Average Error: 10.7 → 10.7
Time: 8.7s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\frac{\sqrt{1 - \left(\sqrt[3]{{\left(\frac{Om}{Omc}\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{Om}{Omc}\right)}^{2}}\right) \cdot \sqrt[3]{{\left(\frac{Om}{Omc}\right)}^{2}}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
\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(\frac{\sqrt{1 - \left(\sqrt[3]{{\left(\frac{Om}{Omc}\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{Om}{Omc}\right)}^{2}}\right) \cdot \sqrt[3]{{\left(\frac{Om}{Omc}\right)}^{2}}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)
double f(double t, double l, double Om, double Omc) {
        double r63454 = 1.0;
        double r63455 = Om;
        double r63456 = Omc;
        double r63457 = r63455 / r63456;
        double r63458 = 2.0;
        double r63459 = pow(r63457, r63458);
        double r63460 = r63454 - r63459;
        double r63461 = t;
        double r63462 = l;
        double r63463 = r63461 / r63462;
        double r63464 = pow(r63463, r63458);
        double r63465 = r63458 * r63464;
        double r63466 = r63454 + r63465;
        double r63467 = r63460 / r63466;
        double r63468 = sqrt(r63467);
        double r63469 = asin(r63468);
        return r63469;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r63470 = 1.0;
        double r63471 = Om;
        double r63472 = Omc;
        double r63473 = r63471 / r63472;
        double r63474 = 2.0;
        double r63475 = pow(r63473, r63474);
        double r63476 = cbrt(r63475);
        double r63477 = r63476 * r63476;
        double r63478 = r63477 * r63476;
        double r63479 = r63470 - r63478;
        double r63480 = sqrt(r63479);
        double r63481 = t;
        double r63482 = l;
        double r63483 = r63481 / r63482;
        double r63484 = pow(r63483, r63474);
        double r63485 = r63474 * r63484;
        double r63486 = r63470 + r63485;
        double r63487 = sqrt(r63486);
        double r63488 = r63480 / r63487;
        double r63489 = asin(r63488);
        return r63489;
}

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 10.7

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

  3. Applied sqrt-div10.7

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

  5. Applied add-cube-cbrt10.7

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

  6. Final simplification10.7

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

Reproduce

herbie shell --seed 2020056 +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)))))))