Average Error: 10.1 → 10.1
Time: 44.8s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{{\left(\frac{\sqrt[3]{-1} \cdot \sqrt[3]{-1}}{\frac{\frac{-\ell}{t}}{\sqrt[3]{-1}}}\right)}^{2} \cdot 2 + 1}}\right)\]
\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{\sqrt[3]{-1} \cdot \sqrt[3]{-1}}{\frac{\frac{-\ell}{t}}{\sqrt[3]{-1}}}\right)}^{2} \cdot 2 + 1}}\right)
double f(double t, double l, double Om, double Omc) {
        double r102311 = 1.0;
        double r102312 = Om;
        double r102313 = Omc;
        double r102314 = r102312 / r102313;
        double r102315 = 2.0;
        double r102316 = pow(r102314, r102315);
        double r102317 = r102311 - r102316;
        double r102318 = t;
        double r102319 = l;
        double r102320 = r102318 / r102319;
        double r102321 = pow(r102320, r102315);
        double r102322 = r102315 * r102321;
        double r102323 = r102311 + r102322;
        double r102324 = r102317 / r102323;
        double r102325 = sqrt(r102324);
        double r102326 = asin(r102325);
        return r102326;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r102327 = 1.0;
        double r102328 = Om;
        double r102329 = Omc;
        double r102330 = r102328 / r102329;
        double r102331 = 2.0;
        double r102332 = pow(r102330, r102331);
        double r102333 = r102327 - r102332;
        double r102334 = -1.0;
        double r102335 = cbrt(r102334);
        double r102336 = r102335 * r102335;
        double r102337 = l;
        double r102338 = -r102337;
        double r102339 = t;
        double r102340 = r102338 / r102339;
        double r102341 = r102340 / r102335;
        double r102342 = r102336 / r102341;
        double r102343 = pow(r102342, r102331);
        double r102344 = r102343 * r102331;
        double r102345 = r102344 + r102327;
        double r102346 = r102333 / r102345;
        double r102347 = sqrt(r102346);
        double r102348 = asin(r102347);
        return r102348;
}

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 *-un-lft-identity10.2

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

  6. Applied add-cube-cbrt10.2

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

  7. Applied times-frac10.2

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

  8. Applied associate-/l*10.2

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

  9. Simplified10.1

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

  10. Final simplification10.1

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

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)))))))