Equirectangular approximation to distance on a great circle

Percentage Accurate: 60.4% → 90.6%

Time: 6.2s

Alternatives: 9

Speedup: 18.6×

• 31.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ R \cdot \sqrt{t\_0 \cdot t\_0 + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))))
   (* R (sqrt (+ (* t_0 t_0) (* (- phi1 phi2) (- phi1 phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	return R * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
}

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
    real(8) :: t_0
    t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0d0))
    code = r * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * Math.cos(((phi1 + phi2) / 2.0));
	return R * Math.sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (lambda1 - lambda2) * math.cos(((phi1 + phi2) / 2.0))
	return R * math.sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0)))
	return Float64(R * sqrt(Float64(Float64(t_0 * t_0) + Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2)))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	tmp = R * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
end

Wolfram

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

Initial Program: 60.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ R \cdot \sqrt{t\_0 \cdot t\_0 + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))))
   (* R (sqrt (+ (* t_0 t_0) (* (- phi1 phi2) (- phi1 phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	return R * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
}

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
    real(8) :: t_0
    t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0d0))
    code = r * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * Math.cos(((phi1 + phi2) / 2.0));
	return R * Math.sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (lambda1 - lambda2) * math.cos(((phi1 + phi2) / 2.0))
	return R * math.sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0)))
	return Float64(R * sqrt(Float64(Float64(t_0 * t_0) + Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2)))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	tmp = R * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
end

Wolfram

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

Alternative 1: 90.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.0012:\\ \;\;\;\;\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 0.0012)
   (* (hypot (* (- lambda1 lambda2) (cos (* 0.5 phi1))) phi1) R)
   (* (hypot (* (cos (* 0.5 phi2)) (- lambda1 lambda2)) phi2) R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 0.0012) {
		tmp = hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), phi1) * R;
	} else {
		tmp = hypot((cos((0.5 * phi2)) * (lambda1 - lambda2)), phi2) * R;
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 0.0012) {
		tmp = Math.hypot(((lambda1 - lambda2) * Math.cos((0.5 * phi1))), phi1) * R;
	} else {
		tmp = Math.hypot((Math.cos((0.5 * phi2)) * (lambda1 - lambda2)), phi2) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 0.0012:
		tmp = math.hypot(((lambda1 - lambda2) * math.cos((0.5 * phi1))), phi1) * R
	else:
		tmp = math.hypot((math.cos((0.5 * phi2)) * (lambda1 - lambda2)), phi2) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 0.0012)
		tmp = Float64(hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi1))), phi1) * R);
	else
		tmp = Float64(hypot(Float64(cos(Float64(0.5 * phi2)) * Float64(lambda1 - lambda2)), phi2) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 0.0012)
		tmp = hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), phi1) * R;
	else
		tmp = hypot((cos((0.5 * phi2)) * (lambda1 - lambda2)), phi2) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 0.0012], N[(N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 0.00119999999999999989

   1. Initial program 62.1%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}} \]

      2. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + {\phi_1}^{2}} \]

      3. unswap-sqr N/A

         \[\leadsto R \cdot \sqrt{\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + {\phi_1}^{2}} \]

      4. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \color{blue}{\phi_1 \cdot \phi_1}} \]

      5. lower-hypot.f64 N/A

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)}, \phi_1\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \]

      9. lower--.f64 79.2

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

   5. Applied rewrites 79.2%

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

   if 0.00119999999999999989 < phi2

   1. Initial program 50.5%

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

   3. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_2}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_2}^{2}} \]

      2. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + {\phi_2}^{2}} \]

      3. unswap-sqr N/A

         \[\leadsto R \cdot \sqrt{\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + {\phi_2}^{2}} \]

      4. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \color{blue}{\phi_2 \cdot \phi_2}} \]

      5. lower-hypot.f64 N/A

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}, \phi_2\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_2\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right) \]

      8. lower-*.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\cos \color{blue}{\left(\frac{1}{2} \cdot \phi_2\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right) \]

      9. lower--.f64 77.1

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

   5. Applied rewrites 77.1%

      \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.0012:\\ \;\;\;\;\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.2 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_2, \phi_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 2.2e+41)
   (* (hypot (* (- lambda1 lambda2) (cos (* 0.5 phi1))) phi1) R)
   (* (hypot (* (cos (* 0.5 phi2)) lambda2) phi2) R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.2e+41) {
		tmp = hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), phi1) * R;
	} else {
		tmp = hypot((cos((0.5 * phi2)) * lambda2), phi2) * R;
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.2e+41) {
		tmp = Math.hypot(((lambda1 - lambda2) * Math.cos((0.5 * phi1))), phi1) * R;
	} else {
		tmp = Math.hypot((Math.cos((0.5 * phi2)) * lambda2), phi2) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 2.2e+41:
		tmp = math.hypot(((lambda1 - lambda2) * math.cos((0.5 * phi1))), phi1) * R
	else:
		tmp = math.hypot((math.cos((0.5 * phi2)) * lambda2), phi2) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 2.2e+41)
		tmp = Float64(hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi1))), phi1) * R);
	else
		tmp = Float64(hypot(Float64(cos(Float64(0.5 * phi2)) * lambda2), phi2) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 2.2e+41)
		tmp = hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), phi1) * R;
	else
		tmp = hypot((cos((0.5 * phi2)) * lambda2), phi2) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 2.2e+41], N[(N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * lambda2), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2.1999999999999999e41

   1. Initial program 61.4%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}} \]

      2. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + {\phi_1}^{2}} \]

      3. unswap-sqr N/A

         \[\leadsto R \cdot \sqrt{\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + {\phi_1}^{2}} \]

      4. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \color{blue}{\phi_1 \cdot \phi_1}} \]

      5. lower-hypot.f64 N/A

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)}, \phi_1\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \]

      9. lower--.f64 77.7

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

   5. Applied rewrites 77.7%

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

   if 2.1999999999999999e41 < phi2

   1. Initial program 51.1%

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

   3. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_2}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_2}^{2}} \]

      2. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + {\phi_2}^{2}} \]

      3. unswap-sqr N/A

         \[\leadsto R \cdot \sqrt{\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + {\phi_2}^{2}} \]

      4. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \color{blue}{\phi_2 \cdot \phi_2}} \]

      5. lower-hypot.f64 N/A

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}, \phi_2\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_2\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right) \]

      8. lower-*.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\cos \color{blue}{\left(\frac{1}{2} \cdot \phi_2\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right) \]

      9. lower--.f64 79.7

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

   5. Applied rewrites 79.7%

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

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} + {\phi_2}^{2}} \]
   7. Step-by-step derivation

   8. Applied rewrites 73.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_2, \color{blue}{\phi_2}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.2 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_2, \phi_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.2 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{hypot}\left(1 \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_2, \phi_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 2.2e+41)
   (* (hypot (* 1.0 (- lambda1 lambda2)) phi1) R)
   (* (hypot (* (cos (* 0.5 phi2)) lambda2) phi2) R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.2e+41) {
		tmp = hypot((1.0 * (lambda1 - lambda2)), phi1) * R;
	} else {
		tmp = hypot((cos((0.5 * phi2)) * lambda2), phi2) * R;
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.2e+41) {
		tmp = Math.hypot((1.0 * (lambda1 - lambda2)), phi1) * R;
	} else {
		tmp = Math.hypot((Math.cos((0.5 * phi2)) * lambda2), phi2) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 2.2e+41:
		tmp = math.hypot((1.0 * (lambda1 - lambda2)), phi1) * R
	else:
		tmp = math.hypot((math.cos((0.5 * phi2)) * lambda2), phi2) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 2.2e+41)
		tmp = Float64(hypot(Float64(1.0 * Float64(lambda1 - lambda2)), phi1) * R);
	else
		tmp = Float64(hypot(Float64(cos(Float64(0.5 * phi2)) * lambda2), phi2) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 2.2e+41)
		tmp = hypot((1.0 * (lambda1 - lambda2)), phi1) * R;
	else
		tmp = hypot((cos((0.5 * phi2)) * lambda2), phi2) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 2.2e+41], N[(N[Sqrt[N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * lambda2), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2.1999999999999999e41

   1. Initial program 61.4%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}} \]

      2. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + {\phi_1}^{2}} \]

      3. unswap-sqr N/A

         \[\leadsto R \cdot \sqrt{\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + {\phi_1}^{2}} \]

      4. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \color{blue}{\phi_1 \cdot \phi_1}} \]

      5. lower-hypot.f64 N/A

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)}, \phi_1\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \]

      9. lower--.f64 77.7

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

   5. Applied rewrites 77.7%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \mathsf{hypot}\left(1 \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 69.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(1 \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \]

   if 2.1999999999999999e41 < phi2

   1. Initial program 51.1%

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

   3. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_2}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_2}^{2}} \]

      2. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + {\phi_2}^{2}} \]

      3. unswap-sqr N/A

         \[\leadsto R \cdot \sqrt{\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + {\phi_2}^{2}} \]

      4. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \color{blue}{\phi_2 \cdot \phi_2}} \]

      5. lower-hypot.f64 N/A

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}, \phi_2\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_2\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right) \]

      8. lower-*.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\cos \color{blue}{\left(\frac{1}{2} \cdot \phi_2\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right) \]

      9. lower--.f64 79.7

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

   5. Applied rewrites 79.7%

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

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} + {\phi_2}^{2}} \]
   7. Step-by-step derivation

   8. Applied rewrites 73.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_2, \color{blue}{\phi_2}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.2 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{hypot}\left(1 \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_2, \phi_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.9 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{hypot}\left(1 \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-R, \phi_1, R \cdot \phi_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 3.9e+71)
   (* (hypot (* 1.0 (- lambda1 lambda2)) phi1) R)
   (fma (- R) phi1 (* R phi2))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 3.9e+71) {
		tmp = hypot((1.0 * (lambda1 - lambda2)), phi1) * R;
	} else {
		tmp = fma(-R, phi1, (R * phi2));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 3.9e+71)
		tmp = Float64(hypot(Float64(1.0 * Float64(lambda1 - lambda2)), phi1) * R);
	else
		tmp = fma(Float64(-R), phi1, Float64(R * phi2));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 3.9e+71], N[(N[Sqrt[N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision] * R), $MachinePrecision], N[((-R) * phi1 + N[(R * phi2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 3.9000000000000001e71

   1. Initial program 61.5%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}} \]

      2. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + {\phi_1}^{2}} \]

      3. unswap-sqr N/A

         \[\leadsto R \cdot \sqrt{\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + {\phi_1}^{2}} \]

      4. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \color{blue}{\phi_1 \cdot \phi_1}} \]

      5. lower-hypot.f64 N/A

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)}, \phi_1\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \]

      9. lower--.f64 77.4

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \mathsf{hypot}\left(1 \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 69.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(1 \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \]

   if 3.9000000000000001e71 < phi2

   1. Initial program 48.9%

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

   3. Taylor expanded in phi1 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\phi_1 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 59.4

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

   5. Applied rewrites 59.4%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto -1 \cdot \left(R \cdot \phi_1\right) + \color{blue}{R \cdot \phi_2} \]
   7. Step-by-step derivation

   8. Applied rewrites 71.7%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.9 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{hypot}\left(1 \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-R, \phi_1, R \cdot \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 46.6% accurate, 13.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -17500000000:\\ \;\;\;\;\left(-\phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq 1.1 \cdot 10^{-194}:\\ \;\;\;\;\left(-\lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -17500000000.0)
   (* (- phi1) R)
   (if (<= phi1 1.1e-194) (* (- lambda1) R) (* R phi2))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -17500000000.0) {
		tmp = -phi1 * R;
	} else if (phi1 <= 1.1e-194) {
		tmp = -lambda1 * R;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this 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) :: tmp
    if (phi1 <= (-17500000000.0d0)) then
        tmp = -phi1 * r
    else if (phi1 <= 1.1d-194) then
        tmp = -lambda1 * r
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -17500000000.0) {
		tmp = -phi1 * R;
	} else if (phi1 <= 1.1e-194) {
		tmp = -lambda1 * R;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -17500000000.0:
		tmp = -phi1 * R
	elif phi1 <= 1.1e-194:
		tmp = -lambda1 * R
	else:
		tmp = R * phi2
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -17500000000.0)
		tmp = Float64(Float64(-phi1) * R);
	elseif (phi1 <= 1.1e-194)
		tmp = Float64(Float64(-lambda1) * R);
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -17500000000.0)
		tmp = -phi1 * R;
	elseif (phi1 <= 1.1e-194)
		tmp = -lambda1 * R;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -17500000000.0], N[((-phi1) * R), $MachinePrecision], If[LessEqual[phi1, 1.1e-194], N[((-lambda1) * R), $MachinePrecision], N[(R * phi2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.75e10

   1. Initial program 58.4%

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

   3. Taylor expanded in phi1 around -inf

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \phi_1\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto R \cdot \color{blue}{\left(\mathsf{neg}\left(\phi_1\right)\right)} \]

      2. lower-neg.f64 62.8

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

   5. Applied rewrites 62.8%

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

   if -1.75e10 < phi1 < 1.1000000000000001e-194

   1. Initial program 64.0%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}} \]

      2. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + {\phi_1}^{2}} \]

      3. unswap-sqr N/A

         \[\leadsto R \cdot \sqrt{\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + {\phi_1}^{2}} \]

      4. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \color{blue}{\phi_1 \cdot \phi_1}} \]

      5. lower-hypot.f64 N/A

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)}, \phi_1\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \]

      9. lower--.f64 68.6

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

   5. Applied rewrites 68.6%

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

   6. Taylor expanded in lambda1 around -inf

      \[\leadsto R \cdot \left(-1 \cdot \color{blue}{\left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 17.7%

      \[\leadsto R \cdot \left(\left(-\lambda_1\right) \cdot \color{blue}{\cos \left(0.5 \cdot \phi_1\right)}\right) \]

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(-1 \cdot \lambda_1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 18.8%

       \[\leadsto R \cdot \left(-\lambda_1\right) \]

   if 1.1000000000000001e-194 < phi1

   1. Initial program 55.7%

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

   3. Taylor expanded in phi2 around inf

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
   4. Step-by-step derivation

      1. lower-*.f64 17.7

         \[\leadsto \color{blue}{R \cdot \phi_2} \]

   5. Applied rewrites 17.7%

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 27.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -17500000000:\\ \;\;\;\;\left(-\phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq 1.1 \cdot 10^{-194}:\\ \;\;\;\;\left(-\lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.7% accurate, 13.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.2 \cdot 10^{+59}:\\ \;\;\;\;\left(-\lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-R, \phi_1, R \cdot \phi_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -1.2e+59) (* (- lambda1) R) (fma (- R) phi1 (* R phi2))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.2e+59) {
		tmp = -lambda1 * R;
	} else {
		tmp = fma(-R, phi1, (R * phi2));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -1.2e+59)
		tmp = Float64(Float64(-lambda1) * R);
	else
		tmp = fma(Float64(-R), phi1, Float64(R * phi2));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -1.2e+59], N[((-lambda1) * R), $MachinePrecision], N[((-R) * phi1 + N[(R * phi2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.2000000000000001e59

   1. Initial program 48.2%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}} \]

      2. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + {\phi_1}^{2}} \]

      3. unswap-sqr N/A

         \[\leadsto R \cdot \sqrt{\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + {\phi_1}^{2}} \]

      4. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \color{blue}{\phi_1 \cdot \phi_1}} \]

      5. lower-hypot.f64 N/A

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)}, \phi_1\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \]

      9. lower--.f64 73.0

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

   5. Applied rewrites 73.0%

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

   6. Taylor expanded in lambda1 around -inf

      \[\leadsto R \cdot \left(-1 \cdot \color{blue}{\left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 41.6%

      \[\leadsto R \cdot \left(\left(-\lambda_1\right) \cdot \color{blue}{\cos \left(0.5 \cdot \phi_1\right)}\right) \]

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(-1 \cdot \lambda_1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 50.4%

       \[\leadsto R \cdot \left(-\lambda_1\right) \]

   if -1.2000000000000001e59 < lambda1

   1. Initial program 62.2%

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

   3. Taylor expanded in phi1 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\phi_1 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 30.4

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

   5. Applied rewrites 30.4%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto -1 \cdot \left(R \cdot \phi_1\right) + \color{blue}{R \cdot \phi_2} \]
   7. Step-by-step derivation

   8. Applied rewrites 32.0%

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

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.2 \cdot 10^{+59}:\\ \;\;\;\;\left(-\lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-R, \phi_1, R \cdot \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.4% accurate, 18.6× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.2 \cdot 10^{+59}:\\ \;\;\;\;\left(-\lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -1.2e+59) (* (- lambda1) R) (* (- phi2 phi1) R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.2e+59) {
		tmp = -lambda1 * R;
	} else {
		tmp = (phi2 - phi1) * R;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this 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) :: tmp
    if (lambda1 <= (-1.2d+59)) then
        tmp = -lambda1 * r
    else
        tmp = (phi2 - phi1) * r
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.2e+59) {
		tmp = -lambda1 * R;
	} else {
		tmp = (phi2 - phi1) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -1.2e+59:
		tmp = -lambda1 * R
	else:
		tmp = (phi2 - phi1) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -1.2e+59)
		tmp = Float64(Float64(-lambda1) * R);
	else
		tmp = Float64(Float64(phi2 - phi1) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -1.2e+59)
		tmp = -lambda1 * R;
	else
		tmp = (phi2 - phi1) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -1.2e+59], N[((-lambda1) * R), $MachinePrecision], N[(N[(phi2 - phi1), $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.2000000000000001e59

   1. Initial program 48.2%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}} \]

      2. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + {\phi_1}^{2}} \]

      3. unswap-sqr N/A

         \[\leadsto R \cdot \sqrt{\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + {\phi_1}^{2}} \]

      4. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \color{blue}{\phi_1 \cdot \phi_1}} \]

      5. lower-hypot.f64 N/A

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)}, \phi_1\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \]

      9. lower--.f64 73.0

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

   5. Applied rewrites 73.0%

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

   6. Taylor expanded in lambda1 around -inf

      \[\leadsto R \cdot \left(-1 \cdot \color{blue}{\left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 41.6%

      \[\leadsto R \cdot \left(\left(-\lambda_1\right) \cdot \color{blue}{\cos \left(0.5 \cdot \phi_1\right)}\right) \]

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(-1 \cdot \lambda_1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 50.4%

       \[\leadsto R \cdot \left(-\lambda_1\right) \]

   if -1.2000000000000001e59 < lambda1

   1. Initial program 62.2%

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

   3. Taylor expanded in phi1 around -inf

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 29.5

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

   5. Applied rewrites 29.5%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(\phi_2 + \color{blue}{-1 \cdot \phi_1}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 32.0%

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

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.2 \cdot 10^{+59}:\\ \;\;\;\;\left(-\lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 38.0% accurate, 19.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.1 \cdot 10^{+20}:\\ \;\;\;\;\left(-\lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 4.1e+20) (* (- lambda1) R) (* R phi2)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 4.1e+20) {
		tmp = -lambda1 * R;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this 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) :: tmp
    if (phi2 <= 4.1d+20) then
        tmp = -lambda1 * r
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 4.1e+20) {
		tmp = -lambda1 * R;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 4.1e+20:
		tmp = -lambda1 * R
	else:
		tmp = R * phi2
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 4.1e+20)
		tmp = Float64(Float64(-lambda1) * R);
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 4.1e+20)
		tmp = -lambda1 * R;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 4.1e+20], N[((-lambda1) * R), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 4.1e20

   1. Initial program 61.5%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}} \]

      2. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + {\phi_1}^{2}} \]

      3. unswap-sqr N/A

         \[\leadsto R \cdot \sqrt{\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} + {\phi_1}^{2}} \]

      4. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \color{blue}{\phi_1 \cdot \phi_1}} \]

      5. lower-hypot.f64 N/A

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)}, \phi_1\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto R \cdot \mathsf{hypot}\left(\cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \]

      9. lower--.f64 77.9

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

   5. Applied rewrites 77.9%

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

   6. Taylor expanded in lambda1 around -inf

      \[\leadsto R \cdot \left(-1 \cdot \color{blue}{\left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 17.0%

      \[\leadsto R \cdot \left(\left(-\lambda_1\right) \cdot \color{blue}{\cos \left(0.5 \cdot \phi_1\right)}\right) \]

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(-1 \cdot \lambda_1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 14.0%

       \[\leadsto R \cdot \left(-\lambda_1\right) \]

   if 4.1e20 < phi2

   1. Initial program 51.1%

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

   3. Taylor expanded in phi2 around inf

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
   4. Step-by-step derivation

      1. lower-*.f64 60.6

         \[\leadsto \color{blue}{R \cdot \phi_2} \]

   5. Applied rewrites 60.6%

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.1 \cdot 10^{+20}:\\ \;\;\;\;\left(-\lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 32.3% accurate, 46.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ R \cdot \phi_2 \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2) :precision binary64 (* R phi2))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * phi2;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this 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
    code = r * phi2
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * phi2;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	return R * phi2

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * phi2)
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * phi2;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * phi2), $MachinePrecision]

Derivation

1. Initial program 59.2%

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

3. Taylor expanded in phi2 around inf

   \[\leadsto \color{blue}{R \cdot \phi_2} \]
4. Step-by-step derivation

   1. lower-*.f64 17.6

      \[\leadsto \color{blue}{R \cdot \phi_2} \]

5. Applied rewrites 17.6%

   \[\leadsto \color{blue}{R \cdot \phi_2} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024332 
(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))))))