Average Error: 10.0 → 10.0
Time: 18.7s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)\]
\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(\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)\right)\right)
double f(double t, double l, double Om, double Omc) {
        double r43172 = 1.0;
        double r43173 = Om;
        double r43174 = Omc;
        double r43175 = r43173 / r43174;
        double r43176 = 2.0;
        double r43177 = pow(r43175, r43176);
        double r43178 = r43172 - r43177;
        double r43179 = t;
        double r43180 = l;
        double r43181 = r43179 / r43180;
        double r43182 = pow(r43181, r43176);
        double r43183 = r43176 * r43182;
        double r43184 = r43172 + r43183;
        double r43185 = r43178 / r43184;
        double r43186 = sqrt(r43185);
        double r43187 = asin(r43186);
        return r43187;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r43188 = 1.0;
        double r43189 = Om;
        double r43190 = Omc;
        double r43191 = r43189 / r43190;
        double r43192 = 2.0;
        double r43193 = pow(r43191, r43192);
        double r43194 = r43188 - r43193;
        double r43195 = t;
        double r43196 = l;
        double r43197 = r43195 / r43196;
        double r43198 = pow(r43197, r43192);
        double r43199 = fma(r43192, r43198, r43188);
        double r43200 = r43194 / r43199;
        double r43201 = sqrt(r43200);
        double r43202 = asin(r43201);
        double r43203 = log1p(r43202);
        double r43204 = expm1(r43203);
        return r43204;
}

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 expm1-log1p-u10.0

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

  5. Final simplification10.0

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

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  :precision binary64
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))