Average Error: 10.1 → 10.1
Time: 27.5s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{2 \cdot {\left(\frac{1}{\frac{\ell}{t}}\right)}^{2} + 1}}\right)\]
\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}}{2 \cdot {\left(\frac{1}{\frac{\ell}{t}}\right)}^{2} + 1}}\right)
double f(double t, double l, double Om, double Omc) {
        double r54760 = 1.0;
        double r54761 = Om;
        double r54762 = Omc;
        double r54763 = r54761 / r54762;
        double r54764 = 2.0;
        double r54765 = pow(r54763, r54764);
        double r54766 = r54760 - r54765;
        double r54767 = t;
        double r54768 = l;
        double r54769 = r54767 / r54768;
        double r54770 = pow(r54769, r54764);
        double r54771 = r54764 * r54770;
        double r54772 = r54760 + r54771;
        double r54773 = r54766 / r54772;
        double r54774 = sqrt(r54773);
        double r54775 = asin(r54774);
        return r54775;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r54776 = 1.0;
        double r54777 = Om;
        double r54778 = Omc;
        double r54779 = r54777 / r54778;
        double r54780 = 2.0;
        double r54781 = pow(r54779, r54780);
        double r54782 = r54776 - r54781;
        double r54783 = 1.0;
        double r54784 = l;
        double r54785 = t;
        double r54786 = r54784 / r54785;
        double r54787 = r54783 / r54786;
        double r54788 = pow(r54787, r54780);
        double r54789 = r54780 * r54788;
        double r54790 = r54789 + r54776;
        double r54791 = r54782 / r54790;
        double r54792 = sqrt(r54791);
        double r54793 = asin(r54792);
        return r54793;
}

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 *-un-lft-identity10.2

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

  7. Applied times-frac10.2

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

  8. Applied associate-/l*10.2

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

  9. Simplified10.1

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

  10. Final simplification10.1

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

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)))))))