Average Error: 0.2 → 0.2
Time: 15.6s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \log \left(e^{\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right) \cdot \sin \phi_1}\right)}\]
\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{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \log \left(e^{\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right) \cdot \sin \phi_1}\right)}
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r107980 = lambda1;
        double r107981 = theta;
        double r107982 = sin(r107981);
        double r107983 = delta;
        double r107984 = sin(r107983);
        double r107985 = r107982 * r107984;
        double r107986 = phi1;
        double r107987 = cos(r107986);
        double r107988 = r107985 * r107987;
        double r107989 = cos(r107983);
        double r107990 = sin(r107986);
        double r107991 = r107990 * r107989;
        double r107992 = r107987 * r107984;
        double r107993 = cos(r107981);
        double r107994 = r107992 * r107993;
        double r107995 = r107991 + r107994;
        double r107996 = asin(r107995);
        double r107997 = sin(r107996);
        double r107998 = r107990 * r107997;
        double r107999 = r107989 - r107998;
        double r108000 = atan2(r107988, r107999);
        double r108001 = r107980 + r108000;
        return r108001;
}
↓
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r108002 = lambda1;
        double r108003 = theta;
        double r108004 = sin(r108003);
        double r108005 = delta;
        double r108006 = sin(r108005);
        double r108007 = r108004 * r108006;
        double r108008 = phi1;
        double r108009 = cos(r108008);
        double r108010 = r108007 * r108009;
        double r108011 = cos(r108005);
        double r108012 = cos(r108003);
        double r108013 = r108009 * r108012;
        double r108014 = sin(r108008);
        double r108015 = r108014 * r108011;
        double r108016 = fma(r108006, r108013, r108015);
        double r108017 = r108016 * r108014;
        double r108018 = exp(r108017);
        double r108019 = log(r108018);
        double r108020 = r108011 - r108019;
        double r108021 = atan2(r108010, r108020);
        double r108022 = r108002 + r108021;
        return r108022;
}

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-log-exp0.2

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

  4. Taylor expanded around inf 0.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (lambda1 phi1 phi2 delta theta)
  :name "Destination given bearing on a great circle"
  :precision binary64
  (+ lambda1 (atan2 (* (* (sin theta) (sin delta)) (cos phi1)) (- (cos delta) (* (sin phi1) (sin (asin (+ (* (sin phi1) (cos delta)) (* (* (cos phi1) (sin delta)) (cos theta))))))))))