Equirectangular approximation to distance on a great circle

Average Error: 38.6 → 19.0

Time: 26.2s

Precision: binary64

Cost: 35652

\[ \begin{array}{c}[phi1, phi2] = \mathsf{sort}([phi1, phi2])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_0 := \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\\ t_1 := \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ t_2 := \left(\lambda_1 - \lambda_2\right) \cdot t_1\\ \mathbf{if}\;t_2 \cdot t_2 + t_0 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;R \cdot \sqrt{t_1 \cdot \left(t_1 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + t_0}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

FPCore

(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 (* (- phi1 phi2) (- phi1 phi2)))
        (t_1 (cos (/ (+ phi1 phi2) 2.0)))
        (t_2 (* (- lambda1 lambda2) t_1)))
   (if (<= (+ (* t_2 t_2) t_0) 5e+306)
     (*
      R
      (sqrt
       (+ (* t_1 (* t_1 (* (- lambda1 lambda2) (- lambda1 lambda2)))) t_0)))
     (* R (- phi2 phi1)))))

C

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 = (phi1 - phi2) * (phi1 - phi2);
	double t_1 = cos(((phi1 + phi2) / 2.0));
	double t_2 = (lambda1 - lambda2) * t_1;
	double tmp;
	if (((t_2 * t_2) + t_0) <= 5e+306) {
		tmp = R * sqrt(((t_1 * (t_1 * ((lambda1 - lambda2) * (lambda1 - lambda2)))) + t_0));
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = r * sqrt(((((lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0d0))) * ((lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0d0)))) + ((phi1 - phi2) * (phi1 - phi2))))
end function

↓

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (phi1 - phi2) * (phi1 - phi2)
    t_1 = cos(((phi1 + phi2) / 2.0d0))
    t_2 = (lambda1 - lambda2) * t_1
    if (((t_2 * t_2) + t_0) <= 5d+306) then
        tmp = r * sqrt(((t_1 * (t_1 * ((lambda1 - lambda2) * (lambda1 - lambda2)))) + t_0))
    else
        tmp = r * (phi2 - phi1)
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * Math.sqrt(((((lambda1 - lambda2) * Math.cos(((phi1 + phi2) / 2.0))) * ((lambda1 - lambda2) * Math.cos(((phi1 + phi2) / 2.0)))) + ((phi1 - phi2) * (phi1 - phi2))));
}

↓

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (phi1 - phi2) * (phi1 - phi2);
	double t_1 = Math.cos(((phi1 + phi2) / 2.0));
	double t_2 = (lambda1 - lambda2) * t_1;
	double tmp;
	if (((t_2 * t_2) + t_0) <= 5e+306) {
		tmp = R * Math.sqrt(((t_1 * (t_1 * ((lambda1 - lambda2) * (lambda1 - lambda2)))) + t_0));
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.sqrt(((((lambda1 - lambda2) * math.cos(((phi1 + phi2) / 2.0))) * ((lambda1 - lambda2) * math.cos(((phi1 + phi2) / 2.0)))) + ((phi1 - phi2) * (phi1 - phi2))))

↓

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (phi1 - phi2) * (phi1 - phi2)
	t_1 = math.cos(((phi1 + phi2) / 2.0))
	t_2 = (lambda1 - lambda2) * t_1
	tmp = 0
	if ((t_2 * t_2) + t_0) <= 5e+306:
		tmp = R * math.sqrt(((t_1 * (t_1 * ((lambda1 - lambda2) * (lambda1 - lambda2)))) + t_0))
	else:
		tmp = R * (phi2 - phi1)
	return tmp

Julia

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 = Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2))
	t_1 = cos(Float64(Float64(phi1 + phi2) / 2.0))
	t_2 = Float64(Float64(lambda1 - lambda2) * t_1)
	tmp = 0.0
	if (Float64(Float64(t_2 * t_2) + t_0) <= 5e+306)
		tmp = Float64(R * sqrt(Float64(Float64(t_1 * Float64(t_1 * Float64(Float64(lambda1 - lambda2) * Float64(lambda1 - lambda2)))) + t_0)));
	else
		tmp = Float64(R * Float64(phi2 - phi1));
	end
	return tmp
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * sqrt(((((lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0))) * ((lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0)))) + ((phi1 - phi2) * (phi1 - phi2))));
end

↓

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (phi1 - phi2) * (phi1 - phi2);
	t_1 = cos(((phi1 + phi2) / 2.0));
	t_2 = (lambda1 - lambda2) * t_1;
	tmp = 0.0;
	if (((t_2 * t_2) + t_0) <= 5e+306)
		tmp = R * sqrt(((t_1 * (t_1 * ((lambda1 - lambda2) * (lambda1 - lambda2)))) + t_0));
	else
		tmp = R * (phi2 - phi1);
	end
	tmp_2 = tmp;
end

Wolfram

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[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(lambda1 - lambda2), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$2 * t$95$2), $MachinePrecision] + t$95$0), $MachinePrecision], 5e+306], N[(R * N[Sqrt[N[(N[(t$95$1 * N[(t$95$1 * N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) 2))) (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) 2)))) (*.f64 (-.f64 phi1 phi2) (-.f64 phi1 phi2))) < 4.99999999999999993e306

   1. Initial program 1.8

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

   2. Simplified 1.8

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

      [Start] 1.8	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational.json-simplify-2 [=>] 1.8	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \color{blue}{\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational.json-simplify-43 [=>] 1.8	\[ R \cdot \sqrt{\color{blue}{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational.json-simplify-43 [<=] 1.8	\[ R \cdot \sqrt{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \color{blue}{\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   if 4.99999999999999993e306 < (+.f64 (*.f64 (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) 2))) (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) 2)))) (*.f64 (-.f64 phi1 phi2) (-.f64 phi1 phi2)))

   1. Initial program 63.8

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

   2. Simplified 63.8

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

      [Start] 63.8	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational.json-simplify-2 [=>] 63.8	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \color{blue}{\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational.json-simplify-43 [=>] 63.8	\[ R \cdot \sqrt{\color{blue}{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      rational.json-simplify-43 [<=] 63.8	\[ R \cdot \sqrt{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \color{blue}{\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   3. Taylor expanded in phi1 around -inf 30.7

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \phi_1 + \phi_2\right)} \]

   4. Simplified 30.7

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 + \left(-\phi_1\right)\right)} \]
      Proof

      [Start] 30.7	\[ R \cdot \left(-1 \cdot \phi_1 + \phi_2\right) \]
      rational.json-simplify-1 [=>] 30.7	\[ R \cdot \color{blue}{\left(\phi_2 + -1 \cdot \phi_1\right)} \]
      rational.json-simplify-2 [=>] 30.7	\[ R \cdot \left(\phi_2 + \color{blue}{\phi_1 \cdot -1}\right) \]
      rational.json-simplify-9 [=>] 30.7	\[ R \cdot \left(\phi_2 + \color{blue}{\left(-\phi_1\right)}\right) \]

   5. Taylor expanded in R around 0 30.7

      \[\leadsto \color{blue}{R \cdot \left(\phi_2 - \phi_1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) \leq 5 \cdot 10^{+306}:\\ \;\;\;\;R \cdot \sqrt{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	28.0
Cost	22620

\[\begin{array}{l} t_0 := R \cdot \left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \left(\lambda_2 + \left(-\lambda_1\right)\right)\right)\\ t_1 := \left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\\ t_2 := \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\\ t_3 := R \cdot \left(\phi_2 - \phi_1\right)\\ t_4 := \cos \left(\phi_2 \cdot 0.5\right)\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -2 \cdot 10^{+169}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{+148}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -2 \cdot 10^{+139}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{+122}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{+111}:\\ \;\;\;\;R \cdot \sqrt{\frac{{t_4}^{2}}{\frac{1}{t_1}} + t_2}\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -2 \cdot 10^{+64}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -400000000000:\\ \;\;\;\;R \cdot \sqrt{t_4 \cdot \left(t_4 \cdot t_1\right) + t_2}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(-\phi_1\right) + R \cdot \phi_2\\ \end{array} \]

Alternative 2
Error	28.0
Cost	22556

\[\begin{array}{l} t_0 := R \cdot \sqrt{\frac{{\cos \left(\phi_2 \cdot 0.5\right)}^{2}}{\frac{1}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ t_2 := R \cdot \left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \left(\lambda_2 + \left(-\lambda_1\right)\right)\right)\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -2 \cdot 10^{+169}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{+148}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -2 \cdot 10^{+139}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{+122}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{+111}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -2 \cdot 10^{+64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -400000000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(-\phi_1\right) + R \cdot \phi_2\\ \end{array} \]

Alternative 3
Error	28.0
Cost	14996

\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 - \phi_1\right)\\ t_1 := R \cdot \left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \left(\lambda_2 + \left(-\lambda_1\right)\right)\right)\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -2 \cdot 10^{+169}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{+148}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -2 \cdot 10^{+139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -2 \cdot 10^{+64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -400000000000:\\ \;\;\;\;R \cdot \sqrt{{\left(\lambda_1 - \lambda_2\right)}^{2} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(-\phi_1\right) + R \cdot \phi_2\\ \end{array} \]

Alternative 4
Error	30.9
Cost	7696

\[\begin{array}{l} t_0 := R \cdot \left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \left(\lambda_2 + \left(-\lambda_1\right)\right)\right)\\ t_1 := R \cdot \left(-\phi_1\right) + R \cdot \phi_2\\ \mathbf{if}\;\phi_1 \leq -51000000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -1 \cdot 10^{-297}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 6 \cdot 10^{-188}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq 1.85 \cdot 10^{-144}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	31.6
Cost	7568

\[\begin{array}{l} t_0 := \cos \left(\phi_1 \cdot -0.5\right) \cdot \left(R \cdot \left(\left(-\lambda_1\right) + \lambda_2\right)\right)\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_2 \leq -2 \cdot 10^{-38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq -1.4 \cdot 10^{-80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq -3.15 \cdot 10^{-118}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 1.05 \cdot 10^{-16}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(-\phi_1\right) + R \cdot \phi_2\\ \end{array} \]

Alternative 6
Error	29.6
Cost	7308

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.2 \cdot 10^{+169}:\\ \;\;\;\;\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 \cdot \left(-R\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq -3.8 \cdot 10^{+63}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_1 \leq -9.4 \cdot 10^{+58}:\\ \;\;\;\;\cos \left(\phi_1 \cdot -0.5\right) \cdot \left(R \cdot \left(-\lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(-\phi_1\right) + R \cdot \phi_2\\ \end{array} \]

Alternative 7
Error	29.4
Cost	7308

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -7.2 \cdot 10^{+168}:\\ \;\;\;\;R \cdot \left(\lambda_1 \cdot \left(-\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq -3.8 \cdot 10^{+63}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_1 \leq -9.7 \cdot 10^{+58}:\\ \;\;\;\;\cos \left(\phi_1 \cdot -0.5\right) \cdot \left(R \cdot \left(-\lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(-\phi_1\right) + R \cdot \phi_2\\ \end{array} \]

Alternative 8
Error	29.4
Cost	7308

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -5.6 \cdot 10^{+168}:\\ \;\;\;\;\lambda_1 \cdot \left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot -0.5\right) \cdot \left(-R\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq -3.8 \cdot 10^{+63}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_1 \leq -9.7 \cdot 10^{+58}:\\ \;\;\;\;\cos \left(\phi_1 \cdot -0.5\right) \cdot \left(R \cdot \left(-\lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(-\phi_1\right) + R \cdot \phi_2\\ \end{array} \]

Alternative 9
Error	29.6
Cost	7044

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -9.6 \cdot 10^{+168}:\\ \;\;\;\;R \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(-\lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(-\phi_1\right) + R \cdot \phi_2\\ \end{array} \]

Alternative 10
Error	29.6
Cost	7044

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.1 \cdot 10^{+169}:\\ \;\;\;\;\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 \cdot \left(-R\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(-\phi_1\right) + R \cdot \phi_2\\ \end{array} \]

Alternative 11
Error	29.3
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 7 \cdot 10^{+130}:\\ \;\;\;\;R \cdot \left(-\phi_1\right) + R \cdot \phi_2\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot R\right)\\ \end{array} \]

Alternative 12
Error	29.3
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 7.6 \cdot 10^{+130}:\\ \;\;\;\;R \cdot \left(-\phi_1\right) + R \cdot \phi_2\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\phi_1 \cdot -0.5\right) \cdot \left(\lambda_2 \cdot R\right)\\ \end{array} \]

Alternative 13
Error	35.1
Cost	652

\[\begin{array}{l} t_0 := R \cdot \left(-\phi_1\right)\\ \mathbf{if}\;\phi_2 \leq 5.9 \cdot 10^{-147}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 1.9 \cdot 10^{-104}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\phi_2 \leq 8.5 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 14
Error	29.6
Cost	644

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.65 \cdot 10^{+169}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(-\phi_1\right) + R \cdot \phi_2\\ \end{array} \]

Alternative 15
Error	29.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.2 \cdot 10^{+169}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternative 16
Error	43.7
Cost	388

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.1 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 17
Error	46.2
Cost	192

\[R \cdot \phi_2 \]

Reproduce

herbie shell --seed 2023065 
(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))))))