Average Error: 0.9 → 0.3
Time: 31.0s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right), \cos \phi_1\right)}\right)\right) + \lambda_1\]
\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)}
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right), \cos \phi_1\right)}\right)\right) + \lambda_1
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r2128316 = lambda1;
        double r2128317 = phi2;
        double r2128318 = cos(r2128317);
        double r2128319 = lambda2;
        double r2128320 = r2128316 - r2128319;
        double r2128321 = sin(r2128320);
        double r2128322 = r2128318 * r2128321;
        double r2128323 = phi1;
        double r2128324 = cos(r2128323);
        double r2128325 = cos(r2128320);
        double r2128326 = r2128318 * r2128325;
        double r2128327 = r2128324 + r2128326;
        double r2128328 = atan2(r2128322, r2128327);
        double r2128329 = r2128316 + r2128328;
        return r2128329;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r2128330 = lambda2;
        double r2128331 = cos(r2128330);
        double r2128332 = lambda1;
        double r2128333 = sin(r2128332);
        double r2128334 = r2128331 * r2128333;
        double r2128335 = cos(r2128332);
        double r2128336 = sin(r2128330);
        double r2128337 = r2128335 * r2128336;
        double r2128338 = r2128334 - r2128337;
        double r2128339 = phi2;
        double r2128340 = cos(r2128339);
        double r2128341 = r2128338 * r2128340;
        double r2128342 = r2128331 * r2128335;
        double r2128343 = fma(r2128333, r2128336, r2128342);
        double r2128344 = phi1;
        double r2128345 = cos(r2128344);
        double r2128346 = fma(r2128340, r2128343, r2128345);
        double r2128347 = atan2(r2128341, r2128346);
        double r2128348 = expm1(r2128347);
        double r2128349 = log1p(r2128348);
        double r2128350 = r2128349 + r2128332;
        return r2128350;
}

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. Simplified0.9

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

  4. Applied sin-diff0.8

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

  6. Applied cos-diff0.2

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

  8. Applied add-log-exp0.2

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

  10. Applied log1p-expm1-u0.3

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \cos \lambda_1 \cdot \cos \lambda_2 + \log \left(e^{\sin \lambda_1 \cdot \sin \lambda_2}\right), \cos \phi_1\right)}\right)\right)} + \lambda_1\]

  11. Simplified0.3

    \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right), \cos \phi_1\right)}\right)}\right) + \lambda_1\]

  12. Final simplification0.3

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right), \cos \phi_1\right)}\right)\right) + \lambda_1\]

Reproduce

herbie shell --seed 2019172 +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)))))))