Midpoint on a great circle

Average Error: 0.8 → 0.3

Time: 33.5s

Precision: 64

Math

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

↓

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

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r767081 = lambda1;
        double r767082 = phi2;
        double r767083 = cos(r767082);
        double r767084 = lambda2;
        double r767085 = r767081 - r767084;
        double r767086 = sin(r767085);
        double r767087 = r767083 * r767086;
        double r767088 = phi1;
        double r767089 = cos(r767088);
        double r767090 = cos(r767085);
        double r767091 = r767083 * r767090;
        double r767092 = r767089 + r767091;
        double r767093 = atan2(r767087, r767092);
        double r767094 = r767081 + r767093;
        return r767094;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r767095 = phi2;
        double r767096 = cos(r767095);
        double r767097 = lambda1;
        double r767098 = sin(r767097);
        double r767099 = lambda2;
        double r767100 = cos(r767099);
        double r767101 = r767098 * r767100;
        double r767102 = cos(r767097);
        double r767103 = sin(r767099);
        double r767104 = r767102 * r767103;
        double r767105 = r767101 - r767104;
        double r767106 = r767096 * r767105;
        double r767107 = r767103 * r767098;
        double r767108 = fma(r767102, r767100, r767107);
        double r767109 = phi1;
        double r767110 = cos(r767109);
        double r767111 = fma(r767096, r767108, r767110);
        double r767112 = r767111 * r767111;
        double r767113 = r767111 * r767112;
        double r767114 = cbrt(r767113);
        double r767115 = atan2(r767106, r767114);
        double r767116 = r767115 + r767097;
        return r767116;
}

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 cos-diff 0.8

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

5. Applied sin-diff 0.2

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

7. Applied add-cbrt-cube 0.3

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

8. Simplified 0.3

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

9. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019152 +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)))))))