Average Error: 17.2 → 3.9
Time: 46.1s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \sqrt[3]{{\left(\sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}^{3}}\right)\right) \cdot R\]
\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
↓
\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 - \sqrt[3]{{\left(\sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}^{3}}\right)\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r24092 = phi1;
        double r24093 = sin(r24092);
        double r24094 = phi2;
        double r24095 = sin(r24094);
        double r24096 = r24093 * r24095;
        double r24097 = cos(r24092);
        double r24098 = cos(r24094);
        double r24099 = r24097 * r24098;
        double r24100 = lambda1;
        double r24101 = lambda2;
        double r24102 = r24100 - r24101;
        double r24103 = cos(r24102);
        double r24104 = r24099 * r24103;
        double r24105 = r24096 + r24104;
        double r24106 = acos(r24105);
        double r24107 = R;
        double r24108 = r24106 * r24107;
        return r24108;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r24109 = phi1;
        double r24110 = sin(r24109);
        double r24111 = phi2;
        double r24112 = sin(r24111);
        double r24113 = r24110 * r24112;
        double r24114 = cos(r24109);
        double r24115 = cos(r24111);
        double r24116 = r24114 * r24115;
        double r24117 = lambda1;
        double r24118 = cos(r24117);
        double r24119 = lambda2;
        double r24120 = cos(r24119);
        double r24121 = r24118 * r24120;
        double r24122 = sin(r24117);
        double r24123 = -r24119;
        double r24124 = sin(r24123);
        double r24125 = r24122 * r24124;
        double r24126 = 3.0;
        double r24127 = pow(r24125, r24126);
        double r24128 = cbrt(r24127);
        double r24129 = r24121 - r24128;
        double r24130 = r24116 * r24129;
        double r24131 = r24113 + r24130;
        double r24132 = acos(r24131);
        double r24133 = R;
        double r24134 = r24132 * r24133;
        return r24134;
}

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 17.2

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

  3. Applied sub-neg17.2

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

  4. Applied cos-sum3.8

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

  5. Simplified3.8

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

  7. Applied add-cbrt-cube3.9

    \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_1 \cdot \color{blue}{\sqrt[3]{\left(\sin \left(-\lambda_2\right) \cdot \sin \left(-\lambda_2\right)\right) \cdot \sin \left(-\lambda_2\right)}}\right)\right) \cdot R\]

  8. Applied add-cbrt-cube3.9

    \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \color{blue}{\sqrt[3]{\left(\sin \lambda_1 \cdot \sin \lambda_1\right) \cdot \sin \lambda_1}} \cdot \sqrt[3]{\left(\sin \left(-\lambda_2\right) \cdot \sin \left(-\lambda_2\right)\right) \cdot \sin \left(-\lambda_2\right)}\right)\right) \cdot R\]

  9. Applied cbrt-unprod3.9

    \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \color{blue}{\sqrt[3]{\left(\left(\sin \lambda_1 \cdot \sin \lambda_1\right) \cdot \sin \lambda_1\right) \cdot \left(\left(\sin \left(-\lambda_2\right) \cdot \sin \left(-\lambda_2\right)\right) \cdot \sin \left(-\lambda_2\right)\right)}}\right)\right) \cdot R\]

  10. Simplified3.9

    \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \sqrt[3]{\color{blue}{{\left(\sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}^{3}}}\right)\right) \cdot R\]

  11. Final simplification3.9

    \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \sqrt[3]{{\left(\sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}^{3}}\right)\right) \cdot R\]

Reproduce

herbie shell --seed 2019306 +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))