Average Error: 10.3 → 8.2
Time: 2.4m
Precision: 64
Internal Precision: 576
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{\frac{1}{2} \cdot (-2 \cdot \left(\log t\right) + \left((\left(\log \ell\right) \cdot 2 + \left(\log \frac{1}{2}\right))_*\right))_*} \cdot \sqrt{(\left(Om \cdot \frac{-1}{Omc}\right) \cdot \left(\frac{1}{Omc} \cdot Om\right) + 1)_*} \le 2.9841571165438344 \cdot 10^{-155}:\\ \;\;\;\;\sin^{-1} \left(e^{\frac{1}{2} \cdot (-2 \cdot \left(\log t\right) + \left((\left(\log \ell\right) \cdot 2 + \left(\log \frac{1}{2}\right))_*\right))_*} \cdot \sqrt{(\left(Om \cdot \frac{-1}{Omc}\right) \cdot \left(\frac{1}{Omc} \cdot Om\right) + 1)_*}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus Omc

Derivation

  1. Split input into 2 regimes

  2. if (* (exp (* (fma -2 (log t) (fma (log l) 2 (log 1/2))) 1/2)) (sqrt (fma (* (/ 1 Omc) (- Om)) (* (/ 1 Omc) Om) 1))) < 2.9841571165438344e-155

    1. Initial program 34.4

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

    3. Applied add-cube-cbrt34.4

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

    4. Taylor expanded around inf 48.2

      \[\leadsto \sin^{-1} \color{blue}{\left(\left(e^{\frac{1}{3} \cdot \left(\left(\log \frac{1}{2} + 2 \cdot \log \left(\frac{1}{t}\right)\right) - 2 \cdot \log \left(\frac{1}{\ell}\right)\right)} \cdot e^{\frac{1}{6} \cdot \left(\left(\log \frac{1}{2} + 2 \cdot \log \left(\frac{1}{t}\right)\right) - 2 \cdot \log \left(\frac{1}{\ell}\right)\right)}\right) \cdot \sqrt{1 - e^{2 \cdot \left(\log \left(\frac{1}{Omc}\right) - \log \left(\frac{1}{Om}\right)\right)}}\right)}\]

    5. Applied simplify4.1

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

    if 2.9841571165438344e-155 < (* (exp (* (fma -2 (log t) (fma (log l) 2 (log 1/2))) 1/2)) (sqrt (fma (* (/ 1 Omc) (- Om)) (* (/ 1 Omc) Om) 1)))

    1. Initial program 8.5

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
  3. Recombined 2 regimes into one program.

  4. Applied simplify8.2

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;e^{\frac{1}{2} \cdot (-2 \cdot \left(\log t\right) + \left((\left(\log \ell\right) \cdot 2 + \left(\log \frac{1}{2}\right))_*\right))_*} \cdot \sqrt{(\left(Om \cdot \frac{-1}{Omc}\right) \cdot \left(\frac{1}{Omc} \cdot Om\right) + 1)_*} \le 2.9841571165438344 \cdot 10^{-155}:\\ \;\;\;\;\sin^{-1} \left(e^{\frac{1}{2} \cdot (-2 \cdot \left(\log t\right) + \left((\left(\log \ell\right) \cdot 2 + \left(\log \frac{1}{2}\right))_*\right))_*} \cdot \sqrt{(\left(Om \cdot \frac{-1}{Omc}\right) \cdot \left(\frac{1}{Omc} \cdot Om\right) + 1)_*}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \end{array}}\]

Runtime

Time bar (total: 2.4m)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' +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)))))))