Average Error: 0.2 → 0.2
Time: 43.2s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{fma}\left(\cos delta, \cos delta, -{\left(\mathsf{fma}\left(\cos \phi_1, \cos theta \cdot \sin delta, \sin \phi_1 \cdot \cos delta\right)\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(\sin \phi_1, \mathsf{fma}\left(\cos delta, \sin \phi_1, \left(\cos theta \cdot \cos \phi_1\right) \cdot \sin delta\right), \cos delta\right)}} + \lambda_1\]
\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{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{fma}\left(\cos delta, \cos delta, -{\left(\mathsf{fma}\left(\cos \phi_1, \cos theta \cdot \sin delta, \sin \phi_1 \cdot \cos delta\right)\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(\sin \phi_1, \mathsf{fma}\left(\cos delta, \sin \phi_1, \left(\cos theta \cdot \cos \phi_1\right) \cdot \sin delta\right), \cos delta\right)}} + \lambda_1
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r96157 = lambda1;
        double r96158 = theta;
        double r96159 = sin(r96158);
        double r96160 = delta;
        double r96161 = sin(r96160);
        double r96162 = r96159 * r96161;
        double r96163 = phi1;
        double r96164 = cos(r96163);
        double r96165 = r96162 * r96164;
        double r96166 = cos(r96160);
        double r96167 = sin(r96163);
        double r96168 = r96167 * r96166;
        double r96169 = r96164 * r96161;
        double r96170 = cos(r96158);
        double r96171 = r96169 * r96170;
        double r96172 = r96168 + r96171;
        double r96173 = asin(r96172);
        double r96174 = sin(r96173);
        double r96175 = r96167 * r96174;
        double r96176 = r96166 - r96175;
        double r96177 = atan2(r96165, r96176);
        double r96178 = r96157 + r96177;
        return r96178;
}
↓
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r96179 = theta;
        double r96180 = sin(r96179);
        double r96181 = delta;
        double r96182 = sin(r96181);
        double r96183 = r96180 * r96182;
        double r96184 = phi1;
        double r96185 = cos(r96184);
        double r96186 = r96183 * r96185;
        double r96187 = cos(r96181);
        double r96188 = cos(r96179);
        double r96189 = r96188 * r96182;
        double r96190 = sin(r96184);
        double r96191 = r96190 * r96187;
        double r96192 = fma(r96185, r96189, r96191);
        double r96193 = 2.0;
        double r96194 = pow(r96192, r96193);
        double r96195 = pow(r96190, r96193);
        double r96196 = r96194 * r96195;
        double r96197 = -r96196;
        double r96198 = fma(r96187, r96187, r96197);
        double r96199 = 1.0;
        double r96200 = r96188 * r96185;
        double r96201 = r96200 * r96182;
        double r96202 = fma(r96187, r96190, r96201);
        double r96203 = fma(r96190, r96202, r96187);
        double r96204 = r96199 / r96203;
        double r96205 = r96198 * r96204;
        double r96206 = atan2(r96186, r96205);
        double r96207 = lambda1;
        double r96208 = r96206 + r96207;
        return r96208;
}

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. Simplified0.1

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

  3. Taylor expanded around inf 0.2

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

  4. Simplified0.2

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

  6. Applied flip--0.2

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

  7. Simplified0.2

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

  8. Simplified0.2

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

  10. Applied fma-neg0.2

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

  11. Simplified0.2

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

  13. Applied div-inv0.2

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

  14. Final simplification0.2

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

Reproduce

herbie shell --seed 2019195 +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))))))))))