Average Error: 0.8 → 0.3
Time: 21.9s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\frac{\cos \phi_1 \cdot {\left(\cos \phi_1\right)}^{2} + {\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}^{3}}{\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) - \cos \phi_1\right) + \cos \phi_1 \cdot \cos \phi_1}}\]
\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{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\frac{\cos \phi_1 \cdot {\left(\cos \phi_1\right)}^{2} + {\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}^{3}}{\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}}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r60200 = lambda1;
        double r60201 = phi2;
        double r60202 = cos(r60201);
        double r60203 = lambda2;
        double r60204 = r60200 - r60203;
        double r60205 = sin(r60204);
        double r60206 = r60202 * r60205;
        double r60207 = phi1;
        double r60208 = cos(r60207);
        double r60209 = cos(r60204);
        double r60210 = r60202 * r60209;
        double r60211 = r60208 + r60210;
        double r60212 = atan2(r60206, r60211);
        double r60213 = r60200 + r60212;
        return r60213;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r60214 = lambda1;
        double r60215 = phi2;
        double r60216 = cos(r60215);
        double r60217 = sin(r60214);
        double r60218 = lambda2;
        double r60219 = cos(r60218);
        double r60220 = r60217 * r60219;
        double r60221 = cos(r60214);
        double r60222 = sin(r60218);
        double r60223 = r60221 * r60222;
        double r60224 = r60220 - r60223;
        double r60225 = r60216 * r60224;
        double r60226 = phi1;
        double r60227 = cos(r60226);
        double r60228 = 2.0;
        double r60229 = pow(r60227, r60228);
        double r60230 = r60227 * r60229;
        double r60231 = r60221 * r60219;
        double r60232 = r60217 * r60222;
        double r60233 = r60231 + r60232;
        double r60234 = r60216 * r60233;
        double r60235 = 3.0;
        double r60236 = pow(r60234, r60235);
        double r60237 = r60230 + r60236;
        double r60238 = r60234 - r60227;
        double r60239 = r60234 * r60238;
        double r60240 = r60227 * r60227;
        double r60241 = r60239 + r60240;
        double r60242 = r60237 / r60241;
        double r60243 = atan2(r60225, r60242);
        double r60244 = r60214 + r60243;
        return r60244;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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-diff0.7

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

  5. Applied cos-diff0.2

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

  7. Applied flip3-+0.3

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

  8. Simplified0.3

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

  10. Applied cube-mult0.3

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

  11. Simplified0.3

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\frac{\cos \phi_1 \cdot \color{blue}{{\left(\cos \phi_1\right)}^{2}} + {\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}^{3}}{\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) - \cos \phi_1\right) + \cos \phi_1 \cdot \cos \phi_1}}\]

  12. Final simplification0.3

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\frac{\cos \phi_1 \cdot {\left(\cos \phi_1\right)}^{2} + {\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}^{3}}{\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) - \cos \phi_1\right) + \cos \phi_1 \cdot \cos \phi_1}}\]

Reproduce

herbie shell --seed 2019304 
(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)))))))