Average Error: 16.8 → 3.5
Time: 34.6s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[e^{\log \left(\log \left(1 \cdot e^{\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right)}\right)\right)} \cdot R\]
\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
↓
e^{\log \left(\log \left(1 \cdot 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 \left(-\lambda_2\right)\right)\right)}\right)\right)} \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r106 = phi1;
        double r107 = sin(r106);
        double r108 = phi2;
        double r109 = sin(r108);
        double r110 = r107 * r109;
        double r111 = cos(r106);
        double r112 = cos(r108);
        double r113 = r111 * r112;
        double r114 = lambda1;
        double r115 = lambda2;
        double r116 = r114 - r115;
        double r117 = cos(r116);
        double r118 = r113 * r117;
        double r119 = r110 + r118;
        double r120 = acos(r119);
        double r121 = R;
        double r122 = r120 * r121;
        return r122;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r123 = 1.0;
        double r124 = phi1;
        double r125 = sin(r124);
        double r126 = phi2;
        double r127 = sin(r126);
        double r128 = r125 * r127;
        double r129 = cos(r124);
        double r130 = cos(r126);
        double r131 = r129 * r130;
        double r132 = lambda1;
        double r133 = cos(r132);
        double r134 = lambda2;
        double r135 = cos(r134);
        double r136 = r133 * r135;
        double r137 = sin(r132);
        double r138 = -r134;
        double r139 = sin(r138);
        double r140 = r137 * r139;
        double r141 = r136 - r140;
        double r142 = r131 * r141;
        double r143 = r128 + r142;
        double r144 = acos(r143);
        double r145 = exp(r144);
        double r146 = r123 * r145;
        double r147 = log(r146);
        double r148 = log(r147);
        double r149 = exp(r148);
        double r150 = R;
        double r151 = r149 * r150;
        return r151;
}

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 16.8

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

  3. Applied sub-neg16.8

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

  4. Applied cos-sum3.5

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

  5. Simplified3.5

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

  7. Applied add-exp-log3.5

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

  9. Applied add-log-exp3.5

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

  11. Applied *-un-lft-identity3.5

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

  12. Final simplification3.5

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

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  :precision binary64
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))