Average Error: 10.3 → 10.5
Time: 23.6s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{\sqrt{1} + {\left(\frac{Om}{Omc}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}} \cdot \frac{\sqrt{1} - {\left(\frac{Om}{Omc}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}\right)\]
\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} + {\left(\frac{Om}{Omc}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}} \cdot \frac{\sqrt{1} - {\left(\frac{Om}{Omc}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}\right)
double f(double t, double l, double Om, double Omc) {
        double r54639 = 1.0;
        double r54640 = Om;
        double r54641 = Omc;
        double r54642 = r54640 / r54641;
        double r54643 = 2.0;
        double r54644 = pow(r54642, r54643);
        double r54645 = r54639 - r54644;
        double r54646 = t;
        double r54647 = l;
        double r54648 = r54646 / r54647;
        double r54649 = pow(r54648, r54643);
        double r54650 = r54643 * r54649;
        double r54651 = r54639 + r54650;
        double r54652 = r54645 / r54651;
        double r54653 = sqrt(r54652);
        double r54654 = asin(r54653);
        return r54654;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r54655 = 1.0;
        double r54656 = sqrt(r54655);
        double r54657 = Om;
        double r54658 = Omc;
        double r54659 = r54657 / r54658;
        double r54660 = 2.0;
        double r54661 = 2.0;
        double r54662 = r54660 / r54661;
        double r54663 = pow(r54659, r54662);
        double r54664 = r54656 + r54663;
        double r54665 = t;
        double r54666 = l;
        double r54667 = r54665 / r54666;
        double r54668 = pow(r54667, r54660);
        double r54669 = r54660 * r54668;
        double r54670 = r54655 + r54669;
        double r54671 = sqrt(r54670);
        double r54672 = r54664 / r54671;
        double r54673 = r54656 - r54663;
        double r54674 = r54673 / r54671;
        double r54675 = r54672 * r54674;
        double r54676 = sqrt(r54675);
        double r54677 = asin(r54676);
        return r54677;
}

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 sqr-pow10.4

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

  5. Applied add-sqr-sqrt10.4

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

  6. Applied difference-of-squares10.5

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

  7. Applied times-frac10.5

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

  8. Final simplification10.5

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

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)))))))