Toniolo and Linder, Equation (2)

Average Error: 16.14% → 1.2%

Time: 14.7s

Precision: binary64

Cost: 26692

Math

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

↓

\[\begin{array}{l} t_1 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{t_1} \cdot \frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (- 1.0 (pow (/ Om Omc) 2.0))))
   (if (<= (/ t l) -2e+14)
     (asin (* (sqrt t_1) (/ (* l (- (sqrt 0.5))) t)))
     (if (<= (/ t l) 5e+153)
       (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (/ (/ t l) (/ l t)))))))
       (asin (* l (/ (sqrt 0.5) t)))))))

C

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

↓

double code(double t, double l, double Om, double Omc) {
	double t_1 = 1.0 - pow((Om / Omc), 2.0);
	double tmp;
	if ((t / l) <= -2e+14) {
		tmp = asin((sqrt(t_1) * ((l * -sqrt(0.5)) / t)));
	} else if ((t / l) <= 5e+153) {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = asin((l * (sqrt(0.5) / t)));
	}
	return tmp;
}

Fortran

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

↓

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 = 1.0d0 - ((om / omc) ** 2.0d0)
    if ((t / l) <= (-2d+14)) then
        tmp = asin((sqrt(t_1) * ((l * -sqrt(0.5d0)) / t)))
    else if ((t / l) <= 5d+153) then
        tmp = asin(sqrt((t_1 / (1.0d0 + (2.0d0 * ((t / l) / (l / t)))))))
    else
        tmp = asin((l * (sqrt(0.5d0) / t)))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double t, double l, double Om, double Omc) {
	double t_1 = 1.0 - Math.pow((Om / Omc), 2.0);
	double tmp;
	if ((t / l) <= -2e+14) {
		tmp = Math.asin((Math.sqrt(t_1) * ((l * -Math.sqrt(0.5)) / t)));
	} else if ((t / l) <= 5e+153) {
		tmp = Math.asin(Math.sqrt((t_1 / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = Math.asin((l * (Math.sqrt(0.5) / t)));
	}
	return tmp;
}

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))))))

↓

def code(t, l, Om, Omc):
	t_1 = 1.0 - math.pow((Om / Omc), 2.0)
	tmp = 0
	if (t / l) <= -2e+14:
		tmp = math.asin((math.sqrt(t_1) * ((l * -math.sqrt(0.5)) / t)))
	elif (t / l) <= 5e+153:
		tmp = math.asin(math.sqrt((t_1 / (1.0 + (2.0 * ((t / l) / (l / t)))))))
	else:
		tmp = math.asin((l * (math.sqrt(0.5) / t)))
	return tmp

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

↓

function code(t, l, Om, Omc)
	t_1 = Float64(1.0 - (Float64(Om / Omc) ^ 2.0))
	tmp = 0.0
	if (Float64(t / l) <= -2e+14)
		tmp = asin(Float64(sqrt(t_1) * Float64(Float64(l * Float64(-sqrt(0.5))) / t)));
	elseif (Float64(t / l) <= 5e+153)
		tmp = asin(sqrt(Float64(t_1 / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) / Float64(l / t)))))));
	else
		tmp = asin(Float64(l * Float64(sqrt(0.5) / t)));
	end
	return tmp
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

↓

function tmp_2 = code(t, l, Om, Omc)
	t_1 = 1.0 - ((Om / Omc) ^ 2.0);
	tmp = 0.0;
	if ((t / l) <= -2e+14)
		tmp = asin((sqrt(t_1) * ((l * -sqrt(0.5)) / t)));
	elseif ((t / l) <= 5e+153)
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	else
		tmp = asin((l * (sqrt(0.5) / t)));
	end
	tmp_2 = tmp;
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]

↓

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -2e+14], N[ArcSin[N[(N[Sqrt[t$95$1], $MachinePrecision] * N[(N[(l * (-N[Sqrt[0.5], $MachinePrecision])), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 5e+153], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(1.0 + N[(2.0 * N[(N[(t / l), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l * N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -2e14

   1. Initial program 32.9

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

   2. Taylor expanded in t around -inf 12.23

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

   3. Simplified 0.5

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

      [Start] 12.23	\[ \sin^{-1} \left(-1 \cdot \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right) \]
      mul-1-neg [=>] 12.23	\[ \sin^{-1} \color{blue}{\left(-\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]
      *-commutative [=>] 12.23	\[ \sin^{-1} \left(-\color{blue}{\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}}\right) \]
      unpow2 [=>] 12.23	\[ \sin^{-1} \left(-\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      unpow2 [=>] 12.23	\[ \sin^{-1} \left(-\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      times-frac [=>] 0.5	\[ \sin^{-1} \left(-\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      unpow2 [<=] 0.5	\[ \sin^{-1} \left(-\sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      *-commutative [=>] 0.5	\[ \sin^{-1} \left(-\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\color{blue}{\ell \cdot \sqrt{0.5}}}{t}\right) \]

   if -2e14 < (/.f64 t l) < 5.00000000000000018e153

   1. Initial program 1.58

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

   2. Applied egg-rr 1.56

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

   if 5.00000000000000018e153 < (/.f64 t l)

   1. Initial program 53.96

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

   2. Applied egg-rr 53.96

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

   3. Taylor expanded in Om around 0 54.04

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

   4. Simplified 54.04

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

      [Start] 54.04	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}\right) \]
      associate-*r/ [=>] 54.04	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + \color{blue}{\frac{2 \cdot {t}^{2}}{{\ell}^{2}}}}}\right) \]
      unpow2 [=>] 54.04	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2 \cdot \color{blue}{\left(t \cdot t\right)}}{{\ell}^{2}}}}\right) \]
      unpow2 [=>] 54.04	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2 \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}}}}\right) \]

   5. Taylor expanded in t around inf 0.81

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

   6. Simplified 0.77

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

      [Start] 0.81	\[ \sin^{-1} \left(\frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      associate-/l* [=>] 2.06	\[ \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)} \]
      associate-/r/ [=>] 0.77	\[ \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5}}{t} \cdot \ell\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.69%
Cost	26624

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

Alternative 2
Error	1.97%
Cost	20872

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+89}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+32}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(t \cdot \frac{\frac{t}{\ell}}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 3
Error	1.22%
Cost	20872

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+137}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{-\ell}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 4
Error	1.89%
Cost	14664

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+94}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+35}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{t}{\ell \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 5
Error	2.64%
Cost	13896

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5000:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.5:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 6
Error	20.2%
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+205}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.5:\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 7
Error	2.95%
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{-\ell}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.5:\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 8
Error	2.94%
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5000:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.5:\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 9
Error	37.78%
Cost	13385

\[\begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{+52} \lor \neg \left(t \leq 1.7 \cdot 10^{+175}\right):\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\\ \end{array} \]

Alternative 10
Error	49.07%
Cost	7104

\[\sin^{-1} \left(1 + -0.5 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right) \]

Alternative 11
Error	49.42%
Cost	6464

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

Reproduce

herbie shell --seed 2023088 
(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)))))))