Average Error: 10.2 → 10.3
Time: 11.3s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(\left(\left(\sqrt[3]{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}}} \cdot \sqrt[3]{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}}}\right) \cdot \sqrt[3]{{\left(\frac{t}{\ell}\right)}^{2}}\right) \cdot \sqrt[3]{{\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right)\]
\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(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(\left(\left(\sqrt[3]{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}}} \cdot \sqrt[3]{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}}}\right) \cdot \sqrt[3]{{\left(\frac{t}{\ell}\right)}^{2}}\right) \cdot \sqrt[3]{{\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right)
double f(double t, double l, double Om, double Omc) {
        double r68159 = 1.0;
        double r68160 = Om;
        double r68161 = Omc;
        double r68162 = r68160 / r68161;
        double r68163 = 2.0;
        double r68164 = pow(r68162, r68163);
        double r68165 = r68159 - r68164;
        double r68166 = t;
        double r68167 = l;
        double r68168 = r68166 / r68167;
        double r68169 = pow(r68168, r68163);
        double r68170 = r68163 * r68169;
        double r68171 = r68159 + r68170;
        double r68172 = r68165 / r68171;
        double r68173 = sqrt(r68172);
        double r68174 = asin(r68173);
        return r68174;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r68175 = 1.0;
        double r68176 = Om;
        double r68177 = Omc;
        double r68178 = r68176 / r68177;
        double r68179 = 2.0;
        double r68180 = pow(r68178, r68179);
        double r68181 = r68175 - r68180;
        double r68182 = t;
        double r68183 = l;
        double r68184 = r68182 / r68183;
        double r68185 = pow(r68184, r68179);
        double r68186 = sqrt(r68185);
        double r68187 = cbrt(r68186);
        double r68188 = r68187 * r68187;
        double r68189 = cbrt(r68185);
        double r68190 = r68188 * r68189;
        double r68191 = r68190 * r68189;
        double r68192 = r68179 * r68191;
        double r68193 = r68175 + r68192;
        double r68194 = r68181 / r68193;
        double r68195 = sqrt(r68194);
        double r68196 = asin(r68195);
        return r68196;
}

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 10.2

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

  3. Applied add-cube-cbrt10.3

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

  5. Applied add-sqr-sqrt10.3

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

  6. Applied cbrt-prod10.3

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

  7. Final simplification10.3

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

Reproduce

herbie shell --seed 2020100 
(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)))))))