Average Error: 0.1 → 0.2
Time: 17.5s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{3} - \sqrt[3]{{\left({\left(\sin \phi_1\right)}^{3} \cdot {\left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)}^{3}\right)}^{3}}}{\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) + \cos delta\right) + \cos delta \cdot \cos delta}}\]
\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}
↓
\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{3} - \sqrt[3]{{\left({\left(\sin \phi_1\right)}^{3} \cdot {\left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)}^{3}\right)}^{3}}}{\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) + \cos delta\right) + \cos delta \cdot \cos delta}}
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r101265 = lambda1;
        double r101266 = theta;
        double r101267 = sin(r101266);
        double r101268 = delta;
        double r101269 = sin(r101268);
        double r101270 = r101267 * r101269;
        double r101271 = phi1;
        double r101272 = cos(r101271);
        double r101273 = r101270 * r101272;
        double r101274 = cos(r101268);
        double r101275 = sin(r101271);
        double r101276 = r101275 * r101274;
        double r101277 = r101272 * r101269;
        double r101278 = cos(r101266);
        double r101279 = r101277 * r101278;
        double r101280 = r101276 + r101279;
        double r101281 = asin(r101280);
        double r101282 = sin(r101281);
        double r101283 = r101275 * r101282;
        double r101284 = r101274 - r101283;
        double r101285 = atan2(r101273, r101284);
        double r101286 = r101265 + r101285;
        return r101286;
}
↓
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r101287 = lambda1;
        double r101288 = theta;
        double r101289 = sin(r101288);
        double r101290 = delta;
        double r101291 = sin(r101290);
        double r101292 = r101289 * r101291;
        double r101293 = phi1;
        double r101294 = cos(r101293);
        double r101295 = r101292 * r101294;
        double r101296 = cos(r101290);
        double r101297 = 3.0;
        double r101298 = pow(r101296, r101297);
        double r101299 = sin(r101293);
        double r101300 = pow(r101299, r101297);
        double r101301 = cos(r101288);
        double r101302 = r101294 * r101301;
        double r101303 = r101291 * r101302;
        double r101304 = r101299 * r101296;
        double r101305 = r101303 + r101304;
        double r101306 = pow(r101305, r101297);
        double r101307 = r101300 * r101306;
        double r101308 = pow(r101307, r101297);
        double r101309 = cbrt(r101308);
        double r101310 = r101298 - r101309;
        double r101311 = r101294 * r101291;
        double r101312 = r101311 * r101301;
        double r101313 = r101304 + r101312;
        double r101314 = asin(r101313);
        double r101315 = sin(r101314);
        double r101316 = r101299 * r101315;
        double r101317 = r101316 + r101296;
        double r101318 = r101316 * r101317;
        double r101319 = r101296 * r101296;
        double r101320 = r101318 + r101319;
        double r101321 = r101310 / r101320;
        double r101322 = atan2(r101295, r101321);
        double r101323 = r101287 + r101322;
        return r101323;
}

Error

Bits error versus lambda1

Bits error versus phi1

Bits error versus phi2

Bits error versus delta

Bits error versus theta

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.1

    \[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
  2. Using strategy rm

  3. Applied flip3--0.1

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\frac{{\left(\cos delta\right)}^{3} - {\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)}^{3}}{\cos delta \cdot \cos delta + \left(\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) + \cos delta \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right)}}}\]

  4. Simplified0.1

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{3} - {\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)}^{3}}{\color{blue}{\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) + \cos delta\right) + \cos delta \cdot \cos delta}}}\]
  5. Using strategy rm

  6. Applied add-cbrt-cube0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{3} - \color{blue}{\sqrt[3]{\left({\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)}^{3} \cdot {\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)}^{3}\right) \cdot {\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)}^{3}}}}{\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) + \cos delta\right) + \cos delta \cdot \cos delta}}\]

  7. Simplified0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{3} - \sqrt[3]{\color{blue}{{\left({\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)}^{3}\right)}^{3}}}}{\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) + \cos delta\right) + \cos delta \cdot \cos delta}}\]

  8. Taylor expanded around inf 0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{3} - \sqrt[3]{{\color{blue}{\left({\left(\sin \phi_1\right)}^{3} \cdot {\left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)}^{3}\right)}}^{3}}}{\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) + \cos delta\right) + \cos delta \cdot \cos delta}}\]

  9. Final simplification0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{3} - \sqrt[3]{{\left({\left(\sin \phi_1\right)}^{3} \cdot {\left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)}^{3}\right)}^{3}}}{\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) + \cos delta\right) + \cos delta \cdot \cos delta}}\]

Reproduce

herbie shell --seed 2020083 
(FPCore (lambda1 phi1 phi2 delta theta)
  :name "Destination given bearing on a great circle"
  :precision binary64
  (+ lambda1 (atan2 (* (* (sin theta) (sin delta)) (cos phi1)) (- (cos delta) (* (sin phi1) (sin (asin (+ (* (sin phi1) (cos delta)) (* (* (cos phi1) (sin delta)) (cos theta))))))))))