Average Error: 10.5 → 10.5
Time: 1.3m
Precision: 64
Internal Precision: 128
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{1}{1 + \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2} - \frac{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2}}\right)\]

Error

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus Omc

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 10.5

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

  2. Initial simplification10.5

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

  4. Applied div-sub10.5

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

  5. Final simplification10.5

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018304 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))