Average Error: 0.2 → 0.2
Time: 1.0m
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\frac{\cos delta \cdot \cos delta - \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right) + \sin \phi_1 \cdot \cos delta\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right) + \sin \phi_1 \cdot \cos delta\right)\right)\right)}{\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right) + \sin \phi_1 \cdot \cos delta\right)\right) + \cos delta}} + \lambda_1\]
\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{\cos delta \cdot \cos delta - \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) \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)}{\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}} + \lambda_1
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r4798255 = lambda1;
        double r4798256 = theta;
        double r4798257 = sin(r4798256);
        double r4798258 = delta;
        double r4798259 = sin(r4798258);
        double r4798260 = r4798257 * r4798259;
        double r4798261 = phi1;
        double r4798262 = cos(r4798261);
        double r4798263 = r4798260 * r4798262;
        double r4798264 = cos(r4798258);
        double r4798265 = sin(r4798261);
        double r4798266 = r4798265 * r4798264;
        double r4798267 = r4798262 * r4798259;
        double r4798268 = cos(r4798256);
        double r4798269 = r4798267 * r4798268;
        double r4798270 = r4798266 + r4798269;
        double r4798271 = asin(r4798270);
        double r4798272 = sin(r4798271);
        double r4798273 = r4798265 * r4798272;
        double r4798274 = r4798264 - r4798273;
        double r4798275 = atan2(r4798263, r4798274);
        double r4798276 = r4798255 + r4798275;
        return r4798276;
}
↓
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r4798277 = phi1;
        double r4798278 = cos(r4798277);
        double r4798279 = delta;
        double r4798280 = sin(r4798279);
        double r4798281 = theta;
        double r4798282 = sin(r4798281);
        double r4798283 = r4798280 * r4798282;
        double r4798284 = r4798278 * r4798283;
        double r4798285 = cos(r4798279);
        double r4798286 = r4798285 * r4798285;
        double r4798287 = sin(r4798277);
        double r4798288 = cos(r4798281);
        double r4798289 = r4798278 * r4798280;
        double r4798290 = r4798288 * r4798289;
        double r4798291 = r4798287 * r4798285;
        double r4798292 = r4798290 + r4798291;
        double r4798293 = asin(r4798292);
        double r4798294 = sin(r4798293);
        double r4798295 = r4798287 * r4798294;
        double r4798296 = r4798295 * r4798295;
        double r4798297 = r4798286 - r4798296;
        double r4798298 = r4798295 + r4798285;
        double r4798299 = r4798297 / r4798298;
        double r4798300 = atan2(r4798284, r4798299);
        double r4798301 = lambda1;
        double r4798302 = r4798300 + r4798301;
        return r4798302;
}

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. Final simplification0.2

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

Reproduce

herbie shell --seed 2019158 
(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))))))))))