Average Error: 16.5 → 3.8
Time: 58.9s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot \log \left(e^{\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right), \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)\]
\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_2 \cdot \cos \phi_1, \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right), \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1469090 = phi1;
        double r1469091 = sin(r1469090);
        double r1469092 = phi2;
        double r1469093 = sin(r1469092);
        double r1469094 = r1469091 * r1469093;
        double r1469095 = cos(r1469090);
        double r1469096 = cos(r1469092);
        double r1469097 = r1469095 * r1469096;
        double r1469098 = lambda1;
        double r1469099 = lambda2;
        double r1469100 = r1469098 - r1469099;
        double r1469101 = cos(r1469100);
        double r1469102 = r1469097 * r1469101;
        double r1469103 = r1469094 + r1469102;
        double r1469104 = acos(r1469103);
        double r1469105 = R;
        double r1469106 = r1469104 * r1469105;
        return r1469106;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1469107 = R;
        double r1469108 = atan2(1.0, 0.0);
        double r1469109 = 2.0;
        double r1469110 = r1469108 / r1469109;
        double r1469111 = phi2;
        double r1469112 = cos(r1469111);
        double r1469113 = phi1;
        double r1469114 = cos(r1469113);
        double r1469115 = r1469112 * r1469114;
        double r1469116 = lambda2;
        double r1469117 = sin(r1469116);
        double r1469118 = lambda1;
        double r1469119 = sin(r1469118);
        double r1469120 = cos(r1469118);
        double r1469121 = cos(r1469116);
        double r1469122 = r1469120 * r1469121;
        double r1469123 = fma(r1469117, r1469119, r1469122);
        double r1469124 = sin(r1469111);
        double r1469125 = sin(r1469113);
        double r1469126 = r1469124 * r1469125;
        double r1469127 = fma(r1469115, r1469123, r1469126);
        double r1469128 = asin(r1469127);
        double r1469129 = r1469110 - r1469128;
        double r1469130 = exp(r1469129);
        double r1469131 = log(r1469130);
        double r1469132 = r1469107 * r1469131;
        return r1469132;
}

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 16.5

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

  3. Applied cos-diff3.7

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

  5. Applied add-log-exp3.8

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

  6. Simplified3.8

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

  8. Applied acos-asin3.8

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

  9. Final simplification3.8

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

Reproduce

herbie shell --seed 2019143 +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))