Average Error: 10.8 → 6.0
Time: 3.1m
Precision: 64
Internal Precision: 384
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right) \le 0.0:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\sqrt{\frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}} + 1}}\right)\\ \mathbf{if}\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right) \le 3.405148864100225 \cdot 10^{-83}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 - \left(\frac{t}{\ell} \cdot \left(2 \cdot 2\right)\right) \cdot {\left(\frac{t}{\ell}\right)}^{3}} \cdot \left(1 - 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}\right)\\ \end{array}\]

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 (asin (/ (sqrt (- 1 (* (/ Om Omc) (/ Om Omc)))) (/ (* t (sqrt 2)) l))) < 0.0

    1. Initial program 16.3

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

    3. Applied sqrt-div16.4

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

    4. Taylor expanded around inf 50.8

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

    5. Applied simplify16.4

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

    if 0.0 < (asin (/ (sqrt (- 1 (* (/ Om Omc) (/ Om Omc)))) (/ (* t (sqrt 2)) l))) < 3.405148864100225e-83

    1. Initial program 39.4

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

    3. Applied sqrt-div39.4

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

    4. Taylor expanded around inf 57.2

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

    5. Applied simplify39.5

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

    6. Taylor expanded around 0 0.5

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

    if 3.405148864100225e-83 < (asin (/ (sqrt (- 1 (* (/ Om Omc) (/ Om Omc)))) (/ (* t (sqrt 2)) l)))

    1. Initial program 1.1

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

    3. Applied flip-+1.4

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

    4. Applied associate-/r/1.4

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

    5. Applied simplify1.4

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

Runtime

Time bar (total: 3.1m)Debug logProfile

herbie shell --seed '#(1070833653 108281690 3330367898 3632331308 3494323072 43156186)' 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))