Destination given bearing on a great circle

Average Error: 0.2 → 0.2

Time: 53.6s

Precision: 64

• unused variable phi2

Math

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

↓

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

C

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r4278164 = lambda1;
        double r4278165 = theta;
        double r4278166 = sin(r4278165);
        double r4278167 = delta;
        double r4278168 = sin(r4278167);
        double r4278169 = r4278166 * r4278168;
        double r4278170 = phi1;
        double r4278171 = cos(r4278170);
        double r4278172 = r4278169 * r4278171;
        double r4278173 = cos(r4278167);
        double r4278174 = sin(r4278170);
        double r4278175 = r4278174 * r4278173;
        double r4278176 = r4278171 * r4278168;
        double r4278177 = cos(r4278165);
        double r4278178 = r4278176 * r4278177;
        double r4278179 = r4278175 + r4278178;
        double r4278180 = asin(r4278179);
        double r4278181 = sin(r4278180);
        double r4278182 = r4278174 * r4278181;
        double r4278183 = r4278173 - r4278182;
        double r4278184 = atan2(r4278172, r4278183);
        double r4278185 = r4278164 + r4278184;
        return r4278185;
}

↓

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r4278186 = phi1;
        double r4278187 = cos(r4278186);
        double r4278188 = delta;
        double r4278189 = sin(r4278188);
        double r4278190 = theta;
        double r4278191 = sin(r4278190);
        double r4278192 = r4278189 * r4278191;
        double r4278193 = r4278187 * r4278192;
        double r4278194 = cos(r4278188);
        double r4278195 = r4278194 * r4278194;
        double r4278196 = sin(r4278186);
        double r4278197 = cos(r4278190);
        double r4278198 = r4278187 * r4278189;
        double r4278199 = r4278197 * r4278198;
        double r4278200 = r4278196 * r4278194;
        double r4278201 = r4278199 + r4278200;
        double r4278202 = asin(r4278201);
        double r4278203 = sin(r4278202);
        double r4278204 = r4278196 * r4278203;
        double r4278205 = r4278204 * r4278204;
        double r4278206 = r4278204 * r4278205;
        double r4278207 = cbrt(r4278206);
        double r4278208 = r4278204 * r4278207;
        double r4278209 = r4278195 - r4278208;
        double r4278210 = r4278194 + r4278207;
        double r4278211 = r4278209 / r4278210;
        double r4278212 = atan2(r4278193, r4278211);
        double r4278213 = lambda1;
        double r4278214 = r4278212 + r4278213;
        return r4278214;
}

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 add-cbrt-cube 0.2

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

5. Applied flip-- 0.2

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

7. Applied pow3 0.2

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

8. Applied rem-cbrt-cube 0.2

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

9. Final simplification 0.2

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

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))))))))))