Toniolo and Linder, Equation (2)

Average Error: 10.3 → 1.0

Time: 5.0min

Precision: binary64

Cost: 27714

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5.565984449169692 \cdot 10^{+160}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 3.7779228963760736 \cdot 10^{+110}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot {\left(1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array}\]

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
 (if (<= (/ t l) -5.565984449169692e+160)
   (asin (* (sqrt (- 1.0 (pow (/ Om Omc) 2.0))) (- (/ (sqrt 0.5) (/ t l)))))
   (if (<= (/ t l) 3.7779228963760736e+110)
     (asin
      (*
       (sqrt (- 1.0 (pow (/ Om Omc) 2.0)))
       (pow (+ 1.0 (* 2.0 (* (/ t l) (/ t l)))) -0.5)))
     (asin (* (sqrt (- 1.0 (pow (/ Om Omc) 2.0))) (/ (* 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 tmp;
	if ((t / l) <= -5.565984449169692e+160) {
		tmp = asin(sqrt(1.0 - pow((Om / Omc), 2.0)) * -(sqrt(0.5) / (t / l)));
	} else if ((t / l) <= 3.7779228963760736e+110) {
		tmp = asin(sqrt(1.0 - pow((Om / Omc), 2.0)) * pow((1.0 + (2.0 * ((t / l) * (t / l)))), -0.5));
	} else {
		tmp = asin(sqrt(1.0 - pow((Om / Omc), 2.0)) * ((l * sqrt(0.5)) / t));
	}
	return tmp;
}

Error

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus Omc

Alternatives

Alternative 1
Error	1.0
Cost	27266

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5.565984449169692 \cdot 10^{+160}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 3.7779228963760736 \cdot 10^{+110}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array}\]

Alternative 2
Error	1.9
Cost	27266

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5.565984449169692 \cdot 10^{+160}:\\ \;\;\;\;\sin^{-1} \left(\left(-\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right) \cdot \sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 3.7779228963760736 \cdot 10^{+110}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array}\]

Alternative 3
Error	3.2
Cost	21250

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5.565984449169692 \cdot 10^{+160}:\\ \;\;\;\;\sin^{-1} \left(\left(-\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right) \cdot \sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5.611029736904483 \cdot 10^{+110}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}} \cdot \sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}}\right)\\ \end{array}\]

Alternative 4
Error	4.0
Cost	21250

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2.646785830037139 \cdot 10^{+68}:\\ \;\;\;\;\sin^{-1} \left(\left(-\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right) \cdot \sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5.611029736904483 \cdot 10^{+110}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(t \cdot \frac{\frac{t}{\ell}}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}} \cdot \sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}}\right)\\ \end{array}\]

Alternative 5
Error	3.7
Cost	21058

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5.565984449169692 \cdot 10^{+160}:\\ \;\;\;\;\sin^{-1} \left(\left(-\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right) \cdot \sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5.611029736904483 \cdot 10^{+110}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}} \cdot \sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}}\right)\\ \end{array}\]

Alternative 6
Error	7.1
Cost	20609

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5.611029736904483 \cdot 10^{+110}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}} \cdot \sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}}\right)\\ \end{array}\]

Alternative 7
Error	10.8
Cost	19840

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

Alternative 8
Error	12.7
Cost	13632

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

Alternative 9
Error	20.3
Cost	14274

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.0930739330951373 \cdot 10^{+70}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\left(1 + \frac{Om}{Omc}\right) \cdot \left(1 - \frac{Om}{Omc}\right)}\right)\\ \mathbf{elif}\;\ell \leq 6.07791287533962 \cdot 10^{+114}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - Om \cdot \frac{\frac{Om}{Omc}}{Omc}}\right)\\ \end{array}\]

Alternative 10
Error	27.9
Cost	14018

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.493781829556217 \cdot 10^{-79}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\left(1 + \frac{Om}{Omc}\right) \cdot \left(1 - \frac{Om}{Omc}\right)}\right)\\ \mathbf{elif}\;\ell \leq 4.2526083201266566 \cdot 10^{-132}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - Om \cdot \frac{\frac{Om}{Omc}}{Omc}}\right)\\ \end{array}\]

Alternative 11
Error	27.9
Cost	13704

\[\begin{array}{l} \mathbf{if}\;\ell \leq -7.290002181617712 \cdot 10^{-80} \lor \neg \left(\ell \leq 5.275550852173152 \cdot 10^{-132}\right):\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - Om \cdot \frac{\frac{Om}{Omc}}{Omc}}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 12
Error	28.1
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\ell \leq -7.290002181617712 \cdot 10^{-80} \lor \neg \left(\ell \leq 4.2526083201266566 \cdot 10^{-132}\right):\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{Om}{Omc}\right)}^{2} \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 13
Error	28.3
Cost	6792

\[\begin{array}{l} \mathbf{if}\;\ell \leq -7.290002181617712 \cdot 10^{-80} \lor \neg \left(\ell \leq 1.0144111720022664 \cdot 10^{-131}\right):\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 14
Error	49.5
Cost	706

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.6236427104905465 \cdot 10^{-79}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 2.740599975692102 \cdot 10^{-46}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Alternative 15
Error	53.9
Cost	64

\[0\]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -5.56598444916969186e160

   1. Initial program 32.9

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

   3. Applied div-inv_binary64_75 32.9

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

   4. Applied sqrt-prod_binary64_94 32.9

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

   5. Taylor expanded around -inf 0.3

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

   6. Simplified 1.2

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

   7. Simplified 1.2

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

   if -5.56598444916969186e160 < (/.f64 t l) < 3.77792289637607363e110

   1. Initial program 1.1

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

   3. Applied div-inv_binary64_75 1.1

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

   4. Applied sqrt-prod_binary64_94 1.1

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

   6. Applied inv-pow_binary64_163 1.1

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

   7. Applied sqrt-pow1_binary64_96 1.1

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

   8. Simplified 1.1

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

   10. Applied add-sqr-sqrt_binary64_100 28.3

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

   11. Applied unpow-prod-down_binary64_157 28.3

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

   12. Simplified 28.3

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

   13. Simplified 1.1

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

   14. Simplified 1.1

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

   if 3.77792289637607363e110 < (/.f64 t l)

   1. Initial program 28.7

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

   3. Applied *-un-lft-identity_binary64_78 28.7

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

   4. Applied add-cube-cbrt_binary64_113 28.9

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

   5. Applied times-frac_binary64_84 28.9

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

   6. Applied unpow-prod-down_binary64_157 33.6

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

   7. Simplified 33.5

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

   8. Taylor expanded around inf 7.0

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

   9. Simplified 0.3

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

   10. Simplified 0.3

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

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5.565984449169692 \cdot 10^{+160}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 3.7779228963760736 \cdot 10^{+110}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot {\left(1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021014 
(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)))))))