Average Error: 10.6 → 10.7
Time: 10.5s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\left|\frac{\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}{\sqrt{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}\right|\right)\]
\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(\left|\frac{\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}{\sqrt{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}\right|\right)
double f(double t, double l, double Om, double Omc) {
        double r75165 = 1.0;
        double r75166 = Om;
        double r75167 = Omc;
        double r75168 = r75166 / r75167;
        double r75169 = 2.0;
        double r75170 = pow(r75168, r75169);
        double r75171 = r75165 - r75170;
        double r75172 = t;
        double r75173 = l;
        double r75174 = r75172 / r75173;
        double r75175 = pow(r75174, r75169);
        double r75176 = r75169 * r75175;
        double r75177 = r75165 + r75176;
        double r75178 = r75171 / r75177;
        double r75179 = sqrt(r75178);
        double r75180 = asin(r75179);
        return r75180;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r75181 = 1.0;
        double r75182 = Om;
        double r75183 = Omc;
        double r75184 = r75182 / r75183;
        double r75185 = 2.0;
        double r75186 = pow(r75184, r75185);
        double r75187 = r75181 - r75186;
        double r75188 = t;
        double r75189 = l;
        double r75190 = r75188 / r75189;
        double r75191 = pow(r75190, r75185);
        double r75192 = r75185 * r75191;
        double r75193 = r75181 + r75192;
        double r75194 = sqrt(r75193);
        double r75195 = r75187 / r75194;
        double r75196 = sqrt(r75195);
        double r75197 = sqrt(r75194);
        double r75198 = r75196 / r75197;
        double r75199 = fabs(r75198);
        double r75200 = asin(r75199);
        return r75200;
}

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 10.6

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

  3. Applied add-sqr-sqrt10.7

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

  4. Applied associate-/r*10.7

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

  6. Applied add-sqr-sqrt10.7

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

  7. Applied sqrt-prod10.7

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

  8. Applied add-sqr-sqrt10.7

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

  9. Applied times-frac10.7

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

  10. Applied rem-sqrt-square10.7

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

  11. Final simplification10.7

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

Reproduce

herbie shell --seed 2020060 
(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)))))))