Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.2% → 96.2%

Time: 6.5s

Alternatives: 9

Speedup: 2.4×

• 30.3% 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: 59.2% 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: 96.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -140000:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, \cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, \cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

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 -140000.0)
   (* (hypot (- phi1 phi2) (* (cos (* 0.5 phi1)) (- lambda1 lambda2))) R)
   (* (hypot (- phi1 phi2) (* (cos (* 0.5 phi2)) (- 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 <= -140000.0) {
		tmp = hypot((phi1 - phi2), (cos((0.5 * phi1)) * (lambda1 - lambda2))) * R;
	} else {
		tmp = hypot((phi1 - phi2), (cos((0.5 * phi2)) * (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 <= -140000.0) {
		tmp = Math.hypot((phi1 - phi2), (Math.cos((0.5 * phi1)) * (lambda1 - lambda2))) * R;
	} else {
		tmp = Math.hypot((phi1 - phi2), (Math.cos((0.5 * phi2)) * (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 <= -140000.0:
		tmp = math.hypot((phi1 - phi2), (math.cos((0.5 * phi1)) * (lambda1 - lambda2))) * R
	else:
		tmp = math.hypot((phi1 - phi2), (math.cos((0.5 * phi2)) * (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 <= -140000.0)
		tmp = Float64(hypot(Float64(phi1 - phi2), Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2))) * R);
	else
		tmp = Float64(hypot(Float64(phi1 - phi2), Float64(cos(Float64(0.5 * phi2)) * 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 <= -140000.0)
		tmp = hypot((phi1 - phi2), (cos((0.5 * phi1)) * (lambda1 - lambda2))) * R;
	else
		tmp = hypot((phi1 - phi2), (cos((0.5 * phi2)) * (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, -140000.0], N[(N[Sqrt[N[(phi1 - phi2), $MachinePrecision] ^ 2 + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(phi1 - phi2), $MachinePrecision] ^ 2 + N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.4e5

   1. Initial program 60.2%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 91.9%

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

   5. Taylor expanded in phi1 around inf

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

      1. lower-*.f64 92.0

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

   7. Applied rewrites 92.0%

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

   if -1.4e5 < 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 97.4%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 90.9

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

   7. Applied rewrites 90.9%

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

Alternative 2: 90.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -2 \cdot 10^{+220}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1, \cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, \cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

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 (<= (- lambda1 lambda2) -2e+220)
   (* (hypot phi1 (* (cos (* 0.5 phi2)) (- lambda1 lambda2))) R)
   (* (hypot (- phi1 phi2) (* (cos (* 0.5 phi1)) (- 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 ((lambda1 - lambda2) <= -2e+220) {
		tmp = hypot(phi1, (cos((0.5 * phi2)) * (lambda1 - lambda2))) * R;
	} else {
		tmp = hypot((phi1 - phi2), (cos((0.5 * phi1)) * (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 ((lambda1 - lambda2) <= -2e+220) {
		tmp = Math.hypot(phi1, (Math.cos((0.5 * phi2)) * (lambda1 - lambda2))) * R;
	} else {
		tmp = Math.hypot((phi1 - phi2), (Math.cos((0.5 * phi1)) * (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 (lambda1 - lambda2) <= -2e+220:
		tmp = math.hypot(phi1, (math.cos((0.5 * phi2)) * (lambda1 - lambda2))) * R
	else:
		tmp = math.hypot((phi1 - phi2), (math.cos((0.5 * phi1)) * (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 (Float64(lambda1 - lambda2) <= -2e+220)
		tmp = Float64(hypot(phi1, Float64(cos(Float64(0.5 * phi2)) * Float64(lambda1 - lambda2))) * R);
	else
		tmp = Float64(hypot(Float64(phi1 - phi2), Float64(cos(Float64(0.5 * phi1)) * 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 ((lambda1 - lambda2) <= -2e+220)
		tmp = hypot(phi1, (cos((0.5 * phi2)) * (lambda1 - lambda2))) * R;
	else
		tmp = hypot((phi1 - phi2), (cos((0.5 * phi1)) * (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[N[(lambda1 - lambda2), $MachinePrecision], -2e+220], N[(N[Sqrt[phi1 ^ 2 + N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(phi1 - phi2), $MachinePrecision] ^ 2 + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < -2e220

   1. Initial program 48.5%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing
   3. 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 92.6%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 82.2

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

   7. Applied rewrites 82.2%

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

   8. Taylor expanded in phi1 around inf

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

   10. Applied rewrites 82.2%

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

   if -2e220 < (-.f64 lambda1 lambda2)

   1. Initial program 61.8%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 96.8%

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

   5. Taylor expanded in phi1 around inf

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

      1. lower-*.f64 92.2

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

   7. Applied rewrites 92.2%

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

Alternative 3: 96.5% accurate, 1.2× speedup

Math

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

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 (- phi1 phi2) (* (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((phi1 - phi2), (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((phi1 - phi2), (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((phi1 - phi2), (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), 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), (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[(phi1 - phi2), $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 59.5%

   \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
2. Add Preprocessing
3. 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 96.1%

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

Alternative 4: 85.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{+174}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1, \cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, 1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

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 (<= (- lambda1 lambda2) -1e+174)
   (* (hypot phi1 (* (cos (* 0.5 phi2)) (- lambda1 lambda2))) R)
   (* (hypot (- phi1 phi2) (* 1.0 (- 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 ((lambda1 - lambda2) <= -1e+174) {
		tmp = hypot(phi1, (cos((0.5 * phi2)) * (lambda1 - lambda2))) * R;
	} else {
		tmp = hypot((phi1 - phi2), (1.0 * (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 ((lambda1 - lambda2) <= -1e+174) {
		tmp = Math.hypot(phi1, (Math.cos((0.5 * phi2)) * (lambda1 - lambda2))) * R;
	} else {
		tmp = Math.hypot((phi1 - phi2), (1.0 * (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 (lambda1 - lambda2) <= -1e+174:
		tmp = math.hypot(phi1, (math.cos((0.5 * phi2)) * (lambda1 - lambda2))) * R
	else:
		tmp = math.hypot((phi1 - phi2), (1.0 * (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 (Float64(lambda1 - lambda2) <= -1e+174)
		tmp = Float64(hypot(phi1, Float64(cos(Float64(0.5 * phi2)) * Float64(lambda1 - lambda2))) * R);
	else
		tmp = Float64(hypot(Float64(phi1 - phi2), Float64(1.0 * 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 ((lambda1 - lambda2) <= -1e+174)
		tmp = hypot(phi1, (cos((0.5 * phi2)) * (lambda1 - lambda2))) * R;
	else
		tmp = hypot((phi1 - phi2), (1.0 * (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[N[(lambda1 - lambda2), $MachinePrecision], -1e+174], N[(N[Sqrt[phi1 ^ 2 + N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(phi1 - phi2), $MachinePrecision] ^ 2 + N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < -1.00000000000000007e174

   1. Initial program 47.3%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 92.8%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 83.1

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

   7. Applied rewrites 83.1%

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

   8. Taylor expanded in phi1 around inf

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

   10. Applied rewrites 80.1%

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

   if -1.00000000000000007e174 < (-.f64 lambda1 lambda2)

   1. Initial program 63.3%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 97.1%

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

   5. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 83.8

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

   7. Applied rewrites 83.8%

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

   8. Taylor expanded in phi2 around 0

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

   10. Applied rewrites 86.4%

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

Alternative 5: 85.1% accurate, 2.4× speedup

Math

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

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 (- phi1 phi2) (* 1.0 (- 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((phi1 - phi2), (1.0 * (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((phi1 - phi2), (1.0 * (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((phi1 - phi2), (1.0 * (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), Float64(1.0 * 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 * (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[(phi1 - phi2), $MachinePrecision] ^ 2 + N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 59.5%

   \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
2. Add Preprocessing
3. 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 96.1%

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

5. Taylor expanded in phi1 around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-*.f64 82.8

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

7. Applied rewrites 82.8%

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

8. Taylor expanded in phi2 around 0

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

10. Applied rewrites 82.6%

    \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, 1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
11. Add Preprocessing

Alternative 6: 58.8% accurate, 8.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8.5 \cdot 10^{+133}:\\ \;\;\;\;\mathsf{fma}\left(-1, \phi_1, \phi_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(-\phi_2\right) \cdot \mathsf{fma}\left(R, \frac{\phi_1}{\phi_2}, -R\right)\\ \end{array} \end{array} \]

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
 (if (<= phi1 -8.5e+133)
   (* (fma -1.0 phi1 phi2) R)
   (* (- phi2) (fma R (/ phi1 phi2) (- R)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -8.5e+133) {
		tmp = fma(-1.0, phi1, phi2) * R;
	} else {
		tmp = -phi2 * fma(R, (phi1 / phi2), -R);
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -8.5e+133)
		tmp = Float64(fma(-1.0, phi1, phi2) * R);
	else
		tmp = Float64(Float64(-phi2) * fma(R, Float64(phi1 / phi2), Float64(-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, -8.5e+133], N[(N[(-1.0 * phi1 + phi2), $MachinePrecision] * R), $MachinePrecision], N[((-phi2) * N[(R * N[(phi1 / phi2), $MachinePrecision] + (-R)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -8.50000000000000044e133

   1. Initial program 60.2%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

         \[\leadsto \left(-\phi_1\right) \cdot \mathsf{fma}\left(\frac{R \cdot \phi_2}{\phi_1}, -1, R\right) \]

      9. *-commutative N/A

         \[\leadsto \left(-\phi_1\right) \cdot \mathsf{fma}\left(\frac{\phi_2 \cdot R}{\phi_1}, -1, R\right) \]

      10. lower-*.f64 74.8

          \[\leadsto \left(-\phi_1\right) \cdot \mathsf{fma}\left(\frac{\phi_2 \cdot R}{\phi_1}, -1, R\right) \]

   5. Applied rewrites 74.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. flip-+ N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 51.2%

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

   8. Taylor expanded in phi1 around 0

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

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot R\right) \cdot \phi_1 + R \cdot \phi_2 \]

      2. mul-1-neg N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(R\right), \phi_1, R \cdot \phi_2\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-R, \phi_1, R \cdot \phi_2\right) \]

      5. lower-*.f64 74.8

         \[\leadsto \mathsf{fma}\left(-R, \phi_1, R \cdot \phi_2\right) \]

   10. Applied rewrites 74.8%

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

   11. Taylor expanded in R around 0

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

       1. *-commutative N/A

          \[\leadsto \left(\phi_2 + -1 \cdot \phi_1\right) \cdot R \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\phi_2 + -1 \cdot \phi_1\right) \cdot R \]

       3. mul-1-neg N/A

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

       4. +-commutative N/A

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

       5. mul-1-neg N/A

          \[\leadsto \left(-1 \cdot \phi_1 + \phi_2\right) \cdot R \]

       6. lower-fma.f64 77.6

          \[\leadsto \mathsf{fma}\left(-1, \phi_1, \phi_2\right) \cdot R \]

   13. Applied rewrites 77.6%

       \[\leadsto \mathsf{fma}\left(-1, \phi_1, \phi_2\right) \cdot R \]

   if -8.50000000000000044e133 < phi1

   1. Initial program 59.4%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

         \[\leadsto \left(-\phi_1\right) \cdot \mathsf{fma}\left(\frac{R \cdot \phi_2}{\phi_1}, -1, R\right) \]

      9. *-commutative N/A

         \[\leadsto \left(-\phi_1\right) \cdot \mathsf{fma}\left(\frac{\phi_2 \cdot R}{\phi_1}, -1, R\right) \]

      10. lower-*.f64 19.4

          \[\leadsto \left(-\phi_1\right) \cdot \mathsf{fma}\left(\frac{\phi_2 \cdot R}{\phi_1}, -1, R\right) \]

   5. Applied rewrites 19.4%

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

   6. Taylor expanded in phi2 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

         \[\leadsto \left(-\phi_2\right) \cdot \mathsf{fma}\left(R, \frac{\phi_1}{\phi_2}, \mathsf{neg}\left(R\right)\right) \]

      10. lower-neg.f64 19.6

          \[\leadsto \left(-\phi_2\right) \cdot \mathsf{fma}\left(R, \frac{\phi_1}{\phi_2}, -R\right) \]

   8. Applied rewrites 19.6%

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

4. Final simplification 27.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8.5 \cdot 10^{+133}:\\ \;\;\;\;\mathsf{fma}\left(-1, \phi_1, \phi_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(-\phi_2\right) \cdot \mathsf{fma}\left(R, \frac{\phi_1}{\phi_2}, -R\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.5% accurate, 19.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.0046:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -0.0046) (* R (- phi1)) (* R phi2)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -0.0046) {
		tmp = R * -phi1;
	} else {
		tmp = R * phi2;
	}
	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) :: tmp
    if (phi1 <= (-0.0046d0)) then
        tmp = r * -phi1
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -0.0046) {
		tmp = R * -phi1;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -0.0046:
		tmp = R * -phi1
	else:
		tmp = R * phi2
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -0.0046)
		tmp = Float64(R * Float64(-phi1));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -0.0046)
		tmp = R * -phi1;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -0.0046], N[(R * (-phi1)), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.0045999999999999999

   1. Initial program 60.1%

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

   3. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 63.0

         \[\leadsto R \cdot \left(-\phi_1\right) \]

   5. Applied rewrites 63.0%

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

   if -0.0045999999999999999 < 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. Taylor expanded in phi2 around inf

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

   5. Applied rewrites 16.6%

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

Alternative 8: 58.0% accurate, 23.3× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \mathsf{fma}\left(-1, \phi_1, \phi_2\right) \cdot R \end{array} \]

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
 (* (fma -1.0 phi1 phi2) R))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return fma(-1.0, phi1, phi2) * R;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(fma(-1.0, phi1, phi2) * 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[(-1.0 * phi1 + phi2), $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 59.5%

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

3. Taylor expanded in phi1 around -inf

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

      \[\leadsto \left(-\phi_1\right) \cdot \mathsf{fma}\left(\frac{R \cdot \phi_2}{\phi_1}, -1, R\right) \]

   9. *-commutative N/A

      \[\leadsto \left(-\phi_1\right) \cdot \mathsf{fma}\left(\frac{\phi_2 \cdot R}{\phi_1}, -1, R\right) \]

   10. lower-*.f64 27.2

       \[\leadsto \left(-\phi_1\right) \cdot \mathsf{fma}\left(\frac{\phi_2 \cdot R}{\phi_1}, -1, R\right) \]

5. Applied rewrites 27.2%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. flip-+ N/A

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

   5. lower-/.f64 N/A

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

7. Applied rewrites 13.4%

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

8. Taylor expanded in phi1 around 0

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

   1. associate-*r* N/A

      \[\leadsto \left(-1 \cdot R\right) \cdot \phi_1 + R \cdot \phi_2 \]

   2. mul-1-neg N/A

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

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(R\right), \phi_1, R \cdot \phi_2\right) \]

   4. lower-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(-R, \phi_1, R \cdot \phi_2\right) \]

   5. lower-*.f64 27.1

      \[\leadsto \mathsf{fma}\left(-R, \phi_1, R \cdot \phi_2\right) \]

10. Applied rewrites 27.1%

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

11. Taylor expanded in R around 0

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

    1. *-commutative N/A

       \[\leadsto \left(\phi_2 + -1 \cdot \phi_1\right) \cdot R \]

    2. lower-*.f64 N/A

       \[\leadsto \left(\phi_2 + -1 \cdot \phi_1\right) \cdot R \]

    3. mul-1-neg N/A

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

    4. +-commutative N/A

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

    5. mul-1-neg N/A

       \[\leadsto \left(-1 \cdot \phi_1 + \phi_2\right) \cdot R \]

    6. lower-fma.f64 26.7

       \[\leadsto \mathsf{fma}\left(-1, \phi_1, \phi_2\right) \cdot R \]

13. Applied rewrites 26.7%

    \[\leadsto \mathsf{fma}\left(-1, \phi_1, \phi_2\right) \cdot R \]
14. Add Preprocessing

Alternative 9: 31.6% accurate, 46.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.5%

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

3. Taylor expanded in phi2 around inf

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

5. Applied rewrites 15.5%

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

Reproduce

herbie shell --seed 2025037 
(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))))))