Toniolo and Linder, Equation (2)

Average Error: 10.1 → 1.0

Time: 22.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\ell \cdot \sqrt{0.5}}{t}\\ t_2 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ t_3 := \sqrt{t_2}\\ \mathbf{if}\;\frac{t}{\ell} \leq -3.022764886422071 \cdot 10^{+163}:\\ \;\;\;\;\sin^{-1} \left(t_3 \cdot \left(-t_1\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 1.0047715027799692 \cdot 10^{+107}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\frac{t_2}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(t_3 \cdot t_1\right)\right)\right)\\ \end{array} \]

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 (/ (* l (sqrt 0.5)) t))
        (t_2 (- 1.0 (pow (/ Om Omc) 2.0)))
        (t_3 (sqrt t_2)))
   (if (<= (/ t l) -3.022764886422071e+163)
     (asin (* t_3 (- t_1)))
     (if (<= (/ t l) 1.0047715027799692e+107)
       (expm1 (log1p (asin (sqrt (/ t_2 (fma 2.0 (pow (/ t l) 2.0) 1.0))))))
       (expm1 (log1p (asin (* t_3 t_1))))))))

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 = (l * sqrt(0.5)) / t;
	double t_2 = 1.0 - pow((Om / Omc), 2.0);
	double t_3 = sqrt(t_2);
	double tmp;
	if ((t / l) <= -3.022764886422071e+163) {
		tmp = asin((t_3 * -t_1));
	} else if ((t / l) <= 1.0047715027799692e+107) {
		tmp = expm1(log1p(asin(sqrt((t_2 / fma(2.0, pow((t / l), 2.0), 1.0))))));
	} else {
		tmp = expm1(log1p(asin((t_3 * t_1))));
	}
	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(Float64(l * sqrt(0.5)) / t)
	t_2 = Float64(1.0 - (Float64(Om / Omc) ^ 2.0))
	t_3 = sqrt(t_2)
	tmp = 0.0
	if (Float64(t / l) <= -3.022764886422071e+163)
		tmp = asin(Float64(t_3 * Float64(-t_1)));
	elseif (Float64(t / l) <= 1.0047715027799692e+107)
		tmp = expm1(log1p(asin(sqrt(Float64(t_2 / fma(2.0, (Float64(t / l) ^ 2.0), 1.0))))));
	else
		tmp = expm1(log1p(asin(Float64(t_3 * t_1))));
	end
	return 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[(N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[t$95$2], $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -3.022764886422071e+163], N[ArcSin[N[(t$95$3 * (-t$95$1)), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 1.0047715027799692e+107], N[(Exp[N[Log[1 + N[ArcSin[N[Sqrt[N[(t$95$2 / N[(2.0 * N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision], N[(Exp[N[Log[1 + N[ArcSin[N[(t$95$3 * t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]]]]]]

Error

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus Omc

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -3.02276488642207099e163

   1. Initial program 32.5

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

   2. Simplified 32.5

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

   3. Taylor expanded in t around -inf 8.5

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

   4. Simplified 0.2

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

   if -3.02276488642207099e163 < (/.f64 t l) < 1.0047715027799692e107

   1. Initial program 1.3

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

   2. Simplified 1.3

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

   3. Applied expm1-log1p-u_binary64 1.3

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)} \]

   if 1.0047715027799692e107 < (/.f64 t l)

   1. Initial program 28.6

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

   2. Simplified 28.6

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

   3. Applied expm1-log1p-u_binary64 28.6

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)} \]

   4. Taylor expanded in t around inf 7.4

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)}\right)\right) \]

   5. Simplified 0.3

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -3.022764886422071 \cdot 10^{+163}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 1.0047715027799692 \cdot 10^{+107}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(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)))))))