Average Error: 16.7 → 3.8
Time: 50.7s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\left(\cos \phi_1 \cdot \cos \phi_2\right), \left(\mathsf{fma}\left(\left(\cos \lambda_2\right), \left(\cos \lambda_1\right), \left(\sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right), \left(\sin \phi_2 \cdot \sin \phi_1\right)\right)\right)\right)\]
\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} - \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_2 \cdot \sin \lambda_1\right)\right)\right), \left(\sin \phi_2 \cdot \sin \phi_1\right)\right)\right)\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1007358 = phi1;
        double r1007359 = sin(r1007358);
        double r1007360 = phi2;
        double r1007361 = sin(r1007360);
        double r1007362 = r1007359 * r1007361;
        double r1007363 = cos(r1007358);
        double r1007364 = cos(r1007360);
        double r1007365 = r1007363 * r1007364;
        double r1007366 = lambda1;
        double r1007367 = lambda2;
        double r1007368 = r1007366 - r1007367;
        double r1007369 = cos(r1007368);
        double r1007370 = r1007365 * r1007369;
        double r1007371 = r1007362 + r1007370;
        double r1007372 = acos(r1007371);
        double r1007373 = R;
        double r1007374 = r1007372 * r1007373;
        return r1007374;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1007375 = R;
        double r1007376 = atan2(1.0, 0.0);
        double r1007377 = 2.0;
        double r1007378 = r1007376 / r1007377;
        double r1007379 = phi1;
        double r1007380 = cos(r1007379);
        double r1007381 = phi2;
        double r1007382 = cos(r1007381);
        double r1007383 = r1007380 * r1007382;
        double r1007384 = lambda2;
        double r1007385 = cos(r1007384);
        double r1007386 = lambda1;
        double r1007387 = cos(r1007386);
        double r1007388 = sin(r1007384);
        double r1007389 = sin(r1007386);
        double r1007390 = r1007388 * r1007389;
        double r1007391 = fma(r1007385, r1007387, r1007390);
        double r1007392 = sin(r1007381);
        double r1007393 = sin(r1007379);
        double r1007394 = r1007392 * r1007393;
        double r1007395 = fma(r1007383, r1007391, r1007394);
        double r1007396 = asin(r1007395);
        double r1007397 = r1007378 - r1007396;
        double r1007398 = r1007375 * r1007397;
        return r1007398;
}

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 16.7

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

  2. Simplified16.7

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

  4. Applied cos-diff3.7

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

  6. Applied acos-asin3.8

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

  7. Simplified3.8

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

  8. Final simplification3.8

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

Reproduce

herbie shell --seed 2019132 +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))