Toniolo and Linder, Equation (2)

Average Error: 10.1 → 3.5

Time: 10.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := 1 - \frac{Om \cdot Om}{Omc \cdot Omc}\\ t_2 := \sqrt{t_1} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\\ t_3 := \sqrt{\frac{1}{t_1}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -3.022764886422071 \cdot 10^{+163}:\\ \;\;\;\;\sin^{-1} \left(-t_2\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2.1581554477552666 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(t_2 - 0.125 \cdot \left(t_3 \cdot \frac{{\ell}^{3}}{\sqrt{0.5} \cdot {t}^{3}} - t_3 \cdot \frac{\left(Om \cdot Om\right) \cdot {\ell}^{3}}{\sqrt{0.5} \cdot \left(\left(Omc \cdot Omc\right) \cdot {t}^{3}\right)}\right)\right)\\ \end{array} \]

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 (- 1.0 (/ (* Om Om) (* Omc Omc))))
        (t_2 (* (sqrt t_1) (/ (* l (sqrt 0.5)) t)))
        (t_3 (sqrt (/ 1.0 t_1))))
   (if (<= (/ t l) -3.022764886422071e+163)
     (asin (- t_2))
     (if (<= (/ t l) 2.1581554477552666e+146)
       (expm1
        (log1p
         (asin
          (sqrt
           (/ (- 1.0 (pow (/ Om Omc) 2.0)) (fma 2.0 (pow (/ t l) 2.0) 1.0))))))
       (asin
        (-
         t_2
         (*
          0.125
          (-
           (* t_3 (/ (pow l 3.0) (* (sqrt 0.5) (pow t 3.0))))
           (*
            t_3
            (/
             (* (* Om Om) (pow l 3.0))
             (* (sqrt 0.5) (* (* Omc Omc) (pow t 3.0)))))))))))))

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 - ((Om * Om) / (Omc * Omc));
	double t_2 = sqrt(t_1) * ((l * sqrt(0.5)) / t);
	double t_3 = sqrt((1.0 / t_1));
	double tmp;
	if ((t / l) <= -3.022764886422071e+163) {
		tmp = asin(-t_2);
	} else if ((t / l) <= 2.1581554477552666e+146) {
		tmp = expm1(log1p(asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / fma(2.0, pow((t / l), 2.0), 1.0))))));
	} else {
		tmp = asin((t_2 - (0.125 * ((t_3 * (pow(l, 3.0) / (sqrt(0.5) * pow(t, 3.0)))) - (t_3 * (((Om * Om) * pow(l, 3.0)) / (sqrt(0.5) * ((Omc * Omc) * pow(t, 3.0)))))))));
	}
	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(Float64(Om * Om) / Float64(Omc * Omc)))
	t_2 = Float64(sqrt(t_1) * Float64(Float64(l * sqrt(0.5)) / t))
	t_3 = sqrt(Float64(1.0 / t_1))
	tmp = 0.0
	if (Float64(t / l) <= -3.022764886422071e+163)
		tmp = asin(Float64(-t_2));
	elseif (Float64(t / l) <= 2.1581554477552666e+146)
		tmp = expm1(log1p(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 - Float64(0.125 * Float64(Float64(t_3 * Float64((l ^ 3.0) / Float64(sqrt(0.5) * (t ^ 3.0)))) - Float64(t_3 * Float64(Float64(Float64(Om * Om) * (l ^ 3.0)) / Float64(sqrt(0.5) * Float64(Float64(Omc * Omc) * (t ^ 3.0)))))))));
	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[(1.0 - N[(N[(Om * Om), $MachinePrecision] / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[t$95$1], $MachinePrecision] * N[(N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -3.022764886422071e+163], N[ArcSin[(-t$95$2)], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 2.1581554477552666e+146], N[(Exp[N[Log[1 + 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]], $MachinePrecision]] - 1), $MachinePrecision], N[ArcSin[N[(t$95$2 - N[(0.125 * N[(N[(t$95$3 * N[(N[Power[l, 3.0], $MachinePrecision] / N[(N[Sqrt[0.5], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$3 * N[(N[(N[(Om * Om), $MachinePrecision] * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[(Omc * Omc), $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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) < -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 8.5

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

   if -3.02276488642207099e163 < (/.f64 t l) < 2.158155447755267e146

   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 1.2

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

   if 2.158155447755267e146 < (/.f64 t l)

   1. Initial program 33.6

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

   2. Simplified 33.6

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

   3. Applied egg-rr 34.1

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

   4. Taylor expanded in t around inf 10.3

      \[\leadsto \sin^{-1} \color{blue}{\left(\left(0.125 \cdot \left(\frac{{Om}^{2} \cdot {\ell}^{3}}{\sqrt{0.5} \cdot \left({t}^{3} \cdot {Omc}^{2}\right)} \cdot \sqrt{\frac{1}{1 - \frac{{Om}^{2}}{{Omc}^{2}}}}\right) + \frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) - 0.125 \cdot \left(\frac{{\ell}^{3}}{\sqrt{0.5} \cdot {t}^{3}} \cdot \sqrt{\frac{1}{1 - \frac{{Om}^{2}}{{Omc}^{2}}}}\right)\right)} \]

   5. Simplified 10.3

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t} - 0.125 \cdot \left(\sqrt{\frac{1}{1 - \frac{Om \cdot Om}{Omc \cdot Omc}}} \cdot \frac{{\ell}^{3}}{\sqrt{0.5} \cdot {t}^{3}} - \sqrt{\frac{1}{1 - \frac{Om \cdot Om}{Omc \cdot Omc}}} \cdot \frac{\left(Om \cdot Om\right) \cdot {\ell}^{3}}{\sqrt{0.5} \cdot \left(\left(Omc \cdot Omc\right) \cdot {t}^{3}\right)}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 3.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -3.022764886422071 \cdot 10^{+163}:\\ \;\;\;\;\sin^{-1} \left(-\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2.1581554477552666 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t} - 0.125 \cdot \left(\sqrt{\frac{1}{1 - \frac{Om \cdot Om}{Omc \cdot Omc}}} \cdot \frac{{\ell}^{3}}{\sqrt{0.5} \cdot {t}^{3}} - \sqrt{\frac{1}{1 - \frac{Om \cdot Om}{Omc \cdot Omc}}} \cdot \frac{\left(Om \cdot Om\right) \cdot {\ell}^{3}}{\sqrt{0.5} \cdot \left(\left(Omc \cdot Omc\right) \cdot {t}^{3}\right)}\right)\right)\\ \end{array} \]

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)))))))