Toniolo and Linder, Equation (2)

Average Error: 10.2 → 0.8

Time: 13.8s

Precision: binary64

Cost: 14664

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+142}:\\ \;\;\;\;\sin^{-1} \left(\frac{\left(-\ell\right) \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 6 \cdot 10^{+131}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

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) -2e+142)
   (asin (/ (* (- l) (sqrt 0.5)) t))
   (if (<= (/ t l) 6e+131)
     (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) <= -2e+142) {
		tmp = asin(((-l * sqrt(0.5)) / t));
	} else if ((t / l) <= 6e+131) {
		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) <= (-2d+142)) then
        tmp = asin(((-l * sqrt(0.5d0)) / t))
    else if ((t / l) <= 6d+131) 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) <= -2e+142) {
		tmp = Math.asin(((-l * Math.sqrt(0.5)) / t));
	} else if ((t / l) <= 6e+131) {
		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) <= -2e+142:
		tmp = math.asin(((-l * math.sqrt(0.5)) / t))
	elif (t / l) <= 6e+131:
		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) <= -2e+142)
		tmp = asin(Float64(Float64(Float64(-l) * sqrt(0.5)) / t));
	elseif (Float64(t / l) <= 6e+131)
		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) <= -2e+142)
		tmp = asin(((-l * sqrt(0.5)) / t));
	elseif ((t / l) <= 6e+131)
		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], -2e+142], N[ArcSin[N[(N[((-l) * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 6e+131], 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) < -2.0000000000000001e142

   1. Initial program 32.8

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

   2. Applied egg-rr 34.9

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

   3. Taylor expanded in Om around 0 34.9

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

   4. Simplified 33.9

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

      [Start] 34.9	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}\right) \]
      unpow2 [=>] 34.9	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{\color{blue}{t \cdot t}}{{\ell}^{2}}}}\right) \]
      unpow2 [=>] 34.9	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}}}\right) \]
      associate-*r/ [<=] 33.9	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \color{blue}{\left(t \cdot \frac{t}{\ell \cdot \ell}\right)}}}\right) \]

   5. Taylor expanded in t around -inf 0.7

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

   6. Simplified 0.7

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

      [Start] 0.7	\[ \sin^{-1} \left(-1 \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      associate-*r/ [=>] 0.7	\[ \sin^{-1} \color{blue}{\left(\frac{-1 \cdot \left(\sqrt{0.5} \cdot \ell\right)}{t}\right)} \]
      *-commutative [=>] 0.7	\[ \sin^{-1} \left(\frac{-1 \cdot \color{blue}{\left(\ell \cdot \sqrt{0.5}\right)}}{t}\right) \]
      associate-*r* [=>] 0.7	\[ \sin^{-1} \left(\frac{\color{blue}{\left(-1 \cdot \ell\right) \cdot \sqrt{0.5}}}{t}\right) \]
      neg-mul-1 [<=] 0.7	\[ \sin^{-1} \left(\frac{\color{blue}{\left(-\ell\right)} \cdot \sqrt{0.5}}{t}\right) \]

   if -2.0000000000000001e142 < (/.f64 t l) < 6.0000000000000003e131

   1. Initial program 0.8

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

   2. Applied egg-rr 0.8

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

   3. Applied egg-rr 0.8

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

   if 6.0000000000000003e131 < (/.f64 t l)

   1. Initial program 32.3

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

   2. Applied egg-rr 36.2

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

   3. Taylor expanded in Om around 0 36.3

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

   4. Simplified 34.8

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

      [Start] 36.3	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}\right) \]
      unpow2 [=>] 36.3	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{\color{blue}{t \cdot t}}{{\ell}^{2}}}}\right) \]
      unpow2 [=>] 36.3	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}}}\right) \]
      associate-*r/ [<=] 34.8	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \color{blue}{\left(t \cdot \frac{t}{\ell \cdot \ell}\right)}}}\right) \]

   5. Taylor expanded in t around inf 0.8

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+142}:\\ \;\;\;\;\sin^{-1} \left(\frac{\left(-\ell\right) \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 6 \cdot 10^{+131}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.0
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
Error	1.6
Cost	19712

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

Alternative 3
Error	1.7
Cost	14152

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+57}:\\ \;\;\;\;\sin^{-1} \left(\frac{\left(-\ell\right) \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4 \cdot 10^{+86}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 4
Error	1.3
Cost	14152

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+142}:\\ \;\;\;\;\sin^{-1} \left(\frac{\left(-\ell\right) \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 6 \cdot 10^{+131}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 5
Error	2.2
Cost	13896

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50000000000:\\ \;\;\;\;\sin^{-1} \left(\frac{\left(-\ell\right) \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4 \cdot 10^{-10}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 6
Error	2.7
Cost	13704

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50000000000:\\ \;\;\;\;\sin^{-1} \left(\frac{-\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4 \cdot 10^{-10}:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 7
Error	2.6
Cost	13704

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50000000000:\\ \;\;\;\;\sin^{-1} \left(\frac{\left(-\ell\right) \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4 \cdot 10^{-10}:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 8
Error	12.9
Cost	13641

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1.2 \cdot 10^{+205} \lor \neg \left(\frac{t}{\ell} \leq 0.7\right):\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 9
Error	13.0
Cost	13640

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

Alternative 10
Error	2.7
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50000000000:\\ \;\;\;\;\sin^{-1} \left(\frac{-\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4 \cdot 10^{-10}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 11
Error	31.4
Cost	6464

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

Reproduce

herbie shell --seed 2023027 
(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)))))))