Average Error: 16.9 → 3.7
Time: 14.4s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \log \left(e^{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)}\right)\right)\right) \cdot R\]
\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
↓
\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \log \left(e^{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)}\right)\right)\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r22191 = phi1;
        double r22192 = sin(r22191);
        double r22193 = phi2;
        double r22194 = sin(r22193);
        double r22195 = r22192 * r22194;
        double r22196 = cos(r22191);
        double r22197 = cos(r22193);
        double r22198 = r22196 * r22197;
        double r22199 = lambda1;
        double r22200 = lambda2;
        double r22201 = r22199 - r22200;
        double r22202 = cos(r22201);
        double r22203 = r22198 * r22202;
        double r22204 = r22195 + r22203;
        double r22205 = acos(r22204);
        double r22206 = R;
        double r22207 = r22205 * r22206;
        return r22207;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r22208 = phi1;
        double r22209 = sin(r22208);
        double r22210 = phi2;
        double r22211 = sin(r22210);
        double r22212 = r22209 * r22211;
        double r22213 = cos(r22208);
        double r22214 = cos(r22210);
        double r22215 = r22213 * r22214;
        double r22216 = lambda1;
        double r22217 = cos(r22216);
        double r22218 = lambda2;
        double r22219 = cos(r22218);
        double r22220 = r22217 * r22219;
        double r22221 = r22215 * r22220;
        double r22222 = sin(r22216);
        double r22223 = sin(r22218);
        double r22224 = r22222 * r22223;
        double r22225 = r22215 * r22224;
        double r22226 = exp(r22225);
        double r22227 = log(r22226);
        double r22228 = r22221 + r22227;
        double r22229 = r22212 + r22228;
        double r22230 = acos(r22229);
        double r22231 = R;
        double r22232 = r22230 * r22231;
        return r22232;
}

Error

Bits error versus R

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 16.9

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

  3. Applied cos-diff3.7

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

  5. Applied distribute-lft-in3.7

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

  7. Applied add-log-exp3.7

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

  8. Final simplification3.7

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

Reproduce

herbie shell --seed 2020046 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  :precision binary64
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))