Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.4% → 96.0%

Time: 5.5s

Alternatives: 15

Speedup: 1.7×

• 31.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 59.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.7 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, \cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, \cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

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 (<= phi2 1.7e-8)
   (* (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 (phi2 <= 1.7e-8) {
		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 (phi2 <= 1.7e-8) {
		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 phi2 <= 1.7e-8:
		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 (phi2 <= 1.7e-8)
		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 (phi2 <= 1.7e-8)
		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[phi2, 1.7e-8], 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 phi2 < 1.7e-8

   1. Initial program 63.5%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. 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 97.9%

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

   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 97.8

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

   6. Applied rewrites 97.8%

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

   if 1.7e-8 < phi2

   1. Initial program 54.5%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. 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 93.8%

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

   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 93.7

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

   6. Applied rewrites 93.7%

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

Alternative 2: 95.9% 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 59.4%

   \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
2. 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 96.0%

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

Alternative 3: 91.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.7 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, \cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, \cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_1\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.7e-8)
   (* (hypot (- phi1 phi2) (* (cos (* 0.5 phi1)) (- lambda1 lambda2))) R)
   (* (hypot (- phi1 phi2) (* (cos (* 0.5 phi2)) lambda1)) 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 <= 1.7e-8) {
		tmp = hypot((phi1 - phi2), (cos((0.5 * phi1)) * (lambda1 - lambda2))) * R;
	} else {
		tmp = hypot((phi1 - phi2), (cos((0.5 * phi2)) * lambda1)) * R;
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.7e-8) {
		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)) * 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 <= 1.7e-8:
		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)) * 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 <= 1.7e-8)
		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)) * lambda1)) * 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 <= 1.7e-8)
		tmp = hypot((phi1 - phi2), (cos((0.5 * phi1)) * (lambda1 - lambda2))) * R;
	else
		tmp = hypot((phi1 - phi2), (cos((0.5 * phi2)) * lambda1)) * 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, 1.7e-8], 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] * lambda1), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.7e-8

   1. Initial program 63.5%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. 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 97.9%

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

   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 97.8

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

   6. Applied rewrites 97.8%

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

   if 1.7e-8 < phi2

   1. Initial program 54.5%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. 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 93.8%

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

   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 93.7

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

   6. Applied rewrites 93.7%

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

   7. Taylor expanded in lambda1 around inf

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

   9. Applied rewrites 83.2%

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

Alternative 4: 89.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 7.8 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1, \cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, \cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_1\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 7.8e-22)
   (* (hypot phi1 (* (cos (* 0.5 phi1)) (- lambda1 lambda2))) R)
   (* (hypot (- phi1 phi2) (* (cos (* 0.5 phi2)) lambda1)) 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 <= 7.8e-22) {
		tmp = hypot(phi1, (cos((0.5 * phi1)) * (lambda1 - lambda2))) * R;
	} else {
		tmp = hypot((phi1 - phi2), (cos((0.5 * phi2)) * lambda1)) * R;
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 7.8e-22) {
		tmp = Math.hypot(phi1, (Math.cos((0.5 * phi1)) * (lambda1 - lambda2))) * R;
	} else {
		tmp = Math.hypot((phi1 - phi2), (Math.cos((0.5 * phi2)) * lambda1)) * 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 <= 7.8e-22:
		tmp = math.hypot(phi1, (math.cos((0.5 * phi1)) * (lambda1 - lambda2))) * R
	else:
		tmp = math.hypot((phi1 - phi2), (math.cos((0.5 * phi2)) * lambda1)) * 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 <= 7.8e-22)
		tmp = Float64(hypot(phi1, Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2))) * R);
	else
		tmp = Float64(hypot(Float64(phi1 - phi2), Float64(cos(Float64(0.5 * phi2)) * lambda1)) * 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 <= 7.8e-22)
		tmp = hypot(phi1, (cos((0.5 * phi1)) * (lambda1 - lambda2))) * R;
	else
		tmp = hypot((phi1 - phi2), (cos((0.5 * phi2)) * lambda1)) * 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, 7.8e-22], N[(N[Sqrt[phi1 ^ 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] * lambda1), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 7.79999999999999996e-22

   1. Initial program 63.4%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. 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 97.9%

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

   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 97.8

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

   6. Applied rewrites 97.8%

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

   7. Taylor expanded in phi1 around inf

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

   9. Applied rewrites 95.9%

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

   if 7.79999999999999996e-22 < phi2

   1. Initial program 54.9%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. 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 93.9%

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

   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 93.5

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

   6. Applied rewrites 93.5%

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

   7. Taylor expanded in lambda1 around inf

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

   9. Applied rewrites 82.8%

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

Alternative 5: 87.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot \phi_1\right)\\ \mathbf{if}\;\phi_2 \leq 7.8 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1, t\_0 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, t\_0 \cdot \lambda_1\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 phi1))))
   (if (<= phi2 7.8e-22)
     (* (hypot phi1 (* t_0 (- lambda1 lambda2))) R)
     (* (hypot (- phi1 phi2) (* t_0 lambda1)) R))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((0.5 * phi1));
	double tmp;
	if (phi2 <= 7.8e-22) {
		tmp = hypot(phi1, (t_0 * (lambda1 - lambda2))) * R;
	} else {
		tmp = hypot((phi1 - phi2), (t_0 * lambda1)) * 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 t_0 = Math.cos((0.5 * phi1));
	double tmp;
	if (phi2 <= 7.8e-22) {
		tmp = Math.hypot(phi1, (t_0 * (lambda1 - lambda2))) * R;
	} else {
		tmp = Math.hypot((phi1 - phi2), (t_0 * lambda1)) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((0.5 * phi1))
	tmp = 0
	if phi2 <= 7.8e-22:
		tmp = math.hypot(phi1, (t_0 * (lambda1 - lambda2))) * R
	else:
		tmp = math.hypot((phi1 - phi2), (t_0 * lambda1)) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(0.5 * phi1))
	tmp = 0.0
	if (phi2 <= 7.8e-22)
		tmp = Float64(hypot(phi1, Float64(t_0 * Float64(lambda1 - lambda2))) * R);
	else
		tmp = Float64(hypot(Float64(phi1 - phi2), Float64(t_0 * lambda1)) * 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)
	t_0 = cos((0.5 * phi1));
	tmp = 0.0;
	if (phi2 <= 7.8e-22)
		tmp = hypot(phi1, (t_0 * (lambda1 - lambda2))) * R;
	else
		tmp = hypot((phi1 - phi2), (t_0 * lambda1)) * 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_] := Block[{t$95$0 = N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, 7.8e-22], N[(N[Sqrt[phi1 ^ 2 + N[(t$95$0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(phi1 - phi2), $MachinePrecision] ^ 2 + N[(t$95$0 * lambda1), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 7.79999999999999996e-22

   1. Initial program 63.4%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. 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 97.9%

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

   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 97.8

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

   6. Applied rewrites 97.8%

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

   7. Taylor expanded in phi1 around inf

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

   9. Applied rewrites 95.9%

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

   if 7.79999999999999996e-22 < phi2

   1. Initial program 54.9%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. 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 93.9%

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

   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 82.1

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

   6. Applied rewrites 82.1%

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

   7. Taylor expanded in lambda1 around inf

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

   9. Applied rewrites 77.0%

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

Alternative 6: 85.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 7.6 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1, \cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-R, \phi_1, R \cdot \phi_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 7.6e+41)
   (* (hypot phi1 (* (cos (* 0.5 phi1)) (- lambda1 lambda2))) R)
   (fma (- R) phi1 (* R phi2))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 7.6e+41) {
		tmp = hypot(phi1, (cos((0.5 * phi1)) * (lambda1 - lambda2))) * R;
	} else {
		tmp = fma(-R, phi1, (R * phi2));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 7.6e+41)
		tmp = Float64(hypot(phi1, Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2))) * R);
	else
		tmp = fma(Float64(-R), phi1, Float64(R * phi2));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 7.6e+41], N[(N[Sqrt[phi1 ^ 2 + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], N[((-R) * phi1 + N[(R * phi2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 7.6000000000000003e41

   1. Initial program 63.5%

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

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

   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 94.6

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

   6. Applied rewrites 94.6%

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

   7. Taylor expanded in phi1 around inf

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

   9. Applied rewrites 91.0%

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

   if 7.6000000000000003e41 < phi2

   1. Initial program 53.2%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. 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 94.3%

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

   4. Taylor expanded in phi2 around inf

      \[\leadsto \color{blue}{\phi_2 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_1}{\phi_2}\right)} \]
   5. 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. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 74.2

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

   6. Applied rewrites 74.2%

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

   7. Taylor expanded in phi1 around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lower-*.f64 77.0

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

   9. Applied rewrites 77.0%

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

Alternative 7: 59.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -9 \cdot 10^{+202}:\\ \;\;\;\;-\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \lambda_1\right) \cdot R\\ \mathbf{elif}\;\lambda_1 \leq -4.2 \cdot 10^{+65}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 \cdot \lambda_1\right) \cdot \mathsf{fma}\left(\cos \left(1 \cdot \left(\phi_2 + \phi_1\right)\right), 0.5, 0.5\right)}\\ \mathbf{elif}\;\lambda_1 \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(-R, \phi_1, R \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \left(\phi_1 \cdot 0.5\right) \cdot \lambda_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -9e+202)
   (- (* (* (cos (* 0.5 (+ phi2 phi1))) lambda1) R))
   (if (<= lambda1 -4.2e+65)
     (*
      R
      (sqrt (* (* lambda1 lambda1) (fma (cos (* 1.0 (+ phi2 phi1))) 0.5 0.5))))
     (if (<= lambda1 2e+14)
       (fma (- R) phi1 (* R phi2))
       (* (* (cos (* phi1 0.5)) 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 <= -9e+202) {
		tmp = -((cos((0.5 * (phi2 + phi1))) * lambda1) * R);
	} else if (lambda1 <= -4.2e+65) {
		tmp = R * sqrt(((lambda1 * lambda1) * fma(cos((1.0 * (phi2 + phi1))), 0.5, 0.5)));
	} else if (lambda1 <= 2e+14) {
		tmp = fma(-R, phi1, (R * phi2));
	} else {
		tmp = (cos((phi1 * 0.5)) * 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 <= -9e+202)
		tmp = Float64(-Float64(Float64(cos(Float64(0.5 * Float64(phi2 + phi1))) * lambda1) * R));
	elseif (lambda1 <= -4.2e+65)
		tmp = Float64(R * sqrt(Float64(Float64(lambda1 * lambda1) * fma(cos(Float64(1.0 * Float64(phi2 + phi1))), 0.5, 0.5))));
	elseif (lambda1 <= 2e+14)
		tmp = fma(Float64(-R), phi1, Float64(R * phi2));
	else
		tmp = Float64(Float64(cos(Float64(phi1 * 0.5)) * 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[lambda1, -9e+202], (-N[(N[(N[Cos[N[(0.5 * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * lambda1), $MachinePrecision] * R), $MachinePrecision]), If[LessEqual[lambda1, -4.2e+65], N[(R * N[Sqrt[N[(N[(lambda1 * lambda1), $MachinePrecision] * N[(N[Cos[N[(1.0 * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda1, 2e+14], N[((-R) * phi1 + N[(R * phi2), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * lambda2), $MachinePrecision] * R), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if lambda1 < -8.99999999999999955e202

   1. Initial program 45.0%

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

   2. Taylor expanded in lambda1 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 50.7

          \[\leadsto -\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \lambda_1\right) \cdot R \]

   4. Applied rewrites 50.7%

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

   if -8.99999999999999955e202 < lambda1 < -4.19999999999999983e65

   1. Initial program 54.2%

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

   2. Taylor expanded in lambda2 around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. sqr-cos-a N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 17.9

          \[\leadsto R \cdot \sqrt{\left(\lambda_2 \cdot \lambda_2\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\right)} \]

   4. Applied rewrites 17.9%

      \[\leadsto R \cdot \sqrt{\color{blue}{\left(\lambda_2 \cdot \lambda_2\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\right)}} \]

   5. Taylor expanded in lambda1 around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. sqr-cos-a-rev N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 40.9

          \[\leadsto R \cdot \sqrt{\left(\lambda_1 \cdot \lambda_1\right) \cdot \mathsf{fma}\left(\cos \left(1 \cdot \left(\phi_2 + \phi_1\right)\right), 0.5, 0.5\right)} \]

   7. Applied rewrites 40.9%

      \[\leadsto R \cdot \sqrt{\color{blue}{\left(\lambda_1 \cdot \lambda_1\right) \cdot \mathsf{fma}\left(\cos \left(1 \cdot \left(\phi_2 + \phi_1\right)\right), 0.5, 0.5\right)}} \]

   if -4.19999999999999983e65 < lambda1 < 2e14

   1. Initial program 65.3%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. 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 97.7%

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

   4. Taylor expanded in phi2 around inf

      \[\leadsto \color{blue}{\phi_2 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_1}{\phi_2}\right)} \]
   5. 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. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 65.2

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

   6. Applied rewrites 65.2%

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

   7. Taylor expanded in phi1 around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lower-*.f64 68.2

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

   9. Applied rewrites 68.2%

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

   if 2e14 < lambda1

   1. Initial program 52.9%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. 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 94.3%

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

   4. Taylor expanded in lambda2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 12.1

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

   6. Applied rewrites 12.1%

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

   7. Taylor expanded in phi1 around inf

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

   9. Applied rewrites 11.8%

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

Alternative 8: 58.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.38 \cdot 10^{+203}:\\ \;\;\;\;-\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \lambda_1\right) \cdot R\\ \mathbf{elif}\;\lambda_1 \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(-R, \phi_1, R \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \left(\phi_1 \cdot 0.5\right) \cdot \lambda_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -1.38e+203)
   (- (* (* (cos (* 0.5 (+ phi2 phi1))) lambda1) R))
   (if (<= lambda1 2e+14)
     (fma (- R) phi1 (* R phi2))
     (* (* (cos (* phi1 0.5)) 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 <= -1.38e+203) {
		tmp = -((cos((0.5 * (phi2 + phi1))) * lambda1) * R);
	} else if (lambda1 <= 2e+14) {
		tmp = fma(-R, phi1, (R * phi2));
	} else {
		tmp = (cos((phi1 * 0.5)) * 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 <= -1.38e+203)
		tmp = Float64(-Float64(Float64(cos(Float64(0.5 * Float64(phi2 + phi1))) * lambda1) * R));
	elseif (lambda1 <= 2e+14)
		tmp = fma(Float64(-R), phi1, Float64(R * phi2));
	else
		tmp = Float64(Float64(cos(Float64(phi1 * 0.5)) * 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[lambda1, -1.38e+203], (-N[(N[(N[Cos[N[(0.5 * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * lambda1), $MachinePrecision] * R), $MachinePrecision]), If[LessEqual[lambda1, 2e+14], N[((-R) * phi1 + N[(R * phi2), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * lambda2), $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -1.3800000000000001e203

   1. Initial program 45.0%

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

   2. Taylor expanded in lambda1 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 50.7

          \[\leadsto -\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \lambda_1\right) \cdot R \]

   4. Applied rewrites 50.7%

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

   if -1.3800000000000001e203 < lambda1 < 2e14

   1. Initial program 63.5%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. 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 97.2%

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

   4. Taylor expanded in phi2 around inf

      \[\leadsto \color{blue}{\phi_2 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_1}{\phi_2}\right)} \]
   5. 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. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 62.5

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

   6. Applied rewrites 62.5%

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

   7. Taylor expanded in phi1 around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lower-*.f64 65.0

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

   9. Applied rewrites 65.0%

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

   if 2e14 < lambda1

   1. Initial program 52.9%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. 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 94.3%

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

   4. Taylor expanded in lambda2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 12.1

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

   6. Applied rewrites 12.1%

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

   7. Taylor expanded in phi1 around inf

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

   9. Applied rewrites 11.8%

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

Alternative 9: 57.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -4 \cdot 10^{+69}:\\ \;\;\;\;\left(\left(\frac{R}{\phi_1} + \frac{-R}{\phi_2}\right) \cdot \phi_1\right) \cdot \phi_2\\ \mathbf{elif}\;\lambda_1 \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(-R, \phi_1, R \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \left(\phi_1 \cdot 0.5\right) \cdot \lambda_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -4e+69)
   (* (* (+ (/ R phi1) (/ (- R) phi2)) phi1) phi2)
   (if (<= lambda1 2e+14)
     (fma (- R) phi1 (* R phi2))
     (* (* (cos (* phi1 0.5)) 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 <= -4e+69) {
		tmp = (((R / phi1) + (-R / phi2)) * phi1) * phi2;
	} else if (lambda1 <= 2e+14) {
		tmp = fma(-R, phi1, (R * phi2));
	} else {
		tmp = (cos((phi1 * 0.5)) * 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 <= -4e+69)
		tmp = Float64(Float64(Float64(Float64(R / phi1) + Float64(Float64(-R) / phi2)) * phi1) * phi2);
	elseif (lambda1 <= 2e+14)
		tmp = fma(Float64(-R), phi1, Float64(R * phi2));
	else
		tmp = Float64(Float64(cos(Float64(phi1 * 0.5)) * 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[lambda1, -4e+69], N[(N[(N[(N[(R / phi1), $MachinePrecision] + N[((-R) / phi2), $MachinePrecision]), $MachinePrecision] * phi1), $MachinePrecision] * phi2), $MachinePrecision], If[LessEqual[lambda1, 2e+14], N[((-R) * phi1 + N[(R * phi2), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * lambda2), $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -4.0000000000000003e69

   1. Initial program 50.1%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. 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 93.0%

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

   4. Taylor expanded in phi2 around inf

      \[\leadsto \color{blue}{\phi_2 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_1}{\phi_2}\right)} \]
   5. 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. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 43.4

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

   6. Applied rewrites 43.4%

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

   7. Taylor expanded in phi1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lift-neg.f64 44.6

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

   9. Applied rewrites 44.6%

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

   if -4.0000000000000003e69 < lambda1 < 2e14

   1. Initial program 65.3%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. 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 97.7%

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

   4. Taylor expanded in phi2 around inf

      \[\leadsto \color{blue}{\phi_2 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_1}{\phi_2}\right)} \]
   5. 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. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 65.2

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

   6. Applied rewrites 65.2%

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

   7. Taylor expanded in phi1 around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lower-*.f64 68.2

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

   9. Applied rewrites 68.2%

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

   if 2e14 < lambda1

   1. Initial program 52.9%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. 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 94.3%

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

   4. Taylor expanded in lambda2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 12.1

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

   6. Applied rewrites 12.1%

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

   7. Taylor expanded in phi1 around inf

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

   9. Applied rewrites 11.8%

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

Alternative 10: 53.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -2 \cdot 10^{+178}:\\ \;\;\;\;\left(\left(\frac{R}{\phi_1} + \frac{-R}{\phi_2}\right) \cdot \phi_1\right) \cdot \phi_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-R, \phi_1, R \cdot \phi_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (- lambda1 lambda2) -2e+178)
   (* (* (+ (/ R phi1) (/ (- R) phi2)) phi1) phi2)
   (fma (- R) phi1 (* R phi2))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda1 - lambda2) <= -2e+178) {
		tmp = (((R / phi1) + (-R / phi2)) * phi1) * phi2;
	} else {
		tmp = fma(-R, phi1, (R * phi2));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= -2e+178)
		tmp = Float64(Float64(Float64(Float64(R / phi1) + Float64(Float64(-R) / phi2)) * phi1) * phi2);
	else
		tmp = fma(Float64(-R), phi1, Float64(R * phi2));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -2e+178], N[(N[(N[(N[(R / phi1), $MachinePrecision] + N[((-R) / phi2), $MachinePrecision]), $MachinePrecision] * phi1), $MachinePrecision] * phi2), $MachinePrecision], N[((-R) * phi1 + N[(R * phi2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < -2.0000000000000001e178

   1. Initial program 43.4%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. 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 91.7%

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

   4. Taylor expanded in phi2 around inf

      \[\leadsto \color{blue}{\phi_2 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_1}{\phi_2}\right)} \]
   5. 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. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 39.0

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

   6. Applied rewrites 39.0%

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

   7. Taylor expanded in phi1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lift-neg.f64 42.4

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

   9. Applied rewrites 42.4%

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

   if -2.0000000000000001e178 < (-.f64 lambda1 lambda2)

   1. Initial program 63.2%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. 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 97.0%

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

   4. Taylor expanded in phi2 around inf

      \[\leadsto \color{blue}{\phi_2 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_1}{\phi_2}\right)} \]
   5. 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. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 61.0

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

   6. Applied rewrites 61.0%

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

   7. Taylor expanded in phi1 around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lower-*.f64 63.3

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

   9. Applied rewrites 63.3%

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

Alternative 11: 51.4% accurate, 5.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -4 \cdot 10^{+256}:\\ \;\;\;\;\left(\frac{-\phi_1 \cdot R}{\phi_2} + R\right) \cdot \phi_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-R, \phi_1, R \cdot \phi_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (- lambda1 lambda2) -4e+256)
   (* (+ (/ (- (* phi1 R)) phi2) R) phi2)
   (fma (- R) phi1 (* R phi2))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda1 - lambda2) <= -4e+256) {
		tmp = ((-(phi1 * R) / phi2) + R) * phi2;
	} else {
		tmp = fma(-R, phi1, (R * phi2));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= -4e+256)
		tmp = Float64(Float64(Float64(Float64(-Float64(phi1 * R)) / phi2) + R) * phi2);
	else
		tmp = fma(Float64(-R), phi1, Float64(R * phi2));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -4e+256], N[(N[(N[((-N[(phi1 * R), $MachinePrecision]) / phi2), $MachinePrecision] + R), $MachinePrecision] * phi2), $MachinePrecision], N[((-R) * phi1 + N[(R * phi2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < -4.0000000000000001e256

   1. Initial program 48.9%

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

   2. 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 31.1

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

   4. Applied rewrites 31.1%

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

   if -4.0000000000000001e256 < (-.f64 lambda1 lambda2)

   1. Initial program 60.3%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. 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 96.5%

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

   4. Taylor expanded in phi2 around inf

      \[\leadsto \color{blue}{\phi_2 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_1}{\phi_2}\right)} \]
   5. 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. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 58.8

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

   6. Applied rewrites 58.8%

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

   7. Taylor expanded in phi1 around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lower-*.f64 60.5

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

   9. Applied rewrites 60.5%

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

Alternative 12: 50.4% accurate, 10.8× speedup

Math

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

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

C

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

Julia

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

Derivation

1. Initial program 59.4%

   \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
2. 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 96.0%

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

4. Taylor expanded in phi2 around inf

   \[\leadsto \color{blue}{\phi_2 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_1}{\phi_2}\right)} \]
5. 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. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

   9. mul-1-neg N/A

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

   10. lower-neg.f64 56.7

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

6. Applied rewrites 56.7%

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

7. Taylor expanded in phi1 around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lift-neg.f64 N/A

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

   5. lower-*.f64 57.8

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

9. Applied rewrites 57.8%

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

Alternative 13: 50.4% accurate, 12.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -680:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -680.0) (* R (- phi1)) (* R phi2)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -680.0) {
		tmp = R * -phi1;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-680.0d0)) then
        tmp = r * -phi1
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -680.0) {
		tmp = R * -phi1;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -680.0:
		tmp = R * -phi1
	else:
		tmp = R * phi2
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -680.0)
		tmp = Float64(R * Float64(-phi1));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -680.0)
		tmp = R * -phi1;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -680.0], N[(R * (-phi1)), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -680

   1. Initial program 54.2%

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

   2. 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 64.3

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

   4. Applied rewrites 64.3%

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

   if -680 < phi1

   1. Initial program 63.4%

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

   2. Taylor expanded in phi2 around inf

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

   4. Applied rewrites 44.6%

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

Alternative 14: 32.1% accurate, 27.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = r * phi2
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.4%

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

2. Taylor expanded in phi2 around inf

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

4. Applied rewrites 32.1%

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

Alternative 15: 3.5% accurate, 27.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = r * phi1
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.4%

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

2. Taylor expanded in phi1 around inf

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

4. Applied rewrites 3.5%

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

Reproduce

herbie shell --seed 2025115 
(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))))))