Average Error: 17.1 → 3.9
Time: 44.9s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot \left(\frac{\pi}{2} - \left(\frac{\pi}{2} - \cos^{-1} \left(\mathsf{fma}\left(\left(\cos \phi_2 \cdot \cos \phi_1\right), \left(\mathsf{fma}\left(\left(\cos \lambda_2\right), \left(\cos \lambda_1\right), \left(\sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right), \left(\sin \phi_2 \cdot \sin \phi_1\right)\right)\right)\right)\right)\]
\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 \left(\frac{\pi}{2} - \left(\frac{\pi}{2} - \cos^{-1} \left(\mathsf{fma}\left(\left(\cos \phi_2 \cdot \cos \phi_1\right), \left(\mathsf{fma}\left(\left(\cos \lambda_2\right), \left(\cos \lambda_1\right), \left(\sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right), \left(\sin \phi_2 \cdot \sin \phi_1\right)\right)\right)\right)\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r369292 = phi1;
        double r369293 = sin(r369292);
        double r369294 = phi2;
        double r369295 = sin(r369294);
        double r369296 = r369293 * r369295;
        double r369297 = cos(r369292);
        double r369298 = cos(r369294);
        double r369299 = r369297 * r369298;
        double r369300 = lambda1;
        double r369301 = lambda2;
        double r369302 = r369300 - r369301;
        double r369303 = cos(r369302);
        double r369304 = r369299 * r369303;
        double r369305 = r369296 + r369304;
        double r369306 = acos(r369305);
        double r369307 = R;
        double r369308 = r369306 * r369307;
        return r369308;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r369309 = R;
        double r369310 = atan2(1.0, 0.0);
        double r369311 = 2.0;
        double r369312 = r369310 / r369311;
        double r369313 = phi2;
        double r369314 = cos(r369313);
        double r369315 = phi1;
        double r369316 = cos(r369315);
        double r369317 = r369314 * r369316;
        double r369318 = lambda2;
        double r369319 = cos(r369318);
        double r369320 = lambda1;
        double r369321 = cos(r369320);
        double r369322 = sin(r369318);
        double r369323 = sin(r369320);
        double r369324 = r369322 * r369323;
        double r369325 = fma(r369319, r369321, r369324);
        double r369326 = sin(r369313);
        double r369327 = sin(r369315);
        double r369328 = r369326 * r369327;
        double r369329 = fma(r369317, r369325, r369328);
        double r369330 = acos(r369329);
        double r369331 = r369312 - r369330;
        double r369332 = r369312 - r369331;
        double r369333 = r369309 * r369332;
        return r369333;
}

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 17.1

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

  3. Applied cos-diff3.9

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

  5. Applied acos-asin4.0

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

  6. Simplified4.0

    \[\leadsto \left(\frac{\pi}{2} - \color{blue}{\sin^{-1} \left(\mathsf{fma}\left(\left(\cos \phi_1 \cdot \cos \phi_2\right), \left(\mathsf{fma}\left(\left(\cos \lambda_2\right), \left(\cos \lambda_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), \left(\sin \phi_2 \cdot \sin \phi_1\right)\right)\right)}\right) \cdot R\]
  7. Using strategy rm

  8. Applied asin-acos3.9

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

  9. Final simplification3.9

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

Reproduce

herbie shell --seed 2019128 +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))