Destination given bearing on a great circle

Average Error: 0.2 → 0.2

Time: 49.1s

Precision: 64

• unused variable phi2

Math

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

↓

\[\tan^{-1}_* \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta\right)\right)}{\frac{\cos delta \cdot \cos delta - \sqrt[3]{\left(\left(\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \sin \phi_1\right) \cdot \left(\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \sin \phi_1 \cdot \cos delta\right)\right)\right) \cdot \left(\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \sin \phi_1 \cdot \cos delta\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \sin \phi_1 \cdot \cos delta\right)\right)\right)\right)\right)\right) \cdot \left(\left(\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \sin \phi_1\right) \cdot \left(\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \sin \phi_1 \cdot \cos delta\right)\right)\right) \cdot \left(\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \sin \phi_1 \cdot \cos delta\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \sin \phi_1 \cdot \cos delta\right)\right)\right)\right)\right)\right)}}{\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \sin \phi_1 \cdot \cos delta\right)\right)\right) \cdot \sin \phi_1 + \cos delta}} + \lambda_1\]

C

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r4096144 = lambda1;
        double r4096145 = theta;
        double r4096146 = sin(r4096145);
        double r4096147 = delta;
        double r4096148 = sin(r4096147);
        double r4096149 = r4096146 * r4096148;
        double r4096150 = phi1;
        double r4096151 = cos(r4096150);
        double r4096152 = r4096149 * r4096151;
        double r4096153 = cos(r4096147);
        double r4096154 = sin(r4096150);
        double r4096155 = r4096154 * r4096153;
        double r4096156 = r4096151 * r4096148;
        double r4096157 = cos(r4096145);
        double r4096158 = r4096156 * r4096157;
        double r4096159 = r4096155 + r4096158;
        double r4096160 = asin(r4096159);
        double r4096161 = sin(r4096160);
        double r4096162 = r4096154 * r4096161;
        double r4096163 = r4096153 - r4096162;
        double r4096164 = atan2(r4096152, r4096163);
        double r4096165 = r4096144 + r4096164;
        return r4096165;
}

↓

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r4096166 = phi1;
        double r4096167 = cos(r4096166);
        double r4096168 = theta;
        double r4096169 = sin(r4096168);
        double r4096170 = r4096167 * r4096169;
        double r4096171 = delta;
        double r4096172 = sin(r4096171);
        double r4096173 = r4096170 * r4096172;
        double r4096174 = log1p(r4096173);
        double r4096175 = expm1(r4096174);
        double r4096176 = cos(r4096171);
        double r4096177 = r4096176 * r4096176;
        double r4096178 = sin(r4096166);
        double r4096179 = r4096178 * r4096178;
        double r4096180 = r4096179 * r4096178;
        double r4096181 = cos(r4096168);
        double r4096182 = r4096167 * r4096172;
        double r4096183 = r4096178 * r4096176;
        double r4096184 = fma(r4096181, r4096182, r4096183);
        double r4096185 = asin(r4096184);
        double r4096186 = sin(r4096185);
        double r4096187 = r4096186 * r4096186;
        double r4096188 = r4096186 * r4096187;
        double r4096189 = r4096180 * r4096188;
        double r4096190 = r4096189 * r4096189;
        double r4096191 = cbrt(r4096190);
        double r4096192 = r4096177 - r4096191;
        double r4096193 = r4096186 * r4096178;
        double r4096194 = r4096193 + r4096176;
        double r4096195 = r4096192 / r4096194;
        double r4096196 = atan2(r4096175, r4096195);
        double r4096197 = lambda1;
        double r4096198 = r4096196 + r4096197;
        return r4096198;
}

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. Simplified 0.2

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

4. Applied expm1-log1p-u 0.2

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)\right)\right)}}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)} + \lambda_1\]
5. Using strategy rm

6. Applied flip-- 0.2

   \[\leadsto \tan^{-1}_* \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)\right)\right)}{\color{blue}{\frac{\cos delta \cdot \cos delta - \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right)}{\cos delta + \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)}}} + \lambda_1\]
7. Using strategy rm

8. Applied add-cbrt-cube 0.2

   \[\leadsto \tan^{-1}_* \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)\right)\right)}{\frac{\cos delta \cdot \cos delta - \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \color{blue}{\sqrt[3]{\left(\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)}}\right)}{\cos delta + \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)}} + \lambda_1\]

9. Applied add-cbrt-cube 0.2

   \[\leadsto \tan^{-1}_* \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)\right)\right)}{\frac{\cos delta \cdot \cos delta - \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right) \cdot \left(\color{blue}{\sqrt[3]{\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \sin \phi_1}} \cdot \sqrt[3]{\left(\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)}\right)}{\cos delta + \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)}} + \lambda_1\]

10. Applied cbrt-unprod 0.2

    \[\leadsto \tan^{-1}_* \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)\right)\right)}{\frac{\cos delta \cdot \cos delta - \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right) \cdot \color{blue}{\sqrt[3]{\left(\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \sin \phi_1\right) \cdot \left(\left(\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right)}}}{\cos delta + \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)}} + \lambda_1\]

11. Applied add-cbrt-cube 0.2

    \[\leadsto \tan^{-1}_* \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)\right)\right)}{\frac{\cos delta \cdot \cos delta - \left(\sin \phi_1 \cdot \color{blue}{\sqrt[3]{\left(\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)}}\right) \cdot \sqrt[3]{\left(\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \sin \phi_1\right) \cdot \left(\left(\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right)}}{\cos delta + \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)}} + \lambda_1\]

12. Applied add-cbrt-cube 0.2

    \[\leadsto \tan^{-1}_* \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)\right)\right)}{\frac{\cos delta \cdot \cos delta - \left(\color{blue}{\sqrt[3]{\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \sin \phi_1}} \cdot \sqrt[3]{\left(\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)}\right) \cdot \sqrt[3]{\left(\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \sin \phi_1\right) \cdot \left(\left(\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right)}}{\cos delta + \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)}} + \lambda_1\]

13. Applied cbrt-unprod 0.2

    \[\leadsto \tan^{-1}_* \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)\right)\right)}{\frac{\cos delta \cdot \cos delta - \color{blue}{\sqrt[3]{\left(\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \sin \phi_1\right) \cdot \left(\left(\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right)}} \cdot \sqrt[3]{\left(\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \sin \phi_1\right) \cdot \left(\left(\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right)}}{\cos delta + \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)}} + \lambda_1\]

14. Applied cbrt-unprod 0.2

    \[\leadsto \tan^{-1}_* \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)\right)\right)}{\frac{\cos delta \cdot \cos delta - \color{blue}{\sqrt[3]{\left(\left(\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \sin \phi_1\right) \cdot \left(\left(\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right)\right) \cdot \left(\left(\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \sin \phi_1\right) \cdot \left(\left(\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)\right)\right)}}}{\cos delta + \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \cos delta \cdot \sin \phi_1\right)\right)\right)}} + \lambda_1\]

15. Final simplification 0.2

    \[\leadsto \tan^{-1}_* \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta\right)\right)}{\frac{\cos delta \cdot \cos delta - \sqrt[3]{\left(\left(\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \sin \phi_1\right) \cdot \left(\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \sin \phi_1 \cdot \cos delta\right)\right)\right) \cdot \left(\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \sin \phi_1 \cdot \cos delta\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \sin \phi_1 \cdot \cos delta\right)\right)\right)\right)\right)\right) \cdot \left(\left(\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \sin \phi_1\right) \cdot \left(\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \sin \phi_1 \cdot \cos delta\right)\right)\right) \cdot \left(\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \sin \phi_1 \cdot \cos delta\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \sin \phi_1 \cdot \cos delta\right)\right)\right)\right)\right)\right)}}{\sin \left(\sin^{-1} \left(\mathsf{fma}\left(\cos theta, \cos \phi_1 \cdot \sin delta, \sin \phi_1 \cdot \cos delta\right)\right)\right) \cdot \sin \phi_1 + \cos delta}} + \lambda_1\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(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))))))))))