Average Error: 10.4 → 10.7
Time: 14.3s
Precision: binary64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left({\left(\sqrt[3]{\frac{t}{\ell}} \cdot \sqrt[3]{\frac{t}{\ell}}\right)}^{2} \cdot {\left(\sqrt[3]{\frac{t}{\ell}}\right)}^{2}\right)}}\right)\]
\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(\sqrt[3]{\frac{t}{\ell}} \cdot \sqrt[3]{\frac{t}{\ell}}\right)}^{2} \cdot {\left(\sqrt[3]{\frac{t}{\ell}}\right)}^{2}\right)}}\right)
double code(double t, double l, double Om, double Omc) {
	return ((double) asin(((double) sqrt((((double) (1.0 - ((double) pow((Om / Omc), 2.0)))) / ((double) (1.0 + ((double) (2.0 * ((double) pow((t / l), 2.0)))))))))));
}
↓
double code(double t, double l, double Om, double Omc) {
	return ((double) asin(((double) sqrt((((double) (1.0 - ((double) pow((Om / Omc), 2.0)))) / ((double) (1.0 + ((double) (2.0 * ((double) (((double) pow(((double) (((double) cbrt((t / l))) * ((double) cbrt((t / l))))), 2.0)) * ((double) pow(((double) cbrt((t / l))), 2.0)))))))))))));
}

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 add-cube-cbrt10.7

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

  4. Applied unpow-prod-down10.7

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

  5. Final simplification10.7

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

Reproduce

herbie shell --seed 2020182 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  :precision binary64
  (asin (sqrt (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t l) 2.0)))))))