Average Error: 10.1 → 10.1
Time: 26.6s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{{\left(\frac{1}{\frac{\ell}{t}}\right)}^{2} \cdot 2 + 1}}\right)\]
\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 - {\left(\frac{Om}{Omc}\right)}^{2}}{{\left(\frac{1}{\frac{\ell}{t}}\right)}^{2} \cdot 2 + 1}}\right)
double f(double t, double l, double Om, double Omc) {
        double r55367 = 1.0;
        double r55368 = Om;
        double r55369 = Omc;
        double r55370 = r55368 / r55369;
        double r55371 = 2.0;
        double r55372 = pow(r55370, r55371);
        double r55373 = r55367 - r55372;
        double r55374 = t;
        double r55375 = l;
        double r55376 = r55374 / r55375;
        double r55377 = pow(r55376, r55371);
        double r55378 = r55371 * r55377;
        double r55379 = r55367 + r55378;
        double r55380 = r55373 / r55379;
        double r55381 = sqrt(r55380);
        double r55382 = asin(r55381);
        return r55382;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r55383 = 1.0;
        double r55384 = Om;
        double r55385 = Omc;
        double r55386 = r55384 / r55385;
        double r55387 = 2.0;
        double r55388 = pow(r55386, r55387);
        double r55389 = r55383 - r55388;
        double r55390 = 1.0;
        double r55391 = l;
        double r55392 = t;
        double r55393 = r55391 / r55392;
        double r55394 = r55390 / r55393;
        double r55395 = pow(r55394, r55387);
        double r55396 = r55395 * r55387;
        double r55397 = r55396 + r55383;
        double r55398 = r55389 / r55397;
        double r55399 = sqrt(r55398);
        double r55400 = asin(r55399);
        return r55400;
}

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 10.1

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

  2. Taylor expanded around -inf 51.2

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

  3. Simplified10.2

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

  5. Applied clear-num10.2

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

  6. Simplified10.1

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

  7. Final simplification10.1

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

Reproduce

herbie shell --seed 2019323 
(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)))))))