Average Error: 16.9 → 3.7
Time: 1.0m
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot \log \left(e^{\cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \log \left(e^{\sin \lambda_2 \cdot \sin \lambda_1}\right)\right) + \sin \phi_2 \cdot \sin \phi_1\right)}\right)\]
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1308110 = phi1;
        double r1308111 = sin(r1308110);
        double r1308112 = phi2;
        double r1308113 = sin(r1308112);
        double r1308114 = r1308111 * r1308113;
        double r1308115 = cos(r1308110);
        double r1308116 = cos(r1308112);
        double r1308117 = r1308115 * r1308116;
        double r1308118 = lambda1;
        double r1308119 = lambda2;
        double r1308120 = r1308118 - r1308119;
        double r1308121 = cos(r1308120);
        double r1308122 = r1308117 * r1308121;
        double r1308123 = r1308114 + r1308122;
        double r1308124 = acos(r1308123);
        double r1308125 = R;
        double r1308126 = r1308124 * r1308125;
        return r1308126;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1308127 = R;
        double r1308128 = phi1;
        double r1308129 = cos(r1308128);
        double r1308130 = phi2;
        double r1308131 = cos(r1308130);
        double r1308132 = r1308129 * r1308131;
        double r1308133 = lambda2;
        double r1308134 = cos(r1308133);
        double r1308135 = lambda1;
        double r1308136 = cos(r1308135);
        double r1308137 = r1308134 * r1308136;
        double r1308138 = sin(r1308133);
        double r1308139 = sin(r1308135);
        double r1308140 = r1308138 * r1308139;
        double r1308141 = exp(r1308140);
        double r1308142 = log(r1308141);
        double r1308143 = r1308137 + r1308142;
        double r1308144 = r1308132 * r1308143;
        double r1308145 = sin(r1308130);
        double r1308146 = sin(r1308128);
        double r1308147 = r1308145 * r1308146;
        double r1308148 = r1308144 + r1308147;
        double r1308149 = acos(r1308148);
        double r1308150 = exp(r1308149);
        double r1308151 = log(r1308150);
        double r1308152 = r1308127 * r1308151;
        return r1308152;
}


\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^{\cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \log \left(e^{\sin \lambda_2 \cdot \sin \lambda_1}\right)\right) + \sin \phi_2 \cdot \sin \phi_1\right)}\right)

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-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-exp3.7

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

  7. Applied add-log-exp3.7

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

  8. Final simplification3.7

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

Reproduce

herbie shell --seed 2019102 +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))