Toniolo and Linder, Equation (2)

Average Error: 10.0 → 1.1

Time: 13.6s

Precision: binary64

Cost: 26692

Math

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

↓

\[\begin{array}{l} t_1 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+166}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{t_1} \cdot \frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+124}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \frac{1}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \]

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)))))))

↓

(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (- 1.0 (pow (/ Om Omc) 2.0))))
   (if (<= (/ t l) -4e+166)
     (asin (* (sqrt t_1) (/ (- l) (/ t (sqrt 0.5)))))
     (if (<= (/ t l) 5e+124)
       (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (/ 1.0 (* (/ l t) (/ l t))))))))
       (asin (/ (/ l t) (sqrt 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))))));
}

↓

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) <= -4e+166) {
		tmp = asin((sqrt(t_1) * (-l / (t / sqrt(0.5)))));
	} else if ((t / l) <= 5e+124) {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * (1.0 / ((l / t) * (l / t))))))));
	} 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
    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) <= (-4d+166)) then
        tmp = asin((sqrt(t_1) * (-l / (t / sqrt(0.5d0)))))
    else if ((t / l) <= 5d+124) then
        tmp = asin(sqrt((t_1 / (1.0d0 + (2.0d0 * (1.0d0 / ((l / t) * (l / t))))))))
    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) {
	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) <= -4e+166) {
		tmp = Math.asin((Math.sqrt(t_1) * (-l / (t / Math.sqrt(0.5)))));
	} else if ((t / l) <= 5e+124) {
		tmp = Math.asin(Math.sqrt((t_1 / (1.0 + (2.0 * (1.0 / ((l / t) * (l / t))))))));
	} else {
		tmp = Math.asin(((l / t) / Math.sqrt(2.0)));
	}
	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) <= -4e+166:
		tmp = math.asin((math.sqrt(t_1) * (-l / (t / math.sqrt(0.5)))))
	elif (t / l) <= 5e+124:
		tmp = math.asin(math.sqrt((t_1 / (1.0 + (2.0 * (1.0 / ((l / t) * (l / t))))))))
	else:
		tmp = math.asin(((l / t) / math.sqrt(2.0)))
	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) <= -4e+166)
		tmp = asin(Float64(sqrt(t_1) * Float64(Float64(-l) / Float64(t / sqrt(0.5)))));
	elseif (Float64(t / l) <= 5e+124)
		tmp = asin(sqrt(Float64(t_1 / Float64(1.0 + Float64(2.0 * Float64(1.0 / Float64(Float64(l / t) * Float64(l / t))))))));
	else
		tmp = asin(Float64(Float64(l / t) / sqrt(2.0)));
	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) <= -4e+166)
		tmp = asin((sqrt(t_1) * (-l / (t / sqrt(0.5)))));
	elseif ((t / l) <= 5e+124)
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * (1.0 / ((l / t) * (l / t))))))));
	else
		tmp = asin(((l / t) / sqrt(2.0)));
	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], -4e+166], N[ArcSin[N[(N[Sqrt[t$95$1], $MachinePrecision] * N[((-l) / N[(t / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 5e+124], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(1.0 + N[(2.0 * N[(1.0 / N[(N[(l / t), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l / t), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -3.99999999999999976e166

   1. Initial program 31.7

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

   2. Taylor expanded in t around -inf 8.1

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

   3. Simplified 0.3

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\right)} \]
      Proof
      (*.f64 (sqrt.f64 (-.f64 1 (pow.f64 (/.f64 Om Omc) 2))) (neg.f64 (/.f64 l (/.f64 t (sqrt.f64 1/2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (-.f64 1 (Rewrite=> unpow2_binary64 (*.f64 (/.f64 Om Omc) (/.f64 Om Omc))))) (neg.f64 (/.f64 l (/.f64 t (sqrt.f64 1/2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (-.f64 1 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 Om Om) (*.f64 Omc Omc))))) (neg.f64 (/.f64 l (/.f64 t (sqrt.f64 1/2))))): 23 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 Om 2)) (*.f64 Omc Omc)))) (neg.f64 (/.f64 l (/.f64 t (sqrt.f64 1/2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (Rewrite<= unpow2_binary64 (pow.f64 Omc 2))))) (neg.f64 (/.f64 l (/.f64 t (sqrt.f64 1/2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2)))) (neg.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l (sqrt.f64 1/2)) t)))): 34 points increase in error, 26 points decrease in error
      (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2)))) (neg.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 1/2) l)) t))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2)))) (/.f64 (*.f64 (sqrt.f64 1/2) l) t)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (*.f64 (sqrt.f64 1/2) l) t) (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (/.f64 (*.f64 (sqrt.f64 1/2) l) t) (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2))))))): 0 points increase in error, 0 points decrease in error

   if -3.99999999999999976e166 < (/.f64 t l) < 4.9999999999999996e124

   1. Initial program 1.4

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

   2. Applied egg-rr 1.4

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

   if 4.9999999999999996e124 < (/.f64 t l)

   1. Initial program 30.4

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

   2. Applied egg-rr 1.2

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

   3. Taylor expanded in t around inf 7.6

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

   4. Simplified 7.6

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \frac{\frac{\ell}{t}}{\sqrt{2}}\right)} \]
      Proof
      (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (*.f64 Om Om) (*.f64 Omc Omc)))) (/.f64 (/.f64 l t) (sqrt.f64 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 Om 2)) (*.f64 Omc Omc)))) (/.f64 (/.f64 l t) (sqrt.f64 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (Rewrite<= unpow2_binary64 (pow.f64 Omc 2))))) (/.f64 (/.f64 l t) (sqrt.f64 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2)))) (Rewrite=> associate-/l/_binary64 (/.f64 l (*.f64 (sqrt.f64 2) t)))): 27 points increase in error, 35 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 l (*.f64 (sqrt.f64 2) t)) (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2)))))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in Om around 0 0.7

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

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+166}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+124}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{1}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \]

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	2.2
Cost	21000

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+166}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \frac{-\frac{\ell}{t}}{\sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+124}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{1}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 3
Error	2.3
Cost	20484

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+166}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \frac{-\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+124}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 4
Error	2.2
Cost	20484

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+166}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \frac{-\frac{\ell}{t}}{\sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+124}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 5
Error	5.3
Cost	14404

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{+124}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 6
Error	12.9
Cost	13896

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+208}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.04:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 7
Error	8.9
Cost	13896

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.04:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 8
Error	13.0
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+208}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.04:\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 9
Error	24.6
Cost	13384

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.8 \cdot 10^{-195}:\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{-66}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 10
Error	31.8
Cost	7104

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

Alternative 11
Error	31.9
Cost	6464

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

Reproduce

herbie shell --seed 2022325 
(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)))))))