Average Error: 10.5 → 10.5
Time: 4.9m
Precision: 64
Internal Precision: 576
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\sqrt{1 + \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2}}}{\sqrt{1 + \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2}}}\right)\]

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

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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 add-sqr-sqrt10.5

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

  5. Applied associate-/r*10.5

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

  6. Final simplification10.5

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

Runtime

Time bar (total: 4.9m)Debug logProfile

herbie shell --seed 2018234 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))