Average Error: 10.3 → 10.4
Time: 10.4s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\frac{1}{\sqrt{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}} \cdot \sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}\right)\right)\right)\]
\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(\frac{1}{\sqrt{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}} \cdot \sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}\right)\right)\right)
double f(double t, double l, double Om, double Omc) {
        double r72139 = 1.0;
        double r72140 = Om;
        double r72141 = Omc;
        double r72142 = r72140 / r72141;
        double r72143 = 2.0;
        double r72144 = pow(r72142, r72143);
        double r72145 = r72139 - r72144;
        double r72146 = t;
        double r72147 = l;
        double r72148 = r72146 / r72147;
        double r72149 = pow(r72148, r72143);
        double r72150 = r72143 * r72149;
        double r72151 = r72139 + r72150;
        double r72152 = r72145 / r72151;
        double r72153 = sqrt(r72152);
        double r72154 = asin(r72153);
        return r72154;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r72155 = 1.0;
        double r72156 = 1.0;
        double r72157 = 2.0;
        double r72158 = t;
        double r72159 = l;
        double r72160 = r72158 / r72159;
        double r72161 = pow(r72160, r72157);
        double r72162 = r72157 * r72161;
        double r72163 = r72156 + r72162;
        double r72164 = sqrt(r72163);
        double r72165 = sqrt(r72164);
        double r72166 = r72155 / r72165;
        double r72167 = Om;
        double r72168 = Omc;
        double r72169 = r72167 / r72168;
        double r72170 = pow(r72169, r72157);
        double r72171 = r72156 - r72170;
        double r72172 = r72171 / r72164;
        double r72173 = sqrt(r72172);
        double r72174 = r72166 * r72173;
        double r72175 = asin(r72174);
        double r72176 = log1p(r72175);
        double r72177 = expm1(r72176);
        return r72177;
}

Error

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus Omc

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 10.3

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

  3. Applied expm1-log1p-u10.3

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right)\right)}\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt10.3

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

  6. Applied *-un-lft-identity10.3

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

  7. Applied times-frac10.3

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

  8. Applied sqrt-prod10.4

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}} \cdot \sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}\right)}\right)\right)\]
  9. Using strategy rm

  10. Applied sqrt-div10.4

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

  11. Simplified10.4

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

  12. Final simplification10.4

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

Reproduce

herbie shell --seed 2020002 +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)))))))