Average Error: 10.0 → 10.1
Time: 20.8s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\left|\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right|\right)\]
\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(\left|\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right|\right)
double f(double t, double l, double Om, double Omc) {
        double r31971 = 1.0;
        double r31972 = Om;
        double r31973 = Omc;
        double r31974 = r31972 / r31973;
        double r31975 = 2.0;
        double r31976 = pow(r31974, r31975);
        double r31977 = r31971 - r31976;
        double r31978 = t;
        double r31979 = l;
        double r31980 = r31978 / r31979;
        double r31981 = pow(r31980, r31975);
        double r31982 = r31975 * r31981;
        double r31983 = r31971 + r31982;
        double r31984 = r31977 / r31983;
        double r31985 = sqrt(r31984);
        double r31986 = asin(r31985);
        return r31986;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r31987 = 1.0;
        double r31988 = Om;
        double r31989 = Omc;
        double r31990 = r31988 / r31989;
        double r31991 = 2.0;
        double r31992 = pow(r31990, r31991);
        double r31993 = r31987 - r31992;
        double r31994 = sqrt(r31993);
        double r31995 = t;
        double r31996 = l;
        double r31997 = r31995 / r31996;
        double r31998 = pow(r31997, r31991);
        double r31999 = fma(r31991, r31998, r31987);
        double r32000 = sqrt(r31999);
        double r32001 = r31994 / r32000;
        double r32002 = fabs(r32001);
        double r32003 = asin(r32002);
        return r32003;
}

Error

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus Omc

Derivation

  1. Initial program 10.0

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

  2. Simplified10.0

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

  4. Applied add-sqr-sqrt10.1

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

  5. Applied add-sqr-sqrt10.1

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

  6. Applied times-frac10.1

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

  7. Applied rem-sqrt-square10.1

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

  8. Final simplification10.1

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

Reproduce

herbie shell --seed 2019194 +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)))))))