Bearing on a great circle

Average Error: 13.1 → 0.2

Time: 17.6s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r89048 = lambda1;
        double r89049 = lambda2;
        double r89050 = r89048 - r89049;
        double r89051 = sin(r89050);
        double r89052 = phi2;
        double r89053 = cos(r89052);
        double r89054 = r89051 * r89053;
        double r89055 = phi1;
        double r89056 = cos(r89055);
        double r89057 = sin(r89052);
        double r89058 = r89056 * r89057;
        double r89059 = sin(r89055);
        double r89060 = r89059 * r89053;
        double r89061 = cos(r89050);
        double r89062 = r89060 * r89061;
        double r89063 = r89058 - r89062;
        double r89064 = atan2(r89054, r89063);
        return r89064;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r89065 = lambda1;
        double r89066 = sin(r89065);
        double r89067 = lambda2;
        double r89068 = cos(r89067);
        double r89069 = r89066 * r89068;
        double r89070 = cos(r89065);
        double r89071 = -r89067;
        double r89072 = sin(r89071);
        double r89073 = r89070 * r89072;
        double r89074 = r89069 + r89073;
        double r89075 = phi2;
        double r89076 = cos(r89075);
        double r89077 = r89074 * r89076;
        double r89078 = phi1;
        double r89079 = cos(r89078);
        double r89080 = sin(r89075);
        double r89081 = r89079 * r89080;
        double r89082 = sin(r89078);
        double r89083 = r89082 * r89076;
        double r89084 = r89070 * r89068;
        double r89085 = r89083 * r89084;
        double r89086 = 3.0;
        double r89087 = pow(r89083, r89086);
        double r89088 = cbrt(r89087);
        double r89089 = sin(r89067);
        double r89090 = r89066 * r89089;
        double r89091 = r89088 * r89090;
        double r89092 = r89085 + r89091;
        double r89093 = r89081 - r89092;
        double r89094 = atan2(r89077, r89093);
        return r89094;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

1. Initial program 13.1

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

3. Applied sub-neg 13.1

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

4. Applied sin-sum 6.8

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

5. Simplified 6.8

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

7. Applied cos-diff 0.2

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

8. Applied distribute-lft-in 0.2

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

10. Applied add-cbrt-cube 0.2

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

11. Applied add-cbrt-cube 0.2

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

12. Applied cbrt-unprod 0.2

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

13. Simplified 0.2

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

14. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020047 
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Bearing on a great circle"
  :precision binary64
  (atan2 (* (sin (- lambda1 lambda2)) (cos phi2)) (- (* (cos phi1) (sin phi2)) (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))))