Average Error: 10.4 → 10.6
Time: 16.6s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\left|\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}} \cdot \left(\left|\sqrt[3]{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right| \cdot \sqrt{\sqrt[3]{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}\right)}\right|\right)\]
\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{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}} \cdot \left(\left|\sqrt[3]{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right| \cdot \sqrt{\sqrt[3]{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}\right)}\right|\right)
double f(double t, double l, double Om, double Omc) {
        double r70004 = 1.0;
        double r70005 = Om;
        double r70006 = Omc;
        double r70007 = r70005 / r70006;
        double r70008 = 2.0;
        double r70009 = pow(r70007, r70008);
        double r70010 = r70004 - r70009;
        double r70011 = t;
        double r70012 = l;
        double r70013 = r70011 / r70012;
        double r70014 = pow(r70013, r70008);
        double r70015 = r70008 * r70014;
        double r70016 = r70004 + r70015;
        double r70017 = r70010 / r70016;
        double r70018 = sqrt(r70017);
        double r70019 = asin(r70018);
        return r70019;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r70020 = 1.0;
        double r70021 = Om;
        double r70022 = Omc;
        double r70023 = r70021 / r70022;
        double r70024 = 2.0;
        double r70025 = pow(r70023, r70024);
        double r70026 = r70020 - r70025;
        double r70027 = sqrt(r70026);
        double r70028 = t;
        double r70029 = l;
        double r70030 = r70028 / r70029;
        double r70031 = pow(r70030, r70024);
        double r70032 = r70024 * r70031;
        double r70033 = r70020 + r70032;
        double r70034 = sqrt(r70033);
        double r70035 = sqrt(r70034);
        double r70036 = cbrt(r70034);
        double r70037 = fabs(r70036);
        double r70038 = sqrt(r70036);
        double r70039 = r70037 * r70038;
        double r70040 = r70035 * r70039;
        double r70041 = r70027 / r70040;
        double r70042 = fabs(r70041);
        double r70043 = asin(r70042);
        return r70043;
}

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 10.4

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

  3. Applied add-sqr-sqrt10.5

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

  4. Applied add-sqr-sqrt10.5

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

  5. Applied times-frac10.5

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

  6. Applied rem-sqrt-square10.5

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

  8. Applied add-sqr-sqrt10.5

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

  9. Applied sqrt-prod10.5

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

  11. Applied add-cube-cbrt10.6

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

  12. Applied sqrt-prod10.6

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

  13. Simplified10.6

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

  14. Final simplification10.6

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

Reproduce

herbie shell --seed 2019350 
(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)))))))