Average Error: 17.1 → 4.0
Time: 54.7s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot e^{\log \left(\log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)\right)}\right)\right)}\]
\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 e^{\log \left(\log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)\right)}\right)\right)}
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1481089 = phi1;
        double r1481090 = sin(r1481089);
        double r1481091 = phi2;
        double r1481092 = sin(r1481091);
        double r1481093 = r1481090 * r1481092;
        double r1481094 = cos(r1481089);
        double r1481095 = cos(r1481091);
        double r1481096 = r1481094 * r1481095;
        double r1481097 = lambda1;
        double r1481098 = lambda2;
        double r1481099 = r1481097 - r1481098;
        double r1481100 = cos(r1481099);
        double r1481101 = r1481096 * r1481100;
        double r1481102 = r1481093 + r1481101;
        double r1481103 = acos(r1481102);
        double r1481104 = R;
        double r1481105 = r1481103 * r1481104;
        return r1481105;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1481106 = R;
        double r1481107 = phi1;
        double r1481108 = sin(r1481107);
        double r1481109 = phi2;
        double r1481110 = sin(r1481109);
        double r1481111 = lambda1;
        double r1481112 = sin(r1481111);
        double r1481113 = lambda2;
        double r1481114 = sin(r1481113);
        double r1481115 = cos(r1481113);
        double r1481116 = cos(r1481111);
        double r1481117 = r1481115 * r1481116;
        double r1481118 = fma(r1481112, r1481114, r1481117);
        double r1481119 = cos(r1481109);
        double r1481120 = cos(r1481107);
        double r1481121 = r1481119 * r1481120;
        double r1481122 = r1481118 * r1481121;
        double r1481123 = fma(r1481108, r1481110, r1481122);
        double r1481124 = acos(r1481123);
        double r1481125 = exp(r1481124);
        double r1481126 = log(r1481125);
        double r1481127 = log(r1481126);
        double r1481128 = exp(r1481127);
        double r1481129 = r1481106 * r1481128;
        return r1481129;
}

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. Simplified17.1

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

  4. Applied cos-diff4.0

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

  6. Applied add-exp-log4.0

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

  7. Simplified4.0

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

  9. Applied add-log-exp4.0

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

  10. Final simplification4.0

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

Reproduce

herbie shell --seed 2019134 +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))