Toniolo and Linder, Equation (2)

Average Accuracy: 84.0% → 97.7%

Time: 14.1s

Precision: binary64

Cost: 14664

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+171}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{-\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+138}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

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
 (if (<= (/ t l) -5e+171)
   (asin (* l (/ (- (sqrt 0.5)) t)))
   (if (<= (/ t l) 1e+138)
     (asin
      (sqrt
       (/
        (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
        (+ 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 tmp;
	if ((t / l) <= -5e+171) {
		tmp = asin((l * (-sqrt(0.5) / t)));
	} else if ((t / l) <= 1e+138) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (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) :: tmp
    if ((t / l) <= (-5d+171)) then
        tmp = asin((l * (-sqrt(0.5d0) / t)))
    else if ((t / l) <= 1d+138) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (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 tmp;
	if ((t / l) <= -5e+171) {
		tmp = Math.asin((l * (-Math.sqrt(0.5) / t)));
	} else if ((t / l) <= 1e+138) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (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):
	tmp = 0
	if (t / l) <= -5e+171:
		tmp = math.asin((l * (-math.sqrt(0.5) / t)))
	elif (t / l) <= 1e+138:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (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)
	tmp = 0.0
	if (Float64(t / l) <= -5e+171)
		tmp = asin(Float64(l * Float64(Float64(-sqrt(0.5)) / t)));
	elseif (Float64(t / l) <= 1e+138)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om))) / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) / Float64(l / t)))))));
	else
		tmp = asin(Float64(Float64(l * 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)
	tmp = 0.0;
	if ((t / l) <= -5e+171)
		tmp = asin((l * (-sqrt(0.5) / t)));
	elseif ((t / l) <= 1e+138)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (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_] := If[LessEqual[N[(t / l), $MachinePrecision], -5e+171], N[ArcSin[N[(l * N[((-N[Sqrt[0.5], $MachinePrecision]) / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 1e+138], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t / l), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -5.0000000000000004e171

   1. Initial program 50.2%

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

   2. Taylor expanded in Om around 0 50.2%

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

   3. Simplified 50.2%

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

      [Start] 50.2	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}\right) \]
      associate-*r/ [=>] 50.2	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + \color{blue}{\frac{2 \cdot {t}^{2}}{{\ell}^{2}}}}}\right) \]
      unpow2 [=>] 50.2	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2 \cdot \color{blue}{\left(t \cdot t\right)}}{{\ell}^{2}}}}\right) \]
      unpow2 [=>] 50.2	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2 \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}}}}\right) \]

   4. Taylor expanded in t around -inf 99.2%

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

   5. Simplified 99.2%

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

      [Start] 99.2	\[ \sin^{-1} \left(-1 \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      mul-1-neg [=>] 99.2	\[ \sin^{-1} \color{blue}{\left(-\frac{\sqrt{0.5} \cdot \ell}{t}\right)} \]
      associate-/l* [=>] 97.3	\[ \sin^{-1} \left(-\color{blue}{\frac{\sqrt{0.5}}{\frac{t}{\ell}}}\right) \]
      associate-/r/ [=>] 99.2	\[ \sin^{-1} \left(-\color{blue}{\frac{\sqrt{0.5}}{t} \cdot \ell}\right) \]

   if -5.0000000000000004e171 < (/.f64 t l) < 1e138

   1. Initial program 97.2%

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

   2. Applied egg-rr 97.2%

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

   3. Applied egg-rr 97.2%

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

   if 1e138 < (/.f64 t l)

   1. Initial program 50.5%

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

   2. Taylor expanded in Om around 0 47.3%

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

   3. Simplified 47.3%

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

      [Start] 47.3	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}\right) \]
      associate-*r/ [=>] 47.3	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + \color{blue}{\frac{2 \cdot {t}^{2}}{{\ell}^{2}}}}}\right) \]
      unpow2 [=>] 47.3	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2 \cdot \color{blue}{\left(t \cdot t\right)}}{{\ell}^{2}}}}\right) \]
      unpow2 [=>] 47.3	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2 \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}}}}\right) \]

   4. Taylor expanded in t around inf 99.2%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+171}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{-\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+138}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.4%
Cost	32832

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

Alternative 2
Accuracy	97.5%
Cost	19712

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

Alternative 3
Accuracy	97.3%
Cost	14152

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+61}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{-\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+89}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{t}{\frac{\ell}{t}} \cdot \frac{2}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \end{array} \]

Alternative 4
Accuracy	98.0%
Cost	14152

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+149}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{-\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+138}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{t}{\ell} \cdot \frac{2 \cdot t}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 5
Accuracy	60.5%
Cost	13915

\[\begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{+114} \lor \neg \left(t \leq -2.85 \cdot 10^{+64}\right) \land \left(t \leq -2.2 \cdot 10^{-13} \lor \neg \left(t \leq 2.05 \cdot 10^{-108}\right) \land \left(t \leq 2.6 \cdot 10^{-78} \lor \neg \left(t \leq 8.4 \cdot 10^{+46}\right)\right)\right):\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 6
Accuracy	60.5%
Cost	13912

\[\begin{array}{l} t_1 := \sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ t_2 := \sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{if}\;t \leq -5.8 \cdot 10^{+115}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.7 \cdot 10^{+66}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;t \leq -1.02 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-108}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-78}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{+46}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Accuracy	78.9%
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+217}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4 \cdot 10^{-12}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 8
Accuracy	96.6%
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -20:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{-\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4 \cdot 10^{-12}:\\ \;\;\;\;\sin^{-1} \left(1 - \frac{\frac{t}{\ell}}{\frac{\ell}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 9
Accuracy	96.6%
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -20:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4 \cdot 10^{-12}:\\ \;\;\;\;\sin^{-1} \left(1 - \frac{\frac{t}{\ell}}{\frac{\ell}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 10
Accuracy	49.5%
Cost	6464

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

Reproduce

herbie shell --seed 2023122 
(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)))))))