Average Error: 9.8 → 9.8
Time: 27.8s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\left|\sqrt{\frac{1 - \frac{\frac{Om}{Omc} \cdot Om}{Omc}}{\frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}} + 1}}\right|\right)\]
\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 - \frac{\frac{Om}{Omc} \cdot Om}{Omc}}{\frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}} + 1}}\right|\right)
double f(double t, double l, double Om, double Omc) {
        double r2818275 = 1.0;
        double r2818276 = Om;
        double r2818277 = Omc;
        double r2818278 = r2818276 / r2818277;
        double r2818279 = 2.0;
        double r2818280 = pow(r2818278, r2818279);
        double r2818281 = r2818275 - r2818280;
        double r2818282 = t;
        double r2818283 = l;
        double r2818284 = r2818282 / r2818283;
        double r2818285 = pow(r2818284, r2818279);
        double r2818286 = r2818279 * r2818285;
        double r2818287 = r2818275 + r2818286;
        double r2818288 = r2818281 / r2818287;
        double r2818289 = sqrt(r2818288);
        double r2818290 = asin(r2818289);
        return r2818290;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r2818291 = 1.0;
        double r2818292 = Om;
        double r2818293 = Omc;
        double r2818294 = r2818292 / r2818293;
        double r2818295 = r2818294 * r2818292;
        double r2818296 = r2818295 / r2818293;
        double r2818297 = r2818291 - r2818296;
        double r2818298 = 2.0;
        double r2818299 = l;
        double r2818300 = t;
        double r2818301 = r2818299 / r2818300;
        double r2818302 = r2818301 * r2818301;
        double r2818303 = r2818298 / r2818302;
        double r2818304 = r2818303 + r2818291;
        double r2818305 = r2818297 / r2818304;
        double r2818306 = sqrt(r2818305);
        double r2818307 = fabs(r2818306);
        double r2818308 = asin(r2818307);
        return r2818308;
}

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 9.8

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

  3. Applied add-sqr-sqrt9.8

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

  4. Applied add-sqr-sqrt9.8

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

  5. Applied times-frac9.8

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

  6. Applied rem-sqrt-square9.8

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

  7. Taylor expanded around inf 26.3

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

  8. Simplified9.8

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

  10. Applied associate-*l/9.8

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

  11. Final simplification9.8

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

Reproduce

herbie shell --seed 2019172 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t l) 2.0)))))))