Average Error: 16.8 → 3.6
Time: 1.1m
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot \cos^{-1} \left((\left(\sin \phi_1\right) \cdot \left(\sin \phi_2\right) + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot (\left(\cos \lambda_2\right) \cdot \left(\cos \lambda_1\right) + \left(\sin \lambda_1 \cdot \sin \lambda_2\right))_*\right))_*\right)\]
\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R
↓
R \cdot \cos^{-1} \left((\left(\sin \phi_1\right) \cdot \left(\sin \phi_2\right) + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot (\left(\cos \lambda_2\right) \cdot \left(\cos \lambda_1\right) + \left(\sin \lambda_1 \cdot \sin \lambda_2\right))_*\right))_*\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1428926 = phi1;
        double r1428927 = sin(r1428926);
        double r1428928 = phi2;
        double r1428929 = sin(r1428928);
        double r1428930 = r1428927 * r1428929;
        double r1428931 = cos(r1428926);
        double r1428932 = cos(r1428928);
        double r1428933 = r1428931 * r1428932;
        double r1428934 = lambda1;
        double r1428935 = lambda2;
        double r1428936 = r1428934 - r1428935;
        double r1428937 = cos(r1428936);
        double r1428938 = r1428933 * r1428937;
        double r1428939 = r1428930 + r1428938;
        double r1428940 = acos(r1428939);
        double r1428941 = R;
        double r1428942 = r1428940 * r1428941;
        return r1428942;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1428943 = R;
        double r1428944 = phi1;
        double r1428945 = sin(r1428944);
        double r1428946 = phi2;
        double r1428947 = sin(r1428946);
        double r1428948 = cos(r1428944);
        double r1428949 = cos(r1428946);
        double r1428950 = r1428948 * r1428949;
        double r1428951 = lambda2;
        double r1428952 = cos(r1428951);
        double r1428953 = lambda1;
        double r1428954 = cos(r1428953);
        double r1428955 = sin(r1428953);
        double r1428956 = sin(r1428951);
        double r1428957 = r1428955 * r1428956;
        double r1428958 = fma(r1428952, r1428954, r1428957);
        double r1428959 = r1428950 * r1428958;
        double r1428960 = fma(r1428945, r1428947, r1428959);
        double r1428961 = acos(r1428960);
        double r1428962 = r1428943 * r1428961;
        return r1428962;
}

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 16.8

    \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]

  2. Simplified16.8

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

  4. Applied cos-diff3.6

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

  5. Taylor expanded around inf 3.6

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

  6. Simplified3.6

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

  7. Final simplification3.6

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

Reproduce

herbie shell --seed 2019119 +o rules:numerics
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))