Toniolo and Linder, Equation (2)

Average Error: 10.3 → 0.9

Time: 9.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+159}:\\ \;\;\;\;\sin^{-1} \left(t_1 \cdot \left(\ell \cdot \frac{-\sqrt{0.5}}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+70}:\\ \;\;\;\;\sin^{-1} \left({\left(\frac{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}{1 - {\left(\frac{Om}{Omc}\right)}^{2}}\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(t_1 \cdot \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\right)\\ \end{array} \]

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 (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc))))))
   (if (<= (/ t l) -1e+159)
     (asin (* t_1 (* l (/ (- (sqrt 0.5)) t))))
     (if (<= (/ t l) 2e+70)
       (asin
        (pow
         (/ (fma 2.0 (pow (/ t l) 2.0) 1.0) (- 1.0 (pow (/ Om Omc) 2.0)))
         -0.5))
       (asin (* t_1 (* 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 t_1 = sqrt((1.0 - ((Om / Omc) * (Om / Omc))));
	double tmp;
	if ((t / l) <= -1e+159) {
		tmp = asin((t_1 * (l * (-sqrt(0.5) / t))));
	} else if ((t / l) <= 2e+70) {
		tmp = asin(pow((fma(2.0, pow((t / l), 2.0), 1.0) / (1.0 - pow((Om / Omc), 2.0))), -0.5));
	} else {
		tmp = asin((t_1 * (l * (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)
	t_1 = sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc))))
	tmp = 0.0
	if (Float64(t / l) <= -1e+159)
		tmp = asin(Float64(t_1 * Float64(l * Float64(Float64(-sqrt(0.5)) / t))));
	elseif (Float64(t / l) <= 2e+70)
		tmp = asin((Float64(fma(2.0, (Float64(t / l) ^ 2.0), 1.0) / Float64(1.0 - (Float64(Om / Omc) ^ 2.0))) ^ -0.5));
	else
		tmp = asin(Float64(t_1 * Float64(l * Float64(sqrt(0.5) / t))));
	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[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -1e+159], N[ArcSin[N[(t$95$1 * N[(l * N[((-N[Sqrt[0.5], $MachinePrecision]) / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 2e+70], N[ArcSin[N[Power[N[(N[(2.0 * N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] / N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(t$95$1 * N[(l * N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $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) < -9.9999999999999993e158

   1. Initial program 33.7

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

   2. Simplified 33.7

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

   3. Taylor expanded in t around -inf 8.2

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

   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 -9.9999999999999993e158 < (/.f64 t l) < 2.00000000000000015e70

   1. Initial program 1.2

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

   2. Simplified 1.2

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

   3. Applied egg-rr 1.3

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

   4. Applied egg-rr 1.2

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

   if 2.00000000000000015e70 < (/.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. Simplified 24.4

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

   3. Taylor expanded in t around inf 8.1

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

   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 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+159}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \left(\ell \cdot \frac{-\sqrt{0.5}}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+70}:\\ \;\;\;\;\sin^{-1} \left({\left(\frac{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}{1 - {\left(\frac{Om}{Omc}\right)}^{2}}\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(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)))))))