Average Error: 0.8 → 0.2
Time: 27.3s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \cos \phi_2\right) \cdot \sin \lambda_1 + \left(-\cos \lambda_1 \cdot \left(\sin \lambda_2 \cdot \cos \phi_2\right)\right)}{\mathsf{fma}\left(\cos \phi_2, \cos \lambda_2 \cdot \cos \lambda_1 + \log \left(e^{\sin \lambda_2 \cdot \sin \lambda_1}\right), \cos \phi_1\right)} + \lambda_1\]
\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)}
↓
\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \cos \phi_2\right) \cdot \sin \lambda_1 + \left(-\cos \lambda_1 \cdot \left(\sin \lambda_2 \cdot \cos \phi_2\right)\right)}{\mathsf{fma}\left(\cos \phi_2, \cos \lambda_2 \cdot \cos \lambda_1 + \log \left(e^{\sin \lambda_2 \cdot \sin \lambda_1}\right), \cos \phi_1\right)} + \lambda_1
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r44535 = lambda1;
        double r44536 = phi2;
        double r44537 = cos(r44536);
        double r44538 = lambda2;
        double r44539 = r44535 - r44538;
        double r44540 = sin(r44539);
        double r44541 = r44537 * r44540;
        double r44542 = phi1;
        double r44543 = cos(r44542);
        double r44544 = cos(r44539);
        double r44545 = r44537 * r44544;
        double r44546 = r44543 + r44545;
        double r44547 = atan2(r44541, r44546);
        double r44548 = r44535 + r44547;
        return r44548;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r44549 = lambda2;
        double r44550 = cos(r44549);
        double r44551 = phi2;
        double r44552 = cos(r44551);
        double r44553 = r44550 * r44552;
        double r44554 = lambda1;
        double r44555 = sin(r44554);
        double r44556 = r44553 * r44555;
        double r44557 = cos(r44554);
        double r44558 = sin(r44549);
        double r44559 = r44558 * r44552;
        double r44560 = r44557 * r44559;
        double r44561 = -r44560;
        double r44562 = r44556 + r44561;
        double r44563 = r44550 * r44557;
        double r44564 = r44558 * r44555;
        double r44565 = exp(r44564);
        double r44566 = log(r44565);
        double r44567 = r44563 + r44566;
        double r44568 = phi1;
        double r44569 = cos(r44568);
        double r44570 = fma(r44552, r44567, r44569);
        double r44571 = atan2(r44562, r44570);
        double r44572 = r44571 + r44554;
        return r44572;
}

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. Simplified0.8

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

  4. Applied cos-diff0.8

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

  6. Applied sub-neg0.8

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

  7. Applied sin-sum0.2

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

  8. Applied distribute-lft-in0.2

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

  9. Simplified0.2

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

  10. Simplified0.2

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

  12. Applied add-log-exp0.2

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

  13. Simplified0.2

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

  14. Final simplification0.2

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

Reproduce

herbie shell --seed 2019195 +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)))))))