Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.6% → 95.8%

Time: 8.4s

Alternatives: 11

Speedup: N/A×

• 29.9% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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: 59.6% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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: 95.8% accurate, N/A× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -900000000000:\\ \;\;\;\;\mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \cos \left(\frac{\phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_2 \cdot \left(\frac{\phi_1}{\phi_2} - 1\right), \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

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 (<= phi1 -900000000000.0)
   (*
    (hypot (pow (- phi1 phi2) 1.0) (* (cos (/ phi1 2.0)) (- lambda1 lambda2)))
    R)
   (*
    (hypot
     (* phi2 (- (/ phi1 phi2) 1.0))
     (* (cos (/ (+ phi2 phi1) 2.0)) (- lambda1 lambda2)))
    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 (phi1 <= -900000000000.0) {
		tmp = hypot(pow((phi1 - phi2), 1.0), (cos((phi1 / 2.0)) * (lambda1 - lambda2))) * R;
	} else {
		tmp = hypot((phi2 * ((phi1 / phi2) - 1.0)), (cos(((phi2 + phi1) / 2.0)) * (lambda1 - lambda2))) * 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 (phi1 <= -900000000000.0) {
		tmp = Math.hypot(Math.pow((phi1 - phi2), 1.0), (Math.cos((phi1 / 2.0)) * (lambda1 - lambda2))) * R;
	} else {
		tmp = Math.hypot((phi2 * ((phi1 / phi2) - 1.0)), (Math.cos(((phi2 + phi1) / 2.0)) * (lambda1 - lambda2))) * 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 phi1 <= -900000000000.0:
		tmp = math.hypot(math.pow((phi1 - phi2), 1.0), (math.cos((phi1 / 2.0)) * (lambda1 - lambda2))) * R
	else:
		tmp = math.hypot((phi2 * ((phi1 / phi2) - 1.0)), (math.cos(((phi2 + phi1) / 2.0)) * (lambda1 - lambda2))) * 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 (phi1 <= -900000000000.0)
		tmp = Float64(hypot((Float64(phi1 - phi2) ^ 1.0), Float64(cos(Float64(phi1 / 2.0)) * Float64(lambda1 - lambda2))) * R);
	else
		tmp = Float64(hypot(Float64(phi2 * Float64(Float64(phi1 / phi2) - 1.0)), Float64(cos(Float64(Float64(phi2 + phi1) / 2.0)) * Float64(lambda1 - lambda2))) * 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 (phi1 <= -900000000000.0)
		tmp = hypot(((phi1 - phi2) ^ 1.0), (cos((phi1 / 2.0)) * (lambda1 - lambda2))) * R;
	else
		tmp = hypot((phi2 * ((phi1 / phi2) - 1.0)), (cos(((phi2 + phi1) / 2.0)) * (lambda1 - lambda2))) * 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[phi1, -900000000000.0], N[(N[Sqrt[N[Power[N[(phi1 - phi2), $MachinePrecision], 1.0], $MachinePrecision] ^ 2 + N[(N[Cos[N[(phi1 / 2.0), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(phi2 * N[(N[(phi1 / phi2), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(N[Cos[N[(N[(phi2 + phi1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -9e11

   1. Initial program 57.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

   4. Applied rewrites 91.5%

      \[\leadsto \color{blue}{\mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R} \]

   5. Taylor expanded in phi1 around inf

      \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \cos \left(\frac{\color{blue}{\phi_1}}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   6. Step-by-step derivation

   7. Applied rewrites 91.4%

      \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \cos \left(\frac{\color{blue}{\phi_1}}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   if -9e11 < phi1

   1. Initial program 57.8%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

   4. Applied rewrites 97.4%

      \[\leadsto \color{blue}{\mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R} \]

   5. Taylor expanded in phi2 around inf

      \[\leadsto \mathsf{hypot}\left(\color{blue}{\phi_2 \cdot \left(\frac{\phi_1}{\phi_2} - 1\right)}, \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   6. Step-by-step derivation

      1. unpow1 N/A

         \[\leadsto \mathsf{hypot}\left(\color{blue}{\phi_2} \cdot \left(\frac{\phi_1}{\phi_2} - 1\right), \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_2 \cdot \color{blue}{\left(\frac{\phi_1}{\phi_2} - 1\right)}, \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. lower--.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_2 \cdot \left(\frac{\phi_1}{\phi_2} - \color{blue}{1}\right), \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower-/.f64 89.3

         \[\leadsto \mathsf{hypot}\left(\phi_2 \cdot \left(\frac{\phi_1}{\phi_2} - 1\right), \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   7. Applied rewrites 89.3%

      \[\leadsto \mathsf{hypot}\left(\color{blue}{\phi_2 \cdot \left(\frac{\phi_1}{\phi_2} - 1\right)}, \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 95.8% accurate, N/A× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \end{array} \]

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
 (*
  (hypot
   (pow (- phi1 phi2) 1.0)
   (* (cos (/ (+ phi2 phi1) 2.0)) (- lambda1 lambda2)))
  R))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return hypot(pow((phi1 - phi2), 1.0), (cos(((phi2 + phi1) / 2.0)) * (lambda1 - lambda2))) * R;
}

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 Math.hypot(Math.pow((phi1 - phi2), 1.0), (Math.cos(((phi2 + phi1) / 2.0)) * (lambda1 - lambda2))) * R;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	return math.hypot(math.pow((phi1 - phi2), 1.0), (math.cos(((phi2 + phi1) / 2.0)) * (lambda1 - lambda2))) * R

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(hypot((Float64(phi1 - phi2) ^ 1.0), Float64(cos(Float64(Float64(phi2 + phi1) / 2.0)) * Float64(lambda1 - lambda2))) * R)
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = hypot(((phi1 - phi2) ^ 1.0), (cos(((phi2 + phi1) / 2.0)) * (lambda1 - lambda2))) * R;
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[(N[Sqrt[N[Power[N[(phi1 - phi2), $MachinePrecision], 1.0], $MachinePrecision] ^ 2 + N[(N[Cos[N[(N[(phi2 + phi1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 57.7%

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

   1. lift-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-+.f64 N/A

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

4. Applied rewrites 96.0%

   \[\leadsto \color{blue}{\mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R} \]
5. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \color{blue}{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. lift-+.f64 N/A

      \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \cos \left(\frac{\color{blue}{\phi_2 + \phi_1}}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   3. lift-/.f64 N/A

      \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \cos \color{blue}{\left(\frac{\phi_2 + \phi_1}{2}\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. sin-+PI/2-rev N/A

      \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \color{blue}{\sin \left(\frac{\phi_2 + \phi_1}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   5. lower-sin.f64 N/A

      \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \color{blue}{\sin \left(\frac{\phi_2 + \phi_1}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   6. lift-/.f64 N/A

      \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(\frac{\phi_2 + \phi_1}{2} + \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   7. lift-PI.f64 N/A

      \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(\frac{\phi_2 + \phi_1}{2} + \frac{\color{blue}{\pi}}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   8. lower-+.f64 N/A

      \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \color{blue}{\left(\frac{\phi_2 + \phi_1}{2} + \frac{\pi}{2}\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   9. lift-/.f64 N/A

      \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(\color{blue}{\frac{\phi_2 + \phi_1}{2}} + \frac{\pi}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   10. lift-+.f64 83.6

       \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(\frac{\color{blue}{\phi_2 + \phi_1}}{2} + \frac{\pi}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

6. Applied rewrites 83.6%

   \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \color{blue}{\sin \left(\frac{\phi_2 + \phi_1}{2} + \frac{\pi}{2}\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
7. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \color{blue}{\sin \left(\frac{\phi_2 + \phi_1}{2} + \frac{\pi}{2}\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   2. lift-+.f64 N/A

      \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \color{blue}{\left(\frac{\phi_2 + \phi_1}{2} + \frac{\pi}{2}\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   3. lift-+.f64 N/A

      \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(\frac{\color{blue}{\phi_2 + \phi_1}}{2} + \frac{\pi}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. lift-/.f64 N/A

      \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(\color{blue}{\frac{\phi_2 + \phi_1}{2}} + \frac{\pi}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   5. lift-PI.f64 N/A

      \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(\frac{\phi_2 + \phi_1}{2} + \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   6. lift-/.f64 N/A

      \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(\frac{\phi_2 + \phi_1}{2} + \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   7. sin-+PI/2-rev N/A

      \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \color{blue}{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   8. lower-cos.f64 N/A

      \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \color{blue}{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   9. lift-/.f64 N/A

      \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \cos \color{blue}{\left(\frac{\phi_2 + \phi_1}{2}\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   10. lift-+.f64 96.0

       \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \cos \left(\frac{\color{blue}{\phi_2 + \phi_1}}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

8. Applied rewrites 96.0%

   \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \color{blue}{\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]
9. Add Preprocessing

Alternative 3: 92.3% accurate, N/A× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.5 \cdot 10^{+163}:\\ \;\;\;\;\mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(\mathsf{fma}\left(0.5, \phi_2, \frac{\pi}{2}\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_2 \cdot \left(\frac{\phi_1}{\phi_2} - 1\right), \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

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 (<= phi1 -4.5e+163)
   (*
    (hypot
     (pow (- phi1 phi2) 1.0)
     (* (sin (fma 0.5 phi2 (/ PI 2.0))) (- lambda1 lambda2)))
    R)
   (*
    (hypot
     (* phi2 (- (/ phi1 phi2) 1.0))
     (* (cos (/ (+ phi2 phi1) 2.0)) (- lambda1 lambda2)))
    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 (phi1 <= -4.5e+163) {
		tmp = hypot(pow((phi1 - phi2), 1.0), (sin(fma(0.5, phi2, (((double) M_PI) / 2.0))) * (lambda1 - lambda2))) * R;
	} else {
		tmp = hypot((phi2 * ((phi1 / phi2) - 1.0)), (cos(((phi2 + phi1) / 2.0)) * (lambda1 - lambda2))) * 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 (phi1 <= -4.5e+163)
		tmp = Float64(hypot((Float64(phi1 - phi2) ^ 1.0), Float64(sin(fma(0.5, phi2, Float64(pi / 2.0))) * Float64(lambda1 - lambda2))) * R);
	else
		tmp = Float64(hypot(Float64(phi2 * Float64(Float64(phi1 / phi2) - 1.0)), Float64(cos(Float64(Float64(phi2 + phi1) / 2.0)) * Float64(lambda1 - lambda2))) * R);
	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[phi1, -4.5e+163], N[(N[Sqrt[N[Power[N[(phi1 - phi2), $MachinePrecision], 1.0], $MachinePrecision] ^ 2 + N[(N[Sin[N[(0.5 * phi2 + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(phi2 * N[(N[(phi1 / phi2), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(N[Cos[N[(N[(phi2 + phi1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -4.49999999999999988e163

   1. Initial program 45.9%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

   4. Applied rewrites 94.5%

      \[\leadsto \color{blue}{\mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R} \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_2\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   6. Step-by-step derivation

      1. sin-+PI/2-rev N/A

         \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(\frac{1}{2} \cdot \phi_2 + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      2. lower-sin.f64 N/A

         \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(\frac{1}{2} \cdot \phi_2 + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. lift-/.f64 N/A

         \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(\frac{1}{2} \cdot \phi_2 + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lift-PI.f64 N/A

         \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(\frac{1}{2} \cdot \phi_2 + \frac{\pi}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      5. lower-fma.f64 83.7

         \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(\mathsf{fma}\left(0.5, \phi_2, \frac{\pi}{2}\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   7. Applied rewrites 83.7%

      \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \color{blue}{\sin \left(\mathsf{fma}\left(0.5, \phi_2, \frac{\pi}{2}\right)\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   if -4.49999999999999988e163 < phi1

   1. Initial program 59.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

   4. Applied rewrites 96.2%

      \[\leadsto \color{blue}{\mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R} \]

   5. Taylor expanded in phi2 around inf

      \[\leadsto \mathsf{hypot}\left(\color{blue}{\phi_2 \cdot \left(\frac{\phi_1}{\phi_2} - 1\right)}, \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   6. Step-by-step derivation

      1. unpow1 N/A

         \[\leadsto \mathsf{hypot}\left(\color{blue}{\phi_2} \cdot \left(\frac{\phi_1}{\phi_2} - 1\right), \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_2 \cdot \color{blue}{\left(\frac{\phi_1}{\phi_2} - 1\right)}, \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. lower--.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_2 \cdot \left(\frac{\phi_1}{\phi_2} - \color{blue}{1}\right), \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower-/.f64 88.7

         \[\leadsto \mathsf{hypot}\left(\phi_2 \cdot \left(\frac{\phi_1}{\phi_2} - 1\right), \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   7. Applied rewrites 88.7%

      \[\leadsto \mathsf{hypot}\left(\color{blue}{\phi_2 \cdot \left(\frac{\phi_1}{\phi_2} - 1\right)}, \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 92.3% accurate, N/A× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.5 \cdot 10^{+163}:\\ \;\;\;\;\mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(0.5 \cdot \pi\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_2 \cdot \left(\frac{\phi_1}{\phi_2} - 1\right), \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

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 (<= phi1 -4.5e+163)
   (*
    (hypot (pow (- phi1 phi2) 1.0) (* (sin (* 0.5 PI)) (- lambda1 lambda2)))
    R)
   (*
    (hypot
     (* phi2 (- (/ phi1 phi2) 1.0))
     (* (cos (/ (+ phi2 phi1) 2.0)) (- lambda1 lambda2)))
    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 (phi1 <= -4.5e+163) {
		tmp = hypot(pow((phi1 - phi2), 1.0), (sin((0.5 * ((double) M_PI))) * (lambda1 - lambda2))) * R;
	} else {
		tmp = hypot((phi2 * ((phi1 / phi2) - 1.0)), (cos(((phi2 + phi1) / 2.0)) * (lambda1 - lambda2))) * 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 (phi1 <= -4.5e+163) {
		tmp = Math.hypot(Math.pow((phi1 - phi2), 1.0), (Math.sin((0.5 * Math.PI)) * (lambda1 - lambda2))) * R;
	} else {
		tmp = Math.hypot((phi2 * ((phi1 / phi2) - 1.0)), (Math.cos(((phi2 + phi1) / 2.0)) * (lambda1 - lambda2))) * 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 phi1 <= -4.5e+163:
		tmp = math.hypot(math.pow((phi1 - phi2), 1.0), (math.sin((0.5 * math.pi)) * (lambda1 - lambda2))) * R
	else:
		tmp = math.hypot((phi2 * ((phi1 / phi2) - 1.0)), (math.cos(((phi2 + phi1) / 2.0)) * (lambda1 - lambda2))) * 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 (phi1 <= -4.5e+163)
		tmp = Float64(hypot((Float64(phi1 - phi2) ^ 1.0), Float64(sin(Float64(0.5 * pi)) * Float64(lambda1 - lambda2))) * R);
	else
		tmp = Float64(hypot(Float64(phi2 * Float64(Float64(phi1 / phi2) - 1.0)), Float64(cos(Float64(Float64(phi2 + phi1) / 2.0)) * Float64(lambda1 - lambda2))) * 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 (phi1 <= -4.5e+163)
		tmp = hypot(((phi1 - phi2) ^ 1.0), (sin((0.5 * pi)) * (lambda1 - lambda2))) * R;
	else
		tmp = hypot((phi2 * ((phi1 / phi2) - 1.0)), (cos(((phi2 + phi1) / 2.0)) * (lambda1 - lambda2))) * 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[phi1, -4.5e+163], N[(N[Sqrt[N[Power[N[(phi1 - phi2), $MachinePrecision], 1.0], $MachinePrecision] ^ 2 + N[(N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(phi2 * N[(N[(phi1 / phi2), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(N[Cos[N[(N[(phi2 + phi1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -4.49999999999999988e163

   1. Initial program 45.9%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

   4. Applied rewrites 94.5%

      \[\leadsto \color{blue}{\mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   6. Step-by-step derivation

      1. sin-+PI/2-rev N/A

         \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(\frac{1}{2} \cdot \phi_1 + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      2. lower-sin.f64 N/A

         \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(\frac{1}{2} \cdot \phi_1 + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. lift-/.f64 N/A

         \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(\frac{1}{2} \cdot \phi_1 + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lift-PI.f64 N/A

         \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(\frac{1}{2} \cdot \phi_1 + \frac{\pi}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      5. lower-fma.f64 83.9

         \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(\mathsf{fma}\left(0.5, \phi_1, \frac{\pi}{2}\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   7. Applied rewrites 83.9%

      \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \color{blue}{\sin \left(\mathsf{fma}\left(0.5, \phi_1, \frac{\pi}{2}\right)\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      2. lift-PI.f64 83.6

         \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(0.5 \cdot \pi\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   10. Applied rewrites 83.6%

       \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(0.5 \cdot \pi\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   if -4.49999999999999988e163 < phi1

   1. Initial program 59.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

   4. Applied rewrites 96.2%

      \[\leadsto \color{blue}{\mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R} \]

   5. Taylor expanded in phi2 around inf

      \[\leadsto \mathsf{hypot}\left(\color{blue}{\phi_2 \cdot \left(\frac{\phi_1}{\phi_2} - 1\right)}, \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   6. Step-by-step derivation

      1. unpow1 N/A

         \[\leadsto \mathsf{hypot}\left(\color{blue}{\phi_2} \cdot \left(\frac{\phi_1}{\phi_2} - 1\right), \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_2 \cdot \color{blue}{\left(\frac{\phi_1}{\phi_2} - 1\right)}, \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. lower--.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_2 \cdot \left(\frac{\phi_1}{\phi_2} - \color{blue}{1}\right), \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower-/.f64 88.7

         \[\leadsto \mathsf{hypot}\left(\phi_2 \cdot \left(\frac{\phi_1}{\phi_2} - 1\right), \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   7. Applied rewrites 88.7%

      \[\leadsto \mathsf{hypot}\left(\color{blue}{\phi_2 \cdot \left(\frac{\phi_1}{\phi_2} - 1\right)}, \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 91.0% accurate, N/A× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.75 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1, \sin \left(\mathsf{fma}\left(0.5, \phi_1, \frac{\pi}{2}\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_2 \cdot \left(\frac{\phi_1}{\phi_2} - 1\right), \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

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 3.75e-121)
   (* (hypot phi1 (* (sin (fma 0.5 phi1 (/ PI 2.0))) (- lambda1 lambda2))) R)
   (*
    (hypot
     (* phi2 (- (/ phi1 phi2) 1.0))
     (* (cos (/ (+ phi2 phi1) 2.0)) (- lambda1 lambda2)))
    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 <= 3.75e-121) {
		tmp = hypot(phi1, (sin(fma(0.5, phi1, (((double) M_PI) / 2.0))) * (lambda1 - lambda2))) * R;
	} else {
		tmp = hypot((phi2 * ((phi1 / phi2) - 1.0)), (cos(((phi2 + phi1) / 2.0)) * (lambda1 - lambda2))) * 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 <= 3.75e-121)
		tmp = Float64(hypot(phi1, Float64(sin(fma(0.5, phi1, Float64(pi / 2.0))) * Float64(lambda1 - lambda2))) * R);
	else
		tmp = Float64(hypot(Float64(phi2 * Float64(Float64(phi1 / phi2) - 1.0)), Float64(cos(Float64(Float64(phi2 + phi1) / 2.0)) * Float64(lambda1 - lambda2))) * R);
	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, 3.75e-121], N[(N[Sqrt[phi1 ^ 2 + N[(N[Sin[N[(0.5 * phi1 + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(phi2 * N[(N[(phi1 / phi2), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(N[Cos[N[(N[(phi2 + phi1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 3.75000000000000013e-121

   1. Initial program 61.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

   4. Applied rewrites 97.5%

      \[\leadsto \color{blue}{\mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   6. Step-by-step derivation

      1. sin-+PI/2-rev N/A

         \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(\frac{1}{2} \cdot \phi_1 + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      2. lower-sin.f64 N/A

         \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(\frac{1}{2} \cdot \phi_1 + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. lift-/.f64 N/A

         \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(\frac{1}{2} \cdot \phi_1 + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lift-PI.f64 N/A

         \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(\frac{1}{2} \cdot \phi_1 + \frac{\pi}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      5. lower-fma.f64 83.0

         \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(\mathsf{fma}\left(0.5, \phi_1, \frac{\pi}{2}\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   7. Applied rewrites 83.0%

      \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \color{blue}{\sin \left(\mathsf{fma}\left(0.5, \phi_1, \frac{\pi}{2}\right)\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   8. Taylor expanded in phi1 around inf

      \[\leadsto \mathsf{hypot}\left(\color{blue}{\phi_1}, \sin \left(\mathsf{fma}\left(\frac{1}{2}, \phi_1, \frac{\pi}{2}\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   9. Step-by-step derivation

      1. unpow1 70.2

         \[\leadsto \mathsf{hypot}\left(\phi_1, \sin \left(\mathsf{fma}\left(0.5, \phi_1, \frac{\pi}{2}\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   10. Applied rewrites 70.2%

       \[\leadsto \mathsf{hypot}\left(\color{blue}{\phi_1}, \sin \left(\mathsf{fma}\left(0.5, \phi_1, \frac{\pi}{2}\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   if 3.75000000000000013e-121 < phi2

   1. Initial program 50.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

   4. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R} \]

   5. Taylor expanded in phi2 around inf

      \[\leadsto \mathsf{hypot}\left(\color{blue}{\phi_2 \cdot \left(\frac{\phi_1}{\phi_2} - 1\right)}, \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   6. Step-by-step derivation

      1. unpow1 N/A

         \[\leadsto \mathsf{hypot}\left(\color{blue}{\phi_2} \cdot \left(\frac{\phi_1}{\phi_2} - 1\right), \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_2 \cdot \color{blue}{\left(\frac{\phi_1}{\phi_2} - 1\right)}, \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. lower--.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_2 \cdot \left(\frac{\phi_1}{\phi_2} - \color{blue}{1}\right), \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower-/.f64 88.6

         \[\leadsto \mathsf{hypot}\left(\phi_2 \cdot \left(\frac{\phi_1}{\phi_2} - 1\right), \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   7. Applied rewrites 88.6%

      \[\leadsto \mathsf{hypot}\left(\color{blue}{\phi_2 \cdot \left(\frac{\phi_1}{\phi_2} - 1\right)}, \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 74.8% accurate, N/A× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ t_1 := \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\\ \mathbf{if}\;\sqrt{t\_0 \cdot t\_0 + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \leq 2 \cdot 10^{+153}:\\ \;\;\;\;R \cdot \sqrt{\mathsf{fma}\left({\left(\lambda_1 - \lambda_2\right)}^{2}, t\_1 \cdot t\_1, {\left(\phi_1 - \phi_2\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1, \sin \left(\mathsf{fma}\left(0.5, \phi_1, \frac{\pi}{2}\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))))
        (t_1 (cos (/ (+ phi2 phi1) 2.0))))
   (if (<= (sqrt (+ (* t_0 t_0) (* (- phi1 phi2) (- phi1 phi2)))) 2e+153)
     (*
      R
      (sqrt
       (fma
        (pow (- lambda1 lambda2) 2.0)
        (* t_1 t_1)
        (pow (- phi1 phi2) 2.0))))
     (*
      (hypot phi1 (* (sin (fma 0.5 phi1 (/ PI 2.0))) (- lambda1 lambda2)))
      R))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	double t_1 = cos(((phi2 + phi1) / 2.0));
	double tmp;
	if (sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2)))) <= 2e+153) {
		tmp = R * sqrt(fma(pow((lambda1 - lambda2), 2.0), (t_1 * t_1), pow((phi1 - phi2), 2.0)));
	} else {
		tmp = hypot(phi1, (sin(fma(0.5, phi1, (((double) M_PI) / 2.0))) * (lambda1 - lambda2))) * R;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0)))
	t_1 = cos(Float64(Float64(phi2 + phi1) / 2.0))
	tmp = 0.0
	if (sqrt(Float64(Float64(t_0 * t_0) + Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2)))) <= 2e+153)
		tmp = Float64(R * sqrt(fma((Float64(lambda1 - lambda2) ^ 2.0), Float64(t_1 * t_1), (Float64(phi1 - phi2) ^ 2.0))));
	else
		tmp = Float64(hypot(phi1, Float64(sin(fma(0.5, phi1, Float64(pi / 2.0))) * Float64(lambda1 - lambda2))) * R);
	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_] := Block[{t$95$0 = N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(N[(phi2 + phi1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2e+153], N[(R * N[Sqrt[N[(N[Power[N[(lambda1 - lambda2), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision] + N[Power[N[(phi1 - phi2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[phi1 ^ 2 + N[(N[Sin[N[(0.5 * phi1 + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (+.f64 (*.f64 (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) #s(literal 2 binary64)))) (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) #s(literal 2 binary64))))) (*.f64 (-.f64 phi1 phi2) (-.f64 phi1 phi2)))) < 2e153

   1. Initial program 98.9%

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

      1. lift-+.f64 N/A

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

   4. Applied rewrites 98.9%

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

   if 2e153 < (sqrt.f64 (+.f64 (*.f64 (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) #s(literal 2 binary64)))) (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) #s(literal 2 binary64))))) (*.f64 (-.f64 phi1 phi2) (-.f64 phi1 phi2))))

   1. Initial program 40.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

   4. Applied rewrites 94.9%

      \[\leadsto \color{blue}{\mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   6. Step-by-step derivation

      1. sin-+PI/2-rev N/A

         \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(\frac{1}{2} \cdot \phi_1 + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      2. lower-sin.f64 N/A

         \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(\frac{1}{2} \cdot \phi_1 + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. lift-/.f64 N/A

         \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(\frac{1}{2} \cdot \phi_1 + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lift-PI.f64 N/A

         \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(\frac{1}{2} \cdot \phi_1 + \frac{\pi}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      5. lower-fma.f64 80.5

         \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \sin \left(\mathsf{fma}\left(0.5, \phi_1, \frac{\pi}{2}\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   7. Applied rewrites 80.5%

      \[\leadsto \mathsf{hypot}\left({\left(\phi_1 - \phi_2\right)}^{1}, \color{blue}{\sin \left(\mathsf{fma}\left(0.5, \phi_1, \frac{\pi}{2}\right)\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   8. Taylor expanded in phi1 around inf

      \[\leadsto \mathsf{hypot}\left(\color{blue}{\phi_1}, \sin \left(\mathsf{fma}\left(\frac{1}{2}, \phi_1, \frac{\pi}{2}\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   9. Step-by-step derivation

      1. unpow1 64.9

         \[\leadsto \mathsf{hypot}\left(\phi_1, \sin \left(\mathsf{fma}\left(0.5, \phi_1, \frac{\pi}{2}\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   10. Applied rewrites 64.9%

       \[\leadsto \mathsf{hypot}\left(\color{blue}{\phi_1}, \sin \left(\mathsf{fma}\left(0.5, \phi_1, \frac{\pi}{2}\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 60.1% accurate, N/A× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\\ \mathbf{if}\;\lambda_2 \leq 3.2 \cdot 10^{+155}:\\ \;\;\;\;R \cdot \sqrt{\mathsf{fma}\left({\left(\lambda_1 - \lambda_2\right)}^{2}, t\_0 \cdot t\_0, {\left(\phi_1 - \phi_2\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (cos (/ (+ phi2 phi1) 2.0))))
   (if (<= lambda2 3.2e+155)
     (*
      R
      (sqrt
       (fma
        (pow (- lambda1 lambda2) 2.0)
        (* t_0 t_0)
        (pow (- phi1 phi2) 2.0))))
     (* (* R lambda2) (cos (* 0.5 (+ phi2 phi1)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(((phi2 + phi1) / 2.0));
	double tmp;
	if (lambda2 <= 3.2e+155) {
		tmp = R * sqrt(fma(pow((lambda1 - lambda2), 2.0), (t_0 * t_0), pow((phi1 - phi2), 2.0)));
	} else {
		tmp = (R * lambda2) * cos((0.5 * (phi2 + phi1)));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(Float64(phi2 + phi1) / 2.0))
	tmp = 0.0
	if (lambda2 <= 3.2e+155)
		tmp = Float64(R * sqrt(fma((Float64(lambda1 - lambda2) ^ 2.0), Float64(t_0 * t_0), (Float64(phi1 - phi2) ^ 2.0))));
	else
		tmp = Float64(Float64(R * lambda2) * cos(Float64(0.5 * Float64(phi2 + phi1))));
	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_] := Block[{t$95$0 = N[Cos[N[(N[(phi2 + phi1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda2, 3.2e+155], N[(R * N[Sqrt[N[(N[Power[N[(lambda1 - lambda2), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision] + N[Power[N[(phi1 - phi2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(R * lambda2), $MachinePrecision] * N[Cos[N[(0.5 * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 3.20000000000000012e155

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

   4. Applied rewrites 61.5%

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

   if 3.20000000000000012e155 < lambda2

   1. Initial program 33.3%

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

   3. Taylor expanded in lambda2 around inf

      \[\leadsto \color{blue}{R \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \color{blue}{\left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)} \]

      4. sin-+PI/2-rev N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      5. lower-sin.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \phi_1 + \phi_2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \phi_2 + \phi_1, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \phi_2 + \phi_1, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \phi_2 + \phi_1, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      10. lower-PI.f64 33.5

          \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right) \]

   5. Applied rewrites 33.5%

      \[\leadsto \color{blue}{\left(R \cdot \lambda_2\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right)} \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right) \]

      2. lift-+.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right) \]

      3. lift-fma.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\phi_2 + \phi_1\right) + \frac{\pi}{2}\right) \]

      4. lift-PI.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\phi_2 + \phi_1\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\phi_2 + \phi_1\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      6. sin-+PI/2 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_2 + \phi_1\right)\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_2 + \phi_1\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_2 + \phi_1\right)\right) \]

      9. lift-+.f64 54.6

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \]

   7. Applied rewrites 54.6%

      \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 60.1% accurate, N/A× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ \mathbf{if}\;\lambda_2 \leq 3.2 \cdot 10^{+155}:\\ \;\;\;\;R \cdot \sqrt{t\_0 \cdot t\_0 + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))))
   (if (<= lambda2 3.2e+155)
     (* R (sqrt (+ (* t_0 t_0) (* (- phi1 phi2) (- phi1 phi2)))))
     (* (* R lambda2) (cos (* 0.5 (+ phi2 phi1)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	double tmp;
	if (lambda2 <= 3.2e+155) {
		tmp = R * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
	} else {
		tmp = (R * lambda2) * cos((0.5 * (phi2 + phi1)));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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) :: tmp
    t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0d0))
    if (lambda2 <= 3.2d+155) then
        tmp = r * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))
    else
        tmp = (r * lambda2) * cos((0.5d0 * (phi2 + phi1)))
    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 t_0 = (lambda1 - lambda2) * Math.cos(((phi1 + phi2) / 2.0));
	double tmp;
	if (lambda2 <= 3.2e+155) {
		tmp = R * Math.sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
	} else {
		tmp = (R * lambda2) * Math.cos((0.5 * (phi2 + phi1)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (lambda1 - lambda2) * math.cos(((phi1 + phi2) / 2.0))
	tmp = 0
	if lambda2 <= 3.2e+155:
		tmp = R * math.sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))
	else:
		tmp = (R * lambda2) * math.cos((0.5 * (phi2 + phi1)))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0)))
	tmp = 0.0
	if (lambda2 <= 3.2e+155)
		tmp = Float64(R * sqrt(Float64(Float64(t_0 * t_0) + Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2)))));
	else
		tmp = Float64(Float64(R * lambda2) * cos(Float64(0.5 * Float64(phi2 + phi1))));
	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)
	t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	tmp = 0.0;
	if (lambda2 <= 3.2e+155)
		tmp = R * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
	else
		tmp = (R * lambda2) * cos((0.5 * (phi2 + phi1)));
	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_] := Block[{t$95$0 = N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, 3.2e+155], 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], N[(N[(R * lambda2), $MachinePrecision] * N[Cos[N[(0.5 * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 3.20000000000000012e155

   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

   if 3.20000000000000012e155 < lambda2

   1. Initial program 33.3%

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

   3. Taylor expanded in lambda2 around inf

      \[\leadsto \color{blue}{R \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \color{blue}{\left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)} \]

      4. sin-+PI/2-rev N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      5. lower-sin.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \phi_1 + \phi_2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \phi_2 + \phi_1, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \phi_2 + \phi_1, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \phi_2 + \phi_1, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      10. lower-PI.f64 33.5

          \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right) \]

   5. Applied rewrites 33.5%

      \[\leadsto \color{blue}{\left(R \cdot \lambda_2\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right)} \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right) \]

      2. lift-+.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right) \]

      3. lift-fma.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\phi_2 + \phi_1\right) + \frac{\pi}{2}\right) \]

      4. lift-PI.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\phi_2 + \phi_1\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\phi_2 + \phi_1\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      6. sin-+PI/2 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_2 + \phi_1\right)\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_2 + \phi_1\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_2 + \phi_1\right)\right) \]

      9. lift-+.f64 54.6

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \]

   7. Applied rewrites 54.6%

      \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 24.4% accurate, N/A× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \left(\mathsf{fma}\left(0.5, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right)\\ \mathbf{if}\;\lambda_1 \leq -2.7 \cdot 10^{-144}:\\ \;\;\;\;\left(-1 \cdot \lambda_1\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(R \cdot \lambda_2, t\_0, \frac{\frac{{\left(\phi_1 - \phi_2\right)}^{2} \cdot R}{\lambda_1}}{\sin \left(\mathsf{fma}\left(0.5, \phi_1, \frac{\pi}{2}\right)\right)} \cdot -0.5\right)}{\lambda_1}, -1, t\_0 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (sin (fma 0.5 (+ phi2 phi1) (/ PI 2.0)))))
   (if (<= lambda1 -2.7e-144)
     (*
      (* -1.0 lambda1)
      (fma
       (/
        (fma
         (* R lambda2)
         t_0
         (*
          (/
           (/ (* (pow (- phi1 phi2) 2.0) R) lambda1)
           (sin (fma 0.5 phi1 (/ PI 2.0))))
          -0.5))
        lambda1)
       -1.0
       (* t_0 R)))
     (* (* R lambda2) (cos (* 0.5 (+ phi2 phi1)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(fma(0.5, (phi2 + phi1), (((double) M_PI) / 2.0)));
	double tmp;
	if (lambda1 <= -2.7e-144) {
		tmp = (-1.0 * lambda1) * fma((fma((R * lambda2), t_0, ((((pow((phi1 - phi2), 2.0) * R) / lambda1) / sin(fma(0.5, phi1, (((double) M_PI) / 2.0)))) * -0.5)) / lambda1), -1.0, (t_0 * R));
	} else {
		tmp = (R * lambda2) * cos((0.5 * (phi2 + phi1)));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(fma(0.5, Float64(phi2 + phi1), Float64(pi / 2.0)))
	tmp = 0.0
	if (lambda1 <= -2.7e-144)
		tmp = Float64(Float64(-1.0 * lambda1) * fma(Float64(fma(Float64(R * lambda2), t_0, Float64(Float64(Float64(Float64((Float64(phi1 - phi2) ^ 2.0) * R) / lambda1) / sin(fma(0.5, phi1, Float64(pi / 2.0)))) * -0.5)) / lambda1), -1.0, Float64(t_0 * R)));
	else
		tmp = Float64(Float64(R * lambda2) * cos(Float64(0.5 * Float64(phi2 + phi1))));
	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_] := Block[{t$95$0 = N[Sin[N[(0.5 * N[(phi2 + phi1), $MachinePrecision] + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda1, -2.7e-144], N[(N[(-1.0 * lambda1), $MachinePrecision] * N[(N[(N[(N[(R * lambda2), $MachinePrecision] * t$95$0 + N[(N[(N[(N[(N[Power[N[(phi1 - phi2), $MachinePrecision], 2.0], $MachinePrecision] * R), $MachinePrecision] / lambda1), $MachinePrecision] / N[Sin[N[(0.5 * phi1 + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / lambda1), $MachinePrecision] * -1.0 + N[(t$95$0 * R), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(R * lambda2), $MachinePrecision] * N[Cos[N[(0.5 * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -2.69999999999999975e-144

   1. Initial program 57.9%

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

   3. Taylor expanded in lambda1 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\lambda_1 \cdot \left(-1 \cdot \frac{\frac{-1}{2} \cdot \frac{R \cdot {\left(\phi_1 - \phi_2\right)}^{2}}{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)} + R \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)}{\lambda_1} + R \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \lambda_1\right) \cdot \color{blue}{\left(-1 \cdot \frac{\frac{-1}{2} \cdot \frac{R \cdot {\left(\phi_1 - \phi_2\right)}^{2}}{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)} + R \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)}{\lambda_1} + R \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \lambda_1\right) \cdot \color{blue}{\left(-1 \cdot \frac{\frac{-1}{2} \cdot \frac{R \cdot {\left(\phi_1 - \phi_2\right)}^{2}}{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)} + R \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)}{\lambda_1} + R \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \lambda_1\right) \cdot \left(\color{blue}{-1 \cdot \frac{\frac{-1}{2} \cdot \frac{R \cdot {\left(\phi_1 - \phi_2\right)}^{2}}{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)} + R \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)}{\lambda_1}} + R \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \lambda_1\right) \cdot \left(\frac{\frac{-1}{2} \cdot \frac{R \cdot {\left(\phi_1 - \phi_2\right)}^{2}}{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)} + R \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)}{\lambda_1} \cdot -1 + \color{blue}{R} \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) \]

   5. Applied rewrites 37.5%

      \[\leadsto \color{blue}{\left(-1 \cdot \lambda_1\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(R \cdot \lambda_2, \sin \left(\mathsf{fma}\left(0.5, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right), \frac{\frac{{\left(\phi_1 - \phi_2\right)}^{2} \cdot R}{\lambda_1}}{\sin \left(\mathsf{fma}\left(0.5, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right)} \cdot -0.5\right)}{\lambda_1}, -1, \sin \left(\mathsf{fma}\left(0.5, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right) \cdot R\right)} \]

   6. Taylor expanded in phi1 around inf

      \[\leadsto \left(-1 \cdot \lambda_1\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(R \cdot \lambda_2, \sin \left(\mathsf{fma}\left(\frac{1}{2}, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right), \frac{\frac{{\left(\phi_1 - \phi_2\right)}^{2} \cdot R}{\lambda_1}}{\sin \left(\mathsf{fma}\left(\frac{1}{2}, \phi_1, \frac{\pi}{2}\right)\right)} \cdot \frac{-1}{2}\right)}{\lambda_1}, -1, \sin \left(\mathsf{fma}\left(\frac{1}{2}, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right) \cdot R\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 43.8%

      \[\leadsto \left(-1 \cdot \lambda_1\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(R \cdot \lambda_2, \sin \left(\mathsf{fma}\left(0.5, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right), \frac{\frac{{\left(\phi_1 - \phi_2\right)}^{2} \cdot R}{\lambda_1}}{\sin \left(\mathsf{fma}\left(0.5, \phi_1, \frac{\pi}{2}\right)\right)} \cdot -0.5\right)}{\lambda_1}, -1, \sin \left(\mathsf{fma}\left(0.5, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right) \cdot R\right) \]

   if -2.69999999999999975e-144 < lambda1

   1. Initial program 57.6%

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

   3. Taylor expanded in lambda2 around inf

      \[\leadsto \color{blue}{R \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \color{blue}{\left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)} \]

      4. sin-+PI/2-rev N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      5. lower-sin.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \phi_1 + \phi_2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \phi_2 + \phi_1, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \phi_2 + \phi_1, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \phi_2 + \phi_1, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      10. lower-PI.f64 12.8

          \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right) \]

   5. Applied rewrites 12.8%

      \[\leadsto \color{blue}{\left(R \cdot \lambda_2\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right)} \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right) \]

      2. lift-+.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right) \]

      3. lift-fma.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\phi_2 + \phi_1\right) + \frac{\pi}{2}\right) \]

      4. lift-PI.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\phi_2 + \phi_1\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\phi_2 + \phi_1\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      6. sin-+PI/2 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_2 + \phi_1\right)\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_2 + \phi_1\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_2 + \phi_1\right)\right) \]

      9. lift-+.f64 17.5

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \]

   7. Applied rewrites 17.5%

      \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 23.3% accurate, N/A× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \left(\mathsf{fma}\left(0.5, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right)\\ t_1 := t\_0 \cdot R\\ \mathbf{if}\;\lambda_2 \leq 3.6 \cdot 10^{+262}:\\ \;\;\;\;\left(-1 \cdot \lambda_1\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(R \cdot \lambda_2, t\_0, \frac{\frac{{\left(\phi_1 - \phi_2\right)}^{2} \cdot R}{\lambda_1}}{\sin \left(\mathsf{fma}\left(0.5, \phi_1, \frac{\pi}{2}\right)\right)} \cdot -0.5\right)}{\lambda_1}, -1, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot \lambda_2\right) \cdot \mathsf{fma}\left(\frac{\left(R \cdot \lambda_1\right) \cdot t\_0}{\lambda_2}, -1, t\_1\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (sin (fma 0.5 (+ phi2 phi1) (/ PI 2.0)))) (t_1 (* t_0 R)))
   (if (<= lambda2 3.6e+262)
     (*
      (* -1.0 lambda1)
      (fma
       (/
        (fma
         (* R lambda2)
         t_0
         (*
          (/
           (/ (* (pow (- phi1 phi2) 2.0) R) lambda1)
           (sin (fma 0.5 phi1 (/ PI 2.0))))
          -0.5))
        lambda1)
       -1.0
       t_1))
     (* (* -1.0 lambda2) (fma (/ (* (* R lambda1) t_0) lambda2) -1.0 t_1)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(fma(0.5, (phi2 + phi1), (((double) M_PI) / 2.0)));
	double t_1 = t_0 * R;
	double tmp;
	if (lambda2 <= 3.6e+262) {
		tmp = (-1.0 * lambda1) * fma((fma((R * lambda2), t_0, ((((pow((phi1 - phi2), 2.0) * R) / lambda1) / sin(fma(0.5, phi1, (((double) M_PI) / 2.0)))) * -0.5)) / lambda1), -1.0, t_1);
	} else {
		tmp = (-1.0 * lambda2) * fma((((R * lambda1) * t_0) / lambda2), -1.0, t_1);
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(fma(0.5, Float64(phi2 + phi1), Float64(pi / 2.0)))
	t_1 = Float64(t_0 * R)
	tmp = 0.0
	if (lambda2 <= 3.6e+262)
		tmp = Float64(Float64(-1.0 * lambda1) * fma(Float64(fma(Float64(R * lambda2), t_0, Float64(Float64(Float64(Float64((Float64(phi1 - phi2) ^ 2.0) * R) / lambda1) / sin(fma(0.5, phi1, Float64(pi / 2.0)))) * -0.5)) / lambda1), -1.0, t_1));
	else
		tmp = Float64(Float64(-1.0 * lambda2) * fma(Float64(Float64(Float64(R * lambda1) * t_0) / lambda2), -1.0, t_1));
	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_] := Block[{t$95$0 = N[Sin[N[(0.5 * N[(phi2 + phi1), $MachinePrecision] + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * R), $MachinePrecision]}, If[LessEqual[lambda2, 3.6e+262], N[(N[(-1.0 * lambda1), $MachinePrecision] * N[(N[(N[(N[(R * lambda2), $MachinePrecision] * t$95$0 + N[(N[(N[(N[(N[Power[N[(phi1 - phi2), $MachinePrecision], 2.0], $MachinePrecision] * R), $MachinePrecision] / lambda1), $MachinePrecision] / N[Sin[N[(0.5 * phi1 + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / lambda1), $MachinePrecision] * -1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * lambda2), $MachinePrecision] * N[(N[(N[(N[(R * lambda1), $MachinePrecision] * t$95$0), $MachinePrecision] / lambda2), $MachinePrecision] * -1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 3.59999999999999992e262

   1. Initial program 59.1%

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

   3. Taylor expanded in lambda1 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\lambda_1 \cdot \left(-1 \cdot \frac{\frac{-1}{2} \cdot \frac{R \cdot {\left(\phi_1 - \phi_2\right)}^{2}}{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)} + R \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)}{\lambda_1} + R \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \lambda_1\right) \cdot \color{blue}{\left(-1 \cdot \frac{\frac{-1}{2} \cdot \frac{R \cdot {\left(\phi_1 - \phi_2\right)}^{2}}{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)} + R \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)}{\lambda_1} + R \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \lambda_1\right) \cdot \color{blue}{\left(-1 \cdot \frac{\frac{-1}{2} \cdot \frac{R \cdot {\left(\phi_1 - \phi_2\right)}^{2}}{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)} + R \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)}{\lambda_1} + R \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \lambda_1\right) \cdot \left(\color{blue}{-1 \cdot \frac{\frac{-1}{2} \cdot \frac{R \cdot {\left(\phi_1 - \phi_2\right)}^{2}}{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)} + R \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)}{\lambda_1}} + R \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \lambda_1\right) \cdot \left(\frac{\frac{-1}{2} \cdot \frac{R \cdot {\left(\phi_1 - \phi_2\right)}^{2}}{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)} + R \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)}{\lambda_1} \cdot -1 + \color{blue}{R} \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) \]

   5. Applied rewrites 25.2%

      \[\leadsto \color{blue}{\left(-1 \cdot \lambda_1\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(R \cdot \lambda_2, \sin \left(\mathsf{fma}\left(0.5, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right), \frac{\frac{{\left(\phi_1 - \phi_2\right)}^{2} \cdot R}{\lambda_1}}{\sin \left(\mathsf{fma}\left(0.5, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right)} \cdot -0.5\right)}{\lambda_1}, -1, \sin \left(\mathsf{fma}\left(0.5, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right) \cdot R\right)} \]

   6. Taylor expanded in phi1 around inf

      \[\leadsto \left(-1 \cdot \lambda_1\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(R \cdot \lambda_2, \sin \left(\mathsf{fma}\left(\frac{1}{2}, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right), \frac{\frac{{\left(\phi_1 - \phi_2\right)}^{2} \cdot R}{\lambda_1}}{\sin \left(\mathsf{fma}\left(\frac{1}{2}, \phi_1, \frac{\pi}{2}\right)\right)} \cdot \frac{-1}{2}\right)}{\lambda_1}, -1, \sin \left(\mathsf{fma}\left(\frac{1}{2}, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right) \cdot R\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 25.9%

      \[\leadsto \left(-1 \cdot \lambda_1\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(R \cdot \lambda_2, \sin \left(\mathsf{fma}\left(0.5, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right), \frac{\frac{{\left(\phi_1 - \phi_2\right)}^{2} \cdot R}{\lambda_1}}{\sin \left(\mathsf{fma}\left(0.5, \phi_1, \frac{\pi}{2}\right)\right)} \cdot -0.5\right)}{\lambda_1}, -1, \sin \left(\mathsf{fma}\left(0.5, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right) \cdot R\right) \]

   if 3.59999999999999992e262 < lambda2

   1. Initial program 29.4%

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

   3. Taylor expanded in lambda2 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\lambda_2 \cdot \left(-1 \cdot \frac{R \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)}{\lambda_2} + R \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \lambda_2\right) \cdot \color{blue}{\left(-1 \cdot \frac{R \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)}{\lambda_2} + R \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \left(\color{blue}{-1 \cdot \frac{R \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)}{\lambda_2}} + R \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \color{blue}{\left(-1 \cdot \frac{R \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)}{\lambda_2} + R \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \lambda_2\right) \cdot \left(\color{blue}{-1 \cdot \frac{R \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)}{\lambda_2}} + R \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \lambda_2\right) \cdot \left(\color{blue}{-1 \cdot \frac{R \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)}{\lambda_2}} + R \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \lambda_2\right) \cdot \left(\frac{R \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)}{\lambda_2} \cdot -1 + \color{blue}{R} \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(-1 \cdot \lambda_2\right) \cdot \mathsf{fma}\left(\frac{R \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)}{\lambda_2}, \color{blue}{-1}, R \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) \]

   5. Applied rewrites 34.1%

      \[\leadsto \color{blue}{\left(-1 \cdot \lambda_2\right) \cdot \mathsf{fma}\left(\frac{\left(R \cdot \lambda_1\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right)}{\lambda_2}, -1, \sin \left(\mathsf{fma}\left(0.5, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right) \cdot R\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 23.0% accurate, N/A× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \left(\mathsf{fma}\left(0.5, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right)\\ \left(-1 \cdot \lambda_1\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(R \cdot \lambda_2, t\_0, \frac{\frac{{\left(\phi_1 - \phi_2\right)}^{2} \cdot R}{\lambda_1}}{\sin \left(\mathsf{fma}\left(0.5, \phi_1, \frac{\pi}{2}\right)\right)} \cdot -0.5\right)}{\lambda_1}, -1, t\_0 \cdot R\right) \end{array} \end{array} \]

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
 (let* ((t_0 (sin (fma 0.5 (+ phi2 phi1) (/ PI 2.0)))))
   (*
    (* -1.0 lambda1)
    (fma
     (/
      (fma
       (* R lambda2)
       t_0
       (*
        (/
         (/ (* (pow (- phi1 phi2) 2.0) R) lambda1)
         (sin (fma 0.5 phi1 (/ PI 2.0))))
        -0.5))
      lambda1)
     -1.0
     (* t_0 R)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(fma(0.5, (phi2 + phi1), (((double) M_PI) / 2.0)));
	return (-1.0 * lambda1) * fma((fma((R * lambda2), t_0, ((((pow((phi1 - phi2), 2.0) * R) / lambda1) / sin(fma(0.5, phi1, (((double) M_PI) / 2.0)))) * -0.5)) / lambda1), -1.0, (t_0 * R));
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(fma(0.5, Float64(phi2 + phi1), Float64(pi / 2.0)))
	return Float64(Float64(-1.0 * lambda1) * fma(Float64(fma(Float64(R * lambda2), t_0, Float64(Float64(Float64(Float64((Float64(phi1 - phi2) ^ 2.0) * R) / lambda1) / sin(fma(0.5, phi1, Float64(pi / 2.0)))) * -0.5)) / lambda1), -1.0, Float64(t_0 * R)))
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_] := Block[{t$95$0 = N[Sin[N[(0.5 * N[(phi2 + phi1), $MachinePrecision] + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(-1.0 * lambda1), $MachinePrecision] * N[(N[(N[(N[(R * lambda2), $MachinePrecision] * t$95$0 + N[(N[(N[(N[(N[Power[N[(phi1 - phi2), $MachinePrecision], 2.0], $MachinePrecision] * R), $MachinePrecision] / lambda1), $MachinePrecision] / N[Sin[N[(0.5 * phi1 + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / lambda1), $MachinePrecision] * -1.0 + N[(t$95$0 * R), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 57.7%

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

3. Taylor expanded in lambda1 around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(\lambda_1 \cdot \left(-1 \cdot \frac{\frac{-1}{2} \cdot \frac{R \cdot {\left(\phi_1 - \phi_2\right)}^{2}}{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)} + R \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)}{\lambda_1} + R \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(-1 \cdot \lambda_1\right) \cdot \color{blue}{\left(-1 \cdot \frac{\frac{-1}{2} \cdot \frac{R \cdot {\left(\phi_1 - \phi_2\right)}^{2}}{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)} + R \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)}{\lambda_1} + R \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(-1 \cdot \lambda_1\right) \cdot \color{blue}{\left(-1 \cdot \frac{\frac{-1}{2} \cdot \frac{R \cdot {\left(\phi_1 - \phi_2\right)}^{2}}{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)} + R \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)}{\lambda_1} + R \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \left(-1 \cdot \lambda_1\right) \cdot \left(\color{blue}{-1 \cdot \frac{\frac{-1}{2} \cdot \frac{R \cdot {\left(\phi_1 - \phi_2\right)}^{2}}{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)} + R \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)}{\lambda_1}} + R \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \left(-1 \cdot \lambda_1\right) \cdot \left(\frac{\frac{-1}{2} \cdot \frac{R \cdot {\left(\phi_1 - \phi_2\right)}^{2}}{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)} + R \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)}{\lambda_1} \cdot -1 + \color{blue}{R} \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) \]

5. Applied rewrites 24.2%

   \[\leadsto \color{blue}{\left(-1 \cdot \lambda_1\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(R \cdot \lambda_2, \sin \left(\mathsf{fma}\left(0.5, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right), \frac{\frac{{\left(\phi_1 - \phi_2\right)}^{2} \cdot R}{\lambda_1}}{\sin \left(\mathsf{fma}\left(0.5, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right)} \cdot -0.5\right)}{\lambda_1}, -1, \sin \left(\mathsf{fma}\left(0.5, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right) \cdot R\right)} \]

6. Taylor expanded in phi1 around inf

   \[\leadsto \left(-1 \cdot \lambda_1\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(R \cdot \lambda_2, \sin \left(\mathsf{fma}\left(\frac{1}{2}, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right), \frac{\frac{{\left(\phi_1 - \phi_2\right)}^{2} \cdot R}{\lambda_1}}{\sin \left(\mathsf{fma}\left(\frac{1}{2}, \phi_1, \frac{\pi}{2}\right)\right)} \cdot \frac{-1}{2}\right)}{\lambda_1}, -1, \sin \left(\mathsf{fma}\left(\frac{1}{2}, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right) \cdot R\right) \]
7. Step-by-step derivation

8. Applied rewrites 24.8%

   \[\leadsto \left(-1 \cdot \lambda_1\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(R \cdot \lambda_2, \sin \left(\mathsf{fma}\left(0.5, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right), \frac{\frac{{\left(\phi_1 - \phi_2\right)}^{2} \cdot R}{\lambda_1}}{\sin \left(\mathsf{fma}\left(0.5, \phi_1, \frac{\pi}{2}\right)\right)} \cdot -0.5\right)}{\lambda_1}, -1, \sin \left(\mathsf{fma}\left(0.5, \phi_2 + \phi_1, \frac{\pi}{2}\right)\right) \cdot R\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(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))))))