Average Error: 16.8 → 3.9
Time: 41.6s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[e^{\log \left(\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \log \left(e^{\mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}\right)\right)\right)\right)} \cdot R\]
\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
↓
e^{\log \left(\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \log \left(e^{\mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}\right)\right)\right)\right)} \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r23280 = phi1;
        double r23281 = sin(r23280);
        double r23282 = phi2;
        double r23283 = sin(r23282);
        double r23284 = r23281 * r23283;
        double r23285 = cos(r23280);
        double r23286 = cos(r23282);
        double r23287 = r23285 * r23286;
        double r23288 = lambda1;
        double r23289 = lambda2;
        double r23290 = r23288 - r23289;
        double r23291 = cos(r23290);
        double r23292 = r23287 * r23291;
        double r23293 = r23284 + r23292;
        double r23294 = acos(r23293);
        double r23295 = R;
        double r23296 = r23294 * r23295;
        return r23296;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r23297 = phi1;
        double r23298 = sin(r23297);
        double r23299 = phi2;
        double r23300 = sin(r23299);
        double r23301 = lambda1;
        double r23302 = sin(r23301);
        double r23303 = lambda2;
        double r23304 = sin(r23303);
        double r23305 = cos(r23301);
        double r23306 = cos(r23303);
        double r23307 = r23305 * r23306;
        double r23308 = fma(r23302, r23304, r23307);
        double r23309 = cos(r23297);
        double r23310 = cos(r23299);
        double r23311 = r23309 * r23310;
        double r23312 = r23308 * r23311;
        double r23313 = exp(r23312);
        double r23314 = log(r23313);
        double r23315 = fma(r23298, r23300, r23314);
        double r23316 = acos(r23315);
        double r23317 = log(r23316);
        double r23318 = exp(r23317);
        double r23319 = R;
        double r23320 = r23318 * r23319;
        return r23320;
}

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. Using strategy rm

  3. Applied cos-diff3.8

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

  5. Applied add-exp-log3.8

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

  6. Simplified3.8

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

  8. Applied add-log-exp3.9

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

  9. Final simplification3.9

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

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(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))