

   2. Taylor expanded in Om around 0 51.4%

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

      1. unpow2 51.4%

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

      2. unpow2 51.4%

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

   4. Simplified 51.4%

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

   5. Taylor expanded in t around -inf 95.2%

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

      1. mul-1-neg 95.2%

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

      2. *-commutative 95.2%

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

      3. associate-*l/ 95.2%

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

      4. *-commutative 95.2%

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

   7. Simplified 95.2%

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

   if -10 < (/.f64 t l) < 1.00000000000000005e-4

   1. Initial program 97.9%

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

   2. Taylor expanded in Om around 0 84.8%

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

      1. unpow2 84.8%

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

      2. unpow2 84.8%

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

   4. Simplified 84.8%

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

   5. Taylor expanded in t around 0 84.8%

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

      1. mul-1-neg 84.8%

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

      2. unpow2 84.8%

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

      3. unpow2 84.8%

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

      4. times-frac 96.4%

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

      5. unpow2 96.4%

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

      6. unsub-neg 96.4%

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

   7. Simplified 96.4%

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

   if 1.00000000000000005e-4 < (/.f64 t l)

   1. Initial program 64.6%

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

   2. Taylor expanded in Om around 0 40.9%

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

      1. unpow2 40.9%

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

      2. unpow2 40.9%

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

   4. Simplified 40.9%

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

      1. expm1-log1p-u 40.9%

         \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)\right)\right)} \]

      2. expm1-udef 23.4%

         \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} - 1\right)} \]

      3. sqrt-div 23.4%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}}\right)} - 1\right) \]

      4. metadata-eval 23.4%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} - 1\right) \]

      5. +-commutative 23.4%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{2 \cdot \frac{t \cdot t}{\ell \cdot \ell} + 1}}}\right)} - 1\right) \]

      6. times-frac 25.0%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2 \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} + 1}}\right)} - 1\right) \]

      7. unpow2 25.0%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2 \cdot \color{blue}{{\left(\frac{t}{\ell}\right)}^{2}} + 1}}\right)} - 1\right) \]

      8. fma-udef 25.0%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}}\right)} - 1\right) \]

   6. Applied egg-rr 25.0%

      \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 64.5%

         \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)} \]

      2. expm1-log1p 64.5%

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

      3. fma-udef 64.5%

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

      4. rem-square-sqrt 64.2%

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

      5. unpow2 64.2%

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

      6. swap-sqr 64.4%

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

      7. *-commutative 64.4%

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

      8. *-commutative 64.4%

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

      9. +-commutative 64.4%

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

      10. hypot-1-def 99.0%

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

      11. *-commutative 99.0%

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

   8. Simplified 99.0%

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

   9. Taylor expanded in t around inf 98.7%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -10:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{-\ell}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0001:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \end{array} \]

Alternative 6: 97.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \sqrt{2}\\ \mathbf{if}\;\frac{t}{\ell} \leq -10:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t_1}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0001:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (* t (sqrt 2.0))))
   (if (<= (/ t l) -10.0)
     (asin (/ (- l) t_1))
     (if (<= (/ t l) 0.0001)
       (asin (- 1.0 (pow (/ t l) 2.0)))
       (asin (/ l t_1))))))

C

double code(double t, double l, double Om, double Omc) {
	double t_1 = t * sqrt(2.0);
	double tmp;
	if ((t / l) <= -10.0) {
		tmp = asin((-l / t_1));
	} else if ((t / l) <= 0.0001) {
		tmp = asin((1.0 - pow((t / l), 2.0)));
	} else {
		tmp = asin((l / t_1));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * sqrt(2.0d0)
    if ((t / l) <= (-10.0d0)) then
        tmp = asin((-l / t_1))
    else if ((t / l) <= 0.0001d0) then
        tmp = asin((1.0d0 - ((t / l) ** 2.0d0)))
    else
        tmp = asin((l / t_1))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double t_1 = t * Math.sqrt(2.0);
	double tmp;
	if ((t / l) <= -10.0) {
		tmp = Math.asin((-l / t_1));
	} else if ((t / l) <= 0.0001) {
		tmp = Math.asin((1.0 - Math.pow((t / l), 2.0)));
	} else {
		tmp = Math.asin((l / t_1));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	t_1 = t * math.sqrt(2.0)
	tmp = 0
	if (t / l) <= -10.0:
		tmp = math.asin((-l / t_1))
	elif (t / l) <= 0.0001:
		tmp = math.asin((1.0 - math.pow((t / l), 2.0)))
	else:
		tmp = math.asin((l / t_1))
	return tmp

Julia

function code(t, l, Om, Omc)
	t_1 = Float64(t * sqrt(2.0))
	tmp = 0.0
	if (Float64(t / l) <= -10.0)
		tmp = asin(Float64(Float64(-l) / t_1));
	elseif (Float64(t / l) <= 0.0001)
		tmp = asin(Float64(1.0 - (Float64(t / l) ^ 2.0)));
	else
		tmp = asin(Float64(l / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	t_1 = t * sqrt(2.0);
	tmp = 0.0;
	if ((t / l) <= -10.0)
		tmp = asin((-l / t_1));
	elseif ((t / l) <= 0.0001)
		tmp = asin((1.0 - ((t / l) ^ 2.0)));
	else
		tmp = asin((l / t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -10.0], N[ArcSin[N[((-l) / t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 0.0001], N[ArcSin[N[(1.0 - N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l / t$95$1), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -10

   1. Initial program 71.3%

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

   2. Taylor expanded in Om around 0 51.4%

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

      1. unpow2 51.4%

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

      2. unpow2 51.4%

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

   4. Simplified 51.4%

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

      1. expm1-log1p-u 51.4%

         \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)\right)\right)} \]

      2. expm1-udef 35.3%

         \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} - 1\right)} \]

      3. sqrt-div 35.3%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}}\right)} - 1\right) \]

      4. metadata-eval 35.3%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} - 1\right) \]

      5. +-commutative 35.3%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{2 \cdot \frac{t \cdot t}{\ell \cdot \ell} + 1}}}\right)} - 1\right) \]

      6. times-frac 36.9%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2 \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} + 1}}\right)} - 1\right) \]

      7. unpow2 36.9%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2 \cdot \color{blue}{{\left(\frac{t}{\ell}\right)}^{2}} + 1}}\right)} - 1\right) \]

      8. fma-udef 36.9%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}}\right)} - 1\right) \]

   6. Applied egg-rr 36.9%

      \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 71.3%

         \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)} \]

      2. expm1-log1p 71.3%

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

      3. fma-udef 71.3%

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

      4. rem-square-sqrt 71.1%

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

      5. unpow2 71.1%

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

      6. swap-sqr 71.3%

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

      7. *-commutative 71.3%

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

      8. *-commutative 71.3%

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

      9. +-commutative 71.3%

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

      10. hypot-1-def 98.4%

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

      11. *-commutative 98.4%

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

   8. Simplified 98.4%

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

   9. Taylor expanded in t around -inf 95.4%

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

       1. associate-*r/ 95.4%

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

       2. mul-1-neg 95.4%

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

   11. Simplified 95.4%

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

   if -10 < (/.f64 t l) < 1.00000000000000005e-4

   1. Initial program 97.9%

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

   2. Taylor expanded in Om around 0 84.8%

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

      1. unpow2 84.8%

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

      2. unpow2 84.8%

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

   4. Simplified 84.8%

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

   5. Taylor expanded in t around 0 84.8%

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

      1. mul-1-neg 84.8%

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

      2. unpow2 84.8%

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

      3. unpow2 84.8%

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

      4. times-frac 96.4%

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

      5. unpow2 96.4%

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

      6. unsub-neg 96.4%

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

   7. Simplified 96.4%

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

   if 1.00000000000000005e-4 < (/.f64 t l)

   1. Initial program 64.6%

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

   2. Taylor expanded in Om around 0 40.9%

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

      1. unpow2 40.9%

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

      2. unpow2 40.9%

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

   4. Simplified 40.9%

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

      1. expm1-log1p-u 40.9%

         \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)\right)\right)} \]

      2. expm1-udef 23.4%

         \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} - 1\right)} \]

      3. sqrt-div 23.4%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}}\right)} - 1\right) \]

      4. metadata-eval 23.4%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} - 1\right) \]

      5. +-commutative 23.4%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{2 \cdot \frac{t \cdot t}{\ell \cdot \ell} + 1}}}\right)} - 1\right) \]

      6. times-frac 25.0%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2 \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} + 1}}\right)} - 1\right) \]

      7. unpow2 25.0%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2 \cdot \color{blue}{{\left(\frac{t}{\ell}\right)}^{2}} + 1}}\right)} - 1\right) \]

      8. fma-udef 25.0%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}}\right)} - 1\right) \]

   6. Applied egg-rr 25.0%

      \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 64.5%

         \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)} \]

      2. expm1-log1p 64.5%

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

      3. fma-udef 64.5%

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

      4. rem-square-sqrt 64.2%

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

      5. unpow2 64.2%

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

      6. swap-sqr 64.4%

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

      7. *-commutative 64.4%

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

      8. *-commutative 64.4%

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

      9. +-commutative 64.4%

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

      10. hypot-1-def 99.0%

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

      11. *-commutative 99.0%

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

   8. Simplified 99.0%

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

   9. Taylor expanded in t around inf 98.7%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -10:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0001:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \end{array} \]

Alternative 7: 80.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+214}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0001:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -2e+214)
   (asin (/ (sqrt 0.5) (/ t l)))
   (if (<= (/ t l) 0.0001) (asin 1.0) (asin (/ l (* t (sqrt 2.0)))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -2e+214) {
		tmp = asin((sqrt(0.5) / (t / l)));
	} else if ((t / l) <= 0.0001) {
		tmp = asin(1.0);
	} else {
		tmp = asin((l / (t * sqrt(2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t / l) <= (-2d+214)) then
        tmp = asin((sqrt(0.5d0) / (t / l)))
    else if ((t / l) <= 0.0001d0) then
        tmp = asin(1.0d0)
    else
        tmp = asin((l / (t * sqrt(2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -2e+214) {
		tmp = Math.asin((Math.sqrt(0.5) / (t / l)));
	} else if ((t / l) <= 0.0001) {
		tmp = Math.asin(1.0);
	} else {
		tmp = Math.asin((l / (t * Math.sqrt(2.0))));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -2e+214:
		tmp = math.asin((math.sqrt(0.5) / (t / l)))
	elif (t / l) <= 0.0001:
		tmp = math.asin(1.0)
	else:
		tmp = math.asin((l / (t * math.sqrt(2.0))))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -2e+214)
		tmp = asin(Float64(sqrt(0.5) / Float64(t / l)));
	elseif (Float64(t / l) <= 0.0001)
		tmp = asin(1.0);
	else
		tmp = asin(Float64(l / Float64(t * sqrt(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -2e+214)
		tmp = asin((sqrt(0.5) / (t / l)));
	elseif ((t / l) <= 0.0001)
		tmp = asin(1.0);
	else
		tmp = asin((l / (t * sqrt(2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -2e+214], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 0.0001], N[ArcSin[1.0], $MachinePrecision], N[ArcSin[N[(l / N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -1.9999999999999999e214

   1. Initial program 68.1%

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

   2. Taylor expanded in Om around 0 68.1%

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

      1. unpow2 68.1%

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

      2. unpow2 68.1%

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

   4. Simplified 68.1%

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

   5. Taylor expanded in t around inf 67.4%

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

      1. associate-/l* 67.6%

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

   7. Simplified 67.6%

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

   if -1.9999999999999999e214 < (/.f64 t l) < 1.00000000000000005e-4

   1. Initial program 93.6%

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

   2. Taylor expanded in Om around 0 77.0%

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

      1. unpow2 77.0%

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

      2. unpow2 77.0%

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

   4. Simplified 77.0%

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

      1. expm1-log1p-u 77.0%

         \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)\right)\right)} \]

      2. expm1-udef 72.3%

         \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} - 1\right)} \]

      3. sqrt-div 72.3%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}}\right)} - 1\right) \]

      4. metadata-eval 72.3%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} - 1\right) \]

      5. +-commutative 72.3%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{2 \cdot \frac{t \cdot t}{\ell \cdot \ell} + 1}}}\right)} - 1\right) \]

      6. times-frac 82.7%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2 \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} + 1}}\right)} - 1\right) \]

      7. unpow2 82.7%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2 \cdot \color{blue}{{\left(\frac{t}{\ell}\right)}^{2}} + 1}}\right)} - 1\right) \]

      8. fma-udef 82.7%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}}\right)} - 1\right) \]

   6. Applied egg-rr 82.7%

      \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 92.7%

         \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)} \]

      2. expm1-log1p 92.7%

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

      3. fma-udef 92.7%

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

      4. rem-square-sqrt 92.6%

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

      5. unpow2 92.6%

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

      6. swap-sqr 92.7%

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

      7. *-commutative 92.7%

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

      8. *-commutative 92.7%

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

      9. +-commutative 92.7%

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

      10. hypot-1-def 97.3%

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

      11. *-commutative 97.3%

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

   8. Simplified 97.3%

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

   9. Taylor expanded in t around 0 79.9%

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

   if 1.00000000000000005e-4 < (/.f64 t l)

   1. Initial program 64.6%

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

   2. Taylor expanded in Om around 0 40.9%

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

      1. unpow2 40.9%

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

      2. unpow2 40.9%

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

   4. Simplified 40.9%

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

      1. expm1-log1p-u 40.9%

         \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)\right)\right)} \]

      2. expm1-udef 23.4%

         \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} - 1\right)} \]

      3. sqrt-div 23.4%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}}\right)} - 1\right) \]

      4. metadata-eval 23.4%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} - 1\right) \]

      5. +-commutative 23.4%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{2 \cdot \frac{t \cdot t}{\ell \cdot \ell} + 1}}}\right)} - 1\right) \]

      6. times-frac 25.0%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2 \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} + 1}}\right)} - 1\right) \]

      7. unpow2 25.0%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2 \cdot \color{blue}{{\left(\frac{t}{\ell}\right)}^{2}} + 1}}\right)} - 1\right) \]

      8. fma-udef 25.0%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}}\right)} - 1\right) \]

   6. Applied egg-rr 25.0%

      \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 64.5%

         \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)} \]

      2. expm1-log1p 64.5%

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

      3. fma-udef 64.5%

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

      4. rem-square-sqrt 64.2%

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

      5. unpow2 64.2%

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

      6. swap-sqr 64.4%

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

      7. *-commutative 64.4%

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

      8. *-commutative 64.4%

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

      9. +-commutative 64.4%

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

      10. hypot-1-def 99.0%

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

      11. *-commutative 99.0%

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

   8. Simplified 99.0%

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

   9. Taylor expanded in t around inf 98.7%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+214}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0001:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \end{array} \]

Alternative 8: 96.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -10:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{-\ell}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0001:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -10.0)
   (asin (* (sqrt 0.5) (/ (- l) t)))
   (if (<= (/ t l) 0.0001) (asin 1.0) (asin (/ l (* t (sqrt 2.0)))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -10.0) {
		tmp = asin((sqrt(0.5) * (-l / t)));
	} else if ((t / l) <= 0.0001) {
		tmp = asin(1.0);
	} else {
		tmp = asin((l / (t * sqrt(2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t / l) <= (-10.0d0)) then
        tmp = asin((sqrt(0.5d0) * (-l / t)))
    else if ((t / l) <= 0.0001d0) then
        tmp = asin(1.0d0)
    else
        tmp = asin((l / (t * sqrt(2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -10.0) {
		tmp = Math.asin((Math.sqrt(0.5) * (-l / t)));
	} else if ((t / l) <= 0.0001) {
		tmp = Math.asin(1.0);
	} else {
		tmp = Math.asin((l / (t * Math.sqrt(2.0))));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -10.0:
		tmp = math.asin((math.sqrt(0.5) * (-l / t)))
	elif (t / l) <= 0.0001:
		tmp = math.asin(1.0)
	else:
		tmp = math.asin((l / (t * math.sqrt(2.0))))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -10.0)
		tmp = asin(Float64(sqrt(0.5) * Float64(Float64(-l) / t)));
	elseif (Float64(t / l) <= 0.0001)
		tmp = asin(1.0);
	else
		tmp = asin(Float64(l / Float64(t * sqrt(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -10.0)
		tmp = asin((sqrt(0.5) * (-l / t)));
	elseif ((t / l) <= 0.0001)
		tmp = asin(1.0);
	else
		tmp = asin((l / (t * sqrt(2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -10.0], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] * N[((-l) / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 0.0001], N[ArcSin[1.0], $MachinePrecision], N[ArcSin[N[(l / N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -10

   1. Initial program 71.3%

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

   2. Taylor expanded in Om around 0 51.4%

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

      1. unpow2 51.4%

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

      2. unpow2 51.4%

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

   4. Simplified 51.4%

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

   5. Taylor expanded in t around -inf 95.2%

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

      1. mul-1-neg 95.2%

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

      2. *-commutative 95.2%

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

      3. associate-*l/ 95.2%

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

      4. *-commutative 95.2%

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

   7. Simplified 95.2%

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

   if -10 < (/.f64 t l) < 1.00000000000000005e-4

   1. Initial program 97.9%

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

   2. Taylor expanded in Om around 0 84.8%

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

      1. unpow2 84.8%

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

      2. unpow2 84.8%

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

   4. Simplified 84.8%

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

      1. expm1-log1p-u 84.8%

         \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)\right)\right)} \]

      2. expm1-udef 84.8%

         \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} - 1\right)} \]

      3. sqrt-div 84.8%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}}\right)} - 1\right) \]

      4. metadata-eval 84.8%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} - 1\right) \]

      5. +-commutative 84.8%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{2 \cdot \frac{t \cdot t}{\ell \cdot \ell} + 1}}}\right)} - 1\right) \]

      6. times-frac 96.8%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2 \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} + 1}}\right)} - 1\right) \]

      7. unpow2 96.8%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2 \cdot \color{blue}{{\left(\frac{t}{\ell}\right)}^{2}} + 1}}\right)} - 1\right) \]

      8. fma-udef 96.8%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}}\right)} - 1\right) \]

   6. Applied egg-rr 96.8%

      \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 96.8%

         \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)} \]

      2. expm1-log1p 96.8%

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

      3. fma-udef 96.8%

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

      4. rem-square-sqrt 96.8%

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

      5. unpow2 96.8%

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

      6. swap-sqr 96.8%

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

      7. *-commutative 96.8%

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

      8. *-commutative 96.8%

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

      9. +-commutative 96.8%

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

      10. hypot-1-def 96.8%

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

      11. *-commutative 96.8%

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

   8. Simplified 96.8%

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

   9. Taylor expanded in t around 0 95.5%

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

   if 1.00000000000000005e-4 < (/.f64 t l)

   1. Initial program 64.6%

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

   2. Taylor expanded in Om around 0 40.9%

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

      1. unpow2 40.9%

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

      2. unpow2 40.9%

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

   4. Simplified 40.9%

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

      1. expm1-log1p-u 40.9%

         \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)\right)\right)} \]

      2. expm1-udef 23.4%

         \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} - 1\right)} \]

      3. sqrt-div 23.4%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}}\right)} - 1\right) \]

      4. metadata-eval 23.4%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} - 1\right) \]

      5. +-commutative 23.4%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{2 \cdot \frac{t \cdot t}{\ell \cdot \ell} + 1}}}\right)} - 1\right) \]

      6. times-frac 25.0%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2 \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} + 1}}\right)} - 1\right) \]

      7. unpow2 25.0%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2 \cdot \color{blue}{{\left(\frac{t}{\ell}\right)}^{2}} + 1}}\right)} - 1\right) \]

      8. fma-udef 25.0%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}}\right)} - 1\right) \]

   6. Applied egg-rr 25.0%

      \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 64.5%

         \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)} \]

      2. expm1-log1p 64.5%

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

      3. fma-udef 64.5%

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

      4. rem-square-sqrt 64.2%

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

      5. unpow2 64.2%

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

      6. swap-sqr 64.4%

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

      7. *-commutative 64.4%

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

      8. *-commutative 64.4%

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

      9. +-commutative 64.4%

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

      10. hypot-1-def 99.0%

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

      11. *-commutative 99.0%

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

   8. Simplified 99.0%

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

   9. Taylor expanded in t around inf 98.7%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -10:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{-\ell}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0001:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \end{array} \]

Alternative 9: 61.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.35 \cdot 10^{-203}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{-19}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= l -2.35e-203)
   (asin 1.0)
   (if (<= l 2.1e-19) (asin (* (sqrt 0.5) (/ l t))) (asin 1.0))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= -2.35e-203) {
		tmp = asin(1.0);
	} else if (l <= 2.1e-19) {
		tmp = asin((sqrt(0.5) * (l / t)));
	} else {
		tmp = asin(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (l <= (-2.35d-203)) then
        tmp = asin(1.0d0)
    else if (l <= 2.1d-19) then
        tmp = asin((sqrt(0.5d0) * (l / t)))
    else
        tmp = asin(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= -2.35e-203) {
		tmp = Math.asin(1.0);
	} else if (l <= 2.1e-19) {
		tmp = Math.asin((Math.sqrt(0.5) * (l / t)));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if l <= -2.35e-203:
		tmp = math.asin(1.0)
	elif l <= 2.1e-19:
		tmp = math.asin((math.sqrt(0.5) * (l / t)))
	else:
		tmp = math.asin(1.0)
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (l <= -2.35e-203)
		tmp = asin(1.0);
	elseif (l <= 2.1e-19)
		tmp = asin(Float64(sqrt(0.5) * Float64(l / t)));
	else
		tmp = asin(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (l <= -2.35e-203)
		tmp = asin(1.0);
	elseif (l <= 2.1e-19)
		tmp = asin((sqrt(0.5) * (l / t)));
	else
		tmp = asin(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[l, -2.35e-203], N[ArcSin[1.0], $MachinePrecision], If[LessEqual[l, 2.1e-19], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -2.35000000000000003e-203 or 2.0999999999999999e-19 < l

   1. Initial program 89.5%

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

   2. Taylor expanded in Om around 0 72.2%

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

      1. unpow2 72.2%

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

      2. unpow2 72.2%

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

   4. Simplified 72.2%

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

      1. expm1-log1p-u 72.2%

         \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)\right)\right)} \]

      2. expm1-udef 66.0%

         \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} - 1\right)} \]

      3. sqrt-div 66.0%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}}\right)} - 1\right) \]

      4. metadata-eval 66.0%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} - 1\right) \]

      5. +-commutative 66.0%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{2 \cdot \frac{t \cdot t}{\ell \cdot \ell} + 1}}}\right)} - 1\right) \]

      6. times-frac 74.2%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2 \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} + 1}}\right)} - 1\right) \]

      7. unpow2 74.2%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2 \cdot \color{blue}{{\left(\frac{t}{\ell}\right)}^{2}} + 1}}\right)} - 1\right) \]

      8. fma-udef 74.2%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}}\right)} - 1\right) \]

   6. Applied egg-rr 74.2%

      \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 89.0%

         \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)} \]

      2. expm1-log1p 89.0%

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

      3. fma-udef 89.0%

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

      4. rem-square-sqrt 88.8%

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

      5. unpow2 88.8%

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

      6. swap-sqr 89.0%

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

      7. *-commutative 89.0%

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

      8. *-commutative 89.0%

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

      9. +-commutative 89.0%

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

      10. hypot-1-def 97.9%

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

      11. *-commutative 97.9%

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

   8. Simplified 97.9%

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

   9. Taylor expanded in t around 0 70.0%

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

   if -2.35000000000000003e-203 < l < 2.0999999999999999e-19

   1. Initial program 70.5%

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

   2. Taylor expanded in Om around 0 52.9%

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

      1. unpow2 52.9%

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

      2. unpow2 52.9%

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

   4. Simplified 52.9%

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

   5. Taylor expanded in t around inf 61.2%

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

      1. *-commutative 61.2%

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

      2. associate-*l/ 61.1%

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

      3. *-commutative 61.1%

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

   7. Simplified 61.1%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.35 \cdot 10^{-203}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{-19}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 10: 61.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -9.8 \cdot 10^{-200}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{-22}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= l -9.8e-200)
   (asin 1.0)
   (if (<= l 8.5e-22) (asin (/ l (* t (sqrt 2.0)))) (asin 1.0))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= -9.8e-200) {
		tmp = asin(1.0);
	} else if (l <= 8.5e-22) {
		tmp = asin((l / (t * sqrt(2.0))));
	} else {
		tmp = asin(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (l <= (-9.8d-200)) then
        tmp = asin(1.0d0)
    else if (l <= 8.5d-22) then
        tmp = asin((l / (t * sqrt(2.0d0))))
    else
        tmp = asin(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= -9.8e-200) {
		tmp = Math.asin(1.0);
	} else if (l <= 8.5e-22) {
		tmp = Math.asin((l / (t * Math.sqrt(2.0))));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if l <= -9.8e-200:
		tmp = math.asin(1.0)
	elif l <= 8.5e-22:
		tmp = math.asin((l / (t * math.sqrt(2.0))))
	else:
		tmp = math.asin(1.0)
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (l <= -9.8e-200)
		tmp = asin(1.0);
	elseif (l <= 8.5e-22)
		tmp = asin(Float64(l / Float64(t * sqrt(2.0))));
	else
		tmp = asin(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (l <= -9.8e-200)
		tmp = asin(1.0);
	elseif (l <= 8.5e-22)
		tmp = asin((l / (t * sqrt(2.0))));
	else
		tmp = asin(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[l, -9.8e-200], N[ArcSin[1.0], $MachinePrecision], If[LessEqual[l, 8.5e-22], N[ArcSin[N[(l / N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -9.7999999999999999e-200 or 8.5000000000000001e-22 < l

   1. Initial program 89.5%

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

   2. Taylor expanded in Om around 0 72.2%

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

      1. unpow2 72.2%

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

      2. unpow2 72.2%

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

   4. Simplified 72.2%

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

      1. expm1-log1p-u 72.2%

         \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)\right)\right)} \]

      2. expm1-udef 66.0%

         \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} - 1\right)} \]

      3. sqrt-div 66.0%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}}\right)} - 1\right) \]

      4. metadata-eval 66.0%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} - 1\right) \]

      5. +-commutative 66.0%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{2 \cdot \frac{t \cdot t}{\ell \cdot \ell} + 1}}}\right)} - 1\right) \]

      6. times-frac 74.2%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2 \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} + 1}}\right)} - 1\right) \]

      7. unpow2 74.2%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2 \cdot \color{blue}{{\left(\frac{t}{\ell}\right)}^{2}} + 1}}\right)} - 1\right) \]

      8. fma-udef 74.2%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}}\right)} - 1\right) \]

   6. Applied egg-rr 74.2%

      \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 89.0%

         \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)} \]

      2. expm1-log1p 89.0%

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

      3. fma-udef 89.0%

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

      4. rem-square-sqrt 88.8%

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

      5. unpow2 88.8%

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

      6. swap-sqr 89.0%

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

      7. *-commutative 89.0%

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

      8. *-commutative 89.0%

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

      9. +-commutative 89.0%

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

      10. hypot-1-def 97.9%

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

      11. *-commutative 97.9%

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

   8. Simplified 97.9%

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

   9. Taylor expanded in t around 0 70.0%

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

   if -9.7999999999999999e-200 < l < 8.5000000000000001e-22

   1. Initial program 70.5%

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

   2. Taylor expanded in Om around 0 52.9%

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

      1. unpow2 52.9%

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

      2. unpow2 52.9%

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

   4. Simplified 52.9%

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

      1. expm1-log1p-u 52.9%

         \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)\right)\right)} \]

      2. expm1-udef 41.2%

         \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} - 1\right)} \]

      3. sqrt-div 41.2%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}}\right)} - 1\right) \]

      4. metadata-eval 41.2%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} - 1\right) \]

      5. +-commutative 41.2%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{2 \cdot \frac{t \cdot t}{\ell \cdot \ell} + 1}}}\right)} - 1\right) \]

      6. times-frac 46.1%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2 \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} + 1}}\right)} - 1\right) \]

      7. unpow2 46.1%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2 \cdot \color{blue}{{\left(\frac{t}{\ell}\right)}^{2}} + 1}}\right)} - 1\right) \]

      8. fma-udef 46.1%

         \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}}\right)} - 1\right) \]

   6. Applied egg-rr 46.1%

      \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 69.7%

         \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)} \]

      2. expm1-log1p 69.6%

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

      3. fma-udef 69.6%

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

      4. rem-square-sqrt 69.5%

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

      5. unpow2 69.5%

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

      6. swap-sqr 69.6%

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

      7. *-commutative 69.6%

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

      8. *-commutative 69.6%

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

      9. +-commutative 69.6%

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

      10. hypot-1-def 97.3%

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

      11. *-commutative 97.3%

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

   8. Simplified 97.3%

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

   9. Taylor expanded in t around inf 61.3%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9.8 \cdot 10^{-200}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{-22}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 11: 50.8% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.6%

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

2. Taylor expanded in Om around 0 66.2%

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

   1. unpow2 66.2%

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

   2. unpow2 66.2%

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

4. Simplified 66.2%

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

   1. expm1-log1p-u 66.2%

      \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)\right)\right)} \]

   2. expm1-udef 58.3%

      \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} - 1\right)} \]

   3. sqrt-div 58.3%

      \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}}\right)} - 1\right) \]

   4. metadata-eval 58.3%

      \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} - 1\right) \]

   5. +-commutative 58.3%

      \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{2 \cdot \frac{t \cdot t}{\ell \cdot \ell} + 1}}}\right)} - 1\right) \]

   6. times-frac 65.4%

      \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2 \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} + 1}}\right)} - 1\right) \]

   7. unpow2 65.4%

      \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2 \cdot \color{blue}{{\left(\frac{t}{\ell}\right)}^{2}} + 1}}\right)} - 1\right) \]

   8. fma-udef 65.4%

      \[\leadsto \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}}\right)} - 1\right) \]

6. Applied egg-rr 65.4%

   \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)} - 1\right)} \]
7. Step-by-step derivation

   1. expm1-def 82.9%

      \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)} \]

   2. expm1-log1p 82.9%

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

   3. fma-udef 82.9%

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

   4. rem-square-sqrt 82.8%

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

   5. unpow2 82.8%

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

   6. swap-sqr 82.9%

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

   7. *-commutative 82.9%

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

   8. *-commutative 82.9%

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

   9. +-commutative 82.9%

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

   10. hypot-1-def 97.7%

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

   11. *-commutative 97.7%

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

8. Simplified 97.7%

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

9. Taylor expanded in t around 0 53.1%

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

10. Final simplification 53.1%

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

Reproduce

herbie shell --seed 2023279 
(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)))))))