Midpoint on a great circle

Average Error: 0.8 → 0.3

Time: 35.1s

Precision: 64

Math

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

↓

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

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r2054478 = lambda1;
        double r2054479 = phi2;
        double r2054480 = cos(r2054479);
        double r2054481 = lambda2;
        double r2054482 = r2054478 - r2054481;
        double r2054483 = sin(r2054482);
        double r2054484 = r2054480 * r2054483;
        double r2054485 = phi1;
        double r2054486 = cos(r2054485);
        double r2054487 = cos(r2054482);
        double r2054488 = r2054480 * r2054487;
        double r2054489 = r2054486 + r2054488;
        double r2054490 = atan2(r2054484, r2054489);
        double r2054491 = r2054478 + r2054490;
        return r2054491;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r2054492 = phi2;
        double r2054493 = cos(r2054492);
        double r2054494 = lambda1;
        double r2054495 = sin(r2054494);
        double r2054496 = lambda2;
        double r2054497 = cos(r2054496);
        double r2054498 = r2054495 * r2054497;
        double r2054499 = cos(r2054494);
        double r2054500 = sin(r2054496);
        double r2054501 = r2054499 * r2054500;
        double r2054502 = r2054498 - r2054501;
        double r2054503 = r2054493 * r2054502;
        double r2054504 = r2054493 * r2054497;
        double r2054505 = phi1;
        double r2054506 = cos(r2054505);
        double r2054507 = fma(r2054504, r2054499, r2054506);
        double r2054508 = r2054507 * r2054507;
        double r2054509 = r2054500 * r2054495;
        double r2054510 = r2054509 * r2054493;
        double r2054511 = r2054510 * r2054510;
        double r2054512 = r2054508 - r2054511;
        double r2054513 = r2054499 * r2054497;
        double r2054514 = fma(r2054513, r2054493, r2054506);
        double r2054515 = r2054514 - r2054510;
        double r2054516 = r2054512 / r2054515;
        double r2054517 = atan2(r2054503, r2054516);
        double r2054518 = r2054517 + r2054494;
        return r2054518;
}

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. Applied distribute-lft-in 0.8

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

5. Applied associate-+r+ 0.8

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

6. Simplified 0.8

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

8. Applied sin-diff 0.2

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

10. Applied flip-+ 0.2

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

11. Taylor expanded around inf 0.3

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

12. Simplified 0.3

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

13. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019162 +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)))))))