Average Error: 16.9 → 4.1
Time: 51.1s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot \log \left(e^{\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right), \cos \phi_2, \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)\]
\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R
↓
R \cdot \log \left(e^{\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right), \cos \phi_2, \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1381061 = phi1;
        double r1381062 = sin(r1381061);
        double r1381063 = phi2;
        double r1381064 = sin(r1381063);
        double r1381065 = r1381062 * r1381064;
        double r1381066 = cos(r1381061);
        double r1381067 = cos(r1381063);
        double r1381068 = r1381066 * r1381067;
        double r1381069 = lambda1;
        double r1381070 = lambda2;
        double r1381071 = r1381069 - r1381070;
        double r1381072 = cos(r1381071);
        double r1381073 = r1381068 * r1381072;
        double r1381074 = r1381065 + r1381073;
        double r1381075 = acos(r1381074);
        double r1381076 = R;
        double r1381077 = r1381075 * r1381076;
        return r1381077;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1381078 = R;
        double r1381079 = atan2(1.0, 0.0);
        double r1381080 = 2.0;
        double r1381081 = r1381079 / r1381080;
        double r1381082 = phi1;
        double r1381083 = cos(r1381082);
        double r1381084 = lambda1;
        double r1381085 = sin(r1381084);
        double r1381086 = lambda2;
        double r1381087 = sin(r1381086);
        double r1381088 = cos(r1381086);
        double r1381089 = cos(r1381084);
        double r1381090 = r1381088 * r1381089;
        double r1381091 = fma(r1381085, r1381087, r1381090);
        double r1381092 = r1381083 * r1381091;
        double r1381093 = phi2;
        double r1381094 = cos(r1381093);
        double r1381095 = sin(r1381093);
        double r1381096 = sin(r1381082);
        double r1381097 = r1381095 * r1381096;
        double r1381098 = fma(r1381092, r1381094, r1381097);
        double r1381099 = asin(r1381098);
        double r1381100 = r1381081 - r1381099;
        double r1381101 = exp(r1381100);
        double r1381102 = log(r1381101);
        double r1381103 = r1381078 * r1381102;
        return r1381103;
}

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 16.9

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

  3. Applied cos-diff4.0

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

  5. Applied add-log-exp4.0

    \[\leadsto \color{blue}{\log \left(e^{\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right)} \cdot R\]

  6. Simplified4.0

    \[\leadsto \log \color{blue}{\left(e^{\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right), \cos \phi_2, \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)} \cdot R\]
  7. Using strategy rm

  8. Applied acos-asin4.1

    \[\leadsto \log \left(e^{\color{blue}{\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right), \cos \phi_2, \sin \phi_2 \cdot \sin \phi_1\right)\right)}}\right) \cdot R\]

  9. Final simplification4.1

    \[\leadsto R \cdot \log \left(e^{\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right), \cos \phi_2, \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))