Average Error: 9.8 → 5.4
Time: 2.4m
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 1.5204539847207166 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\mathsf{fma}\left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right), 2, 1\right)}}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]
\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 1.5204539847207166 \cdot 10^{+108}:\\
\;\;\;\;\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}:\\
\;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\

\end{array}
double f(double t, double l, double Om, double Omc) {
        double r7558527 = 1.0;
        double r7558528 = Om;
        double r7558529 = Omc;
        double r7558530 = r7558528 / r7558529;
        double r7558531 = 2.0;
        double r7558532 = pow(r7558530, r7558531);
        double r7558533 = r7558527 - r7558532;
        double r7558534 = t;
        double r7558535 = l;
        double r7558536 = r7558534 / r7558535;
        double r7558537 = pow(r7558536, r7558531);
        double r7558538 = r7558531 * r7558537;
        double r7558539 = r7558527 + r7558538;
        double r7558540 = r7558533 / r7558539;
        double r7558541 = sqrt(r7558540);
        double r7558542 = asin(r7558541);
        return r7558542;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r7558543 = t;
        double r7558544 = l;
        double r7558545 = r7558543 / r7558544;
        double r7558546 = 1.5204539847207166e+108;
        bool r7558547 = r7558545 <= r7558546;
        double r7558548 = 1.0;
        double r7558549 = Om;
        double r7558550 = Omc;
        double r7558551 = r7558549 / r7558550;
        double r7558552 = r7558551 * r7558551;
        double r7558553 = r7558548 - r7558552;
        double r7558554 = r7558545 * r7558545;
        double r7558555 = 2.0;
        double r7558556 = fma(r7558554, r7558555, r7558548);
        double r7558557 = r7558553 / r7558556;
        double r7558558 = sqrt(r7558557);
        double r7558559 = asin(r7558558);
        double r7558560 = log1p(r7558559);
        double r7558561 = expm1(r7558560);
        double r7558562 = sqrt(r7558553);
        double r7558563 = sqrt(r7558555);
        double r7558564 = r7558543 * r7558563;
        double r7558565 = r7558564 / r7558544;
        double r7558566 = r7558562 / r7558565;
        double r7558567 = asin(r7558566);
        double r7558568 = r7558547 ? r7558561 : r7558567;
        return r7558568;
}

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) < 1.5204539847207166e+108

    1. Initial program 6.3

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

    2. Simplified6.3

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

    4. Applied expm1-log1p-u6.3

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

    if 1.5204539847207166e+108 < (/ t l)

    1. Initial program 27.8

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

    2. Simplified27.8

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

    4. Applied sqrt-div27.8

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

    5. Taylor expanded around inf 1.0

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

  4. Final simplification5.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 1.5204539847207166 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\mathsf{fma}\left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right), 2, 1\right)}}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 +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)))))))