Average Error: 16.7 → 3.8
Time: 40.7s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[\cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \log \left(e^{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R\]
\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(\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) + \log \left(e^{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r424572 = phi1;
        double r424573 = sin(r424572);
        double r424574 = phi2;
        double r424575 = sin(r424574);
        double r424576 = r424573 * r424575;
        double r424577 = cos(r424572);
        double r424578 = cos(r424574);
        double r424579 = r424577 * r424578;
        double r424580 = lambda1;
        double r424581 = lambda2;
        double r424582 = r424580 - r424581;
        double r424583 = cos(r424582);
        double r424584 = r424579 * r424583;
        double r424585 = r424576 + r424584;
        double r424586 = acos(r424585);
        double r424587 = R;
        double r424588 = r424586 * r424587;
        return r424588;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r424589 = phi1;
        double r424590 = cos(r424589);
        double r424591 = phi2;
        double r424592 = cos(r424591);
        double r424593 = r424590 * r424592;
        double r424594 = lambda1;
        double r424595 = cos(r424594);
        double r424596 = lambda2;
        double r424597 = cos(r424596);
        double r424598 = r424595 * r424597;
        double r424599 = sin(r424594);
        double r424600 = sin(r424596);
        double r424601 = r424599 * r424600;
        double r424602 = r424598 + r424601;
        double r424603 = r424593 * r424602;
        double r424604 = sin(r424591);
        double r424605 = sin(r424589);
        double r424606 = r424604 * r424605;
        double r424607 = exp(r424606);
        double r424608 = log(r424607);
        double r424609 = r424603 + r424608;
        double r424610 = acos(r424609);
        double r424611 = R;
        double r424612 = r424610 * r424611;
        return r424612;
}

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 16.7

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

  3. Applied cos-diff3.8

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

  5. Applied add-log-exp3.8

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

  6. Final simplification3.8

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

Reproduce

herbie shell --seed 2019156 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))