Average Error: 10.1 → 10.1
Time: 8.5s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\left|\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right|\right)\]
\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(\left|\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right|\right)
double f(double t, double l, double Om, double Omc) {
        double r65266 = 1.0;
        double r65267 = Om;
        double r65268 = Omc;
        double r65269 = r65267 / r65268;
        double r65270 = 2.0;
        double r65271 = pow(r65269, r65270);
        double r65272 = r65266 - r65271;
        double r65273 = t;
        double r65274 = l;
        double r65275 = r65273 / r65274;
        double r65276 = pow(r65275, r65270);
        double r65277 = r65270 * r65276;
        double r65278 = r65266 + r65277;
        double r65279 = r65272 / r65278;
        double r65280 = sqrt(r65279);
        double r65281 = asin(r65280);
        return r65281;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r65282 = 1.0;
        double r65283 = Om;
        double r65284 = Omc;
        double r65285 = r65283 / r65284;
        double r65286 = 2.0;
        double r65287 = pow(r65285, r65286);
        double r65288 = r65282 - r65287;
        double r65289 = sqrt(r65288);
        double r65290 = t;
        double r65291 = l;
        double r65292 = r65290 / r65291;
        double r65293 = pow(r65292, r65286);
        double r65294 = r65286 * r65293;
        double r65295 = r65282 + r65294;
        double r65296 = sqrt(r65295);
        double r65297 = r65289 / r65296;
        double r65298 = fabs(r65297);
        double r65299 = asin(r65298);
        return r65299;
}

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 10.1

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

  3. Applied add-sqr-sqrt10.1

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

  4. Applied add-sqr-sqrt10.1

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

  5. Applied times-frac10.1

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

  6. Applied rem-sqrt-square10.1

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

  7. Final simplification10.1

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

Reproduce

herbie shell --seed 2020062 
(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)))))))