Average Error: 16.6 → 3.8
Time: 1.1m
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot e^{\log_* (1 + (e^{\log \left(\cos^{-1} \left((\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\log \left(e^{\sin \lambda_1 \cdot \sin \lambda_2}\right) + \cos \lambda_2 \cdot \cos \lambda_1\right) + \left(\sin \phi_1 \cdot \sin \phi_2\right))_*\right)\right)} - 1)^*)}\]
\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 e^{\log_* (1 + (e^{\log \left(\cos^{-1} \left((\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\log \left(e^{\sin \lambda_1 \cdot \sin \lambda_2}\right) + \cos \lambda_2 \cdot \cos \lambda_1\right) + \left(\sin \phi_1 \cdot \sin \phi_2\right))_*\right)\right)} - 1)^*)}
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1630565 = phi1;
        double r1630566 = sin(r1630565);
        double r1630567 = phi2;
        double r1630568 = sin(r1630567);
        double r1630569 = r1630566 * r1630568;
        double r1630570 = cos(r1630565);
        double r1630571 = cos(r1630567);
        double r1630572 = r1630570 * r1630571;
        double r1630573 = lambda1;
        double r1630574 = lambda2;
        double r1630575 = r1630573 - r1630574;
        double r1630576 = cos(r1630575);
        double r1630577 = r1630572 * r1630576;
        double r1630578 = r1630569 + r1630577;
        double r1630579 = acos(r1630578);
        double r1630580 = R;
        double r1630581 = r1630579 * r1630580;
        return r1630581;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1630582 = R;
        double r1630583 = phi2;
        double r1630584 = cos(r1630583);
        double r1630585 = phi1;
        double r1630586 = cos(r1630585);
        double r1630587 = r1630584 * r1630586;
        double r1630588 = lambda1;
        double r1630589 = sin(r1630588);
        double r1630590 = lambda2;
        double r1630591 = sin(r1630590);
        double r1630592 = r1630589 * r1630591;
        double r1630593 = exp(r1630592);
        double r1630594 = log(r1630593);
        double r1630595 = cos(r1630590);
        double r1630596 = cos(r1630588);
        double r1630597 = r1630595 * r1630596;
        double r1630598 = r1630594 + r1630597;
        double r1630599 = sin(r1630585);
        double r1630600 = sin(r1630583);
        double r1630601 = r1630599 * r1630600;
        double r1630602 = fma(r1630587, r1630598, r1630601);
        double r1630603 = acos(r1630602);
        double r1630604 = log(r1630603);
        double r1630605 = expm1(r1630604);
        double r1630606 = log1p(r1630605);
        double r1630607 = exp(r1630606);
        double r1630608 = r1630582 * r1630607;
        return r1630608;
}

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 16.6

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

  2. Simplified16.6

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

  4. Applied cos-diff3.8

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

  6. Applied add-log-exp3.8

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

  8. Applied add-exp-log3.8

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

  10. Applied log1p-expm1-u3.8

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

  11. Final simplification3.8

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

Reproduce

herbie shell --seed 2019104 +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))