Average Error: 0.9 → 0.3
Time: 10.4s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\log \left(e^{\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1\right)}\right) + \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}\]
\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{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\log \left(e^{\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1\right)}\right) + \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r50087 = lambda1;
        double r50088 = phi2;
        double r50089 = cos(r50088);
        double r50090 = lambda2;
        double r50091 = r50087 - r50090;
        double r50092 = sin(r50091);
        double r50093 = r50089 * r50092;
        double r50094 = phi1;
        double r50095 = cos(r50094);
        double r50096 = cos(r50091);
        double r50097 = r50089 * r50096;
        double r50098 = r50095 + r50097;
        double r50099 = atan2(r50093, r50098);
        double r50100 = r50087 + r50099;
        return r50100;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r50101 = lambda1;
        double r50102 = phi2;
        double r50103 = cos(r50102);
        double r50104 = sin(r50101);
        double r50105 = lambda2;
        double r50106 = cos(r50105);
        double r50107 = r50104 * r50106;
        double r50108 = cos(r50101);
        double r50109 = sin(r50105);
        double r50110 = r50108 * r50109;
        double r50111 = r50107 - r50110;
        double r50112 = r50103 * r50111;
        double r50113 = r50106 * r50103;
        double r50114 = phi1;
        double r50115 = cos(r50114);
        double r50116 = fma(r50108, r50113, r50115);
        double r50117 = exp(r50116);
        double r50118 = log(r50117);
        double r50119 = r50104 * r50109;
        double r50120 = r50119 * r50103;
        double r50121 = r50118 + r50120;
        double r50122 = atan2(r50112, r50121);
        double r50123 = r50101 + r50122;
        return r50123;
}

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. Using strategy rm

  3. Applied sin-diff0.9

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

  5. Applied cos-diff0.2

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

  6. Applied distribute-rgt-in0.2

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

  7. Applied associate-+r+0.2

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

  8. Simplified0.2

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

  10. Applied add-log-exp0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Midpoint on a great circle"
  :precision binary64
  (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))