Average Error: 10.4 → 10.4
Time: 17.1s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{{1}^{3} - {\left({\left(\frac{Om}{Omc}\right)}^{2}\right)}^{3}}{\left({\left(\frac{Om}{Omc}\right)}^{2} \cdot \left({\left(\frac{Om}{Omc}\right)}^{2} + 1\right) + 1 \cdot 1\right) \cdot \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}}\right)\]
\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(\sqrt{\frac{{1}^{3} - {\left({\left(\frac{Om}{Omc}\right)}^{2}\right)}^{3}}{\left({\left(\frac{Om}{Omc}\right)}^{2} \cdot \left({\left(\frac{Om}{Omc}\right)}^{2} + 1\right) + 1 \cdot 1\right) \cdot \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}}\right)
double f(double t, double l, double Om, double Omc) {
        double r95082 = 1.0;
        double r95083 = Om;
        double r95084 = Omc;
        double r95085 = r95083 / r95084;
        double r95086 = 2.0;
        double r95087 = pow(r95085, r95086);
        double r95088 = r95082 - r95087;
        double r95089 = t;
        double r95090 = l;
        double r95091 = r95089 / r95090;
        double r95092 = pow(r95091, r95086);
        double r95093 = r95086 * r95092;
        double r95094 = r95082 + r95093;
        double r95095 = r95088 / r95094;
        double r95096 = sqrt(r95095);
        double r95097 = asin(r95096);
        return r95097;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r95098 = 1.0;
        double r95099 = 3.0;
        double r95100 = pow(r95098, r95099);
        double r95101 = Om;
        double r95102 = Omc;
        double r95103 = r95101 / r95102;
        double r95104 = 2.0;
        double r95105 = pow(r95103, r95104);
        double r95106 = pow(r95105, r95099);
        double r95107 = r95100 - r95106;
        double r95108 = r95105 + r95098;
        double r95109 = r95105 * r95108;
        double r95110 = r95098 * r95098;
        double r95111 = r95109 + r95110;
        double r95112 = t;
        double r95113 = l;
        double r95114 = r95112 / r95113;
        double r95115 = pow(r95114, r95104);
        double r95116 = r95104 * r95115;
        double r95117 = r95098 + r95116;
        double r95118 = r95111 * r95117;
        double r95119 = r95107 / r95118;
        double r95120 = sqrt(r95119);
        double r95121 = asin(r95120);
        return r95121;
}

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 10.4

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

  3. Applied flip3--10.4

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

  4. Applied associate-/l/10.4

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

  5. Simplified10.4

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

  6. Final simplification10.4

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

Reproduce

herbie shell --seed 2020018 +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)))))))