Average Error: 10.5 → 8.2
Time: 2.6m
Precision: 64
Internal Precision: 576
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{-1}{Omc}}{\frac{-1}{Om}} \cdot \frac{\frac{-1}{Omc}}{\frac{-1}{Om}}} \cdot e^{(\left(\log \left(\frac{-1}{t}\right) - \log \left(\frac{-1}{\ell}\right)\right) \cdot 2 + \left(\log \frac{1}{2}\right))_* \cdot \left(\frac{1}{6} + \frac{1}{3}\right)}\right) \le 4.583463697946747 \cdot 10^{-155}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{-1}{Omc}}{\frac{-1}{Om}} \cdot \frac{\frac{-1}{Omc}}{\frac{-1}{Om}}} \cdot e^{(\left(\log \left(\frac{-1}{t}\right) - \log \left(\frac{-1}{\ell}\right)\right) \cdot 2 + \left(\log \frac{1}{2}\right))_* \cdot \left(\frac{1}{6} + \frac{1}{3}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left((e^{\log_* (1 + \sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}})} - 1)^*\right)\\ \end{array}\]

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 (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))))) < 4.583463697946747e-155

    1. Initial program 35.2

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

    3. Applied add-cube-cbrt35.2

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

    4. Taylor expanded around -inf 48.3

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

    5. Applied simplify4.1

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

    if 4.583463697946747e-155 < (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 8.5

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

    3. Applied expm1-log1p-u8.5

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

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed 2018201 +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)))))))