Average Error: 0.2 → 0.2
Time: 55.0s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\frac{1}{\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right) + \sin \phi_1 \cdot \cos delta\right)\right) + \cos delta} \cdot \left(\cos delta \cdot \cos delta - \sin \phi_1 \cdot \left(\sin \left(\sin^{-1} \left(\cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right) + \sin \phi_1 \cdot \cos delta\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right) + \sin \phi_1 \cdot \cos delta\right)\right)\right)\right)\right)} + \lambda_1\]
\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)}
↓
\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\frac{1}{\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right) + \sin \phi_1 \cdot \cos delta\right)\right) + \cos delta} \cdot \left(\cos delta \cdot \cos delta - \sin \phi_1 \cdot \left(\sin \left(\sin^{-1} \left(\cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right) + \sin \phi_1 \cdot \cos delta\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right) + \sin \phi_1 \cdot \cos delta\right)\right)\right)\right)\right)} + \lambda_1
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r3281299 = lambda1;
        double r3281300 = theta;
        double r3281301 = sin(r3281300);
        double r3281302 = delta;
        double r3281303 = sin(r3281302);
        double r3281304 = r3281301 * r3281303;
        double r3281305 = phi1;
        double r3281306 = cos(r3281305);
        double r3281307 = r3281304 * r3281306;
        double r3281308 = cos(r3281302);
        double r3281309 = sin(r3281305);
        double r3281310 = r3281309 * r3281308;
        double r3281311 = r3281306 * r3281303;
        double r3281312 = cos(r3281300);
        double r3281313 = r3281311 * r3281312;
        double r3281314 = r3281310 + r3281313;
        double r3281315 = asin(r3281314);
        double r3281316 = sin(r3281315);
        double r3281317 = r3281309 * r3281316;
        double r3281318 = r3281308 - r3281317;
        double r3281319 = atan2(r3281307, r3281318);
        double r3281320 = r3281299 + r3281319;
        return r3281320;
}
↓
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r3281321 = phi1;
        double r3281322 = cos(r3281321);
        double r3281323 = delta;
        double r3281324 = sin(r3281323);
        double r3281325 = theta;
        double r3281326 = sin(r3281325);
        double r3281327 = r3281324 * r3281326;
        double r3281328 = r3281322 * r3281327;
        double r3281329 = 1.0;
        double r3281330 = sin(r3281321);
        double r3281331 = cos(r3281325);
        double r3281332 = r3281322 * r3281324;
        double r3281333 = r3281331 * r3281332;
        double r3281334 = cos(r3281323);
        double r3281335 = r3281330 * r3281334;
        double r3281336 = r3281333 + r3281335;
        double r3281337 = asin(r3281336);
        double r3281338 = sin(r3281337);
        double r3281339 = r3281330 * r3281338;
        double r3281340 = r3281339 + r3281334;
        double r3281341 = r3281329 / r3281340;
        double r3281342 = r3281334 * r3281334;
        double r3281343 = r3281338 * r3281339;
        double r3281344 = r3281330 * r3281343;
        double r3281345 = r3281342 - r3281344;
        double r3281346 = r3281341 * r3281345;
        double r3281347 = atan2(r3281328, r3281346);
        double r3281348 = lambda1;
        double r3281349 = r3281347 + r3281348;
        return r3281349;
}

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 0.2

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

  5. Applied div-inv0.2

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

  7. Applied associate-*l*0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019152 
(FPCore (lambda1 phi1 phi2 delta theta)
  :name "Destination given bearing on a great circle"
  (+ lambda1 (atan2 (* (* (sin theta) (sin delta)) (cos phi1)) (- (cos delta) (* (sin phi1) (sin (asin (+ (* (sin phi1) (cos delta)) (* (* (cos phi1) (sin delta)) (cos theta))))))))))