Average Error: 10.3 → 5.5
Time: 1.3m
Precision: 64
Internal Precision: 128
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 3.523980809076174 \cdot 10^{+137}:\\ \;\;\;\;(e^{\log_* (1 + \sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}}\right))} - 1)^*\\ \mathbf{else}:\\ \;\;\;\;(e^{\log_* (1 + \sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right))} - 1)^*\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus Omc

Derivation

  1. Split input into 2 regimes

  2. if (/ t l) < 3.523980809076174e+137

    1. Initial program 6.4

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

    2. Simplified6.4

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

    4. Applied expm1-log1p-u6.4

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

    if 3.523980809076174e+137 < (/ t l)

    1. Initial program 32.0

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

    2. Simplified32.0

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

    4. Applied expm1-log1p-u32.0

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

    6. Applied sqrt-div32.0

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

    7. Taylor expanded around inf 1.0

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

  4. Final simplification5.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 3.523980809076174 \cdot 10^{+137}:\\ \;\;\;\;(e^{\log_* (1 + \sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}}\right))} - 1)^*\\ \mathbf{else}:\\ \;\;\;\;(e^{\log_* (1 + \sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right))} - 1)^*\\ \end{array}\]

Reproduce

herbie shell --seed 2019089 +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)))))))