Average Error: 10.9 → 1.0
Time: 19.4s
Precision: 64
Internal precision: 384
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^2}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^2}}\right)\]
⬇
\[\begin{array}{l} \mathbf{if}\;\frac{1 - {\left(\frac{Om}{Omc}\right)}^2}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^2} \le 1.1249976801847743 \cdot 10^{-288}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^2}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \log \left(e^{{\left(\frac{Om}{Omc}\right)}^2}\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^2}}\right)\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus Omc

Derivation

  1. Split input into 2 regimes.

  2. if (/ (- 1 (sqr (/ Om Omc))) (+ 1 (* 2 (sqr (/ t l))))) < 1.1249976801847743e-288

    1. Initial program 34.9

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

    3. Applied sqrt-div 34.9

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

    4. Applied taylor 1.2

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

    5. Taylor expanded around inf 1.2

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

    if 1.1249976801847743e-288 < (/ (- 1 (sqr (/ Om Omc))) (+ 1 (* 2 (sqr (/ t l)))))

    1. Initial program 0.9

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

    3. Applied add-log-exp 0.9

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

Runtime

Time bar (total: 19.4s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1064524629 4159152179 2999149171 575749698 4006532819 692958815)'
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (sqr (/ Om Omc))) (+ 1 (* 2 (sqr (/ t l))))))))