Toniolo and Linder, Equation (2)

Average Error: 9.9 → 0.7

Time: 18.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \ell \cdot \frac{\sqrt{0.5}}{t}\\ t_2 := \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -2.756749336531866 \cdot 10^{+78}:\\ \;\;\;\;\sin^{-1} \left(t_2 \cdot \left(-t_1\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 1.9491841797855502 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(t_2 \cdot t_1\right)\\ \end{array} \]

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 (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc))))))
   (if (<= (/ t l) -2.756749336531866e+78)
     (asin (* t_2 (- t_1)))
     (if (<= (/ t l) 1.9491841797855502e+132)
       (log1p
        (expm1
         (asin
          (sqrt
           (/ (- 1.0 (pow (/ Om Omc) 2.0)) (fma 2.0 (pow (/ t l) 2.0) 1.0))))))
       (asin (* t_2 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 = sqrt((1.0 - ((Om / Omc) * (Om / Omc))));
	double tmp;
	if ((t / l) <= -2.756749336531866e+78) {
		tmp = asin((t_2 * -t_1));
	} else if ((t / l) <= 1.9491841797855502e+132) {
		tmp = log1p(expm1(asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / fma(2.0, pow((t / l), 2.0), 1.0))))));
	} else {
		tmp = asin((t_2 * 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(l * Float64(sqrt(0.5) / t))
	t_2 = sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc))))
	tmp = 0.0
	if (Float64(t / l) <= -2.756749336531866e+78)
		tmp = asin(Float64(t_2 * Float64(-t_1)));
	elseif (Float64(t / l) <= 1.9491841797855502e+132)
		tmp = log1p(expm1(asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / fma(2.0, (Float64(t / l) ^ 2.0), 1.0))))));
	else
		tmp = asin(Float64(t_2 * 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[(l * N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -2.756749336531866e+78], N[ArcSin[N[(t$95$2 * (-t$95$1)), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 1.9491841797855502e+132], N[Log[1 + N[(Exp[N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(t$95$2 * t$95$1), $MachinePrecision]], $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) < -2.7567493365318659e78

   1. Initial program 24.3

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

   2. Simplified 24.3

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

   3. Taylor expanded in t around -inf 7.8

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

   4. Simplified 0.3

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

   if -2.7567493365318659e78 < (/.f64 t l) < 1.9491841797855502e132

   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. Simplified 0.9

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

   3. Applied egg-rr 0.9

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

   if 1.9491841797855502e132 < (/.f64 t l)

   1. Initial program 31.0

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

   2. Simplified 31.0

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

   3. Taylor expanded in t around inf 7.9

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

   4. Simplified 0.3

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2.756749336531866 \cdot 10^{+78}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \left(-\ell \cdot \frac{\sqrt{0.5}}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 1.9491841797855502 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022148 
(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)))))))