Average Error: 0.2 → 0.2
Time: 38.8s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\frac{{\left(\mathsf{fma}\left(-\sin \phi_1, \left(\cos theta \cdot \sin delta\right) \cdot \cos \phi_1, \cos delta\right)\right)}^{3} - {\left({\left(\sin \phi_1\right)}^{2} \cdot \cos delta\right)}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos delta, {\left(\sin \phi_1\right)}^{2}, \mathsf{fma}\left(-\sin \phi_1, \left(\cos theta \cdot \sin delta\right) \cdot \cos \phi_1, \cos delta\right)\right), {\left(\sin \phi_1\right)}^{2} \cdot \cos delta, \mathsf{fma}\left(-\sin \phi_1, \left(\cos theta \cdot \sin delta\right) \cdot \cos \phi_1, \cos delta\right) \cdot \mathsf{fma}\left(-\sin \phi_1, \left(\cos theta \cdot \sin delta\right) \cdot \cos \phi_1, \cos delta\right)\right)}}\]
\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)}
↓
\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\frac{{\left(\mathsf{fma}\left(-\sin \phi_1, \left(\cos theta \cdot \sin delta\right) \cdot \cos \phi_1, \cos delta\right)\right)}^{3} - {\left({\left(\sin \phi_1\right)}^{2} \cdot \cos delta\right)}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos delta, {\left(\sin \phi_1\right)}^{2}, \mathsf{fma}\left(-\sin \phi_1, \left(\cos theta \cdot \sin delta\right) \cdot \cos \phi_1, \cos delta\right)\right), {\left(\sin \phi_1\right)}^{2} \cdot \cos delta, \mathsf{fma}\left(-\sin \phi_1, \left(\cos theta \cdot \sin delta\right) \cdot \cos \phi_1, \cos delta\right) \cdot \mathsf{fma}\left(-\sin \phi_1, \left(\cos theta \cdot \sin delta\right) \cdot \cos \phi_1, \cos delta\right)\right)}}
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r78045 = lambda1;
        double r78046 = theta;
        double r78047 = sin(r78046);
        double r78048 = delta;
        double r78049 = sin(r78048);
        double r78050 = r78047 * r78049;
        double r78051 = phi1;
        double r78052 = cos(r78051);
        double r78053 = r78050 * r78052;
        double r78054 = cos(r78048);
        double r78055 = sin(r78051);
        double r78056 = r78055 * r78054;
        double r78057 = r78052 * r78049;
        double r78058 = cos(r78046);
        double r78059 = r78057 * r78058;
        double r78060 = r78056 + r78059;
        double r78061 = asin(r78060);
        double r78062 = sin(r78061);
        double r78063 = r78055 * r78062;
        double r78064 = r78054 - r78063;
        double r78065 = atan2(r78053, r78064);
        double r78066 = r78045 + r78065;
        return r78066;
}
↓
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r78067 = lambda1;
        double r78068 = delta;
        double r78069 = sin(r78068);
        double r78070 = theta;
        double r78071 = sin(r78070);
        double r78072 = phi1;
        double r78073 = cos(r78072);
        double r78074 = r78071 * r78073;
        double r78075 = r78069 * r78074;
        double r78076 = sin(r78072);
        double r78077 = -r78076;
        double r78078 = cos(r78070);
        double r78079 = r78078 * r78069;
        double r78080 = r78079 * r78073;
        double r78081 = cos(r78068);
        double r78082 = fma(r78077, r78080, r78081);
        double r78083 = 3.0;
        double r78084 = pow(r78082, r78083);
        double r78085 = 2.0;
        double r78086 = pow(r78076, r78085);
        double r78087 = r78086 * r78081;
        double r78088 = pow(r78087, r78083);
        double r78089 = r78084 - r78088;
        double r78090 = fma(r78081, r78086, r78082);
        double r78091 = r78082 * r78082;
        double r78092 = fma(r78090, r78087, r78091);
        double r78093 = r78089 / r78092;
        double r78094 = atan2(r78075, r78093);
        double r78095 = r78067 + r78094;
        return r78095;
}

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

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

  3. Taylor expanded around inf 0.1

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

  4. Simplified0.2

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

  6. Applied fma-udef0.2

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

  7. Applied distribute-lft-in0.2

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

  8. Applied associate--r+0.2

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

  9. Simplified0.2

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

  11. Applied associate-*r*0.2

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

  12. Simplified0.2

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

  14. Applied flip3--0.2

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

  15. Simplified0.2

    \[\leadsto \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\frac{\color{blue}{{\left(\mathsf{fma}\left(-\sin \phi_1, \left(\cos theta \cdot \sin delta\right) \cdot \cos \phi_1, \cos delta\right)\right)}^{3} - {\left({\left(\sin \phi_1\right)}^{2} \cdot \cos delta\right)}^{3}}}{\mathsf{fma}\left(-\sin \phi_1, \left(\sin delta \cdot \cos theta\right) \cdot \cos \phi_1, \cos delta\right) \cdot \mathsf{fma}\left(-\sin \phi_1, \left(\sin delta \cdot \cos theta\right) \cdot \cos \phi_1, \cos delta\right) + \left(\left(\sin \phi_1 \cdot \left(\cos delta \cdot \sin \phi_1\right)\right) \cdot \left(\sin \phi_1 \cdot \left(\cos delta \cdot \sin \phi_1\right)\right) + \mathsf{fma}\left(-\sin \phi_1, \left(\sin delta \cdot \cos theta\right) \cdot \cos \phi_1, \cos delta\right) \cdot \left(\sin \phi_1 \cdot \left(\cos delta \cdot \sin \phi_1\right)\right)\right)}} + \lambda_1\]

  16. Simplified0.2

    \[\leadsto \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\frac{{\left(\mathsf{fma}\left(-\sin \phi_1, \left(\cos theta \cdot \sin delta\right) \cdot \cos \phi_1, \cos delta\right)\right)}^{3} - {\left({\left(\sin \phi_1\right)}^{2} \cdot \cos delta\right)}^{3}}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos delta, {\left(\sin \phi_1\right)}^{2}, \mathsf{fma}\left(-\sin \phi_1, \left(\cos theta \cdot \sin delta\right) \cdot \cos \phi_1, \cos delta\right)\right), {\left(\sin \phi_1\right)}^{2} \cdot \cos delta, \mathsf{fma}\left(-\sin \phi_1, \left(\cos theta \cdot \sin delta\right) \cdot \cos \phi_1, \cos delta\right) \cdot \mathsf{fma}\left(-\sin \phi_1, \left(\cos theta \cdot \sin delta\right) \cdot \cos \phi_1, \cos delta\right)\right)}}} + \lambda_1\]

  17. Final simplification0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\frac{{\left(\mathsf{fma}\left(-\sin \phi_1, \left(\cos theta \cdot \sin delta\right) \cdot \cos \phi_1, \cos delta\right)\right)}^{3} - {\left({\left(\sin \phi_1\right)}^{2} \cdot \cos delta\right)}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos delta, {\left(\sin \phi_1\right)}^{2}, \mathsf{fma}\left(-\sin \phi_1, \left(\cos theta \cdot \sin delta\right) \cdot \cos \phi_1, \cos delta\right)\right), {\left(\sin \phi_1\right)}^{2} \cdot \cos delta, \mathsf{fma}\left(-\sin \phi_1, \left(\cos theta \cdot \sin delta\right) \cdot \cos \phi_1, \cos delta\right) \cdot \mathsf{fma}\left(-\sin \phi_1, \left(\cos theta \cdot \sin delta\right) \cdot \cos \phi_1, \cos delta\right)\right)}}\]

Reproduce

herbie shell --seed 2019194 +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))))))))))