Average Error: 10.1 → 1.3
Time: 21.3s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le -4.0820958753315573 \cdot 10^{+161}:\\ \;\;\;\;\sin^{-1} \left(\left|\frac{\sqrt{1 - \log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}}{\frac{\sqrt{2} \cdot t}{\ell}}\right|\right)\\ \mathbf{elif}\;\frac{t}{\ell} \le 1.504437824641995 \cdot 10^{+83}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left|\frac{\sqrt{1 - \log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}}{\frac{\sqrt{2} \cdot t}{\ell}}\right|\right)\\ \end{array}\]
\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}\;\frac{t}{\ell} \le -4.0820958753315573 \cdot 10^{+161}:\\
\;\;\;\;\sin^{-1} \left(\left|\frac{\sqrt{1 - \log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}}{\frac{\sqrt{2} \cdot t}{\ell}}\right|\right)\\

\mathbf{elif}\;\frac{t}{\ell} \le 1.504437824641995 \cdot 10^{+83}:\\
\;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)\\

\mathbf{else}:\\
\;\;\;\;\sin^{-1} \left(\left|\frac{\sqrt{1 - \log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}}{\frac{\sqrt{2} \cdot t}{\ell}}\right|\right)\\

\end{array}
double f(double t, double l, double Om, double Omc) {
        double r2068128 = 1.0;
        double r2068129 = Om;
        double r2068130 = Omc;
        double r2068131 = r2068129 / r2068130;
        double r2068132 = 2.0;
        double r2068133 = pow(r2068131, r2068132);
        double r2068134 = r2068128 - r2068133;
        double r2068135 = t;
        double r2068136 = l;
        double r2068137 = r2068135 / r2068136;
        double r2068138 = pow(r2068137, r2068132);
        double r2068139 = r2068132 * r2068138;
        double r2068140 = r2068128 + r2068139;
        double r2068141 = r2068134 / r2068140;
        double r2068142 = sqrt(r2068141);
        double r2068143 = asin(r2068142);
        return r2068143;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r2068144 = t;
        double r2068145 = l;
        double r2068146 = r2068144 / r2068145;
        double r2068147 = -4.0820958753315573e+161;
        bool r2068148 = r2068146 <= r2068147;
        double r2068149 = 1.0;
        double r2068150 = Om;
        double r2068151 = Omc;
        double r2068152 = r2068150 / r2068151;
        double r2068153 = r2068152 * r2068152;
        double r2068154 = exp(r2068153);
        double r2068155 = log(r2068154);
        double r2068156 = r2068149 - r2068155;
        double r2068157 = sqrt(r2068156);
        double r2068158 = 2.0;
        double r2068159 = sqrt(r2068158);
        double r2068160 = r2068159 * r2068144;
        double r2068161 = r2068160 / r2068145;
        double r2068162 = r2068157 / r2068161;
        double r2068163 = fabs(r2068162);
        double r2068164 = asin(r2068163);
        double r2068165 = 1.504437824641995e+83;
        bool r2068166 = r2068146 <= r2068165;
        double r2068167 = r2068146 * r2068146;
        double r2068168 = fma(r2068158, r2068167, r2068149);
        double r2068169 = r2068156 / r2068168;
        double r2068170 = sqrt(r2068169);
        double r2068171 = asin(r2068170);
        double r2068172 = r2068166 ? r2068171 : r2068164;
        double r2068173 = r2068148 ? r2068164 : r2068172;
        return r2068173;
}

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 (/ t l) < -4.0820958753315573e+161 or 1.504437824641995e+83 < (/ t l)

    1. Initial program 28.7

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

    2. Simplified28.7

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

    4. Applied add-log-exp28.7

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \color{blue}{\log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt28.8

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

    7. Applied add-sqr-sqrt28.8

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

    8. Applied times-frac28.7

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

    9. Applied rem-sqrt-square28.7

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

    10. Taylor expanded around inf 1.1

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

    if -4.0820958753315573e+161 < (/ t l) < 1.504437824641995e+83

    1. Initial program 1.3

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

    2. Simplified1.3

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

    4. Applied add-log-exp1.3

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le -4.0820958753315573 \cdot 10^{+161}:\\ \;\;\;\;\sin^{-1} \left(\left|\frac{\sqrt{1 - \log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}}{\frac{\sqrt{2} \cdot t}{\ell}}\right|\right)\\ \mathbf{elif}\;\frac{t}{\ell} \le 1.504437824641995 \cdot 10^{+83}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left|\frac{\sqrt{1 - \log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}}{\frac{\sqrt{2} \cdot t}{\ell}}\right|\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +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)))))))