Average Error: 36.8 → 0.1
Time: 32.0s
Precision: 64
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\mathsf{hypot}\left(\mathsf{fma}\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \lambda_1, \cos \left(\phi_2 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \lambda_2\right)\right) - \mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), \lambda_1 \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\lambda_2 \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right), \phi_1 - \phi_2\right) \cdot R\]
R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}
↓
\mathsf{hypot}\left(\mathsf{fma}\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \lambda_1, \cos \left(\phi_2 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \lambda_2\right)\right) - \mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), \lambda_1 \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\lambda_2 \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right), \phi_1 - \phi_2\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r2270498 = R;
        double r2270499 = lambda1;
        double r2270500 = lambda2;
        double r2270501 = r2270499 - r2270500;
        double r2270502 = phi1;
        double r2270503 = phi2;
        double r2270504 = r2270502 + r2270503;
        double r2270505 = 2.0;
        double r2270506 = r2270504 / r2270505;
        double r2270507 = cos(r2270506);
        double r2270508 = r2270501 * r2270507;
        double r2270509 = r2270508 * r2270508;
        double r2270510 = r2270502 - r2270503;
        double r2270511 = r2270510 * r2270510;
        double r2270512 = r2270509 + r2270511;
        double r2270513 = sqrt(r2270512);
        double r2270514 = r2270498 * r2270513;
        return r2270514;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r2270515 = phi1;
        double r2270516 = 0.5;
        double r2270517 = r2270515 * r2270516;
        double r2270518 = cos(r2270517);
        double r2270519 = lambda1;
        double r2270520 = r2270518 * r2270519;
        double r2270521 = phi2;
        double r2270522 = r2270521 * r2270516;
        double r2270523 = cos(r2270522);
        double r2270524 = sin(r2270517);
        double r2270525 = sin(r2270522);
        double r2270526 = lambda2;
        double r2270527 = r2270525 * r2270526;
        double r2270528 = r2270524 * r2270527;
        double r2270529 = fma(r2270520, r2270523, r2270528);
        double r2270530 = r2270519 * r2270524;
        double r2270531 = r2270526 * r2270523;
        double r2270532 = r2270518 * r2270531;
        double r2270533 = fma(r2270525, r2270530, r2270532);
        double r2270534 = r2270529 - r2270533;
        double r2270535 = r2270515 - r2270521;
        double r2270536 = hypot(r2270534, r2270535);
        double r2270537 = R;
        double r2270538 = r2270536 * r2270537;
        return r2270538;
}

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 36.8

    \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

  2. Simplified3.5

    \[\leadsto \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right) \cdot R}\]

  3. Taylor expanded around inf 3.5

    \[\leadsto \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}, \phi_1 - \phi_2\right) \cdot R\]
  4. Using strategy rm

  5. Applied distribute-rgt-in3.5

    \[\leadsto \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2} + \phi_2 \cdot \frac{1}{2}\right)}, \phi_1 - \phi_2\right) \cdot R\]

  6. Applied cos-sum0.1

    \[\leadsto \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}, \phi_1 - \phi_2\right) \cdot R\]

  7. Taylor expanded around inf 0.1

    \[\leadsto \mathsf{hypot}\left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \lambda_1\right) + \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \lambda_2\right)\right) - \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right) + \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \lambda_1\right)\right)}, \phi_1 - \phi_2\right) \cdot R\]

  8. Simplified0.1

    \[\leadsto \mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \lambda_1, \cos \left(\frac{1}{2} \cdot \phi_2\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \lambda_2\right)\right) - \mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \phi_2\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \lambda_1, \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}, \phi_1 - \phi_2\right) \cdot R\]

  9. Final simplification0.1

    \[\leadsto \mathsf{hypot}\left(\mathsf{fma}\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \lambda_1, \cos \left(\phi_2 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \lambda_2\right)\right) - \mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), \lambda_1 \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\lambda_2 \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right), \phi_1 - \phi_2\right) \cdot R\]

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Equirectangular approximation to distance on a great circle"
  (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2))))))