Average Error: 16.4 → 3.8
Time: 50.0s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot \cos^{-1} \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right), \sin \phi_2 \cdot \sin \phi_1\right)\right)\right)\right)\]
\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 \cos^{-1} \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right), \sin \phi_2 \cdot \sin \phi_1\right)\right)\right)\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1313553 = phi1;
        double r1313554 = sin(r1313553);
        double r1313555 = phi2;
        double r1313556 = sin(r1313555);
        double r1313557 = r1313554 * r1313556;
        double r1313558 = cos(r1313553);
        double r1313559 = cos(r1313555);
        double r1313560 = r1313558 * r1313559;
        double r1313561 = lambda1;
        double r1313562 = lambda2;
        double r1313563 = r1313561 - r1313562;
        double r1313564 = cos(r1313563);
        double r1313565 = r1313560 * r1313564;
        double r1313566 = r1313557 + r1313565;
        double r1313567 = acos(r1313566);
        double r1313568 = R;
        double r1313569 = r1313567 * r1313568;
        return r1313569;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1313570 = R;
        double r1313571 = phi2;
        double r1313572 = cos(r1313571);
        double r1313573 = phi1;
        double r1313574 = cos(r1313573);
        double r1313575 = r1313572 * r1313574;
        double r1313576 = lambda2;
        double r1313577 = sin(r1313576);
        double r1313578 = lambda1;
        double r1313579 = sin(r1313578);
        double r1313580 = cos(r1313578);
        double r1313581 = cos(r1313576);
        double r1313582 = r1313580 * r1313581;
        double r1313583 = fma(r1313577, r1313579, r1313582);
        double r1313584 = sin(r1313571);
        double r1313585 = sin(r1313573);
        double r1313586 = r1313584 * r1313585;
        double r1313587 = fma(r1313575, r1313583, r1313586);
        double r1313588 = log1p(r1313587);
        double r1313589 = expm1(r1313588);
        double r1313590 = acos(r1313589);
        double r1313591 = r1313570 * r1313590;
        return r1313591;
}

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 16.4

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

  2. Simplified16.4

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

  4. Applied cos-diff3.7

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

  6. Applied expm1-log1p-u3.8

    \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2, \sin \phi_2 \cdot \sin \phi_1\right)\right)\right)\right)}\]

  7. Simplified3.8

    \[\leadsto R \cdot \cos^{-1} \left(\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\right)\]

  8. Final simplification3.8

    \[\leadsto R \cdot \cos^{-1} \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right), \sin \phi_2 \cdot \sin \phi_1\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2019163 +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))