Average Error: 10.7 → 10.8
Time: 2.0m
Precision: 64
Internal Precision: 576
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}} \cdot \sqrt{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}\right)\]

Error

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus Omc

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 10.7

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

  3. Applied sqrt-div10.8

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

  5. Applied add-sqr-sqrt10.8

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

  6. Applied sqrt-prod10.8

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

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed 2018208 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))