Toniolo and Linder, Equation (2)

Average Error: 10.6 → 0.8

Time: 14.8s

Precision: binary64

Cost: 20872

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+154}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{-\ell}{\sqrt{2}}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+57}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

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) -4e+154)
   (asin (/ (/ (- l) (sqrt 2.0)) t))
   (if (<= (/ t l) 5e+57)
     (asin
      (sqrt
       (/
        (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
        (+ 1.0 (* 2.0 (pow (/ t l) 2.0))))))
     (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) <= -4e+154) {
		tmp = asin(((-l / sqrt(2.0)) / t));
	} else if ((t / l) <= 5e+57) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * pow((t / l), 2.0))))));
	} 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) <= (-4d+154)) then
        tmp = asin(((-l / sqrt(2.0d0)) / t))
    else if ((t / l) <= 5d+57) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * ((t / l) ** 2.0d0))))))
    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) <= -4e+154) {
		tmp = Math.asin(((-l / Math.sqrt(2.0)) / t));
	} else if ((t / l) <= 5e+57) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * Math.pow((t / l), 2.0))))));
	} 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) <= -4e+154:
		tmp = math.asin(((-l / math.sqrt(2.0)) / t))
	elif (t / l) <= 5e+57:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * math.pow((t / l), 2.0))))))
	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) <= -4e+154)
		tmp = asin(Float64(Float64(Float64(-l) / sqrt(2.0)) / t));
	elseif (Float64(t / l) <= 5e+57)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om))) / Float64(1.0 + Float64(2.0 * (Float64(t / l) ^ 2.0))))));
	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) <= -4e+154)
		tmp = asin(((-l / sqrt(2.0)) / t));
	elseif ((t / l) <= 5e+57)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) ^ 2.0))))));
	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], -4e+154], N[ArcSin[N[(N[((-l) / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 5e+57], 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[Power[N[(t / l), $MachinePrecision], 2.0], $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) < -4.00000000000000015e154

   1. Initial program 34.3

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

   2. Taylor expanded in Om around 0 34.3

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

   3. Simplified 34.3

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

      [Start] 34.3	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}\right) \]
      associate-*r/ [=>] 34.3	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + \color{blue}{\frac{2 \cdot {t}^{2}}{{\ell}^{2}}}}}\right) \]
      unpow2 [=>] 34.3	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2 \cdot \color{blue}{\left(t \cdot t\right)}}{{\ell}^{2}}}}\right) \]
      unpow2 [=>] 34.3	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2 \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}}}}\right) \]

   4. Applied egg-rr 34.3

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

   5. Simplified 1.6

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

      [Start] 34.3	\[ \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(1, \frac{t \cdot \sqrt{2}}{\ell}\right)}\right)} - 1\right) \]
      expm1-def [=>] 1.6	\[ \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(1, \frac{t \cdot \sqrt{2}}{\ell}\right)}\right)\right)\right)} \]
      expm1-log1p [=>] 1.6	\[ \sin^{-1} \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(1, \frac{t \cdot \sqrt{2}}{\ell}\right)}\right)} \]
      associate-/l* [=>] 1.6	\[ \sin^{-1} \left(\frac{1}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\frac{\ell}{\sqrt{2}}}}\right)}\right) \]

   6. Taylor expanded in t around -inf 0.5

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

   7. Simplified 0.5

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

      [Start] 0.5	\[ \sin^{-1} \left(-1 \cdot \frac{\ell}{\sqrt{2} \cdot t}\right) \]
      mul-1-neg [=>] 0.5	\[ \sin^{-1} \color{blue}{\left(-\frac{\ell}{\sqrt{2} \cdot t}\right)} \]
      associate-/r* [=>] 0.5	\[ \sin^{-1} \left(-\color{blue}{\frac{\frac{\ell}{\sqrt{2}}}{t}}\right) \]

   if -4.00000000000000015e154 < (/.f64 t l) < 4.99999999999999972e57

   1. Initial program 0.9

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

   2. Applied egg-rr 0.9

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

   if 4.99999999999999972e57 < (/.f64 t l)

   1. Initial program 24.4

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

   2. Taylor expanded in Om around 0 36.2

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

   3. Simplified 36.2

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

      [Start] 36.2	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}\right) \]
      associate-*r/ [=>] 36.2	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + \color{blue}{\frac{2 \cdot {t}^{2}}{{\ell}^{2}}}}}\right) \]
      unpow2 [=>] 36.2	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2 \cdot \color{blue}{\left(t \cdot t\right)}}{{\ell}^{2}}}}\right) \]
      unpow2 [=>] 36.2	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2 \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}}}}\right) \]

   4. Taylor expanded in t around inf 0.9

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+154}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{-\ell}{\sqrt{2}}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+57}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

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

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

Alternative 3
Error	1.2
Cost	14280

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+134}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{-\ell}{\sqrt{2}}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+19}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{t}{\ell} \cdot \frac{2 \cdot t}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{\left(1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\right) \cdot 0.5}\right)\\ \end{array} \]

Alternative 4
Error	1.5
Cost	14152

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+32}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{-\ell}{\sqrt{2}}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+57}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{t}{\frac{\ell}{t}} \cdot \frac{2}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 5
Error	1.3
Cost	14152

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+134}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{-\ell}{\sqrt{2}}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+57}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{t}{\ell} \cdot \frac{2 \cdot t}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 6
Error	1.9
Cost	13896

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -200:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{-\ell}{\sqrt{2}}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 7
Error	13.6
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+213}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \end{array} \]

Alternative 8
Error	13.6
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+213}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 9
Error	2.4
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -200:\\ \;\;\;\;\sin^{-1} \left(\frac{-\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\sin^{-1} \left(1 - \frac{\frac{t}{\ell}}{\frac{\ell}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 10
Error	2.2
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -200:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{-\ell}{\sqrt{2}}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\sin^{-1} \left(1 - \frac{\frac{t}{\ell}}{\frac{\ell}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 11
Error	24.0
Cost	13385

\[\begin{array}{l} \mathbf{if}\;t \leq -5.7 \cdot 10^{+66} \lor \neg \left(t \leq 7.4 \cdot 10^{+111}\right):\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 12
Error	24.1
Cost	13385

\[\begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{+69} \lor \neg \left(t \leq 3.2 \cdot 10^{+102}\right):\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 13
Error	31.7
Cost	6464

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

Reproduce

herbie shell --seed 2023059 
(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)))))))