Average Error: 10.3 → 10.3
Time: 9.5s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left({\left(\frac{t}{\ell}\right)}^{2}, 2, 1\right)}}\right)\]
\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{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left({\left(\frac{t}{\ell}\right)}^{2}, 2, 1\right)}}\right)
double code(double t, double l, double Om, double Omc) {
	return asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t / l), 2.0))))));
}
↓
double code(double t, double l, double Om, double Omc) {
	return asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) * (1.0 / fma(pow((t / l), 2.0), 2.0, 1.0)))));
}

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 10.3

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

  3. Applied div-inv10.3

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

  4. Simplified10.3

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

  5. Final simplification10.3

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

Reproduce

herbie shell --seed 2020106 +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)))))))