Average Error: 10.5 → 5.8
Time: 23.0s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 7.639427664533362 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\mathsf{fma}\left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right), 2, 1\right)}}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\right)\right)\right)\right)\\ \end{array}\]
\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 7.639427664533362 \cdot 10^{+141}:\\
\;\;\;\;\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\mathsf{fma}\left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right), 2, 1\right)}}\right)\right)\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\right)\right)\right)\right)\\

\end{array}
double f(double t, double l, double Om, double Omc) {
        double r1183135 = 1.0;
        double r1183136 = Om;
        double r1183137 = Omc;
        double r1183138 = r1183136 / r1183137;
        double r1183139 = 2.0;
        double r1183140 = pow(r1183138, r1183139);
        double r1183141 = r1183135 - r1183140;
        double r1183142 = t;
        double r1183143 = l;
        double r1183144 = r1183142 / r1183143;
        double r1183145 = pow(r1183144, r1183139);
        double r1183146 = r1183139 * r1183145;
        double r1183147 = r1183135 + r1183146;
        double r1183148 = r1183141 / r1183147;
        double r1183149 = sqrt(r1183148);
        double r1183150 = asin(r1183149);
        return r1183150;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r1183151 = t;
        double r1183152 = l;
        double r1183153 = r1183151 / r1183152;
        double r1183154 = 7.639427664533362e+141;
        bool r1183155 = r1183153 <= r1183154;
        double r1183156 = 1.0;
        double r1183157 = Om;
        double r1183158 = Omc;
        double r1183159 = r1183157 / r1183158;
        double r1183160 = r1183159 * r1183159;
        double r1183161 = r1183156 - r1183160;
        double r1183162 = r1183153 * r1183153;
        double r1183163 = 2.0;
        double r1183164 = fma(r1183162, r1183163, r1183156);
        double r1183165 = r1183161 / r1183164;
        double r1183166 = sqrt(r1183165);
        double r1183167 = asin(r1183166);
        double r1183168 = log1p(r1183167);
        double r1183169 = expm1(r1183168);
        double r1183170 = sqrt(r1183161);
        double r1183171 = sqrt(r1183163);
        double r1183172 = r1183151 * r1183171;
        double r1183173 = r1183172 / r1183152;
        double r1183174 = r1183170 / r1183173;
        double r1183175 = asin(r1183174);
        double r1183176 = log1p(r1183175);
        double r1183177 = expm1(r1183176);
        double r1183178 = r1183155 ? r1183169 : r1183177;
        return r1183178;
}

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) < 7.639427664533362e+141

    1. Initial program 6.6

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

    2. Simplified6.6

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

    4. Applied expm1-log1p-u6.6

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\mathsf{fma}\left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right), 2, 1\right)}}\right)\right)\right)\right)\right)}\]

    if 7.639427664533362e+141 < (/ t l)

    1. Initial program 32.2

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

    2. Simplified32.2

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

    4. Applied expm1-log1p-u32.2

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\mathsf{fma}\left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right), 2, 1\right)}}\right)\right)\right)\right)\right)}\]
    5. Using strategy rm

    6. Applied sqrt-div32.3

      \[\leadsto \mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\sqrt{\mathsf{fma}\left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right), 2, 1\right)}}\right)}\right)\right)\right)\right)\]

    7. Taylor expanded around inf 1.5

      \[\leadsto \mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\color{blue}{\frac{t \cdot \sqrt{2}}{\ell}}}\right)\right)\right)\right)\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification5.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 7.639427664533362 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\mathsf{fma}\left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right), 2, 1\right)}}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +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)))))))