Average Error: 0.1 → 0.2
Time: 44.0s
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 r3153147 = lambda1;
        double r3153148 = theta;
        double r3153149 = sin(r3153148);
        double r3153150 = delta;
        double r3153151 = sin(r3153150);
        double r3153152 = r3153149 * r3153151;
        double r3153153 = phi1;
        double r3153154 = cos(r3153153);
        double r3153155 = r3153152 * r3153154;
        double r3153156 = cos(r3153150);
        double r3153157 = sin(r3153153);
        double r3153158 = r3153157 * r3153156;
        double r3153159 = r3153154 * r3153151;
        double r3153160 = cos(r3153148);
        double r3153161 = r3153159 * r3153160;
        double r3153162 = r3153158 + r3153161;
        double r3153163 = asin(r3153162);
        double r3153164 = sin(r3153163);
        double r3153165 = r3153157 * r3153164;
        double r3153166 = r3153156 - r3153165;
        double r3153167 = atan2(r3153155, r3153166);
        double r3153168 = r3153147 + r3153167;
        return r3153168;
}
↓
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r3153169 = phi1;
        double r3153170 = cos(r3153169);
        double r3153171 = delta;
        double r3153172 = sin(r3153171);
        double r3153173 = theta;
        double r3153174 = sin(r3153173);
        double r3153175 = r3153172 * r3153174;
        double r3153176 = r3153170 * r3153175;
        double r3153177 = cos(r3153171);
        double r3153178 = sin(r3153169);
        double r3153179 = cos(r3153173);
        double r3153180 = r3153179 * r3153170;
        double r3153181 = r3153177 * r3153178;
        double r3153182 = fma(r3153172, r3153180, r3153181);
        double r3153183 = -1.0;
        double r3153184 = fma(r3153178, r3153182, r3153183);
        double r3153185 = r3153177 - r3153184;
        double r3153186 = log(r3153185);
        double r3153187 = expm1(r3153186);
        double r3153188 = atan2(r3153176, r3153187);
        double r3153189 = lambda1;
        double r3153190 = r3153188 + r3153189;
        return r3153190;
}

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))))))))))