Equirectangular approximation to distance on a great circle

Percentage Accurate: 58.7% → 95.8%

Time: 5.5s

Alternatives: 12

Speedup: 2.6×

• 30.5% 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: 58.7% 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, 1.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.025:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, \cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, \cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

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 -0.025)
   (* (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 <= -0.025) {
		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 <= -0.025) {
		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 <= -0.025:
		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 <= -0.025)
		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 <= -0.025)
		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, -0.025], 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 < -0.025000000000000001

   1. Initial program 58.7%

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

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

   3. Applied rewrites 95.8%

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

   4. Taylor expanded in phi1 around inf

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

      1. lower-*.f64 90.3

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

   6. Applied rewrites 90.3%

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

   if -0.025000000000000001 < phi1

   1. Initial program 58.7%

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

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

   3. Applied rewrites 95.8%

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

   4. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 90.1

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

   6. Applied rewrites 90.1%

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

Alternative 2: 95.7% accurate, 1.7× 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 58.7%

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

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

3. Applied rewrites 95.8%

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

Alternative 3: 90.3% accurate, 1.8× speedup

Math

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

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 (* 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) {
	return hypot((phi1 - phi2), (cos((0.5 * phi1)) * (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((0.5 * phi1)) * (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((0.5 * phi1)) * (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(0.5 * phi1)) * 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((0.5 * phi1)) * (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[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 58.7%

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

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

3. Applied rewrites 95.8%

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

4. Taylor expanded in phi1 around inf

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

   1. lower-*.f64 90.3

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

6. Applied rewrites 90.3%

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

Alternative 4: 81.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.8 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{hypot}\left(-\phi_2, \cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, \cos \left(0.5 \cdot \phi_2\right) \cdot \left(-\lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

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 -1.8e+66)
   (* (hypot (- phi2) (* (cos (* 0.5 phi1)) (- lambda1 lambda2))) R)
   (* (hypot (- phi1 phi2) (* (cos (* 0.5 phi2)) (- 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 <= -1.8e+66) {
		tmp = hypot(-phi2, (cos((0.5 * phi1)) * (lambda1 - lambda2))) * R;
	} else {
		tmp = hypot((phi1 - phi2), (cos((0.5 * phi2)) * -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 <= -1.8e+66) {
		tmp = Math.hypot(-phi2, (Math.cos((0.5 * phi1)) * (lambda1 - lambda2))) * R;
	} else {
		tmp = Math.hypot((phi1 - phi2), (Math.cos((0.5 * phi2)) * -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 <= -1.8e+66:
		tmp = math.hypot(-phi2, (math.cos((0.5 * phi1)) * (lambda1 - lambda2))) * R
	else:
		tmp = math.hypot((phi1 - phi2), (math.cos((0.5 * phi2)) * -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 (lambda1 <= -1.8e+66)
		tmp = Float64(hypot(Float64(-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(-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 <= -1.8e+66)
		tmp = hypot(-phi2, (cos((0.5 * phi1)) * (lambda1 - lambda2))) * R;
	else
		tmp = hypot((phi1 - phi2), (cos((0.5 * phi2)) * -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[lambda1, -1.8e+66], N[(N[Sqrt[(-phi2) ^ 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] * (-lambda2)), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.8e66

   1. Initial program 58.7%

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

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

   3. Applied rewrites 95.8%

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

   4. Taylor expanded in phi1 around inf

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

      1. lower-*.f64 90.3

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

   6. Applied rewrites 90.3%

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

   7. Taylor expanded in phi1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 69.9

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

   9. Applied rewrites 69.9%

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

   if -1.8e66 < lambda1

   1. Initial program 58.7%

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

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

   3. Applied rewrites 95.8%

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

   4. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 90.1

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

   6. Applied rewrites 90.1%

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

   7. Taylor expanded in lambda1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 74.9

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

   9. Applied rewrites 74.9%

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

Alternative 5: 80.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.1 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(R, \frac{\phi_2}{\phi_1}, -R\right) \cdot \phi_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(-\phi_2, \cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

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 (<= phi1 -1.1e+108)
   (* (fma R (/ phi2 phi1) (- R)) phi1)
   (* (hypot (- 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 (phi1 <= -1.1e+108) {
		tmp = fma(R, (phi2 / phi1), -R) * phi1;
	} else {
		tmp = hypot(-phi2, (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 (phi1 <= -1.1e+108)
		tmp = Float64(fma(R, Float64(phi2 / phi1), Float64(-R)) * phi1);
	else
		tmp = Float64(hypot(Float64(-phi2), Float64(cos(Float64(0.5 * phi1)) * 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, -1.1e+108], N[(N[(R * N[(phi2 / phi1), $MachinePrecision] + (-R)), $MachinePrecision] * phi1), $MachinePrecision], N[(N[Sqrt[(-phi2) ^ 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 phi1 < -1.1000000000000001e108

   1. Initial program 58.7%

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

   2. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 55.9

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

   4. Applied rewrites 55.9%

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

   5. Taylor expanded in R around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 54.2

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

   7. Applied rewrites 54.2%

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

   8. Taylor expanded in phi1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 54.1

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

   10. Applied rewrites 54.1%

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

   11. Taylor expanded in phi1 around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. associate-/l* N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. mul-1-neg N/A

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

       8. lower-neg.f64 53.6

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

   13. Applied rewrites 53.6%

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

   if -1.1000000000000001e108 < phi1

   1. Initial program 58.7%

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

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

   3. Applied rewrites 95.8%

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

   4. Taylor expanded in phi1 around inf

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

      1. lower-*.f64 90.3

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

   6. Applied rewrites 90.3%

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

   7. Taylor expanded in phi1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 69.9

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

   9. Applied rewrites 69.9%

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

Alternative 6: 80.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.75 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(R, \frac{\phi_2}{\phi_1}, -R\right) \cdot \phi_1\\ \mathbf{elif}\;\phi_1 \leq 8.2 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, \mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.125, 1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(-\phi_2, \cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

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 -1.75e+59)
   (* (fma R (/ phi2 phi1) (- R)) phi1)
   (if (<= phi1 8.2e+14)
     (*
      (hypot
       (- phi1 phi2)
       (* (fma (* phi1 phi1) -0.125 1.0) (- lambda1 lambda2)))
      R)
     (* (hypot (- phi2) (* (cos (* 0.5 phi1)) (- 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 <= -1.75e+59) {
		tmp = fma(R, (phi2 / phi1), -R) * phi1;
	} else if (phi1 <= 8.2e+14) {
		tmp = hypot((phi1 - phi2), (fma((phi1 * phi1), -0.125, 1.0) * (lambda1 - lambda2))) * R;
	} else {
		tmp = hypot(-phi2, (cos((0.5 * phi1)) * -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 <= -1.75e+59)
		tmp = Float64(fma(R, Float64(phi2 / phi1), Float64(-R)) * phi1);
	elseif (phi1 <= 8.2e+14)
		tmp = Float64(hypot(Float64(phi1 - phi2), Float64(fma(Float64(phi1 * phi1), -0.125, 1.0) * Float64(lambda1 - lambda2))) * R);
	else
		tmp = Float64(hypot(Float64(-phi2), Float64(cos(Float64(0.5 * phi1)) * Float64(-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, -1.75e+59], N[(N[(R * N[(phi2 / phi1), $MachinePrecision] + (-R)), $MachinePrecision] * phi1), $MachinePrecision], If[LessEqual[phi1, 8.2e+14], N[(N[Sqrt[N[(phi1 - phi2), $MachinePrecision] ^ 2 + N[(N[(N[(phi1 * phi1), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[(-phi2) ^ 2 + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * (-lambda2)), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.75e59

   1. Initial program 58.7%

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

   2. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 55.9

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

   4. Applied rewrites 55.9%

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

   5. Taylor expanded in R around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 54.2

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

   7. Applied rewrites 54.2%

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

   8. Taylor expanded in phi1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 54.1

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

   10. Applied rewrites 54.1%

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

   11. Taylor expanded in phi1 around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. associate-/l* N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. mul-1-neg N/A

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

       8. lower-neg.f64 53.6

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

   13. Applied rewrites 53.6%

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

   if -1.75e59 < phi1 < 8.2e14

   1. Initial program 58.7%

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

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

   3. Applied rewrites 95.8%

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

   4. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \phi_1 \cdot \left(\frac{-1}{8} \cdot \left(\phi_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) - \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \color{blue}{\phi_1 \cdot \left(\frac{-1}{8} \cdot \left(\phi_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) - \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \color{blue}{\phi_1} \cdot \left(\frac{-1}{8} \cdot \left(\phi_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) - \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \phi_1 \cdot \left(\frac{-1}{8} \cdot \left(\phi_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) - \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \phi_1 \cdot \color{blue}{\left(\frac{-1}{8} \cdot \left(\phi_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) - \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \phi_1 \cdot \left(\frac{-1}{8} \cdot \left(\phi_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) - \color{blue}{\frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)}\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \phi_1 \cdot \left(\frac{-1}{8} \cdot \left(\phi_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) - \color{blue}{\frac{1}{2}} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \phi_1 \cdot \left(\frac{-1}{8} \cdot \left(\phi_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) - \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      8. lower-cos.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \phi_1 \cdot \left(\frac{-1}{8} \cdot \left(\phi_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) - \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \phi_1 \cdot \left(\frac{-1}{8} \cdot \left(\phi_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) - \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \phi_1 \cdot \left(\frac{-1}{8} \cdot \left(\phi_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) - \frac{1}{2} \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_2\right)}\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      11. lower-sin.f64 N/A

          \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \phi_1 \cdot \left(\frac{-1}{8} \cdot \left(\phi_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) - \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      12. lower-*.f64 72.1

          \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\cos \left(0.5 \cdot \phi_2\right) + \phi_1 \cdot \left(-0.125 \cdot \left(\phi_1 \cdot \cos \left(0.5 \cdot \phi_2\right)\right) - 0.5 \cdot \sin \left(0.5 \cdot \phi_2\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   6. Applied rewrites 72.1%

      \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \color{blue}{\left(\cos \left(0.5 \cdot \phi_2\right) + \phi_1 \cdot \left(-0.125 \cdot \left(\phi_1 \cdot \cos \left(0.5 \cdot \phi_2\right)\right) - 0.5 \cdot \sin \left(0.5 \cdot \phi_2\right)\right)\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   7. Taylor expanded in phi1 around -inf

      \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left({\phi_1}^{2} \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\cos \left(\frac{1}{2} \cdot \phi_2\right)}{\phi_1} + \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)}{\phi_1} + \frac{-1}{8} \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left({\phi_1}^{2} \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\cos \left(\frac{1}{2} \cdot \phi_2\right)}{\phi_1} + \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)}{\phi_1} + \color{blue}{\frac{-1}{8} \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      2. unpow2 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\left(\phi_1 \cdot \phi_1\right) \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\cos \left(\frac{1}{2} \cdot \phi_2\right)}{\phi_1} + \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)}{\phi_1} + \color{blue}{\frac{-1}{8}} \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\left(\phi_1 \cdot \phi_1\right) \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\cos \left(\frac{1}{2} \cdot \phi_2\right)}{\phi_1} + \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)}{\phi_1} + \color{blue}{\frac{-1}{8}} \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\left(\phi_1 \cdot \phi_1\right) \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\cos \left(\frac{1}{2} \cdot \phi_2\right)}{\phi_1} + \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)}{\phi_1} - \left(\mathsf{neg}\left(\frac{-1}{8}\right)\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_2\right)}\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\left(\phi_1 \cdot \phi_1\right) \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\cos \left(\frac{1}{2} \cdot \phi_2\right)}{\phi_1} + \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)}{\phi_1} - \frac{1}{8} \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\left(\phi_1 \cdot \phi_1\right) \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\cos \left(\frac{1}{2} \cdot \phi_2\right)}{\phi_1} + \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)}{\phi_1} - \frac{1}{8} \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_2\right)}\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   9. Applied rewrites 48.0%

      \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\left(\phi_1 \cdot \phi_1\right) \cdot \color{blue}{\left(\left(-\frac{\left(-\frac{\cos \left(0.5 \cdot \phi_2\right)}{\phi_1}\right) - -0.5 \cdot \sin \left(0.5 \cdot \phi_2\right)}{\phi_1}\right) - 0.125 \cdot \cos \left(0.5 \cdot \phi_2\right)\right)}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   10. Taylor expanded in phi2 around 0

       \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(1 + \color{blue}{\frac{-1}{8} \cdot {\phi_1}^{2}}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\frac{-1}{8} \cdot {\phi_1}^{2} + 1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

       2. *-commutative N/A

          \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left({\phi_1}^{2} \cdot \frac{-1}{8} + 1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

       3. lower-fma.f64 N/A

          \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \mathsf{fma}\left({\phi_1}^{2}, \frac{-1}{8}, 1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

       4. pow2 N/A

          \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \mathsf{fma}\left(\phi_1 \cdot \phi_1, \frac{-1}{8}, 1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

       5. lift-*.f64 66.5

          \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.125, 1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   12. Applied rewrites 66.5%

       \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \mathsf{fma}\left(\phi_1 \cdot \phi_1, \color{blue}{-0.125}, 1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   if 8.2e14 < phi1

   1. Initial program 58.7%

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

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

   3. Applied rewrites 95.8%

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

   4. Taylor expanded in phi1 around inf

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

      1. lower-*.f64 90.3

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

   6. Applied rewrites 90.3%

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

   7. Taylor expanded in phi1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 69.9

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

   9. Applied rewrites 69.9%

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

   10. Taylor expanded in lambda1 around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 53.3

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

   12. Applied rewrites 53.3%

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

Alternative 7: 78.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.75 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(R, \frac{\phi_2}{\phi_1}, -R\right) \cdot \phi_1\\ \mathbf{elif}\;\phi_1 \leq 1.45 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, \mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.125, 1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \end{array} \]

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 -1.75e+59)
   (* (fma R (/ phi2 phi1) (- R)) phi1)
   (if (<= phi1 1.45e+15)
     (*
      (hypot
       (- phi1 phi2)
       (* (fma (* phi1 phi1) -0.125 1.0) (- lambda1 lambda2)))
      R)
     (* 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 <= -1.75e+59) {
		tmp = fma(R, (phi2 / phi1), -R) * phi1;
	} else if (phi1 <= 1.45e+15) {
		tmp = hypot((phi1 - phi2), (fma((phi1 * phi1), -0.125, 1.0) * (lambda1 - lambda2))) * R;
	} else {
		tmp = 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 <= -1.75e+59)
		tmp = Float64(fma(R, Float64(phi2 / phi1), Float64(-R)) * phi1);
	elseif (phi1 <= 1.45e+15)
		tmp = Float64(hypot(Float64(phi1 - phi2), Float64(fma(Float64(phi1 * phi1), -0.125, 1.0) * Float64(lambda1 - lambda2))) * R);
	else
		tmp = Float64(phi2 * 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, -1.75e+59], N[(N[(R * N[(phi2 / phi1), $MachinePrecision] + (-R)), $MachinePrecision] * phi1), $MachinePrecision], If[LessEqual[phi1, 1.45e+15], N[(N[Sqrt[N[(phi1 - phi2), $MachinePrecision] ^ 2 + N[(N[(N[(phi1 * phi1), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(phi2 * R), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.75e59

   1. Initial program 58.7%

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

   2. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 55.9

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

   4. Applied rewrites 55.9%

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

   5. Taylor expanded in R around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 54.2

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

   7. Applied rewrites 54.2%

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

   8. Taylor expanded in phi1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 54.1

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

   10. Applied rewrites 54.1%

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

   11. Taylor expanded in phi1 around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. associate-/l* N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. mul-1-neg N/A

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

       8. lower-neg.f64 53.6

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

   13. Applied rewrites 53.6%

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

   if -1.75e59 < phi1 < 1.45e15

   1. Initial program 58.7%

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

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

   3. Applied rewrites 95.8%

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

   4. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \phi_1 \cdot \left(\frac{-1}{8} \cdot \left(\phi_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) - \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \color{blue}{\phi_1 \cdot \left(\frac{-1}{8} \cdot \left(\phi_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) - \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \color{blue}{\phi_1} \cdot \left(\frac{-1}{8} \cdot \left(\phi_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) - \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \phi_1 \cdot \left(\frac{-1}{8} \cdot \left(\phi_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) - \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \phi_1 \cdot \color{blue}{\left(\frac{-1}{8} \cdot \left(\phi_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) - \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \phi_1 \cdot \left(\frac{-1}{8} \cdot \left(\phi_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) - \color{blue}{\frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)}\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \phi_1 \cdot \left(\frac{-1}{8} \cdot \left(\phi_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) - \color{blue}{\frac{1}{2}} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \phi_1 \cdot \left(\frac{-1}{8} \cdot \left(\phi_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) - \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      8. lower-cos.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \phi_1 \cdot \left(\frac{-1}{8} \cdot \left(\phi_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) - \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \phi_1 \cdot \left(\frac{-1}{8} \cdot \left(\phi_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) - \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \phi_1 \cdot \left(\frac{-1}{8} \cdot \left(\phi_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) - \frac{1}{2} \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_2\right)}\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      11. lower-sin.f64 N/A

          \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \phi_1 \cdot \left(\frac{-1}{8} \cdot \left(\phi_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) - \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      12. lower-*.f64 72.1

          \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\cos \left(0.5 \cdot \phi_2\right) + \phi_1 \cdot \left(-0.125 \cdot \left(\phi_1 \cdot \cos \left(0.5 \cdot \phi_2\right)\right) - 0.5 \cdot \sin \left(0.5 \cdot \phi_2\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   6. Applied rewrites 72.1%

      \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \color{blue}{\left(\cos \left(0.5 \cdot \phi_2\right) + \phi_1 \cdot \left(-0.125 \cdot \left(\phi_1 \cdot \cos \left(0.5 \cdot \phi_2\right)\right) - 0.5 \cdot \sin \left(0.5 \cdot \phi_2\right)\right)\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   7. Taylor expanded in phi1 around -inf

      \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left({\phi_1}^{2} \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\cos \left(\frac{1}{2} \cdot \phi_2\right)}{\phi_1} + \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)}{\phi_1} + \frac{-1}{8} \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left({\phi_1}^{2} \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\cos \left(\frac{1}{2} \cdot \phi_2\right)}{\phi_1} + \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)}{\phi_1} + \color{blue}{\frac{-1}{8} \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      2. unpow2 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\left(\phi_1 \cdot \phi_1\right) \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\cos \left(\frac{1}{2} \cdot \phi_2\right)}{\phi_1} + \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)}{\phi_1} + \color{blue}{\frac{-1}{8}} \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\left(\phi_1 \cdot \phi_1\right) \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\cos \left(\frac{1}{2} \cdot \phi_2\right)}{\phi_1} + \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)}{\phi_1} + \color{blue}{\frac{-1}{8}} \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\left(\phi_1 \cdot \phi_1\right) \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\cos \left(\frac{1}{2} \cdot \phi_2\right)}{\phi_1} + \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)}{\phi_1} - \left(\mathsf{neg}\left(\frac{-1}{8}\right)\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_2\right)}\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\left(\phi_1 \cdot \phi_1\right) \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\cos \left(\frac{1}{2} \cdot \phi_2\right)}{\phi_1} + \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)}{\phi_1} - \frac{1}{8} \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\left(\phi_1 \cdot \phi_1\right) \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\cos \left(\frac{1}{2} \cdot \phi_2\right)}{\phi_1} + \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)}{\phi_1} - \frac{1}{8} \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_2\right)}\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   9. Applied rewrites 48.0%

      \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\left(\phi_1 \cdot \phi_1\right) \cdot \color{blue}{\left(\left(-\frac{\left(-\frac{\cos \left(0.5 \cdot \phi_2\right)}{\phi_1}\right) - -0.5 \cdot \sin \left(0.5 \cdot \phi_2\right)}{\phi_1}\right) - 0.125 \cdot \cos \left(0.5 \cdot \phi_2\right)\right)}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   10. Taylor expanded in phi2 around 0

       \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(1 + \color{blue}{\frac{-1}{8} \cdot {\phi_1}^{2}}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\frac{-1}{8} \cdot {\phi_1}^{2} + 1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

       2. *-commutative N/A

          \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left({\phi_1}^{2} \cdot \frac{-1}{8} + 1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

       3. lower-fma.f64 N/A

          \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \mathsf{fma}\left({\phi_1}^{2}, \frac{-1}{8}, 1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

       4. pow2 N/A

          \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \mathsf{fma}\left(\phi_1 \cdot \phi_1, \frac{-1}{8}, 1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

       5. lift-*.f64 66.5

          \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.125, 1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   12. Applied rewrites 66.5%

       \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \mathsf{fma}\left(\phi_1 \cdot \phi_1, \color{blue}{-0.125}, 1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   if 1.45e15 < phi1

   1. Initial program 58.7%

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

   2. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 31.5

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

   4. Applied rewrites 31.5%

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

Alternative 8: 57.7% accurate, 5.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -2 \cdot 10^{+141}:\\ \;\;\;\;\left(\frac{-\phi_1 \cdot R}{\phi_2} + R\right) \cdot \phi_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\phi_1, R, R \cdot \phi_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (- lambda1 lambda2) -2e+141)
   (* (+ (/ (- (* phi1 R)) phi2) R) phi2)
   (fma (- phi1) R (* R phi2))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda1 - lambda2) <= -2e+141) {
		tmp = ((-(phi1 * R) / phi2) + R) * phi2;
	} else {
		tmp = fma(-phi1, R, (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 (Float64(lambda1 - lambda2) <= -2e+141)
		tmp = Float64(Float64(Float64(Float64(-Float64(phi1 * R)) / phi2) + R) * phi2);
	else
		tmp = fma(Float64(-phi1), R, Float64(R * phi2));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -2e+141], N[(N[(N[((-N[(phi1 * R), $MachinePrecision]) / phi2), $MachinePrecision] + R), $MachinePrecision] * phi2), $MachinePrecision], N[((-phi1) * R + N[(R * phi2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < -2.00000000000000003e141

   1. Initial program 58.7%

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

   2. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 55.9

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

   4. Applied rewrites 55.9%

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

   if -2.00000000000000003e141 < (-.f64 lambda1 lambda2)

   1. Initial program 58.7%

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

   2. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 55.9

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

   4. Applied rewrites 55.9%

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

   5. Taylor expanded in R around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 54.2

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

   7. Applied rewrites 54.2%

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

   8. Taylor expanded in phi1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 54.1

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

   10. Applied rewrites 54.1%

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

   11. Taylor expanded in phi1 around 0

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

       3. distribute-lft-neg-in N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-neg.f64 N/A

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

       6. lift-*.f64 56.9

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

   13. Applied rewrites 56.9%

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

Alternative 9: 56.9% accurate, 10.8× speedup

Math

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

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 (- phi1) R (* R phi2)))

C

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

Julia

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

Derivation

1. Initial program 58.7%

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

2. Taylor expanded in phi2 around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. associate-*r/ N/A

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

   6. lower-/.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 55.9

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

4. Applied rewrites 55.9%

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

5. Taylor expanded in R around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 54.2

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

7. Applied rewrites 54.2%

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

8. Taylor expanded in phi1 around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 54.1

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

10. Applied rewrites 54.1%

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

11. Taylor expanded in phi1 around 0

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

    1. mul-1-neg N/A

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

    2. *-commutative N/A

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

    3. distribute-lft-neg-in N/A

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

    4. lower-fma.f64 N/A

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

    5. lower-neg.f64 N/A

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

    6. lift-*.f64 56.9

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

13. Applied rewrites 56.9%

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

Alternative 10: 52.1% accurate, 12.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.65 \cdot 10^{+44}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 2.65e+44) (* 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 (phi2 <= 2.65e+44) {
		tmp = R * -phi1;
	} else {
		tmp = phi2 * R;
	}
	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 (phi2 <= 2.65d+44) then
        tmp = r * -phi1
    else
        tmp = phi2 * r
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.65e+44) {
		tmp = R * -phi1;
	} else {
		tmp = phi2 * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 2.65e+44:
		tmp = R * -phi1
	else:
		tmp = phi2 * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 2.65e+44)
		tmp = Float64(R * Float64(-phi1));
	else
		tmp = Float64(phi2 * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 2.65e+44)
		tmp = R * -phi1;
	else
		tmp = phi2 * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 2.65e+44], N[(R * (-phi1)), $MachinePrecision], N[(phi2 * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2.65e44

   1. Initial program 58.7%

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

   2. Taylor expanded in phi1 around -inf

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \phi_1\right)} \]
   3. 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 31.4

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

   4. Applied rewrites 31.4%

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

   if 2.65e44 < phi2

   1. Initial program 58.7%

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

   2. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 31.5

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

   4. Applied rewrites 31.5%

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

Alternative 11: 31.5% accurate, 27.0× speedup

Math

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

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 (* 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 phi2 * R;
}

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 = phi2 * r
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 phi2 * R;
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.7%

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

2. Taylor expanded in phi2 around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 31.5

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

4. Applied rewrites 31.5%

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

Alternative 12: 3.4% accurate, 27.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \phi_1 \cdot R \end{array} \]

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 (* phi1 R))

C

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

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 = phi1 * r
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 phi1 * R;
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.7%

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

2. Taylor expanded in phi1 around inf

   \[\leadsto \color{blue}{R \cdot \phi_1} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \phi_1 \cdot \color{blue}{R} \]

   2. lower-*.f64 3.4

      \[\leadsto \phi_1 \cdot \color{blue}{R} \]

4. Applied rewrites 3.4%

   \[\leadsto \color{blue}{\phi_1 \cdot R} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025139 
(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))))))