Average Error: 9.9 → 10.0
Time: 15.2s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{\sqrt{{1}^{3} - {\left({\left(\frac{Om}{Omc}\right)}^{2}\right)}^{3}}}{{\left(\frac{Om}{Omc}\right)}^{2} \cdot \left({\left(\frac{Om}{Omc}\right)}^{2} + 1\right) + 1 \cdot 1}} \cdot \sqrt{\frac{\sqrt{{1}^{3} - {\left({\left(\frac{Om}{Omc}\right)}^{2}\right)}^{3}}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
\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}^{3} - {\left({\left(\frac{Om}{Omc}\right)}^{2}\right)}^{3}}}{{\left(\frac{Om}{Omc}\right)}^{2} \cdot \left({\left(\frac{Om}{Omc}\right)}^{2} + 1\right) + 1 \cdot 1}} \cdot \sqrt{\frac{\sqrt{{1}^{3} - {\left({\left(\frac{Om}{Omc}\right)}^{2}\right)}^{3}}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)
double f(double t, double l, double Om, double Omc) {
        double r91573 = 1.0;
        double r91574 = Om;
        double r91575 = Omc;
        double r91576 = r91574 / r91575;
        double r91577 = 2.0;
        double r91578 = pow(r91576, r91577);
        double r91579 = r91573 - r91578;
        double r91580 = t;
        double r91581 = l;
        double r91582 = r91580 / r91581;
        double r91583 = pow(r91582, r91577);
        double r91584 = r91577 * r91583;
        double r91585 = r91573 + r91584;
        double r91586 = r91579 / r91585;
        double r91587 = sqrt(r91586);
        double r91588 = asin(r91587);
        return r91588;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r91589 = 1.0;
        double r91590 = 3.0;
        double r91591 = pow(r91589, r91590);
        double r91592 = Om;
        double r91593 = Omc;
        double r91594 = r91592 / r91593;
        double r91595 = 2.0;
        double r91596 = pow(r91594, r91595);
        double r91597 = pow(r91596, r91590);
        double r91598 = r91591 - r91597;
        double r91599 = sqrt(r91598);
        double r91600 = r91596 + r91589;
        double r91601 = r91596 * r91600;
        double r91602 = r91589 * r91589;
        double r91603 = r91601 + r91602;
        double r91604 = r91599 / r91603;
        double r91605 = sqrt(r91604);
        double r91606 = t;
        double r91607 = l;
        double r91608 = r91606 / r91607;
        double r91609 = pow(r91608, r91595);
        double r91610 = r91595 * r91609;
        double r91611 = r91589 + r91610;
        double r91612 = r91599 / r91611;
        double r91613 = sqrt(r91612);
        double r91614 = r91605 * r91613;
        double r91615 = asin(r91614);
        return r91615;
}

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 9.9

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

  3. Applied flip3--9.9

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

  4. Applied associate-/l/9.9

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

  5. Simplified9.9

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

  7. Applied add-sqr-sqrt10.0

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

  8. Applied times-frac10.0

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

  9. Applied sqrt-prod10.0

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

  10. Final simplification10.0

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

Reproduce

herbie shell --seed 2020083 
(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)))))))