Midpoint on a great circle

Average Error: 0.9 → 0.3

Time: 14.4s

Precision: 64

Math

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

↓

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

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r69208 = lambda1;
        double r69209 = phi2;
        double r69210 = cos(r69209);
        double r69211 = lambda2;
        double r69212 = r69208 - r69211;
        double r69213 = sin(r69212);
        double r69214 = r69210 * r69213;
        double r69215 = phi1;
        double r69216 = cos(r69215);
        double r69217 = cos(r69212);
        double r69218 = r69210 * r69217;
        double r69219 = r69216 + r69218;
        double r69220 = atan2(r69214, r69219);
        double r69221 = r69208 + r69220;
        return r69221;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r69222 = lambda1;
        double r69223 = phi2;
        double r69224 = cos(r69223);
        double r69225 = sin(r69222);
        double r69226 = r69224 * r69225;
        double r69227 = lambda2;
        double r69228 = cos(r69227);
        double r69229 = r69226 * r69228;
        double r69230 = cos(r69222);
        double r69231 = -r69227;
        double r69232 = sin(r69231);
        double r69233 = r69230 * r69232;
        double r69234 = r69224 * r69233;
        double r69235 = r69229 + r69234;
        double r69236 = 1.0;
        double r69237 = r69230 * r69228;
        double r69238 = sin(r69227);
        double r69239 = r69225 * r69238;
        double r69240 = r69237 + r69239;
        double r69241 = r69224 * r69240;
        double r69242 = phi1;
        double r69243 = cos(r69242);
        double r69244 = r69241 - r69243;
        double r69245 = r69241 * r69244;
        double r69246 = r69243 * r69243;
        double r69247 = r69245 + r69246;
        double r69248 = 3.0;
        double r69249 = pow(r69243, r69248);
        double r69250 = pow(r69241, r69248);
        double r69251 = r69249 + r69250;
        double r69252 = r69247 / r69251;
        double r69253 = r69236 / r69252;
        double r69254 = atan2(r69235, r69253);
        double r69255 = r69222 + r69254;
        return r69255;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

1. Initial program 0.9

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

3. Applied sub-neg 0.9

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

4. Applied sin-sum 0.8

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

5. Applied distribute-lft-in 0.8

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

6. Simplified 0.8

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

8. Applied cos-diff 0.2

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

10. Applied flip3-+ 0.3

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

11. Simplified 0.3

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

13. Applied clear-num 0.3

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

14. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020065 
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Midpoint on a great circle"
  :precision binary64
  (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))