Average Error: 9.8 → 9.8
Time: 28.0s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\left|\sqrt{\frac{1 - \frac{\frac{Om}{Omc} \cdot Om}{Omc}}{\frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}} + 1}}\right|\right)\]
\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|\sqrt{\frac{1 - \frac{\frac{Om}{Omc} \cdot Om}{Omc}}{\frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}} + 1}}\right|\right)
double f(double t, double l, double Om, double Omc) {
        double r2849227 = 1.0;
        double r2849228 = Om;
        double r2849229 = Omc;
        double r2849230 = r2849228 / r2849229;
        double r2849231 = 2.0;
        double r2849232 = pow(r2849230, r2849231);
        double r2849233 = r2849227 - r2849232;
        double r2849234 = t;
        double r2849235 = l;
        double r2849236 = r2849234 / r2849235;
        double r2849237 = pow(r2849236, r2849231);
        double r2849238 = r2849231 * r2849237;
        double r2849239 = r2849227 + r2849238;
        double r2849240 = r2849233 / r2849239;
        double r2849241 = sqrt(r2849240);
        double r2849242 = asin(r2849241);
        return r2849242;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r2849243 = 1.0;
        double r2849244 = Om;
        double r2849245 = Omc;
        double r2849246 = r2849244 / r2849245;
        double r2849247 = r2849246 * r2849244;
        double r2849248 = r2849247 / r2849245;
        double r2849249 = r2849243 - r2849248;
        double r2849250 = 2.0;
        double r2849251 = l;
        double r2849252 = t;
        double r2849253 = r2849251 / r2849252;
        double r2849254 = r2849253 * r2849253;
        double r2849255 = r2849250 / r2849254;
        double r2849256 = r2849255 + r2849243;
        double r2849257 = r2849249 / r2849256;
        double r2849258 = sqrt(r2849257);
        double r2849259 = fabs(r2849258);
        double r2849260 = asin(r2849259);
        return r2849260;
}

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 9.8

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

  3. Applied add-sqr-sqrt9.8

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

  4. Applied add-sqr-sqrt9.8

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

  5. Applied times-frac9.8

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

  6. Applied rem-sqrt-square9.8

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

  7. Taylor expanded around inf 26.3

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

  8. Simplified9.8

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

  10. Applied associate-*l/9.8

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

  11. Final simplification9.8

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

Reproduce

herbie shell --seed 2019172 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t l) 2.0)))))))