Average Error: 0 → 0
Time: 8.1m
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{(\left(\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right)\right) + \left(\cos \phi_1\right))_*}\]
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r2984404 = lambda1;
        double r2984405 = phi2;
        double r2984406 = cos(r2984405);
        double r2984407 = lambda2;
        double r2984408 = r2984404 - r2984407;
        double r2984409 = sin(r2984408);
        double r2984410 = r2984406 * r2984409;
        double r2984411 = phi1;
        double r2984412 = cos(r2984411);
        double r2984413 = cos(r2984408);
        double r2984414 = r2984406 * r2984413;
        double r2984415 = r2984412 + r2984414;
        double r2984416 = atan2(r2984410, r2984415);
        double r2984417 = r2984404 + r2984416;
        return r2984417;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r2984418 = lambda1;
        double r2984419 = lambda2;
        double r2984420 = r2984418 - r2984419;
        double r2984421 = sin(r2984420);
        double r2984422 = phi2;
        double r2984423 = cos(r2984422);
        double r2984424 = r2984421 * r2984423;
        double r2984425 = cos(r2984420);
        double r2984426 = phi1;
        double r2984427 = cos(r2984426);
        double r2984428 = fma(r2984423, r2984425, r2984427);
        double r2984429 = atan2(r2984424, r2984428);
        double r2984430 = r2984418 + r2984429;
        return r2984430;
}


\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{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{(\left(\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right)\right) + \left(\cos \phi_1\right))_*}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 0

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

  2. Simplified0

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

  3. Final simplification0

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

Reproduce

herbie shell --seed 2019101 +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)))))))