Average Error: 0.9 → 0.3
Time: 26.3s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\left(-\sin \lambda_2 \cdot \cos \lambda_1\right) + \cos \lambda_2 \cdot \sin \lambda_1\right) \cdot \cos \phi_2}{\sqrt[3]{{\left(\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \cos \phi_2, \cos \phi_1\right)\right)}^{3}}}\]
\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(\left(-\sin \lambda_2 \cdot \cos \lambda_1\right) + \cos \lambda_2 \cdot \sin \lambda_1\right) \cdot \cos \phi_2}{\sqrt[3]{{\left(\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \cos \phi_2, \cos \phi_1\right)\right)}^{3}}}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r52129 = lambda1;
        double r52130 = phi2;
        double r52131 = cos(r52130);
        double r52132 = lambda2;
        double r52133 = r52129 - r52132;
        double r52134 = sin(r52133);
        double r52135 = r52131 * r52134;
        double r52136 = phi1;
        double r52137 = cos(r52136);
        double r52138 = cos(r52133);
        double r52139 = r52131 * r52138;
        double r52140 = r52137 + r52139;
        double r52141 = atan2(r52135, r52140);
        double r52142 = r52129 + r52141;
        return r52142;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r52143 = lambda1;
        double r52144 = lambda2;
        double r52145 = sin(r52144);
        double r52146 = cos(r52143);
        double r52147 = r52145 * r52146;
        double r52148 = -r52147;
        double r52149 = cos(r52144);
        double r52150 = sin(r52143);
        double r52151 = r52149 * r52150;
        double r52152 = r52148 + r52151;
        double r52153 = phi2;
        double r52154 = cos(r52153);
        double r52155 = r52152 * r52154;
        double r52156 = r52145 * r52150;
        double r52157 = fma(r52149, r52146, r52156);
        double r52158 = phi1;
        double r52159 = cos(r52158);
        double r52160 = fma(r52157, r52154, r52159);
        double r52161 = 3.0;
        double r52162 = pow(r52160, r52161);
        double r52163 = cbrt(r52162);
        double r52164 = atan2(r52155, r52163);
        double r52165 = r52143 + r52164;
        return r52165;
}

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 sub-neg0.9

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

  5. Applied cos-sum0.9

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

  6. Simplified0.9

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

  7. Simplified0.9

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

  9. Applied sub-neg0.9

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

  10. Applied sin-sum0.2

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

  11. Simplified0.2

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

  12. Simplified0.2

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

  14. Applied add-cbrt-cube0.3

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

  15. Simplified0.3

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

  16. Final simplification0.3

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

Reproduce

herbie shell --seed 2019179 +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)))))))