Average Error: 10.4 → 5.5
Time: 31.5s
Precision: 64
Internal Precision: 128
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 1.5669942029752657 \cdot 10^{+150}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\left|\sqrt{\frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}} + 1}\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\frac{\sqrt{2} \cdot t}{\ell}}\right)\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus Omc

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (/ t l) < 1.5669942029752657e+150

    1. Initial program 6.1

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

    3. Applied sqrt-div6.2

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

    5. Applied add-cube-cbrt6.3

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

    7. Applied add-sqr-sqrt6.3

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

    8. Applied rem-sqrt-square6.3

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

    9. Simplified6.2

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

    if 1.5669942029752657e+150 < (/ t l)

    1. Initial program 35.3

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

    3. Applied sqrt-div35.3

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

    5. Applied add-cube-cbrt35.3

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

    6. Taylor expanded around inf 1.3

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

  4. Final simplification5.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 1.5669942029752657 \cdot 10^{+150}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\left|\sqrt{\frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}} + 1}\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\frac{\sqrt{2} \cdot t}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019026 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))