Average Error: 17.1 → 4.0
Time: 1.9m
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot \log \left(\frac{\sqrt{e^{\pi}}}{e^{\sin^{-1} \left(\mathsf{fma}\left(\left(\cos \phi_2 \cdot \cos \phi_1\right), \left(\mathsf{fma}\left(\left(\cos \lambda_2\right), \left(\cos \lambda_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), \left(\sin \phi_2 \cdot \sin \phi_1\right)\right)\right)}}\right)\]
\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 \log \left(\frac{\sqrt{e^{\pi}}}{e^{\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_2 \cdot \sin \phi_1\right)\right)\right)}}\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r3966387 = phi1;
        double r3966388 = sin(r3966387);
        double r3966389 = phi2;
        double r3966390 = sin(r3966389);
        double r3966391 = r3966388 * r3966390;
        double r3966392 = cos(r3966387);
        double r3966393 = cos(r3966389);
        double r3966394 = r3966392 * r3966393;
        double r3966395 = lambda1;
        double r3966396 = lambda2;
        double r3966397 = r3966395 - r3966396;
        double r3966398 = cos(r3966397);
        double r3966399 = r3966394 * r3966398;
        double r3966400 = r3966391 + r3966399;
        double r3966401 = acos(r3966400);
        double r3966402 = R;
        double r3966403 = r3966401 * r3966402;
        return r3966403;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r3966404 = R;
        double r3966405 = atan2(1.0, 0.0);
        double r3966406 = exp(r3966405);
        double r3966407 = sqrt(r3966406);
        double r3966408 = phi2;
        double r3966409 = cos(r3966408);
        double r3966410 = phi1;
        double r3966411 = cos(r3966410);
        double r3966412 = r3966409 * r3966411;
        double r3966413 = lambda2;
        double r3966414 = cos(r3966413);
        double r3966415 = lambda1;
        double r3966416 = cos(r3966415);
        double r3966417 = sin(r3966415);
        double r3966418 = sin(r3966413);
        double r3966419 = r3966417 * r3966418;
        double r3966420 = fma(r3966414, r3966416, r3966419);
        double r3966421 = sin(r3966408);
        double r3966422 = sin(r3966410);
        double r3966423 = r3966421 * r3966422;
        double r3966424 = fma(r3966412, r3966420, r3966423);
        double r3966425 = asin(r3966424);
        double r3966426 = exp(r3966425);
        double r3966427 = r3966407 / r3966426;
        double r3966428 = log(r3966427);
        double r3966429 = r3966404 * r3966428;
        return r3966429;
}

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 add-log-exp3.9

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

  6. Simplified3.9

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

  8. Applied acos-asin4.0

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

  9. Applied exp-diff4.0

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

  10. Simplified4.0

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

  11. Final simplification4.0

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

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))