Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.0% → 95.9%

Time: 6.1s

Alternatives: 16

Speedup: 2.0×

• 30.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 59.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.8 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, \cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, \cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -4.8e-7)
   (* (hypot (- phi1 phi2) (* (cos (* 0.5 phi1)) (- lambda1 lambda2))) R)
   (* (hypot (- phi1 phi2) (* (cos (* 0.5 phi2)) (- lambda1 lambda2))) R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -4.8e-7) {
		tmp = hypot((phi1 - phi2), (cos((0.5 * phi1)) * (lambda1 - lambda2))) * R;
	} else {
		tmp = hypot((phi1 - phi2), (cos((0.5 * phi2)) * (lambda1 - lambda2))) * R;
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -4.8e-7) {
		tmp = Math.hypot((phi1 - phi2), (Math.cos((0.5 * phi1)) * (lambda1 - lambda2))) * R;
	} else {
		tmp = Math.hypot((phi1 - phi2), (Math.cos((0.5 * phi2)) * (lambda1 - lambda2))) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -4.8e-7:
		tmp = math.hypot((phi1 - phi2), (math.cos((0.5 * phi1)) * (lambda1 - lambda2))) * R
	else:
		tmp = math.hypot((phi1 - phi2), (math.cos((0.5 * phi2)) * (lambda1 - lambda2))) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -4.8e-7)
		tmp = Float64(hypot(Float64(phi1 - phi2), Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2))) * R);
	else
		tmp = Float64(hypot(Float64(phi1 - phi2), Float64(cos(Float64(0.5 * phi2)) * Float64(lambda1 - lambda2))) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -4.8e-7)
		tmp = hypot((phi1 - phi2), (cos((0.5 * phi1)) * (lambda1 - lambda2))) * R;
	else
		tmp = hypot((phi1 - phi2), (cos((0.5 * phi2)) * (lambda1 - lambda2))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -4.8e-7], N[(N[Sqrt[N[(phi1 - phi2), $MachinePrecision] ^ 2 + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(phi1 - phi2), $MachinePrecision] ^ 2 + N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -4.79999999999999957e-7

   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 93.0%

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

   4. Taylor expanded in 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 92.9

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

   6. Applied rewrites 92.9%

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

   if -4.79999999999999957e-7 < phi1

   1. Initial program 62.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

   3. Applied rewrites 98.2%

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

   4. Taylor expanded in 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 98.2

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

   6. Applied rewrites 98.2%

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

Alternative 2: 95.8% 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 59.0%

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

   1. lift-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-+.f64 N/A

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

3. Applied rewrites 95.9%

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

Alternative 3: 90.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	return math.hypot((phi1 - phi2), (math.cos((0.5 * phi1)) * (lambda1 - lambda2))) * R

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(hypot(Float64(phi1 - phi2), Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2))) * R)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.0%

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

   1. lift-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-+.f64 N/A

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

3. Applied rewrites 95.9%

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

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 90.2

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

6. Applied rewrites 90.2%

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

Alternative 4: 87.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.5 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1, \cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 1.55 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1, \cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\left(\frac{-\phi_1}{\phi_2} + 1\right) \cdot \phi_2\right)\\ \end{array} \end{array} \]

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 2.5e-13)
   (* (hypot phi1 (* (cos (* 0.5 phi1)) (- lambda1 lambda2))) R)
   (if (<= phi2 1.55e+138)
     (* (hypot phi1 (* (cos (* 0.5 phi2)) (- lambda1 lambda2))) R)
     (* R (* (+ (/ (- phi1) phi2) 1.0) 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 <= 2.5e-13) {
		tmp = hypot(phi1, (cos((0.5 * phi1)) * (lambda1 - lambda2))) * R;
	} else if (phi2 <= 1.55e+138) {
		tmp = hypot(phi1, (cos((0.5 * phi2)) * (lambda1 - lambda2))) * R;
	} else {
		tmp = R * (((-phi1 / phi2) + 1.0) * phi2);
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.5e-13) {
		tmp = Math.hypot(phi1, (Math.cos((0.5 * phi1)) * (lambda1 - lambda2))) * R;
	} else if (phi2 <= 1.55e+138) {
		tmp = Math.hypot(phi1, (Math.cos((0.5 * phi2)) * (lambda1 - lambda2))) * R;
	} else {
		tmp = R * (((-phi1 / phi2) + 1.0) * phi2);
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 2.5e-13:
		tmp = math.hypot(phi1, (math.cos((0.5 * phi1)) * (lambda1 - lambda2))) * R
	elif phi2 <= 1.55e+138:
		tmp = math.hypot(phi1, (math.cos((0.5 * phi2)) * (lambda1 - lambda2))) * R
	else:
		tmp = R * (((-phi1 / phi2) + 1.0) * 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 <= 2.5e-13)
		tmp = Float64(hypot(phi1, Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2))) * R);
	elseif (phi2 <= 1.55e+138)
		tmp = Float64(hypot(phi1, Float64(cos(Float64(0.5 * phi2)) * Float64(lambda1 - lambda2))) * R);
	else
		tmp = Float64(R * Float64(Float64(Float64(Float64(-phi1) / phi2) + 1.0) * phi2));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 2.5e-13)
		tmp = hypot(phi1, (cos((0.5 * phi1)) * (lambda1 - lambda2))) * R;
	elseif (phi2 <= 1.55e+138)
		tmp = hypot(phi1, (cos((0.5 * phi2)) * (lambda1 - lambda2))) * R;
	else
		tmp = R * (((-phi1 / phi2) + 1.0) * phi2);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 2.5e-13], N[(N[Sqrt[phi1 ^ 2 + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi2, 1.55e+138], N[(N[Sqrt[phi1 ^ 2 + N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(R * N[(N[(N[((-phi1) / phi2), $MachinePrecision] + 1.0), $MachinePrecision] * phi2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 2.49999999999999995e-13

   1. Initial program 62.9%

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

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

   6. Applied rewrites 97.9%

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

   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 95.8%

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

   if 2.49999999999999995e-13 < phi2 < 1.5499999999999999e138

   1. Initial program 67.2%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 91.4%

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

   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 91.0

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

   6. Applied rewrites 91.0%

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

   7. Taylor expanded in phi1 around inf

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

   9. Applied rewrites 67.4%

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

   if 1.5499999999999999e138 < phi2

   1. Initial program 44.9%

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

   2. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 84.1

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

   4. Applied rewrites 84.1%

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

Alternative 5: 84.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.7 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1, \cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 + -1 \cdot \phi_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 2.7e+18)
   (* (hypot phi1 (* (cos (* 0.5 phi1)) (- lambda1 lambda2))) R)
   (* R (+ phi2 (* -1.0 phi1)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.7e+18) {
		tmp = hypot(phi1, (cos((0.5 * phi1)) * (lambda1 - lambda2))) * R;
	} else {
		tmp = R * (phi2 + (-1.0 * phi1));
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.7e+18) {
		tmp = Math.hypot(phi1, (Math.cos((0.5 * phi1)) * (lambda1 - lambda2))) * R;
	} else {
		tmp = R * (phi2 + (-1.0 * phi1));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 2.7e+18:
		tmp = math.hypot(phi1, (math.cos((0.5 * phi1)) * (lambda1 - lambda2))) * R
	else:
		tmp = R * (phi2 + (-1.0 * phi1))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 2.7e+18)
		tmp = Float64(hypot(phi1, Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2))) * R);
	else
		tmp = Float64(R * Float64(phi2 + Float64(-1.0 * phi1)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 2.7e+18)
		tmp = hypot(phi1, (cos((0.5 * phi1)) * (lambda1 - lambda2))) * R;
	else
		tmp = R * (phi2 + (-1.0 * phi1));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 2.7e+18], 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 * N[(phi2 + N[(-1.0 * phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2.7e18

   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 97.6%

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

   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 96.4

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

   6. Applied rewrites 96.4%

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

   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 93.4%

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

   if 2.7e18 < 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. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 62.0

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

   4. Applied rewrites 62.0%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 72.8

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

   7. Applied rewrites 72.8%

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

Alternative 6: 62.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sqrt{0.5 + 0.5 \cdot \cos \left(\phi_1 + \phi_2\right)}\\ \mathbf{if}\;\lambda_1 \leq -1.5 \cdot 10^{+118}:\\ \;\;\;\;-1 \cdot \left(\left(R \cdot \lambda_1\right) \cdot t\_0\right)\\ \mathbf{elif}\;\lambda_1 \leq 2.6 \cdot 10^{-150}:\\ \;\;\;\;R \cdot \left(\phi_2 + -1 \cdot \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot \lambda_2\right) \cdot t\_0\\ \end{array} \end{array} \]

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 (sqrt (+ 0.5 (* 0.5 (cos (+ phi1 phi2)))))))
   (if (<= lambda1 -1.5e+118)
     (* -1.0 (* (* R lambda1) t_0))
     (if (<= lambda1 2.6e-150)
       (* R (+ phi2 (* -1.0 phi1)))
       (* (* R lambda2) t_0)))))

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 = sqrt((0.5 + (0.5 * cos((phi1 + phi2)))));
	double tmp;
	if (lambda1 <= -1.5e+118) {
		tmp = -1.0 * ((R * lambda1) * t_0);
	} else if (lambda1 <= 2.6e-150) {
		tmp = R * (phi2 + (-1.0 * phi1));
	} else {
		tmp = (R * lambda2) * t_0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((0.5d0 + (0.5d0 * cos((phi1 + phi2)))))
    if (lambda1 <= (-1.5d+118)) then
        tmp = (-1.0d0) * ((r * lambda1) * t_0)
    else if (lambda1 <= 2.6d-150) then
        tmp = r * (phi2 + ((-1.0d0) * phi1))
    else
        tmp = (r * lambda2) * t_0
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sqrt((0.5 + (0.5 * Math.cos((phi1 + phi2)))));
	double tmp;
	if (lambda1 <= -1.5e+118) {
		tmp = -1.0 * ((R * lambda1) * t_0);
	} else if (lambda1 <= 2.6e-150) {
		tmp = R * (phi2 + (-1.0 * phi1));
	} else {
		tmp = (R * lambda2) * t_0;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sqrt((0.5 + (0.5 * math.cos((phi1 + phi2)))))
	tmp = 0
	if lambda1 <= -1.5e+118:
		tmp = -1.0 * ((R * lambda1) * t_0)
	elif lambda1 <= 2.6e-150:
		tmp = R * (phi2 + (-1.0 * phi1))
	else:
		tmp = (R * lambda2) * t_0
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sqrt(Float64(0.5 + Float64(0.5 * cos(Float64(phi1 + phi2)))))
	tmp = 0.0
	if (lambda1 <= -1.5e+118)
		tmp = Float64(-1.0 * Float64(Float64(R * lambda1) * t_0));
	elseif (lambda1 <= 2.6e-150)
		tmp = Float64(R * Float64(phi2 + Float64(-1.0 * phi1)));
	else
		tmp = Float64(Float64(R * lambda2) * t_0);
	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 = sqrt((0.5 + (0.5 * cos((phi1 + phi2)))));
	tmp = 0.0;
	if (lambda1 <= -1.5e+118)
		tmp = -1.0 * ((R * lambda1) * t_0);
	elseif (lambda1 <= 2.6e-150)
		tmp = R * (phi2 + (-1.0 * phi1));
	else
		tmp = (R * lambda2) * t_0;
	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[Sqrt[N[(0.5 + N[(0.5 * N[Cos[N[(phi1 + phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda1, -1.5e+118], N[(-1.0 * N[(N[(R * lambda1), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda1, 2.6e-150], N[(R * N[(phi2 + N[(-1.0 * phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(R * lambda2), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -1.5e118

   1. Initial program 45.0%

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

   3. Applied rewrites 44.5%

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

   4. Taylor expanded in lambda1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-+.f64 67.6

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

   6. Applied rewrites 67.6%

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

   if -1.5e118 < lambda1 < 2.5999999999999998e-150

   1. Initial program 65.4%

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

   2. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 61.4

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

   4. Applied rewrites 61.4%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 66.8

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

   7. Applied rewrites 66.8%

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

   if 2.5999999999999998e-150 < lambda1

   1. Initial program 56.4%

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

   3. Applied rewrites 54.8%

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

   4. Taylor expanded in lambda2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-+.f64 14.9

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

   6. Applied rewrites 14.9%

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

Alternative 7: 61.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 41000000000:\\ \;\;\;\;R \cdot \left(\phi_2 + -1 \cdot \phi_1\right)\\ \mathbf{elif}\;\lambda_2 \leq 6.4 \cdot 10^{+176}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \left(\left(-\frac{\phi_2 \cdot R}{\phi_1}\right) + R\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot \lambda_2\right) \cdot \sqrt{0.5 + 0.5 \cdot \cos \phi_1}\\ \end{array} \end{array} \]

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 (<= lambda2 41000000000.0)
   (* R (+ phi2 (* -1.0 phi1)))
   (if (<= lambda2 6.4e+176)
     (* (- phi1) (+ (- (/ (* phi2 R) phi1)) R))
     (* (* R lambda2) (sqrt (+ 0.5 (* 0.5 (cos phi1))))))))

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 (lambda2 <= 41000000000.0) {
		tmp = R * (phi2 + (-1.0 * phi1));
	} else if (lambda2 <= 6.4e+176) {
		tmp = -phi1 * (-((phi2 * R) / phi1) + R);
	} else {
		tmp = (R * lambda2) * sqrt((0.5 + (0.5 * cos(phi1))));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= 41000000000.0d0) then
        tmp = r * (phi2 + ((-1.0d0) * phi1))
    else if (lambda2 <= 6.4d+176) then
        tmp = -phi1 * (-((phi2 * r) / phi1) + r)
    else
        tmp = (r * lambda2) * sqrt((0.5d0 + (0.5d0 * cos(phi1))))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 41000000000.0) {
		tmp = R * (phi2 + (-1.0 * phi1));
	} else if (lambda2 <= 6.4e+176) {
		tmp = -phi1 * (-((phi2 * R) / phi1) + R);
	} else {
		tmp = (R * lambda2) * Math.sqrt((0.5 + (0.5 * Math.cos(phi1))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 41000000000.0:
		tmp = R * (phi2 + (-1.0 * phi1))
	elif lambda2 <= 6.4e+176:
		tmp = -phi1 * (-((phi2 * R) / phi1) + R)
	else:
		tmp = (R * lambda2) * math.sqrt((0.5 + (0.5 * math.cos(phi1))))
	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 (lambda2 <= 41000000000.0)
		tmp = Float64(R * Float64(phi2 + Float64(-1.0 * phi1)));
	elseif (lambda2 <= 6.4e+176)
		tmp = Float64(Float64(-phi1) * Float64(Float64(-Float64(Float64(phi2 * R) / phi1)) + R));
	else
		tmp = Float64(Float64(R * lambda2) * sqrt(Float64(0.5 + Float64(0.5 * cos(phi1)))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 41000000000.0)
		tmp = R * (phi2 + (-1.0 * phi1));
	elseif (lambda2 <= 6.4e+176)
		tmp = -phi1 * (-((phi2 * R) / phi1) + R);
	else
		tmp = (R * lambda2) * sqrt((0.5 + (0.5 * cos(phi1))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 41000000000.0], N[(R * N[(phi2 + N[(-1.0 * phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 6.4e+176], N[((-phi1) * N[((-N[(N[(phi2 * R), $MachinePrecision] / phi1), $MachinePrecision]) + R), $MachinePrecision]), $MachinePrecision], N[(N[(R * lambda2), $MachinePrecision] * N[Sqrt[N[(0.5 + N[(0.5 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < 4.1e10

   1. Initial program 60.9%

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

   2. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 56.3

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

   4. Applied rewrites 56.3%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 61.2

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

   7. Applied rewrites 61.2%

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

   if 4.1e10 < lambda2 < 6.3999999999999996e176

   1. Initial program 61.0%

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

   2. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 54.2

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

   4. Applied rewrites 54.2%

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

   if 6.3999999999999996e176 < lambda2

   1. Initial program 44.2%

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

   3. Applied rewrites 44.2%

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

   4. Taylor expanded in lambda2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-+.f64 76.8

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

   6. Applied rewrites 76.8%

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

   7. Taylor expanded in phi1 around inf

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

   9. Applied rewrites 68.9%

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

Alternative 8: 58.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 7.6 \cdot 10^{+90}:\\ \;\;\;\;R \cdot \left(\phi_2 + -1 \cdot \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot \lambda_2\right) \cdot \sqrt{0.5 + 0.5 \cdot \cos \left(\phi_1 + \phi_2\right)}\\ \end{array} \end{array} \]

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 (<= lambda2 7.6e+90)
   (* R (+ phi2 (* -1.0 phi1)))
   (* (* R lambda2) (sqrt (+ 0.5 (* 0.5 (cos (+ phi1 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 (lambda2 <= 7.6e+90) {
		tmp = R * (phi2 + (-1.0 * phi1));
	} else {
		tmp = (R * lambda2) * sqrt((0.5 + (0.5 * cos((phi1 + 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 (lambda2 <= 7.6d+90) then
        tmp = r * (phi2 + ((-1.0d0) * phi1))
    else
        tmp = (r * lambda2) * sqrt((0.5d0 + (0.5d0 * cos((phi1 + 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 (lambda2 <= 7.6e+90) {
		tmp = R * (phi2 + (-1.0 * phi1));
	} else {
		tmp = (R * lambda2) * Math.sqrt((0.5 + (0.5 * Math.cos((phi1 + 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 lambda2 <= 7.6e+90:
		tmp = R * (phi2 + (-1.0 * phi1))
	else:
		tmp = (R * lambda2) * math.sqrt((0.5 + (0.5 * math.cos((phi1 + 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 (lambda2 <= 7.6e+90)
		tmp = Float64(R * Float64(phi2 + Float64(-1.0 * phi1)));
	else
		tmp = Float64(Float64(R * lambda2) * sqrt(Float64(0.5 + Float64(0.5 * cos(Float64(phi1 + 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 (lambda2 <= 7.6e+90)
		tmp = R * (phi2 + (-1.0 * phi1));
	else
		tmp = (R * lambda2) * sqrt((0.5 + (0.5 * cos((phi1 + 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[lambda2, 7.6e+90], N[(R * N[(phi2 + N[(-1.0 * phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(R * lambda2), $MachinePrecision] * N[Sqrt[N[(0.5 + N[(0.5 * N[Cos[N[(phi1 + phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 7.6000000000000002e90

   1. Initial program 61.3%

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

   2. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 56.1

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 61.1

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

   7. Applied rewrites 61.1%

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

   if 7.6000000000000002e90 < lambda2

   1. Initial program 49.0%

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

   3. Applied rewrites 48.2%

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

   4. Taylor expanded in lambda2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-+.f64 66.9

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

   6. Applied rewrites 66.9%

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

Alternative 9: 58.1% accurate, 4.3× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{+203}:\\ \;\;\;\;\left(\phi_1 \cdot \mathsf{fma}\left(-1, \frac{R}{\phi_2}, \frac{R}{\phi_1}\right)\right) \cdot \phi_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 + -1 \cdot \phi_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (- lambda1 lambda2) -1e+203)
   (* (* phi1 (fma -1.0 (/ R phi2) (/ R phi1))) phi2)
   (* R (+ phi2 (* -1.0 phi1)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda1 - lambda2) <= -1e+203) {
		tmp = (phi1 * fma(-1.0, (R / phi2), (R / phi1))) * phi2;
	} else {
		tmp = R * (phi2 + (-1.0 * phi1));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= -1e+203)
		tmp = Float64(Float64(phi1 * fma(-1.0, Float64(R / phi2), Float64(R / phi1))) * phi2);
	else
		tmp = Float64(R * Float64(phi2 + Float64(-1.0 * phi1)));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -1e+203], N[(N[(phi1 * N[(-1.0 * N[(R / phi2), $MachinePrecision] + N[(R / phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * phi2), $MachinePrecision], N[(R * N[(phi2 + N[(-1.0 * phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < -9.9999999999999999e202

   1. Initial program 45.3%

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

   2. 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 35.9

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

   4. Applied rewrites 35.9%

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

   5. Taylor expanded in phi1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 40.2

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

   7. Applied rewrites 40.2%

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

   if -9.9999999999999999e202 < (-.f64 lambda1 lambda2)

   1. Initial program 61.6%

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

   2. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 56.6

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

   4. Applied rewrites 56.6%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 61.8

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

   7. Applied rewrites 61.8%

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

Alternative 10: 57.6% accurate, 5.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.05 \cdot 10^{+61}:\\ \;\;\;\;R \cdot \left(\phi_2 + -1 \cdot \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot \left(R \cdot \left(\frac{\phi_1}{\phi_2} - 1\right)\right)\right) \cdot \phi_2\\ \end{array} \end{array} \]

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 -1.05e+61)
   (* R (+ phi2 (* -1.0 phi1)))
   (* (* -1.0 (* R (- (/ phi1 phi2) 1.0))) 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 <= -1.05e+61) {
		tmp = R * (phi2 + (-1.0 * phi1));
	} else {
		tmp = (-1.0 * (R * ((phi1 / phi2) - 1.0))) * 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 <= (-1.05d+61)) then
        tmp = r * (phi2 + ((-1.0d0) * phi1))
    else
        tmp = ((-1.0d0) * (r * ((phi1 / phi2) - 1.0d0))) * 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 <= -1.05e+61) {
		tmp = R * (phi2 + (-1.0 * phi1));
	} else {
		tmp = (-1.0 * (R * ((phi1 / phi2) - 1.0))) * 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 <= -1.05e+61:
		tmp = R * (phi2 + (-1.0 * phi1))
	else:
		tmp = (-1.0 * (R * ((phi1 / phi2) - 1.0))) * 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 <= -1.05e+61)
		tmp = Float64(R * Float64(phi2 + Float64(-1.0 * phi1)));
	else
		tmp = Float64(Float64(-1.0 * Float64(R * Float64(Float64(phi1 / phi2) - 1.0))) * 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 <= -1.05e+61)
		tmp = R * (phi2 + (-1.0 * phi1));
	else
		tmp = (-1.0 * (R * ((phi1 / phi2) - 1.0))) * 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, -1.05e+61], N[(R * N[(phi2 + N[(-1.0 * phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * N[(R * N[(N[(phi1 / phi2), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.0500000000000001e61

   1. Initial program 52.2%

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

   2. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 77.6

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

   4. Applied rewrites 77.6%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 77.6

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

   7. Applied rewrites 77.6%

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

   if -1.0500000000000001e61 < phi1

   1. Initial program 63.0%

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

   2. Taylor expanded in 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 45.5

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

   4. Applied rewrites 45.5%

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

   5. Taylor expanded in R around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 46.8

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

   7. Applied rewrites 46.8%

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

Alternative 11: 57.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1.35 \cdot 10^{+44}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \left(\left(-\frac{\phi_2 \cdot R}{\phi_1}\right) + R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 + -1 \cdot \phi_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (- lambda1 lambda2) -1.35e+44)
   (* (- phi1) (+ (- (/ (* phi2 R) phi1)) R))
   (* R (+ phi2 (* -1.0 phi1)))))

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) <= -1.35e+44) {
		tmp = -phi1 * (-((phi2 * R) / phi1) + R);
	} else {
		tmp = R * (phi2 + (-1.0 * phi1));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if ((lambda1 - lambda2) <= (-1.35d+44)) then
        tmp = -phi1 * (-((phi2 * r) / phi1) + r)
    else
        tmp = r * (phi2 + ((-1.0d0) * phi1))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda1 - lambda2) <= -1.35e+44) {
		tmp = -phi1 * (-((phi2 * R) / phi1) + R);
	} else {
		tmp = R * (phi2 + (-1.0 * phi1));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (lambda1 - lambda2) <= -1.35e+44:
		tmp = -phi1 * (-((phi2 * R) / phi1) + R)
	else:
		tmp = R * (phi2 + (-1.0 * phi1))
	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) <= -1.35e+44)
		tmp = Float64(Float64(-phi1) * Float64(Float64(-Float64(Float64(phi2 * R) / phi1)) + R));
	else
		tmp = Float64(R * Float64(phi2 + Float64(-1.0 * phi1)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((lambda1 - lambda2) <= -1.35e+44)
		tmp = -phi1 * (-((phi2 * R) / phi1) + R);
	else
		tmp = R * (phi2 + (-1.0 * phi1));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -1.35e+44], N[((-phi1) * N[((-N[(N[(phi2 * R), $MachinePrecision] / phi1), $MachinePrecision]) + R), $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 + N[(-1.0 * phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < -1.35e44

   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. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 45.7

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

   4. Applied rewrites 45.7%

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

   if -1.35e44 < (-.f64 lambda1 lambda2)

   1. Initial program 62.1%

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

   2. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 58.3

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

   4. Applied rewrites 58.3%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 63.4

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

   7. Applied rewrites 63.4%

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

Alternative 12: 57.3% accurate, 6.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.3 \cdot 10^{+62}:\\ \;\;\;\;R \cdot \left(\phi_2 + -1 \cdot \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-\phi_1 \cdot R}{\phi_2} + R\right) \cdot \phi_2\\ \end{array} \end{array} \]

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 -1.3e+62)
   (* R (+ phi2 (* -1.0 phi1)))
   (* (+ (/ (- (* phi1 R)) phi2) 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 <= -1.3e+62) {
		tmp = R * (phi2 + (-1.0 * phi1));
	} else {
		tmp = ((-(phi1 * R) / phi2) + 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 <= (-1.3d+62)) then
        tmp = r * (phi2 + ((-1.0d0) * phi1))
    else
        tmp = ((-(phi1 * r) / phi2) + 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 <= -1.3e+62) {
		tmp = R * (phi2 + (-1.0 * phi1));
	} else {
		tmp = ((-(phi1 * R) / phi2) + 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 <= -1.3e+62:
		tmp = R * (phi2 + (-1.0 * phi1))
	else:
		tmp = ((-(phi1 * R) / phi2) + 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 <= -1.3e+62)
		tmp = Float64(R * Float64(phi2 + Float64(-1.0 * phi1)));
	else
		tmp = Float64(Float64(Float64(Float64(-Float64(phi1 * R)) / phi2) + 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 <= -1.3e+62)
		tmp = R * (phi2 + (-1.0 * phi1));
	else
		tmp = ((-(phi1 * R) / phi2) + 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, -1.3e+62], N[(R * N[(phi2 + N[(-1.0 * phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-N[(phi1 * R), $MachinePrecision]) / phi2), $MachinePrecision] + R), $MachinePrecision] * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.29999999999999992e62

   1. Initial program 52.1%

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

   2. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 77.6

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

   4. Applied rewrites 77.6%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 77.6

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

   7. Applied rewrites 77.6%

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

   if -1.29999999999999992e62 < phi1

   1. Initial program 63.0%

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

   2. Taylor expanded in 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 45.5

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

   4. Applied rewrites 45.5%

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

Alternative 13: 51.9% accurate, 11.1× speedup

Math

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

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
 (* R (+ phi2 (* -1.0 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 * (phi2 + (-1.0 * 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 * (phi2 + ((-1.0d0) * 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 * (phi2 + (-1.0 * phi1));
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	return R * (phi2 + (-1.0 * phi1))

Julia

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

Derivation

1. Initial program 59.0%

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

2. Taylor expanded in phi1 around -inf

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 N/A

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

   7. associate-*r/ N/A

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

   8. mul-1-neg N/A

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

   9. lower-/.f64 N/A

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

   10. lower-neg.f64 52.9

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

4. Applied rewrites 52.9%

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

5. Taylor expanded in phi1 around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 57.4

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

7. Applied rewrites 57.4%

   \[\leadsto R \cdot \left(\phi_2 + \color{blue}{-1 \cdot \phi_1}\right) \]
8. Add Preprocessing

Alternative 14: 47.8% accurate, 12.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 10^{-32}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

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 (<= phi2 1e-32) (* 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 <= 1e-32) {
		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 (phi2 <= 1d-32) 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 (phi2 <= 1e-32) {
		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 phi2 <= 1e-32:
		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 (phi2 <= 1e-32)
		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 (phi2 <= 1e-32)
		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[phi2, 1e-32], N[(R * (-phi1)), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.00000000000000006e-32

   1. Initial program 62.5%

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

   2. 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 45.3

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

   4. Applied rewrites 45.3%

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

   if 1.00000000000000006e-32 < phi2

   1. Initial program 55.3%

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

   2. Taylor expanded in phi2 around inf

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

   4. Applied rewrites 59.1%

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

Alternative 15: 30.9% 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 59.0%

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

2. Taylor expanded in phi2 around inf

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

4. Applied rewrites 30.9%

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

Alternative 16: 3.3% 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 59.0%

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

2. Taylor expanded in phi1 around inf

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

4. Applied rewrites 3.3%

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

Reproduce

herbie shell --seed 2025116 
(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))))))