Midpoint on a great circle

Average Error: 0.9 → 0.3

Time: 34.2s

Precision: 64

Math

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

↓

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

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r2094565 = lambda1;
        double r2094566 = phi2;
        double r2094567 = cos(r2094566);
        double r2094568 = lambda2;
        double r2094569 = r2094565 - r2094568;
        double r2094570 = sin(r2094569);
        double r2094571 = r2094567 * r2094570;
        double r2094572 = phi1;
        double r2094573 = cos(r2094572);
        double r2094574 = cos(r2094569);
        double r2094575 = r2094567 * r2094574;
        double r2094576 = r2094573 + r2094575;
        double r2094577 = atan2(r2094571, r2094576);
        double r2094578 = r2094565 + r2094577;
        return r2094578;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r2094579 = lambda1;
        double r2094580 = sin(r2094579);
        double r2094581 = lambda2;
        double r2094582 = cos(r2094581);
        double r2094583 = r2094580 * r2094582;
        double r2094584 = cos(r2094579);
        double r2094585 = sin(r2094581);
        double r2094586 = r2094584 * r2094585;
        double r2094587 = r2094583 - r2094586;
        double r2094588 = phi2;
        double r2094589 = cos(r2094588);
        double r2094590 = r2094587 * r2094589;
        double r2094591 = phi1;
        double r2094592 = cos(r2094591);
        double r2094593 = r2094592 * r2094592;
        double r2094594 = r2094585 * r2094580;
        double r2094595 = fma(r2094582, r2094584, r2094594);
        double r2094596 = r2094595 * r2094589;
        double r2094597 = r2094596 * r2094596;
        double r2094598 = r2094597 * r2094596;
        double r2094599 = fma(r2094592, r2094593, r2094598);
        double r2094600 = r2094592 - r2094596;
        double r2094601 = fma(r2094600, r2094592, r2094597);
        double r2094602 = r2094599 / r2094601;
        double r2094603 = atan2(r2094590, r2094602);
        double r2094604 = r2094603 + r2094579;
        return r2094604;
}

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. Simplified 0.9

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

4. Applied sin-diff 0.9

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

6. Applied cos-diff 0.2

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

8. Applied fma-udef 0.2

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

9. Simplified 0.2

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

11. Applied flip3-+ 0.3

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

12. Simplified 0.3

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

13. Simplified 0.3

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

14. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019163 +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)))))))