Average Error: 10.5 → 5.8
Time: 1.9m
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 r6475082 = 1.0;
        double r6475083 = Om;
        double r6475084 = Omc;
        double r6475085 = r6475083 / r6475084;
        double r6475086 = 2.0;
        double r6475087 = pow(r6475085, r6475086);
        double r6475088 = r6475082 - r6475087;
        double r6475089 = t;
        double r6475090 = l;
        double r6475091 = r6475089 / r6475090;
        double r6475092 = pow(r6475091, r6475086);
        double r6475093 = r6475086 * r6475092;
        double r6475094 = r6475082 + r6475093;
        double r6475095 = r6475088 / r6475094;
        double r6475096 = sqrt(r6475095);
        double r6475097 = asin(r6475096);
        return r6475097;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r6475098 = t;
        double r6475099 = l;
        double r6475100 = r6475098 / r6475099;
        double r6475101 = 7.639427664533362e+141;
        bool r6475102 = r6475100 <= r6475101;
        double r6475103 = 1.0;
        double r6475104 = Om;
        double r6475105 = Omc;
        double r6475106 = r6475104 / r6475105;
        double r6475107 = r6475106 * r6475106;
        double r6475108 = r6475103 - r6475107;
        double r6475109 = r6475100 * r6475100;
        double r6475110 = 2.0;
        double r6475111 = fma(r6475109, r6475110, r6475103);
        double r6475112 = r6475108 / r6475111;
        double r6475113 = sqrt(r6475112);
        double r6475114 = asin(r6475113);
        double r6475115 = log1p(r6475114);
        double r6475116 = expm1(r6475115);
        double r6475117 = sqrt(r6475108);
        double r6475118 = sqrt(r6475110);
        double r6475119 = r6475098 * r6475118;
        double r6475120 = r6475119 / r6475099;
        double r6475121 = r6475117 / r6475120;
        double r6475122 = asin(r6475121);
        double r6475123 = log1p(r6475122);
        double r6475124 = expm1(r6475123);
        double r6475125 = r6475102 ? r6475116 : r6475124;
        return r6475125;
}

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)))))))