Average Error: 0.8 → 0.3
Time: 11.7s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\frac{{\left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1\right)\right)}^{3} + {\left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \lambda_2, \cos \phi_1\right), \log \left(e^{\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \lambda_2, \cos \phi_1\right)}\right), \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) - \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1\right)\right)\right)}}\]
\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{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\frac{{\left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1\right)\right)}^{3} + {\left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \lambda_2, \cos \phi_1\right), \log \left(e^{\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \lambda_2, \cos \phi_1\right)}\right), \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) - \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1\right)\right)\right)}}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r58387 = lambda1;
        double r58388 = phi2;
        double r58389 = cos(r58388);
        double r58390 = lambda2;
        double r58391 = r58387 - r58390;
        double r58392 = sin(r58391);
        double r58393 = r58389 * r58392;
        double r58394 = phi1;
        double r58395 = cos(r58394);
        double r58396 = cos(r58391);
        double r58397 = r58389 * r58396;
        double r58398 = r58395 + r58397;
        double r58399 = atan2(r58393, r58398);
        double r58400 = r58387 + r58399;
        return r58400;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r58401 = lambda1;
        double r58402 = phi2;
        double r58403 = cos(r58402);
        double r58404 = sin(r58401);
        double r58405 = lambda2;
        double r58406 = cos(r58405);
        double r58407 = r58404 * r58406;
        double r58408 = cos(r58401);
        double r58409 = sin(r58405);
        double r58410 = r58408 * r58409;
        double r58411 = r58407 - r58410;
        double r58412 = r58403 * r58411;
        double r58413 = r58406 * r58403;
        double r58414 = phi1;
        double r58415 = cos(r58414);
        double r58416 = fma(r58408, r58413, r58415);
        double r58417 = 3.0;
        double r58418 = pow(r58416, r58417);
        double r58419 = r58404 * r58409;
        double r58420 = r58403 * r58419;
        double r58421 = pow(r58420, r58417);
        double r58422 = r58418 + r58421;
        double r58423 = r58403 * r58406;
        double r58424 = fma(r58408, r58423, r58415);
        double r58425 = exp(r58424);
        double r58426 = log(r58425);
        double r58427 = r58420 - r58416;
        double r58428 = r58420 * r58427;
        double r58429 = fma(r58424, r58426, r58428);
        double r58430 = r58422 / r58429;
        double r58431 = atan2(r58412, r58430);
        double r58432 = r58401 + r58431;
        return r58432;
}

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. Using strategy rm

  3. Applied cos-diff0.8

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

  4. Applied distribute-lft-in0.8

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

  5. Applied associate-+r+0.8

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

  6. Simplified0.8

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

  8. Applied sin-diff0.2

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

  10. Applied flip3-+0.3

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

  11. Simplified0.3

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

  13. Applied add-log-exp0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2020089 +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)))))))