Average Error: 10.3 → 10.4
Time: 9.1s
Precision: binary64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{1}{\sqrt[3]{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt[3]{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}} \cdot \frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\sqrt[3]{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}} \cdot \sqrt[3]{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right)\]
\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}{\sqrt[3]{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt[3]{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}} \cdot \frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\sqrt[3]{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}} \cdot \sqrt[3]{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right)
double code(double t, double l, double Om, double Omc) {
	return ((double) asin(((double) sqrt(((double) (((double) (1.0 - ((double) pow(((double) (Om / Omc)), 2.0)))) / ((double) (1.0 + ((double) (2.0 * ((double) pow(((double) (t / l)), 2.0))))))))))));
}
↓
double code(double t, double l, double Om, double Omc) {
	return ((double) asin(((double) sqrt(((double) (((double) (1.0 / ((double) (((double) cbrt(((double) (1.0 + ((double) (2.0 * ((double) pow(((double) (t / l)), 2.0)))))))) * ((double) cbrt(((double) (1.0 + ((double) (2.0 * ((double) pow(((double) (t / l)), 2.0)))))))))))) * ((double) (((double) (1.0 - ((double) pow(((double) (Om / Omc)), 2.0)))) / ((double) (((double) cbrt(((double) sqrt(((double) (1.0 + ((double) (2.0 * ((double) pow(((double) (t / l)), 2.0)))))))))) * ((double) cbrt(((double) sqrt(((double) (1.0 + ((double) (2.0 * ((double) pow(((double) (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 10.3

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

  3. Applied add-cube-cbrt10.4

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

  4. Applied *-un-lft-identity10.4

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

  5. Applied times-frac10.4

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

  7. Applied add-sqr-sqrt10.4

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

  8. Applied cbrt-prod10.4

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

  9. Final simplification10.4

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

Reproduce

herbie shell --seed 2020184 
(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)))))))