Average Error: 10.6 → 10.7
Time: 14.0s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\left|\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right|\right)\]
\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|\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right|\right)
double f(double t, double l, double Om, double Omc) {
        double r74179 = 1.0;
        double r74180 = Om;
        double r74181 = Omc;
        double r74182 = r74180 / r74181;
        double r74183 = 2.0;
        double r74184 = pow(r74182, r74183);
        double r74185 = r74179 - r74184;
        double r74186 = t;
        double r74187 = l;
        double r74188 = r74186 / r74187;
        double r74189 = pow(r74188, r74183);
        double r74190 = r74183 * r74189;
        double r74191 = r74179 + r74190;
        double r74192 = r74185 / r74191;
        double r74193 = sqrt(r74192);
        double r74194 = asin(r74193);
        return r74194;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r74195 = 1.0;
        double r74196 = Om;
        double r74197 = Omc;
        double r74198 = r74196 / r74197;
        double r74199 = 2.0;
        double r74200 = pow(r74198, r74199);
        double r74201 = r74195 - r74200;
        double r74202 = sqrt(r74201);
        double r74203 = t;
        double r74204 = l;
        double r74205 = r74203 / r74204;
        double r74206 = pow(r74205, r74199);
        double r74207 = r74199 * r74206;
        double r74208 = r74195 + r74207;
        double r74209 = sqrt(r74208);
        double r74210 = r74202 / r74209;
        double r74211 = fabs(r74210);
        double r74212 = asin(r74211);
        return r74212;
}

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 10.6

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

  3. Applied add-sqr-sqrt10.7

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

  4. Applied add-sqr-sqrt10.7

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

  5. Applied times-frac10.7

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

  6. Applied rem-sqrt-square10.7

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

  7. Final simplification10.7

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

Reproduce

herbie shell --seed 2020045 
(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)))))))