Average Error: 9.8 → 9.8
Time: 29.2s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left({\left(\frac{t}{\ell}\right)}^{2}, 2, 1\right)}}\right)\]
\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)
↓
\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left({\left(\frac{t}{\ell}\right)}^{2}, 2, 1\right)}}\right)
double f(double t, double l, double Om, double Omc) {
        double r2639439 = 1.0;
        double r2639440 = Om;
        double r2639441 = Omc;
        double r2639442 = r2639440 / r2639441;
        double r2639443 = 2.0;
        double r2639444 = pow(r2639442, r2639443);
        double r2639445 = r2639439 - r2639444;
        double r2639446 = t;
        double r2639447 = l;
        double r2639448 = r2639446 / r2639447;
        double r2639449 = pow(r2639448, r2639443);
        double r2639450 = r2639443 * r2639449;
        double r2639451 = r2639439 + r2639450;
        double r2639452 = r2639445 / r2639451;
        double r2639453 = sqrt(r2639452);
        double r2639454 = asin(r2639453);
        return r2639454;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r2639455 = 1.0;
        double r2639456 = Om;
        double r2639457 = Omc;
        double r2639458 = r2639456 / r2639457;
        double r2639459 = 2.0;
        double r2639460 = pow(r2639458, r2639459);
        double r2639461 = r2639455 - r2639460;
        double r2639462 = t;
        double r2639463 = l;
        double r2639464 = r2639462 / r2639463;
        double r2639465 = pow(r2639464, r2639459);
        double r2639466 = fma(r2639465, r2639459, r2639455);
        double r2639467 = r2639461 / r2639466;
        double r2639468 = sqrt(r2639467);
        double r2639469 = asin(r2639468);
        return r2639469;
}

Error

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus Omc

Derivation

  1. Initial program 9.8

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

  2. Simplified9.8

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

  3. Final simplification9.8

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t l) 2.0)))))))