Average Error: 10.2 → 10.2
Time: 1.1m
Precision: 64
Internal Precision: 128
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[(e^{\log_* (1 + \sin^{-1} \left(\sqrt{(e^{\log_* (1 + \frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*})} - 1)^*}\right))} - 1)^*\]

Error

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus Omc

Derivation

  1. Initial program 10.2

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

  2. Initial simplification10.2

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

  4. Applied expm1-log1p-u10.2

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

  6. Applied expm1-log1p-u10.2

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

  7. Final simplification10.2

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018346 +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)))))))