Average Error: 10.6 → 10.3
Time: 2.9m
Precision: 64
Internal Precision: 384
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 2.28804701721715 \cdot 10^{+160}:\\ \;\;\;\;(e^{\log_* (1 + \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right))} - 1)^*\\ \mathbf{if}\;\frac{t}{\ell} \le 2.884017844939365 \cdot 10^{+206}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{-1}{Omc}}{\frac{-1}{Om}} \cdot \frac{\frac{-1}{Omc}}{\frac{-1}{Om}}} \cdot e^{(\left(\log \left(\frac{-1}{t}\right) - \log \left(\frac{-1}{\ell}\right)\right) \cdot 2 + \left(\log \frac{1}{2}\right))_* \cdot \left(\frac{1}{6} + \frac{1}{3}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(e^{\left(\frac{1}{6} + \frac{1}{3}\right) \cdot \left((\left(\log \ell\right) \cdot 2 + \left(\log \frac{1}{2}\right))_* - 2 \cdot \log t\right)} \cdot \sqrt{(\left(\frac{1}{Omc} \cdot Om\right) \cdot \left(\left(-Om\right) \cdot \frac{1}{Omc}\right) + 1)_*}\right)\\ \end{array}\]

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 (/ t l) < 2.28804701721715e+160

    1. Initial program 6.8

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

    3. Applied expm1-log1p-u6.8

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

    if 2.28804701721715e+160 < (/ t l) < 2.884017844939365e+206

    1. Initial program 60.7

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

    3. Applied add-cube-cbrt60.7

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

    4. Applied add-cube-cbrt60.7

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

    5. Applied times-frac60.7

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

    6. Taylor expanded around -inf 57.4

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

    7. Applied simplify35.6

      \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{1 - \frac{\frac{-1}{Omc}}{\frac{-1}{Om}} \cdot \frac{\frac{-1}{Omc}}{\frac{-1}{Om}}} \cdot e^{(\left(\log \left(\frac{-1}{t}\right) - \log \left(\frac{-1}{\ell}\right)\right) \cdot 2 + \left(\log \frac{1}{2}\right))_* \cdot \left(\frac{1}{6} + \frac{1}{3}\right)}\right)}\]

    if 2.884017844939365e+206 < (/ t l)

    1. Initial program 27.7

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

    3. Applied add-cube-cbrt27.7

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

    4. Applied add-cube-cbrt27.7

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

    5. Applied times-frac27.7

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

    6. Taylor expanded around inf 55.2

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

    7. Applied simplify31.8

      \[\leadsto \color{blue}{\sin^{-1} \left(e^{\left(\frac{1}{6} + \frac{1}{3}\right) \cdot \left((\left(\log \ell\right) \cdot 2 + \left(\log \frac{1}{2}\right))_* - 2 \cdot \log t\right)} \cdot \sqrt{(\left(\frac{1}{Omc} \cdot Om\right) \cdot \left(\left(-Om\right) \cdot \frac{1}{Omc}\right) + 1)_*}\right)}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 2.9m)Debug logProfile

herbie shell --seed '#(1070609872 3456127585 2380521889 2328837196 1765472538 734540918)' +o rules:numerics
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))