Average Error: 16.7 → 3.6
Time: 45.6s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \cos \lambda_2 \cdot \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R\]
\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(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \cos \lambda_2 \cdot \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r547283 = phi1;
        double r547284 = sin(r547283);
        double r547285 = phi2;
        double r547286 = sin(r547285);
        double r547287 = r547284 * r547286;
        double r547288 = cos(r547283);
        double r547289 = cos(r547285);
        double r547290 = r547288 * r547289;
        double r547291 = lambda1;
        double r547292 = lambda2;
        double r547293 = r547291 - r547292;
        double r547294 = cos(r547293);
        double r547295 = r547290 * r547294;
        double r547296 = r547287 + r547295;
        double r547297 = acos(r547296);
        double r547298 = R;
        double r547299 = r547297 * r547298;
        return r547299;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r547300 = phi2;
        double r547301 = cos(r547300);
        double r547302 = phi1;
        double r547303 = cos(r547302);
        double r547304 = r547301 * r547303;
        double r547305 = lambda1;
        double r547306 = sin(r547305);
        double r547307 = lambda2;
        double r547308 = sin(r547307);
        double r547309 = r547306 * r547308;
        double r547310 = expm1(r547309);
        double r547311 = log1p(r547310);
        double r547312 = cos(r547307);
        double r547313 = cos(r547305);
        double r547314 = r547312 * r547313;
        double r547315 = r547311 + r547314;
        double r547316 = sin(r547302);
        double r547317 = sin(r547300);
        double r547318 = r547316 * r547317;
        double r547319 = fma(r547304, r547315, r547318);
        double r547320 = acos(r547319);
        double r547321 = R;
        double r547322 = r547320 * r547321;
        return r547322;
}

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. Simplified16.6

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

  4. Applied cos-diff3.6

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

  6. Applied log1p-expm1-u3.6

    \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1 \cdot \cos \lambda_2 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}, \sin \phi_2 \cdot \sin \phi_1\right)\right)\]

  7. Final simplification3.6

    \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \cos \lambda_2 \cdot \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R\]

Reproduce

herbie shell --seed 2019151 +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))