Average Error: 10.3 → 10.3
Time: 18.6s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{1 - \log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}{\frac{t}{\ell} \cdot \left(\frac{t}{\ell} + \frac{t}{\ell}\right) + 1}}\right)\]
\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 - \log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}{\frac{t}{\ell} \cdot \left(\frac{t}{\ell} + \frac{t}{\ell}\right) + 1}}\right)
double f(double t, double l, double Om, double Omc) {
        double r906152 = 1.0;
        double r906153 = Om;
        double r906154 = Omc;
        double r906155 = r906153 / r906154;
        double r906156 = 2.0;
        double r906157 = pow(r906155, r906156);
        double r906158 = r906152 - r906157;
        double r906159 = t;
        double r906160 = l;
        double r906161 = r906159 / r906160;
        double r906162 = pow(r906161, r906156);
        double r906163 = r906156 * r906162;
        double r906164 = r906152 + r906163;
        double r906165 = r906158 / r906164;
        double r906166 = sqrt(r906165);
        double r906167 = asin(r906166);
        return r906167;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r906168 = 1.0;
        double r906169 = Om;
        double r906170 = Omc;
        double r906171 = r906169 / r906170;
        double r906172 = r906171 * r906171;
        double r906173 = exp(r906172);
        double r906174 = log(r906173);
        double r906175 = r906168 - r906174;
        double r906176 = t;
        double r906177 = l;
        double r906178 = r906176 / r906177;
        double r906179 = r906178 + r906178;
        double r906180 = r906178 * r906179;
        double r906181 = r906180 + r906168;
        double r906182 = r906175 / r906181;
        double r906183 = sqrt(r906182);
        double r906184 = asin(r906183);
        return r906184;
}

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. Simplified10.3

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

  4. Applied add-log-exp10.3

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

  5. Final simplification10.3

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

Reproduce

herbie shell --seed 2019154 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))