Toniolo and Linder, Equation (2)

Average Error: 10.2 → 1.0

Time: 15.1s

Precision: binary64

Cost: 34434

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1.4397610098885724 \cdot 10^{+75}:\\ \;\;\;\;\sin^{-1} \left(-\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4.0699161400542827 \cdot 10^{+55}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 - 4 \cdot {\left(\frac{t}{\ell}\right)}^{4}} \cdot \left(1 - 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array}\]

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) -1.4397610098885724e+75)
   (asin (- (* (sqrt (- 1.0 (pow (/ Om Omc) 2.0))) (/ (sqrt 0.5) (/ t l)))))
   (if (<= (/ t l) 4.0699161400542827e+55)
     (asin
      (sqrt
       (*
        (/ (- 1.0 (pow (/ Om Omc) 2.0)) (- 1.0 (* 4.0 (pow (/ t l) 4.0))))
        (- 1.0 (* 2.0 (pow (/ t l) 2.0))))))
     (asin (* (sqrt (- 1.0 (pow (/ Om Omc) 2.0))) (/ (sqrt 0.5) (/ t l)))))))

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) <= -1.4397610098885724e+75) {
		tmp = asin(-(sqrt(1.0 - pow((Om / Omc), 2.0)) * (sqrt(0.5) / (t / l))));
	} else if ((t / l) <= 4.0699161400542827e+55) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 - (4.0 * pow((t / l), 4.0)))) * (1.0 - (2.0 * pow((t / l), 2.0)))));
	} else {
		tmp = asin(sqrt(1.0 - pow((Om / Omc), 2.0)) * (sqrt(0.5) / (t / l)));
	}
	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

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -3.5189165217235146 \cdot 10^{+142}:\\ \;\;\;\;\sin^{-1} \left(-\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2.7098674073710237 \cdot 10^{+62}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\frac{t}{\ell} \cdot \sqrt{2}}\right)\\ \end{array}\]

Alternative 2
Error	1.0
Cost	27266

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2.4191898762670895 \cdot 10^{+119}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{-\frac{t}{\ell} \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 3.945702929768114 \cdot 10^{+68}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\frac{t}{\ell} \cdot \sqrt{2}}\right)\\ \end{array}\]

Alternative 3
Error	5.9
Cost	26817

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 4.0408429634684533 \cdot 10^{+68}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\frac{t}{\ell} \cdot \sqrt{2}}\right)\\ \end{array}\]

Alternative 4
Error	10.2
Cost	20352

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

Alternative 5
Error	10.7
Cost	19840

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

Alternative 6
Error	27.1
Cost	20226

\[\begin{array}{l} \mathbf{if}\;t \leq -3.8979961988787077 \cdot 10^{+174}:\\ \;\;\;\;0\\ \mathbf{elif}\;t \leq 9.812738210921705 \cdot 10^{+151}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 7
Error	49.6
Cost	706

\[\begin{array}{l} \mathbf{if}\;t \leq -4.132669693900505 \cdot 10^{+38}:\\ \;\;\;\;0\\ \mathbf{elif}\;t \leq 3.2969184450764058 \cdot 10^{+112}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 8
Error	55.9
Cost	64

\[1\]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -1.4397610098885724e75

   1. Initial program 25.2

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

   2. Taylor expanded around -inf 7.1

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

   3. Simplified 1.1

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

   4. Simplified 1.1

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

   if -1.4397610098885724e75 < (/.f64 t l) < 4.0699161400542827e55

   1. Initial program 1.0

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

   3. Applied flip-+_binary64_52 1.0

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

   4. Applied associate-/r/_binary64_24 1.0

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

   5. Simplified 1.0

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 - 4 \cdot {\left(\frac{t}{\ell}\right)}^{4}}} \cdot \left(1 - 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}\right)\]

   6. Simplified 1.0

      \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 - 4 \cdot {\left(\frac{t}{\ell}\right)}^{4}} \cdot \left(1 - 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}\right)}\]

   if 4.0699161400542827e55 < (/.f64 t l)

   1. Initial program 22.7

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

   2. Taylor expanded around inf 8.5

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

   3. Simplified 1.0

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

   4. Simplified 1.0

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

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1.4397610098885724 \cdot 10^{+75}:\\ \;\;\;\;\sin^{-1} \left(-\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4.0699161400542827 \cdot 10^{+55}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 - 4 \cdot {\left(\frac{t}{\ell}\right)}^{4}} \cdot \left(1 - 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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)))))))