Destination given bearing on a great circle

Average Error: 0.2 → 0.2

Time: 45.5s

Precision: 64

• unused variable phi2

Math

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

↓

\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\frac{{\left({\left(\cos delta\right)}^{2}\right)}^{3} - {\left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)\right) + \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right) + {\left({\left(\sin \phi_1\right)}^{\left(\sqrt{4}\right)}\right)}^{\left(\sqrt{4}\right)} \cdot {\left(\cos delta\right)}^{2}\right)\right)}^{3}}{\left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)\right) + \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right) + {\left({\left(\sin \phi_1\right)}^{\left(\sqrt{4}\right)}\right)}^{\left(\sqrt{4}\right)} \cdot {\left(\cos delta\right)}^{2}\right)\right) \cdot \left(\left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)\right) + \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right) + {\left({\left(\sin \phi_1\right)}^{\left(\sqrt{4}\right)}\right)}^{\left(\sqrt{4}\right)} \cdot {\left(\cos delta\right)}^{2}\right)\right) + {\left(\cos delta\right)}^{2}\right) + {\left(\cos delta\right)}^{4}}}{\cos delta + \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}}\]

C

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r85260 = lambda1;
        double r85261 = theta;
        double r85262 = sin(r85261);
        double r85263 = delta;
        double r85264 = sin(r85263);
        double r85265 = r85262 * r85264;
        double r85266 = phi1;
        double r85267 = cos(r85266);
        double r85268 = r85265 * r85267;
        double r85269 = cos(r85263);
        double r85270 = sin(r85266);
        double r85271 = r85270 * r85269;
        double r85272 = r85267 * r85264;
        double r85273 = cos(r85261);
        double r85274 = r85272 * r85273;
        double r85275 = r85271 + r85274;
        double r85276 = asin(r85275);
        double r85277 = sin(r85276);
        double r85278 = r85270 * r85277;
        double r85279 = r85269 - r85278;
        double r85280 = atan2(r85268, r85279);
        double r85281 = r85260 + r85280;
        return r85281;
}

↓

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r85282 = lambda1;
        double r85283 = theta;
        double r85284 = sin(r85283);
        double r85285 = delta;
        double r85286 = sin(r85285);
        double r85287 = r85284 * r85286;
        double r85288 = phi1;
        double r85289 = cos(r85288);
        double r85290 = r85287 * r85289;
        double r85291 = cos(r85285);
        double r85292 = 2.0;
        double r85293 = pow(r85291, r85292);
        double r85294 = 3.0;
        double r85295 = pow(r85293, r85294);
        double r85296 = pow(r85286, r85292);
        double r85297 = pow(r85289, r85292);
        double r85298 = cos(r85283);
        double r85299 = pow(r85298, r85292);
        double r85300 = sin(r85288);
        double r85301 = pow(r85300, r85292);
        double r85302 = r85299 * r85301;
        double r85303 = r85297 * r85302;
        double r85304 = r85296 * r85303;
        double r85305 = pow(r85300, r85294);
        double r85306 = r85291 * r85298;
        double r85307 = r85305 * r85306;
        double r85308 = r85289 * r85307;
        double r85309 = r85286 * r85308;
        double r85310 = r85292 * r85309;
        double r85311 = 4.0;
        double r85312 = sqrt(r85311);
        double r85313 = pow(r85300, r85312);
        double r85314 = pow(r85313, r85312);
        double r85315 = r85314 * r85293;
        double r85316 = r85310 + r85315;
        double r85317 = r85304 + r85316;
        double r85318 = pow(r85317, r85294);
        double r85319 = r85295 - r85318;
        double r85320 = r85317 + r85293;
        double r85321 = r85317 * r85320;
        double r85322 = pow(r85291, r85311);
        double r85323 = r85321 + r85322;
        double r85324 = r85319 / r85323;
        double r85325 = r85300 * r85291;
        double r85326 = r85289 * r85286;
        double r85327 = r85326 * r85298;
        double r85328 = r85325 + r85327;
        double r85329 = asin(r85328);
        double r85330 = sin(r85329);
        double r85331 = r85300 * r85330;
        double r85332 = r85291 + r85331;
        double r85333 = r85324 / r85332;
        double r85334 = atan2(r85290, r85333);
        double r85335 = r85282 + r85334;
        return r85335;
}

Error

Bits error versus lambda1

Bits error versus phi1

Bits error versus phi2

Bits error versus delta

Bits error versus theta

Derivation

1. Initial program 0.2

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

3. Applied flip-- 0.2

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

4. Taylor expanded around inf 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\color{blue}{{\left(\cos delta\right)}^{2} - \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)\right) + \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right) + {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right)}}{\cos delta + \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}}\]
5. Using strategy rm

6. Applied add-sqr-sqrt 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{2} - \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)\right) + \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right) + {\left(\sin \phi_1\right)}^{\color{blue}{\left(\sqrt{4} \cdot \sqrt{4}\right)}} \cdot {\left(\cos delta\right)}^{2}\right)\right)}{\cos delta + \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}}\]

7. Applied pow-unpow 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{2} - \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)\right) + \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right) + \color{blue}{{\left({\left(\sin \phi_1\right)}^{\left(\sqrt{4}\right)}\right)}^{\left(\sqrt{4}\right)}} \cdot {\left(\cos delta\right)}^{2}\right)\right)}{\cos delta + \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}}\]
8. Using strategy rm

9. Applied flip3-- 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\color{blue}{\frac{{\left({\left(\cos delta\right)}^{2}\right)}^{3} - {\left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)\right) + \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right) + {\left({\left(\sin \phi_1\right)}^{\left(\sqrt{4}\right)}\right)}^{\left(\sqrt{4}\right)} \cdot {\left(\cos delta\right)}^{2}\right)\right)}^{3}}{{\left(\cos delta\right)}^{2} \cdot {\left(\cos delta\right)}^{2} + \left(\left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)\right) + \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right) + {\left({\left(\sin \phi_1\right)}^{\left(\sqrt{4}\right)}\right)}^{\left(\sqrt{4}\right)} \cdot {\left(\cos delta\right)}^{2}\right)\right) \cdot \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)\right) + \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right) + {\left({\left(\sin \phi_1\right)}^{\left(\sqrt{4}\right)}\right)}^{\left(\sqrt{4}\right)} \cdot {\left(\cos delta\right)}^{2}\right)\right) + {\left(\cos delta\right)}^{2} \cdot \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)\right) + \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right) + {\left({\left(\sin \phi_1\right)}^{\left(\sqrt{4}\right)}\right)}^{\left(\sqrt{4}\right)} \cdot {\left(\cos delta\right)}^{2}\right)\right)\right)}}}{\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)}}\]

10. Simplified 0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\frac{{\left({\left(\cos delta\right)}^{2}\right)}^{3} - {\left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)\right) + \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right) + {\left({\left(\sin \phi_1\right)}^{\left(\sqrt{4}\right)}\right)}^{\left(\sqrt{4}\right)} \cdot {\left(\cos delta\right)}^{2}\right)\right)}^{3}}{\color{blue}{\left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)\right) + \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right) + {\left({\left(\sin \phi_1\right)}^{\left(\sqrt{4}\right)}\right)}^{\left(\sqrt{4}\right)} \cdot {\left(\cos delta\right)}^{2}\right)\right) \cdot \left(\left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)\right) + \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right) + {\left({\left(\sin \phi_1\right)}^{\left(\sqrt{4}\right)}\right)}^{\left(\sqrt{4}\right)} \cdot {\left(\cos delta\right)}^{2}\right)\right) + {\left(\cos delta\right)}^{2}\right) + {\left(\cos delta\right)}^{4}}}}{\cos delta + \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}}\]

11. Final simplification 0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\frac{{\left({\left(\cos delta\right)}^{2}\right)}^{3} - {\left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)\right) + \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right) + {\left({\left(\sin \phi_1\right)}^{\left(\sqrt{4}\right)}\right)}^{\left(\sqrt{4}\right)} \cdot {\left(\cos delta\right)}^{2}\right)\right)}^{3}}{\left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)\right) + \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right) + {\left({\left(\sin \phi_1\right)}^{\left(\sqrt{4}\right)}\right)}^{\left(\sqrt{4}\right)} \cdot {\left(\cos delta\right)}^{2}\right)\right) \cdot \left(\left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)\right) + \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right) + {\left({\left(\sin \phi_1\right)}^{\left(\sqrt{4}\right)}\right)}^{\left(\sqrt{4}\right)} \cdot {\left(\cos delta\right)}^{2}\right)\right) + {\left(\cos delta\right)}^{2}\right) + {\left(\cos delta\right)}^{4}}}{\cos delta + \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}}\]

Reproduce

herbie shell --seed 2019303 
(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))))))))))