Average Error: 10.3 → 10.3
Time: 15.7s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\left|\sqrt{\frac{1 - \log \left(e^{{\left(\frac{Om}{Omc}\right)}^{2}}\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right|\right)\]
\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|\sqrt{\frac{1 - \log \left(e^{{\left(\frac{Om}{Omc}\right)}^{2}}\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right|\right)
double f(double t, double l, double Om, double Omc) {
        double r161 = 1.0;
        double r162 = Om;
        double r163 = Omc;
        double r164 = r162 / r163;
        double r165 = 2.0;
        double r166 = pow(r164, r165);
        double r167 = r161 - r166;
        double r168 = t;
        double r169 = l;
        double r170 = r168 / r169;
        double r171 = pow(r170, r165);
        double r172 = r165 * r171;
        double r173 = r161 + r172;
        double r174 = r167 / r173;
        double r175 = sqrt(r174);
        double r176 = asin(r175);
        return r176;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r177 = 1.0;
        double r178 = Om;
        double r179 = Omc;
        double r180 = r178 / r179;
        double r181 = 2.0;
        double r182 = pow(r180, r181);
        double r183 = exp(r182);
        double r184 = log(r183);
        double r185 = r177 - r184;
        double r186 = t;
        double r187 = l;
        double r188 = r186 / r187;
        double r189 = pow(r188, r181);
        double r190 = r181 * r189;
        double r191 = r177 + r190;
        double r192 = r185 / r191;
        double r193 = sqrt(r192);
        double r194 = fabs(r193);
        double r195 = asin(r194);
        return r195;
}

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 add-sqr-sqrt10.4

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

  4. Applied add-sqr-sqrt10.4

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

  5. Applied times-frac10.4

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

  6. Applied rem-sqrt-square10.4

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

  8. Applied add-log-exp10.4

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

  10. Applied sqrt-undiv10.3

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

  11. Final simplification10.3

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

Reproduce

herbie shell --seed 2020025 
(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)))))))