Average Error: 10.9 → 10.9
Time: 10.6s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
\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{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)
double f(double t, double l, double Om, double Omc) {
        double r68626 = 1.0;
        double r68627 = Om;
        double r68628 = Omc;
        double r68629 = r68627 / r68628;
        double r68630 = 2.0;
        double r68631 = pow(r68629, r68630);
        double r68632 = r68626 - r68631;
        double r68633 = t;
        double r68634 = l;
        double r68635 = r68633 / r68634;
        double r68636 = pow(r68635, r68630);
        double r68637 = r68630 * r68636;
        double r68638 = r68626 + r68637;
        double r68639 = r68632 / r68638;
        double r68640 = sqrt(r68639);
        double r68641 = asin(r68640);
        return r68641;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r68642 = 1.0;
        double r68643 = Om;
        double r68644 = Omc;
        double r68645 = r68643 / r68644;
        double r68646 = 2.0;
        double r68647 = pow(r68645, r68646);
        double r68648 = r68642 - r68647;
        double r68649 = t;
        double r68650 = l;
        double r68651 = r68649 / r68650;
        double r68652 = pow(r68651, r68646);
        double r68653 = r68646 * r68652;
        double r68654 = r68642 + r68653;
        double r68655 = r68648 / r68654;
        double r68656 = sqrt(r68655);
        double r68657 = asin(r68656);
        return r68657;
}

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 10.9

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

  2. Final simplification10.9

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

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(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)))))))