Average Error: 0.9 → 0.3
Time: 38.7s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{log1p}\left(\log \left(e^{\mathsf{expm1}\left(\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_1\right)\right)}\right)\right)}\]
\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}
↓
\lambda_1 + \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{log1p}\left(\log \left(e^{\mathsf{expm1}\left(\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_1\right)\right)}\right)\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r1430130 = lambda1;
        double r1430131 = phi2;
        double r1430132 = cos(r1430131);
        double r1430133 = lambda2;
        double r1430134 = r1430130 - r1430133;
        double r1430135 = sin(r1430134);
        double r1430136 = r1430132 * r1430135;
        double r1430137 = phi1;
        double r1430138 = cos(r1430137);
        double r1430139 = cos(r1430134);
        double r1430140 = r1430132 * r1430139;
        double r1430141 = r1430138 + r1430140;
        double r1430142 = atan2(r1430136, r1430141);
        double r1430143 = r1430130 + r1430142;
        return r1430143;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r1430144 = lambda1;
        double r1430145 = sin(r1430144);
        double r1430146 = lambda2;
        double r1430147 = cos(r1430146);
        double r1430148 = r1430145 * r1430147;
        double r1430149 = cos(r1430144);
        double r1430150 = sin(r1430146);
        double r1430151 = r1430149 * r1430150;
        double r1430152 = r1430148 - r1430151;
        double r1430153 = phi2;
        double r1430154 = cos(r1430153);
        double r1430155 = r1430152 * r1430154;
        double r1430156 = r1430149 * r1430147;
        double r1430157 = fma(r1430145, r1430150, r1430156);
        double r1430158 = phi1;
        double r1430159 = cos(r1430158);
        double r1430160 = fma(r1430154, r1430157, r1430159);
        double r1430161 = expm1(r1430160);
        double r1430162 = exp(r1430161);
        double r1430163 = log(r1430162);
        double r1430164 = log1p(r1430163);
        double r1430165 = atan2(r1430155, r1430164);
        double r1430166 = r1430144 + r1430165;
        return r1430166;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 0.9

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

  2. Simplified0.9

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

  4. Applied cos-diff0.9

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

  6. Applied sin-diff0.2

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

  8. Applied log1p-expm1-u0.2

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

  9. Simplified0.2

    \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_1\right)\right)}\right)} + \lambda_1\]
  10. Using strategy rm

  11. Applied add-log-exp0.3

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

  12. Final simplification0.3

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{log1p}\left(\log \left(e^{\mathsf{expm1}\left(\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_1\right)\right)}\right)\right)}\]

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Midpoint on a great circle"
  (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))