Average Error: 10.3 → 10.4
Time: 31.9s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\sqrt{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2} + 1}} \cdot \sqrt{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2} + 1}}}}{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2} + 1}}}\right)\]
\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{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\sqrt{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2} + 1}} \cdot \sqrt{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2} + 1}}}}{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2} + 1}}}\right)
double f(double t, double l, double Om, double Omc) {
        double r3428248 = 1.0;
        double r3428249 = Om;
        double r3428250 = Omc;
        double r3428251 = r3428249 / r3428250;
        double r3428252 = 2.0;
        double r3428253 = pow(r3428251, r3428252);
        double r3428254 = r3428248 - r3428253;
        double r3428255 = t;
        double r3428256 = l;
        double r3428257 = r3428255 / r3428256;
        double r3428258 = pow(r3428257, r3428252);
        double r3428259 = r3428252 * r3428258;
        double r3428260 = r3428248 + r3428259;
        double r3428261 = r3428254 / r3428260;
        double r3428262 = sqrt(r3428261);
        double r3428263 = asin(r3428262);
        return r3428263;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r3428264 = 1.0;
        double r3428265 = Om;
        double r3428266 = Omc;
        double r3428267 = r3428265 / r3428266;
        double r3428268 = 2.0;
        double r3428269 = pow(r3428267, r3428268);
        double r3428270 = r3428264 - r3428269;
        double r3428271 = t;
        double r3428272 = l;
        double r3428273 = r3428271 / r3428272;
        double r3428274 = pow(r3428273, r3428268);
        double r3428275 = r3428268 * r3428274;
        double r3428276 = r3428275 + r3428264;
        double r3428277 = sqrt(r3428276);
        double r3428278 = sqrt(r3428277);
        double r3428279 = r3428278 * r3428278;
        double r3428280 = r3428270 / r3428279;
        double r3428281 = r3428280 / r3428277;
        double r3428282 = sqrt(r3428281);
        double r3428283 = asin(r3428282);
        return r3428283;
}

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 associate-/r*10.4

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

  6. Applied *-un-lft-identity10.4

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

  8. Applied add-sqr-sqrt10.4

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

  9. Applied sqrt-prod10.4

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

  10. Final simplification10.4

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

Reproduce

herbie shell --seed 2019171 
(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)))))))