Average Error: 0 → 0
Time: 6.5m
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\left(\cos \phi_2\right), \left(\cos \left(\lambda_1 - \lambda_2\right)\right), \left(\cos \phi_1\right)\right)}\]
\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}{\mathsf{fma}\left(\left(\cos \phi_2\right), \left(\cos \left(\lambda_1 - \lambda_2\right)\right), \left(\cos \phi_1\right)\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r795190 = lambda1;
        double r795191 = phi2;
        double r795192 = cos(r795191);
        double r795193 = lambda2;
        double r795194 = r795190 - r795193;
        double r795195 = sin(r795194);
        double r795196 = r795192 * r795195;
        double r795197 = phi1;
        double r795198 = cos(r795197);
        double r795199 = cos(r795194);
        double r795200 = r795192 * r795199;
        double r795201 = r795198 + r795200;
        double r795202 = atan2(r795196, r795201);
        double r795203 = r795190 + r795202;
        return r795203;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r795204 = lambda1;
        double r795205 = lambda2;
        double r795206 = r795204 - r795205;
        double r795207 = sin(r795206);
        double r795208 = phi2;
        double r795209 = cos(r795208);
        double r795210 = r795207 * r795209;
        double r795211 = cos(r795206);
        double r795212 = phi1;
        double r795213 = cos(r795212);
        double r795214 = fma(r795209, r795211, r795213);
        double r795215 = atan2(r795210, r795214);
        double r795216 = r795204 + r795215;
        return r795216;
}

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

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

  3. Final simplification0

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

Reproduce

herbie shell --seed 2019133 +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)))))))