Average Error: 10.5 → 10.6
Time: 10.2s
Precision: binary64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{1 + \frac{Om}{Omc}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \left(1 - \frac{Om}{Omc}\right)}\right)\]
\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 + \frac{Om}{Omc}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \left(1 - \frac{Om}{Omc}\right)}\right)
(FPCore (t l Om Omc)
 :precision binary64
 (asin
  (sqrt (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t l) 2.0)))))))
↓
(FPCore (t l Om Omc)
 :precision binary64
 (asin
  (sqrt
   (*
    (/ (+ 1.0 (/ Om Omc)) (+ 1.0 (* 2.0 (pow (/ t l) 2.0))))
    (- 1.0 (/ Om Omc))))))
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 + (Om / Omc)) / (1.0 + (2.0 * pow((t / l), 2.0)))) * (1.0 - (Om / Omc))));
}

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 10.5

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

  3. Applied unpow2_binary64_14310.5

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

  4. Applied *-un-lft-identity_binary64_7810.5

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

  5. Applied difference-of-squares_binary64_4710.6

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

  6. Applied associate-/l*_binary64_2310.6

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

  8. Applied associate-/r/_binary64_2410.6

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

  9. Final simplification10.6

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

Reproduce

herbie shell --seed 2020346 
(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)))))))