Average Error: 0.1 → 0.2
Time: 50.0s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\mathsf{expm1}\left(\log \left(\cos delta - \mathsf{fma}\left(\sin \phi_1, \mathsf{fma}\left(\sin delta, \cos theta \cdot \cos \phi_1, \cos delta \cdot \sin \phi_1\right), -1\right)\right)\right)} + \lambda_1\]
\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{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\mathsf{expm1}\left(\log \left(\cos delta - \mathsf{fma}\left(\sin \phi_1, \mathsf{fma}\left(\sin delta, \cos theta \cdot \cos \phi_1, \cos delta \cdot \sin \phi_1\right), -1\right)\right)\right)} + \lambda_1
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r3547795 = lambda1;
        double r3547796 = theta;
        double r3547797 = sin(r3547796);
        double r3547798 = delta;
        double r3547799 = sin(r3547798);
        double r3547800 = r3547797 * r3547799;
        double r3547801 = phi1;
        double r3547802 = cos(r3547801);
        double r3547803 = r3547800 * r3547802;
        double r3547804 = cos(r3547798);
        double r3547805 = sin(r3547801);
        double r3547806 = r3547805 * r3547804;
        double r3547807 = r3547802 * r3547799;
        double r3547808 = cos(r3547796);
        double r3547809 = r3547807 * r3547808;
        double r3547810 = r3547806 + r3547809;
        double r3547811 = asin(r3547810);
        double r3547812 = sin(r3547811);
        double r3547813 = r3547805 * r3547812;
        double r3547814 = r3547804 - r3547813;
        double r3547815 = atan2(r3547803, r3547814);
        double r3547816 = r3547795 + r3547815;
        return r3547816;
}
↓
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r3547817 = phi1;
        double r3547818 = cos(r3547817);
        double r3547819 = delta;
        double r3547820 = sin(r3547819);
        double r3547821 = theta;
        double r3547822 = sin(r3547821);
        double r3547823 = r3547820 * r3547822;
        double r3547824 = r3547818 * r3547823;
        double r3547825 = cos(r3547819);
        double r3547826 = sin(r3547817);
        double r3547827 = cos(r3547821);
        double r3547828 = r3547827 * r3547818;
        double r3547829 = r3547825 * r3547826;
        double r3547830 = fma(r3547820, r3547828, r3547829);
        double r3547831 = -1.0;
        double r3547832 = fma(r3547826, r3547830, r3547831);
        double r3547833 = r3547825 - r3547832;
        double r3547834 = log(r3547833);
        double r3547835 = expm1(r3547834);
        double r3547836 = atan2(r3547824, r3547835);
        double r3547837 = lambda1;
        double r3547838 = r3547836 + r3547837;
        return r3547838;
}

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 0.1

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

  2. Taylor expanded around inf 0.1

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

  3. Simplified0.1

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

  5. Applied expm1-log1p-u0.2

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

  7. Applied log1p-udef0.2

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

  8. Simplified0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019162 +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))))))))))