Midpoint on a great circle

Average Error: 0.8 → 0.2

Time: 16.2s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r59044 = lambda1;
        double r59045 = phi2;
        double r59046 = cos(r59045);
        double r59047 = lambda2;
        double r59048 = r59044 - r59047;
        double r59049 = sin(r59048);
        double r59050 = r59046 * r59049;
        double r59051 = phi1;
        double r59052 = cos(r59051);
        double r59053 = cos(r59048);
        double r59054 = r59046 * r59053;
        double r59055 = r59052 + r59054;
        double r59056 = atan2(r59050, r59055);
        double r59057 = r59044 + r59056;
        return r59057;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r59058 = lambda1;
        double r59059 = phi2;
        double r59060 = cos(r59059);
        double r59061 = sin(r59058);
        double r59062 = lambda2;
        double r59063 = cos(r59062);
        double r59064 = r59061 * r59063;
        double r59065 = cos(r59058);
        double r59066 = sin(r59062);
        double r59067 = r59065 * r59066;
        double r59068 = r59064 - r59067;
        double r59069 = r59060 * r59068;
        double r59070 = r59065 * r59063;
        double r59071 = r59060 * r59070;
        double r59072 = r59061 * r59066;
        double r59073 = phi1;
        double r59074 = cos(r59073);
        double r59075 = fma(r59060, r59072, r59074);
        double r59076 = r59071 + r59075;
        double r59077 = atan2(r59069, r59076);
        double r59078 = r59058 + r59077;
        return r59078;
}

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. Simplified 0.8

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

4. Applied cos-diff 0.8

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

6. Applied sin-diff 0.2

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

8. Applied fma-udef 0.2

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

9. Simplified 0.2

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

11. Applied fma-udef 0.2

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

12. Applied distribute-lft-in 0.2

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

13. Applied associate-+l+ 0.2

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

14. Simplified 0.2

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

15. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(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)))))))