Average Error: 37.1 → 0.1
Time: 51.2s
Precision: 64
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\mathsf{hypot}\left(\left(\mathsf{fma}\left(\left(\cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \left(\lambda_1 \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right)\right), \left(\left(\lambda_2 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right) - \mathsf{fma}\left(\left(\sin \left(\frac{1}{2} \cdot \phi_2\right)\right), \left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \lambda_1\right), \left(\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \lambda_2\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)\right)\right)\right), \left(\phi_1 - \phi_2\right)\right) \cdot R\]
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(\left(\mathsf{fma}\left(\left(\cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \left(\lambda_1 \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right)\right), \left(\left(\lambda_2 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right) - \mathsf{fma}\left(\left(\sin \left(\frac{1}{2} \cdot \phi_2\right)\right), \left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \lambda_1\right), \left(\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \lambda_2\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)\right)\right)\right), \left(\phi_1 - \phi_2\right)\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r4418999 = R;
        double r4419000 = lambda1;
        double r4419001 = lambda2;
        double r4419002 = r4419000 - r4419001;
        double r4419003 = phi1;
        double r4419004 = phi2;
        double r4419005 = r4419003 + r4419004;
        double r4419006 = 2.0;
        double r4419007 = r4419005 / r4419006;
        double r4419008 = cos(r4419007);
        double r4419009 = r4419002 * r4419008;
        double r4419010 = r4419009 * r4419009;
        double r4419011 = r4419003 - r4419004;
        double r4419012 = r4419011 * r4419011;
        double r4419013 = r4419010 + r4419012;
        double r4419014 = sqrt(r4419013);
        double r4419015 = r4418999 * r4419014;
        return r4419015;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r4419016 = 0.5;
        double r4419017 = phi2;
        double r4419018 = r4419016 * r4419017;
        double r4419019 = cos(r4419018);
        double r4419020 = lambda1;
        double r4419021 = phi1;
        double r4419022 = r4419021 * r4419016;
        double r4419023 = cos(r4419022);
        double r4419024 = r4419020 * r4419023;
        double r4419025 = lambda2;
        double r4419026 = sin(r4419018);
        double r4419027 = r4419025 * r4419026;
        double r4419028 = sin(r4419022);
        double r4419029 = r4419027 * r4419028;
        double r4419030 = r4419028 * r4419020;
        double r4419031 = r4419019 * r4419025;
        double r4419032 = r4419031 * r4419023;
        double r4419033 = fma(r4419026, r4419030, r4419032);
        double r4419034 = r4419029 - r4419033;
        double r4419035 = fma(r4419019, r4419024, r4419034);
        double r4419036 = r4419021 - r4419017;
        double r4419037 = hypot(r4419035, r4419036);
        double r4419038 = R;
        double r4419039 = r4419037 * r4419038;
        return r4419039;
}

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 37.1

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

  2. Simplified3.7

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

  4. Applied log1p-expm1-u3.8

    \[\leadsto \mathsf{hypot}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)\right)\right)\right)}\right), \left(\phi_1 - \phi_2\right)\right) \cdot R\]

  5. Taylor expanded around inf 3.7

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

  7. Applied distribute-rgt-in3.7

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

  8. Applied cos-sum0.1

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

  9. Taylor expanded around -inf 0.1

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

  10. Simplified0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2019132 +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))))))