Average Error: 16.9 → 3.9
Time: 25.4s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[e^{\log \left(\log \left(e^{\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2\right) \cdot \cos \phi_1\right)}\right)\right)} \cdot R\]
\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(e^{\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2\right) \cdot \cos \phi_1\right)}\right)\right)} \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r25076 = phi1;
        double r25077 = sin(r25076);
        double r25078 = phi2;
        double r25079 = sin(r25078);
        double r25080 = r25077 * r25079;
        double r25081 = cos(r25076);
        double r25082 = cos(r25078);
        double r25083 = r25081 * r25082;
        double r25084 = lambda1;
        double r25085 = lambda2;
        double r25086 = r25084 - r25085;
        double r25087 = cos(r25086);
        double r25088 = r25083 * r25087;
        double r25089 = r25080 + r25088;
        double r25090 = acos(r25089);
        double r25091 = R;
        double r25092 = r25090 * r25091;
        return r25092;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r25093 = phi1;
        double r25094 = sin(r25093);
        double r25095 = phi2;
        double r25096 = sin(r25095);
        double r25097 = r25094 * r25096;
        double r25098 = lambda1;
        double r25099 = sin(r25098);
        double r25100 = lambda2;
        double r25101 = sin(r25100);
        double r25102 = r25099 * r25101;
        double r25103 = cos(r25098);
        double r25104 = cos(r25100);
        double r25105 = r25103 * r25104;
        double r25106 = r25102 + r25105;
        double r25107 = cos(r25095);
        double r25108 = r25106 * r25107;
        double r25109 = cos(r25093);
        double r25110 = r25108 * r25109;
        double r25111 = r25097 + r25110;
        double r25112 = acos(r25111);
        double r25113 = exp(r25112);
        double r25114 = log(r25113);
        double r25115 = log(r25114);
        double r25116 = exp(r25115);
        double r25117 = R;
        double r25118 = r25116 * r25117;
        return r25118;
}

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 16.9

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

  3. Applied sub-neg16.9

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

  4. Applied cos-sum3.9

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

  5. Simplified3.9

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

  7. Applied add-exp-log3.9

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

  8. Simplified3.9

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

  10. Applied add-log-exp3.9

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

  11. Final simplification3.9

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

Reproduce

herbie shell --seed 2020045 
(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))