Midpoint on a great circle

Average Error: 0.8 → 0.2

Time: 39.3s

Precision: 64

Math

\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]

↓

\[\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\frac{\mathsf{fma}\left(\cos \phi_1, \cos \phi_1 \cdot \cos \phi_1, \left(\left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)}{\mathsf{fma}\left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right) - \cos \phi_1, \cos \phi_1 \cdot \cos \phi_1\right)}} + \lambda_1\]

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r1965199 = lambda1;
        double r1965200 = phi2;
        double r1965201 = cos(r1965200);
        double r1965202 = lambda2;
        double r1965203 = r1965199 - r1965202;
        double r1965204 = sin(r1965203);
        double r1965205 = r1965201 * r1965204;
        double r1965206 = phi1;
        double r1965207 = cos(r1965206);
        double r1965208 = cos(r1965203);
        double r1965209 = r1965201 * r1965208;
        double r1965210 = r1965207 + r1965209;
        double r1965211 = atan2(r1965205, r1965210);
        double r1965212 = r1965199 + r1965211;
        return r1965212;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r1965213 = phi2;
        double r1965214 = cos(r1965213);
        double r1965215 = lambda1;
        double r1965216 = sin(r1965215);
        double r1965217 = lambda2;
        double r1965218 = cos(r1965217);
        double r1965219 = r1965216 * r1965218;
        double r1965220 = cos(r1965215);
        double r1965221 = sin(r1965217);
        double r1965222 = r1965220 * r1965221;
        double r1965223 = r1965219 - r1965222;
        double r1965224 = r1965214 * r1965223;
        double r1965225 = phi1;
        double r1965226 = cos(r1965225);
        double r1965227 = r1965226 * r1965226;
        double r1965228 = r1965220 * r1965218;
        double r1965229 = fma(r1965216, r1965221, r1965228);
        double r1965230 = r1965214 * r1965229;
        double r1965231 = r1965230 * r1965230;
        double r1965232 = r1965231 * r1965230;
        double r1965233 = fma(r1965226, r1965227, r1965232);
        double r1965234 = r1965230 - r1965226;
        double r1965235 = fma(r1965230, r1965234, r1965227);
        double r1965236 = r1965233 / r1965235;
        double r1965237 = atan2(r1965224, r1965236);
        double r1965238 = r1965237 + r1965215;
        return r1965238;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

1. Initial program 0.8

   \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
2. Using strategy rm

3. Applied sin-diff 0.8

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
4. Using strategy rm

5. Applied cos-diff 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}}\]
6. Using strategy rm

7. Applied flip3-+ 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\color{blue}{\frac{{\left(\cos \phi_1\right)}^{3} + {\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}^{3}}{\cos \phi_1 \cdot \cos \phi_1 + \left(\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)}}}\]

8. Simplified 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\frac{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_1 \cdot \cos \phi_1, \left(\mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2\right) \cdot \left(\left(\mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2\right) \cdot \left(\mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2\right)\right)\right)}}{\cos \phi_1 \cdot \cos \phi_1 + \left(\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)}}\]

9. Simplified 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\frac{\mathsf{fma}\left(\cos \phi_1, \cos \phi_1 \cdot \cos \phi_1, \left(\mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2\right) \cdot \left(\left(\mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2\right) \cdot \left(\mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2\right)\right)\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2 - \cos \phi_1, \cos \phi_1 \cdot \cos \phi_1\right)}}}\]

10. Final simplification 0.2

    \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\frac{\mathsf{fma}\left(\cos \phi_1, \cos \phi_1 \cdot \cos \phi_1, \left(\left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)}{\mathsf{fma}\left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right) - \cos \phi_1, \cos \phi_1 \cdot \cos \phi_1\right)}} + \lambda_1\]

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Midpoint on a great circle"
  (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))