Average Error: 39.2 → 0.1
Time: 15.8s
Precision: binary64
\[ \begin{array}{c}[lambda1, lambda2] = \mathsf{sort}([lambda1, lambda2])\\ [phi1, phi2] = \mathsf{sort}([phi1, phi2])\\ \end{array} \]
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
↓
\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \phi_2\right)\\ t_1 := \sin \left(0.5 \cdot \phi_1\right)\\ R \cdot \mathsf{hypot}\left(\mathsf{fma}\left(t_0 \cdot t_1, \lambda_2, \cos \left(0.5 \cdot \phi_1\right) \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) - t_0 \cdot \left(t_1 \cdot \lambda_1\right)\right), \phi_1 - \phi_2\right) \end{array} \]
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (sqrt
   (+
    (*
     (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))
     (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))))
    (* (- phi1 phi2) (- phi1 phi2))))))
↓
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 phi2))) (t_1 (sin (* 0.5 phi1))))
   (*
    R
    (hypot
     (fma
      (* t_0 t_1)
      lambda2
      (-
       (* (cos (* 0.5 phi1)) (* (cos (* 0.5 phi2)) (- lambda1 lambda2)))
       (* t_0 (* t_1 lambda1))))
     (- phi1 phi2)))))
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * sqrt(((((lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0))) * ((lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0)))) + ((phi1 - phi2) * (phi1 - phi2))));
}
↓
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * phi2));
	double t_1 = sin((0.5 * phi1));
	return R * hypot(fma((t_0 * t_1), lambda2, ((cos((0.5 * phi1)) * (cos((0.5 * phi2)) * (lambda1 - lambda2))) - (t_0 * (t_1 * lambda1)))), (phi1 - phi2));
}

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * sqrt(Float64(Float64(Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0))) * Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0)))) + Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2)))))
end
↓
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * phi2))
	t_1 = sin(Float64(0.5 * phi1))
	return Float64(R * hypot(fma(Float64(t_0 * t_1), lambda2, Float64(Float64(cos(Float64(0.5 * phi1)) * Float64(cos(Float64(0.5 * phi2)) * Float64(lambda1 - lambda2))) - Float64(t_0 * Float64(t_1 * lambda1)))), Float64(phi1 - phi2)))
end

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Sqrt[N[(N[(N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]
↓
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, N[(R * N[Sqrt[N[(N[(t$95$0 * t$95$1), $MachinePrecision] * lambda2 + N[(N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[(t$95$1 * lambda1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]]

R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}
↓
\begin{array}{l}
t_0 := \sin \left(0.5 \cdot \phi_2\right)\\
t_1 := \sin \left(0.5 \cdot \phi_1\right)\\
R \cdot \mathsf{hypot}\left(\mathsf{fma}\left(t_0 \cdot t_1, \lambda_2, \cos \left(0.5 \cdot \phi_1\right) \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) - t_0 \cdot \left(t_1 \cdot \lambda_1\right)\right), \phi_1 - \phi_2\right)
\end{array}

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 39.2

    \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

  2. Simplified3.5

    \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

  3. Applied egg-rr3.5

    \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sqrt[3]{{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)}^{3}}}, \phi_1 - \phi_2\right) \]

  4. Applied egg-rr3.6

    \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt[3]{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)\right)\right)\right)}}^{3}}, \phi_1 - \phi_2\right) \]

  5. Applied egg-rr0.1

    \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}, \phi_1 - \phi_2\right) \]

  6. Taylor expanded in lambda1 around 0 0.1

    \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(\lambda_2 \cdot \left(\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right) + \cos \left(0.5 \cdot \phi_1\right) \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_1\right)\right) - \left(\sin \left(0.5 \cdot \phi_1\right) \cdot \left(\sin \left(0.5 \cdot \phi_2\right) \cdot \lambda_1\right) + \cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}, \phi_1 - \phi_2\right) \]

  7. Simplified0.1

    \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(\sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right), \lambda_2, \cos \left(0.5 \cdot \phi_1\right) \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \phi_1\right) \cdot \lambda_1\right)\right)}, \phi_1 - \phi_2\right) \]

  8. Final simplification0.1

    \[\leadsto R \cdot \mathsf{hypot}\left(\mathsf{fma}\left(\sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right), \lambda_2, \cos \left(0.5 \cdot \phi_1\right) \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \phi_1\right) \cdot \lambda_1\right)\right), \phi_1 - \phi_2\right) \]

Reproduce

herbie shell --seed 2022155 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Equirectangular approximation to distance on a great circle"
  :precision binary64
  (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2))))))