Average Error: 0.9 → 0.3
Time: 26.9s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_1\right)}\right)\right)\]
\lambda_1 + \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 + \mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\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)
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r1918103 = lambda1;
        double r1918104 = phi2;
        double r1918105 = cos(r1918104);
        double r1918106 = lambda2;
        double r1918107 = r1918103 - r1918106;
        double r1918108 = sin(r1918107);
        double r1918109 = r1918105 * r1918108;
        double r1918110 = phi1;
        double r1918111 = cos(r1918110);
        double r1918112 = cos(r1918107);
        double r1918113 = r1918105 * r1918112;
        double r1918114 = r1918111 + r1918113;
        double r1918115 = atan2(r1918109, r1918114);
        double r1918116 = r1918103 + r1918115;
        return r1918116;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r1918117 = lambda1;
        double r1918118 = sin(r1918117);
        double r1918119 = lambda2;
        double r1918120 = cos(r1918119);
        double r1918121 = r1918118 * r1918120;
        double r1918122 = sin(r1918119);
        double r1918123 = cos(r1918117);
        double r1918124 = r1918122 * r1918123;
        double r1918125 = r1918121 - r1918124;
        double r1918126 = phi2;
        double r1918127 = cos(r1918126);
        double r1918128 = r1918125 * r1918127;
        double r1918129 = r1918123 * r1918120;
        double r1918130 = fma(r1918118, r1918122, r1918129);
        double r1918131 = phi1;
        double r1918132 = cos(r1918131);
        double r1918133 = fma(r1918127, r1918130, r1918132);
        double r1918134 = atan2(r1918128, r1918133);
        double r1918135 = expm1(r1918134);
        double r1918136 = log1p(r1918135);
        double r1918137 = r1918117 + r1918136;
        return r1918137;
}

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 sin-diff0.8

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

  6. Applied cos-diff0.2

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

  8. Applied add-log-exp0.2

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

  10. Applied log1p-expm1-u0.3

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

  11. Simplified0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019172 +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)))))))