Average Error: 17.3 → 4.1
Time: 22.5s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \mathsf{log1p}\left(\mathsf{expm1}\left(\left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \cos \phi_2\right)\right)\right)\right) \cdot R\]
\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
↓
\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \mathsf{log1p}\left(\mathsf{expm1}\left(\left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \cos \phi_2\right)\right)\right)\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r26126 = phi1;
        double r26127 = sin(r26126);
        double r26128 = phi2;
        double r26129 = sin(r26128);
        double r26130 = r26127 * r26129;
        double r26131 = cos(r26126);
        double r26132 = cos(r26128);
        double r26133 = r26131 * r26132;
        double r26134 = lambda1;
        double r26135 = lambda2;
        double r26136 = r26134 - r26135;
        double r26137 = cos(r26136);
        double r26138 = r26133 * r26137;
        double r26139 = r26130 + r26138;
        double r26140 = acos(r26139);
        double r26141 = R;
        double r26142 = r26140 * r26141;
        return r26142;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r26143 = phi1;
        double r26144 = sin(r26143);
        double r26145 = phi2;
        double r26146 = sin(r26145);
        double r26147 = cos(r26143);
        double r26148 = lambda1;
        double r26149 = cos(r26148);
        double r26150 = lambda2;
        double r26151 = cos(r26150);
        double r26152 = sin(r26148);
        double r26153 = sin(r26150);
        double r26154 = r26152 * r26153;
        double r26155 = fma(r26149, r26151, r26154);
        double r26156 = r26147 * r26155;
        double r26157 = cos(r26145);
        double r26158 = r26156 * r26157;
        double r26159 = expm1(r26158);
        double r26160 = log1p(r26159);
        double r26161 = fma(r26144, r26146, r26160);
        double r26162 = acos(r26161);
        double r26163 = R;
        double r26164 = r26162 * r26163;
        return r26164;
}

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 17.3

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

  2. Simplified17.3

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

  4. Applied sub-neg17.3

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

  5. Applied cos-sum4.0

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

  6. Simplified4.0

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

  8. Applied log1p-expm1-u4.1

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

  9. Simplified4.1

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

  10. Final simplification4.1

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

Reproduce

herbie shell --seed 2020047 +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))