Average Error: 10.3 → 10.4
Time: 1.9m
Precision: 64
Internal Precision: 128
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\left|\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + \frac{\frac{2}{\frac{\ell}{t}}}{\frac{\ell}{t}}}}\right|\right)\]

Error

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus Omc

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Results

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Derivation

  1. Initial program 10.3

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

  2. Initial simplification10.3

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

  4. Applied add-sqr-sqrt10.4

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

  5. Applied associate-/r*10.4

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

  7. Applied add-sqr-sqrt10.4

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

  8. Applied rem-sqrt-square10.4

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

  9. Simplified10.4

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

  10. Final simplification10.4

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

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed 2018278 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))