Average Error: 10.3 → 10.5
Time: 26.0s
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 r58078 = 1.0;
        double r58079 = Om;
        double r58080 = Omc;
        double r58081 = r58079 / r58080;
        double r58082 = 2.0;
        double r58083 = pow(r58081, r58082);
        double r58084 = r58078 - r58083;
        double r58085 = t;
        double r58086 = l;
        double r58087 = r58085 / r58086;
        double r58088 = pow(r58087, r58082);
        double r58089 = r58082 * r58088;
        double r58090 = r58078 + r58089;
        double r58091 = r58084 / r58090;
        double r58092 = sqrt(r58091);
        double r58093 = asin(r58092);
        return r58093;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r58094 = 1.0;
        double r58095 = sqrt(r58094);
        double r58096 = Om;
        double r58097 = Omc;
        double r58098 = r58096 / r58097;
        double r58099 = 2.0;
        double r58100 = pow(r58098, r58099);
        double r58101 = sqrt(r58100);
        double r58102 = r58095 + r58101;
        double r58103 = t;
        double r58104 = l;
        double r58105 = r58103 / r58104;
        double r58106 = pow(r58105, r58099);
        double r58107 = r58099 * r58106;
        double r58108 = r58094 + r58107;
        double r58109 = sqrt(r58108);
        double r58110 = r58102 / r58109;
        double r58111 = r58095 - r58101;
        double r58112 = r58111 / r58109;
        double r58113 = r58110 * r58112;
        double r58114 = sqrt(r58113);
        double r58115 = asin(r58114);
        return r58115;
}

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)))))))