Average Error: 10.5 → 10.5
Time: 3.0m
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\sqrt{1 + \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2}}}{\sqrt{1 + \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2}}}\right)\]
\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{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\sqrt{1 + \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2}}}{\sqrt{1 + \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2}}}\right)
double f(double t, double l, double Om, double Omc) {
        double r10489187 = 1.0;
        double r10489188 = Om;
        double r10489189 = Omc;
        double r10489190 = r10489188 / r10489189;
        double r10489191 = 2.0;
        double r10489192 = pow(r10489190, r10489191);
        double r10489193 = r10489187 - r10489192;
        double r10489194 = t;
        double r10489195 = l;
        double r10489196 = r10489194 / r10489195;
        double r10489197 = pow(r10489196, r10489191);
        double r10489198 = r10489191 * r10489197;
        double r10489199 = r10489187 + r10489198;
        double r10489200 = r10489193 / r10489199;
        double r10489201 = sqrt(r10489200);
        double r10489202 = asin(r10489201);
        return r10489202;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r10489203 = 1.0;
        double r10489204 = Om;
        double r10489205 = Omc;
        double r10489206 = r10489204 / r10489205;
        double r10489207 = r10489206 * r10489206;
        double r10489208 = r10489203 - r10489207;
        double r10489209 = t;
        double r10489210 = l;
        double r10489211 = r10489209 / r10489210;
        double r10489212 = r10489211 * r10489211;
        double r10489213 = 2.0;
        double r10489214 = r10489212 * r10489213;
        double r10489215 = r10489203 + r10489214;
        double r10489216 = sqrt(r10489215);
        double r10489217 = r10489208 / r10489216;
        double r10489218 = r10489217 / r10489216;
        double r10489219 = sqrt(r10489218);
        double r10489220 = asin(r10489219);
        return r10489220;
}

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 10.5

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

  2. Simplified10.5

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

  4. Applied add-sqr-sqrt10.5

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

  5. Applied associate-/r*10.5

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

  6. Final simplification10.5

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

Reproduce

herbie shell --seed 2019112 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))