Average Error: 0.9 → 0.3
Time: 1.0m
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\log \left(e^{\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \cos \phi_1}\right) + \left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \cos \phi_2} + \lambda_1\]
\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{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\log \left(e^{\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \cos \phi_1}\right) + \left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \cos \phi_2} + \lambda_1
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r3651930 = lambda1;
        double r3651931 = phi2;
        double r3651932 = cos(r3651931);
        double r3651933 = lambda2;
        double r3651934 = r3651930 - r3651933;
        double r3651935 = sin(r3651934);
        double r3651936 = r3651932 * r3651935;
        double r3651937 = phi1;
        double r3651938 = cos(r3651937);
        double r3651939 = cos(r3651934);
        double r3651940 = r3651932 * r3651939;
        double r3651941 = r3651938 + r3651940;
        double r3651942 = atan2(r3651936, r3651941);
        double r3651943 = r3651930 + r3651942;
        return r3651943;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r3651944 = phi2;
        double r3651945 = cos(r3651944);
        double r3651946 = lambda1;
        double r3651947 = sin(r3651946);
        double r3651948 = lambda2;
        double r3651949 = cos(r3651948);
        double r3651950 = r3651947 * r3651949;
        double r3651951 = cos(r3651946);
        double r3651952 = sin(r3651948);
        double r3651953 = r3651951 * r3651952;
        double r3651954 = r3651950 - r3651953;
        double r3651955 = r3651945 * r3651954;
        double r3651956 = r3651951 * r3651949;
        double r3651957 = r3651945 * r3651956;
        double r3651958 = phi1;
        double r3651959 = cos(r3651958);
        double r3651960 = r3651957 + r3651959;
        double r3651961 = exp(r3651960);
        double r3651962 = log(r3651961);
        double r3651963 = r3651952 * r3651947;
        double r3651964 = r3651963 * r3651945;
        double r3651965 = r3651962 + r3651964;
        double r3651966 = atan2(r3651955, r3651965);
        double r3651967 = r3651966 + r3651946;
        return r3651967;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied sin-diff0.8

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

  5. Applied cos-diff0.2

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

  6. Applied distribute-lft-in0.2

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

  7. Applied associate-+r+0.2

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

  9. Applied add-log-exp0.3

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

  10. Applied add-log-exp0.3

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

  11. Applied sum-log0.3

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

  12. Simplified0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019168 
(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)))))))