Average Error: 10.0 → 8.7
Time: 2.9m
Precision: 64
Internal Precision: 576
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{-1}{Omc}}{\frac{-1}{Om}} \cdot \frac{\frac{-1}{Omc}}{\frac{-1}{Om}}} \cdot e^{(\left(\log \left(\frac{-1}{t}\right) - \log \left(\frac{-1}{\ell}\right)\right) \cdot 2 + \left(\log \frac{1}{2}\right))_* \cdot \left(\frac{1}{6} + \frac{1}{3}\right)}\right) \le 3.1414898372282937 \cdot 10^{-208}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{-1}{Omc}}{\frac{-1}{Om}} \cdot \frac{\frac{-1}{Omc}}{\frac{-1}{Om}}} \cdot e^{(\left(\log \left(\frac{-1}{t}\right) - \log \left(\frac{-1}{\ell}\right)\right) \cdot 2 + \left(\log \frac{1}{2}\right))_* \cdot \left(\frac{1}{6} + \frac{1}{3}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;(e^{\log_* (1 + \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right))} - 1)^*\\ \end{array}\]

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 (asin (* (sqrt (- 1 (* (/ (/ -1 Omc) (/ -1 Om)) (/ (/ -1 Omc) (/ -1 Om))))) (exp (* (fma (- (log (/ -1 t)) (log (/ -1 l))) 2 (log 1/2)) (+ 1/6 1/3))))) < 3.1414898372282937e-208

    1. Initial program 27.2

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

    3. Applied add-cube-cbrt27.2

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

    4. Applied add-cube-cbrt27.2

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

    5. Applied times-frac27.2

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

    6. Taylor expanded around -inf 47.8

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

    7. Applied simplify3.3

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

    if 3.1414898372282937e-208 < (asin (* (sqrt (- 1 (* (/ (/ -1 Omc) (/ -1 Om)) (/ (/ -1 Omc) (/ -1 Om))))) (exp (* (fma (- (log (/ -1 t)) (log (/ -1 l))) 2 (log 1/2)) (+ 1/6 1/3)))))

    1. Initial program 9.0

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

    3. Applied expm1-log1p-u9.0

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

Runtime

Time bar (total: 2.9m)Debug logProfile

herbie shell --seed '#(1072330854 3074818769 591214268 3603999196 3863745332 3332387116)' +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)))))))