Average Error: 10.6 → 10.7
Time: 11.3s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\left|\sqrt[3]{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right| \cdot \sqrt{\sqrt[3]{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right)}\right)\]
\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{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\left|\sqrt[3]{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right| \cdot \sqrt{\sqrt[3]{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\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(expm1(log1p((((1.0 - pow((Om / Omc), 2.0)) / (fabs(cbrt((1.0 + (2.0 * pow((t / l), 2.0))))) * sqrt(cbrt((1.0 + (2.0 * pow((t / l), 2.0))))))) / 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 10.6

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

  3. Applied expm1-log1p-u10.6

    \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)}}\right)\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt10.7

    \[\leadsto \sin^{-1} \left(\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\color{blue}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}\right)\right)}\right)\]

  6. Applied associate-/r*10.7

    \[\leadsto \sin^{-1} \left(\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}\right)\right)}\right)\]
  7. Using strategy rm

  8. Applied add-cube-cbrt10.7

    \[\leadsto \sin^{-1} \left(\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\sqrt{\color{blue}{\left(\sqrt[3]{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt[3]{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right) \cdot \sqrt[3]{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right)}\right)\]

  9. Applied sqrt-prod10.7

    \[\leadsto \sin^{-1} \left(\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\color{blue}{\sqrt{\sqrt[3]{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt[3]{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}} \cdot \sqrt{\sqrt[3]{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right)}\right)\]

  10. Simplified10.7

    \[\leadsto \sin^{-1} \left(\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\color{blue}{\left|\sqrt[3]{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right|} \cdot \sqrt{\sqrt[3]{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right)}\right)\]

  11. Final simplification10.7

    \[\leadsto \sin^{-1} \left(\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\left|\sqrt[3]{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right| \cdot \sqrt{\sqrt[3]{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right)}\right)\]

Reproduce

herbie shell --seed 2020060 +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)))))))