Average Error: 10.5 → 10.6
Time: 9.9s
Precision: binary64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{1 + \frac{Om}{Omc}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}} \cdot \frac{1 - \frac{Om}{Omc}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}\right)\]
\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)
↓
\sin^{-1} \left(\sqrt{\frac{1 + \frac{Om}{Omc}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}} \cdot \frac{1 - \frac{Om}{Omc}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}\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)) (sqrt (+ 1.0 (* 2.0 (pow (/ t l) 2.0)))))
    (/ (- 1.0 (/ Om Omc)) (sqrt (+ 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 + (2.0 * pow((t / l), 2.0)))));
}
↓
double code(double t, double l, double Om, double Omc) {
	return asin(sqrt(((1.0 + (Om / Omc)) / sqrt(1.0 + (2.0 * pow((t / l), 2.0)))) * ((1.0 - (Om / Omc)) / sqrt(1.0 + (2.0 * pow((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 10.5

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

  3. Applied add-sqr-sqrt_binary64_10010.6

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

  4. Applied add-sqr-sqrt_binary64_10031.1

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

  5. Applied unpow-prod-down_binary64_15731.1

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

  6. Applied add-sqr-sqrt_binary64_10031.1

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

  7. Applied difference-of-squares_binary64_4731.1

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

  8. Applied times-frac_binary64_8431.1

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

  9. Simplified31.1

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

  10. Simplified10.6

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

  11. Final simplification10.6

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

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)))))))