Average Error: 10.5 → 5.7
Time: 1.3m
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 7.528201376140965 \cdot 10^{+69}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\mathsf{fma}\left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right), 2, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]
\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 7.528201376140965 \cdot 10^{+69}:\\
\;\;\;\;\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)\\

\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 r3126090 = 1.0;
        double r3126091 = Om;
        double r3126092 = Omc;
        double r3126093 = r3126091 / r3126092;
        double r3126094 = 2.0;
        double r3126095 = pow(r3126093, r3126094);
        double r3126096 = r3126090 - r3126095;
        double r3126097 = t;
        double r3126098 = l;
        double r3126099 = r3126097 / r3126098;
        double r3126100 = pow(r3126099, r3126094);
        double r3126101 = r3126094 * r3126100;
        double r3126102 = r3126090 + r3126101;
        double r3126103 = r3126096 / r3126102;
        double r3126104 = sqrt(r3126103);
        double r3126105 = asin(r3126104);
        return r3126105;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r3126106 = t;
        double r3126107 = l;
        double r3126108 = r3126106 / r3126107;
        double r3126109 = 7.528201376140965e+69;
        bool r3126110 = r3126108 <= r3126109;
        double r3126111 = 1.0;
        double r3126112 = Om;
        double r3126113 = Omc;
        double r3126114 = r3126112 / r3126113;
        double r3126115 = r3126114 * r3126114;
        double r3126116 = r3126111 - r3126115;
        double r3126117 = r3126108 * r3126108;
        double r3126118 = 2.0;
        double r3126119 = fma(r3126117, r3126118, r3126111);
        double r3126120 = r3126116 / r3126119;
        double r3126121 = sqrt(r3126120);
        double r3126122 = asin(r3126121);
        double r3126123 = sqrt(r3126116);
        double r3126124 = sqrt(r3126118);
        double r3126125 = r3126106 * r3126124;
        double r3126126 = r3126125 / r3126107;
        double r3126127 = r3126123 / r3126126;
        double r3126128 = asin(r3126127);
        double r3126129 = r3126110 ? r3126122 : r3126128;
        return r3126129;
}

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) < 7.528201376140965e+69

    1. Initial program 6.8

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

    2. Simplified6.8

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

    3. Taylor expanded around inf 23.0

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

    4. Simplified6.8

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

    if 7.528201376140965e+69 < (/ t l)

    1. Initial program 25.8

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

    2. Simplified25.8

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

    4. Applied sqrt-div25.8

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

    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 simplification5.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 7.528201376140965 \cdot 10^{+69}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\mathsf{fma}\left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right), 2, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 +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)))))))