Average Error: 10.3 → 10.5
Time: 26.3s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{\sqrt{1} + \sqrt{{\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}} \cdot \frac{\sqrt{1} - \sqrt{{\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}\right)\]
\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{\sqrt{1} + \sqrt{{\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}} \cdot \frac{\sqrt{1} - \sqrt{{\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}\right)
double f(double t, double l, double Om, double Omc) {
        double r58047 = 1.0;
        double r58048 = Om;
        double r58049 = Omc;
        double r58050 = r58048 / r58049;
        double r58051 = 2.0;
        double r58052 = pow(r58050, r58051);
        double r58053 = r58047 - r58052;
        double r58054 = t;
        double r58055 = l;
        double r58056 = r58054 / r58055;
        double r58057 = pow(r58056, r58051);
        double r58058 = r58051 * r58057;
        double r58059 = r58047 + r58058;
        double r58060 = r58053 / r58059;
        double r58061 = sqrt(r58060);
        double r58062 = asin(r58061);
        return r58062;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r58063 = 1.0;
        double r58064 = sqrt(r58063);
        double r58065 = Om;
        double r58066 = Omc;
        double r58067 = r58065 / r58066;
        double r58068 = 2.0;
        double r58069 = pow(r58067, r58068);
        double r58070 = sqrt(r58069);
        double r58071 = r58064 + r58070;
        double r58072 = t;
        double r58073 = l;
        double r58074 = r58072 / r58073;
        double r58075 = pow(r58074, r58068);
        double r58076 = r58068 * r58075;
        double r58077 = r58063 + r58076;
        double r58078 = sqrt(r58077);
        double r58079 = r58071 / r58078;
        double r58080 = r58064 - r58070;
        double r58081 = r58080 / r58078;
        double r58082 = r58079 * r58081;
        double r58083 = sqrt(r58082);
        double r58084 = asin(r58083);
        return r58084;
}

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. Using strategy rm

  3. Applied add-sqr-sqrt10.4

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

  4. Applied add-sqr-sqrt10.4

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

  5. Applied add-sqr-sqrt10.4

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

  6. Applied difference-of-squares10.5

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

  7. Applied times-frac10.5

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

  8. Final simplification10.5

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

Reproduce

herbie shell --seed 2019303 
(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)))))))