Average Error: 0.9 → 0.3
Time: 29.7s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_2, \cos \phi_1\right)}\right)\right)\]
\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 + \mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_2, \cos \phi_1\right)}\right)\right)
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r724949 = lambda1;
        double r724950 = phi2;
        double r724951 = cos(r724950);
        double r724952 = lambda2;
        double r724953 = r724949 - r724952;
        double r724954 = sin(r724953);
        double r724955 = r724951 * r724954;
        double r724956 = phi1;
        double r724957 = cos(r724956);
        double r724958 = cos(r724953);
        double r724959 = r724951 * r724958;
        double r724960 = r724957 + r724959;
        double r724961 = atan2(r724955, r724960);
        double r724962 = r724949 + r724961;
        return r724962;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r724963 = lambda1;
        double r724964 = sin(r724963);
        double r724965 = lambda2;
        double r724966 = cos(r724965);
        double r724967 = r724964 * r724966;
        double r724968 = sin(r724965);
        double r724969 = cos(r724963);
        double r724970 = r724968 * r724969;
        double r724971 = r724967 - r724970;
        double r724972 = phi2;
        double r724973 = cos(r724972);
        double r724974 = r724971 * r724973;
        double r724975 = r724969 * r724966;
        double r724976 = fma(r724968, r724964, r724975);
        double r724977 = phi1;
        double r724978 = cos(r724977);
        double r724979 = fma(r724976, r724973, r724978);
        double r724980 = atan2(r724974, r724979);
        double r724981 = expm1(r724980);
        double r724982 = log1p(r724981);
        double r724983 = r724963 + r724982;
        return r724983;
}

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 log1p-expm1-u0.3

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

  9. Simplified0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019153 +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)))))))