Average Error: 10.3 → 10.6
Time: 25.2s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\left(\frac{\sqrt[3]{t}}{\ell} \cdot \left(\sqrt[3]{t} \cdot \left(\frac{t}{\ell} \cdot \sqrt[3]{t}\right)\right) + \frac{t}{\ell} \cdot \frac{t}{\ell}\right) + 1}}\right)\]
\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 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\left(\frac{\sqrt[3]{t}}{\ell} \cdot \left(\sqrt[3]{t} \cdot \left(\frac{t}{\ell} \cdot \sqrt[3]{t}\right)\right) + \frac{t}{\ell} \cdot \frac{t}{\ell}\right) + 1}}\right)
double f(double t, double l, double Om, double Omc) {
        double r1639247 = 1.0;
        double r1639248 = Om;
        double r1639249 = Omc;
        double r1639250 = r1639248 / r1639249;
        double r1639251 = 2.0;
        double r1639252 = pow(r1639250, r1639251);
        double r1639253 = r1639247 - r1639252;
        double r1639254 = t;
        double r1639255 = l;
        double r1639256 = r1639254 / r1639255;
        double r1639257 = pow(r1639256, r1639251);
        double r1639258 = r1639251 * r1639257;
        double r1639259 = r1639247 + r1639258;
        double r1639260 = r1639253 / r1639259;
        double r1639261 = sqrt(r1639260);
        double r1639262 = asin(r1639261);
        return r1639262;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r1639263 = 1.0;
        double r1639264 = Om;
        double r1639265 = Omc;
        double r1639266 = r1639264 / r1639265;
        double r1639267 = r1639266 * r1639266;
        double r1639268 = r1639263 - r1639267;
        double r1639269 = t;
        double r1639270 = cbrt(r1639269);
        double r1639271 = l;
        double r1639272 = r1639270 / r1639271;
        double r1639273 = r1639269 / r1639271;
        double r1639274 = r1639273 * r1639270;
        double r1639275 = r1639270 * r1639274;
        double r1639276 = r1639272 * r1639275;
        double r1639277 = r1639273 * r1639273;
        double r1639278 = r1639276 + r1639277;
        double r1639279 = r1639278 + r1639263;
        double r1639280 = r1639268 / r1639279;
        double r1639281 = sqrt(r1639280);
        double r1639282 = asin(r1639281);
        return r1639282;
}

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

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

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

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

  5. Applied add-cube-cbrt10.4

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

  6. Applied times-frac10.4

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

  7. Applied associate-*r*10.6

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

  8. Simplified10.6

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

  9. Final simplification10.6

    \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\left(\frac{\sqrt[3]{t}}{\ell} \cdot \left(\sqrt[3]{t} \cdot \left(\frac{t}{\ell} \cdot \sqrt[3]{t}\right)\right) + \frac{t}{\ell} \cdot \frac{t}{\ell}\right) + 1}}\right)\]

Reproduce

herbie shell --seed 2019132 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))