Average Error: 16.7 → 3.9
Time: 1.2m
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot e^{\log_* (1 + (e^{\log \left(\cos^{-1} \left((\left(\sin \phi_1\right) \cdot \left(\sin \phi_2\right) + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot (\left(\cos \lambda_2\right) \cdot \left(\cos \lambda_1\right) + \left(\sin \lambda_1 \cdot \sin \lambda_2\right))_*\right))_*\right)\right)} - 1)^*)}\]
\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 e^{\log_* (1 + (e^{\log \left(\cos^{-1} \left((\left(\sin \phi_1\right) \cdot \left(\sin \phi_2\right) + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot (\left(\cos \lambda_2\right) \cdot \left(\cos \lambda_1\right) + \left(\sin \lambda_1 \cdot \sin \lambda_2\right))_*\right))_*\right)\right)} - 1)^*)}
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1237050 = phi1;
        double r1237051 = sin(r1237050);
        double r1237052 = phi2;
        double r1237053 = sin(r1237052);
        double r1237054 = r1237051 * r1237053;
        double r1237055 = cos(r1237050);
        double r1237056 = cos(r1237052);
        double r1237057 = r1237055 * r1237056;
        double r1237058 = lambda1;
        double r1237059 = lambda2;
        double r1237060 = r1237058 - r1237059;
        double r1237061 = cos(r1237060);
        double r1237062 = r1237057 * r1237061;
        double r1237063 = r1237054 + r1237062;
        double r1237064 = acos(r1237063);
        double r1237065 = R;
        double r1237066 = r1237064 * r1237065;
        return r1237066;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1237067 = R;
        double r1237068 = phi1;
        double r1237069 = sin(r1237068);
        double r1237070 = phi2;
        double r1237071 = sin(r1237070);
        double r1237072 = cos(r1237068);
        double r1237073 = cos(r1237070);
        double r1237074 = r1237072 * r1237073;
        double r1237075 = lambda2;
        double r1237076 = cos(r1237075);
        double r1237077 = lambda1;
        double r1237078 = cos(r1237077);
        double r1237079 = sin(r1237077);
        double r1237080 = sin(r1237075);
        double r1237081 = r1237079 * r1237080;
        double r1237082 = fma(r1237076, r1237078, r1237081);
        double r1237083 = r1237074 * r1237082;
        double r1237084 = fma(r1237069, r1237071, r1237083);
        double r1237085 = acos(r1237084);
        double r1237086 = log(r1237085);
        double r1237087 = expm1(r1237086);
        double r1237088 = log1p(r1237087);
        double r1237089 = exp(r1237088);
        double r1237090 = r1237067 * r1237089;
        return r1237090;
}

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 16.7

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

  2. Simplified16.7

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

  4. Applied cos-diff3.9

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

  5. Taylor expanded around inf 3.9

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

  6. Simplified3.9

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

  8. Applied add-exp-log3.9

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

  10. Applied log1p-expm1-u3.9

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

  11. Final simplification3.9

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

Reproduce

herbie shell --seed 2019112 +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))