Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.2% → 95.9%

Time: 6.4s

Alternatives: 10

Speedup: 2.5×

• 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: 59.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

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

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

Julia

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

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

Wolfram

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 63.1%

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-hypot.f64 N/A

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

   5. lower--.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lower--.f64 96.7

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

4. Applied rewrites 96.7%

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

Alternative 2: 73.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\\ t_1 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ \mathbf{if}\;R \cdot \sqrt{t\_1 \cdot t\_1 + t\_0} \leq 5 \cdot 10^{+297}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \frac{\phi_2 + \phi_1}{2}\right)\right) + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \lambda_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (- phi1 phi2) (- phi1 phi2)))
        (t_1 (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))))
   (if (<= (* R (sqrt (+ (* t_1 t_1) t_0))) 5e+297)
     (*
      R
      (sqrt
       (+
        (*
         (* (- lambda1 lambda2) (- lambda1 lambda2))
         (+ 0.5 (* 0.5 (cos (* 2.0 (/ (+ phi2 phi1) 2.0))))))
        t_0)))
     (* (hypot (- phi1 phi2) (- lambda1 lambda2)) R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (phi1 - phi2) * (phi1 - phi2);
	double t_1 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	double tmp;
	if ((R * sqrt(((t_1 * t_1) + t_0))) <= 5e+297) {
		tmp = R * sqrt(((((lambda1 - lambda2) * (lambda1 - lambda2)) * (0.5 + (0.5 * cos((2.0 * ((phi2 + phi1) / 2.0)))))) + t_0));
	} else {
		tmp = hypot((phi1 - phi2), (lambda1 - lambda2)) * R;
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (phi1 - phi2) * (phi1 - phi2);
	double t_1 = (lambda1 - lambda2) * Math.cos(((phi1 + phi2) / 2.0));
	double tmp;
	if ((R * Math.sqrt(((t_1 * t_1) + t_0))) <= 5e+297) {
		tmp = R * Math.sqrt(((((lambda1 - lambda2) * (lambda1 - lambda2)) * (0.5 + (0.5 * Math.cos((2.0 * ((phi2 + phi1) / 2.0)))))) + t_0));
	} else {
		tmp = Math.hypot((phi1 - phi2), (lambda1 - lambda2)) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (phi1 - phi2) * (phi1 - phi2)
	t_1 = (lambda1 - lambda2) * math.cos(((phi1 + phi2) / 2.0))
	tmp = 0
	if (R * math.sqrt(((t_1 * t_1) + t_0))) <= 5e+297:
		tmp = R * math.sqrt(((((lambda1 - lambda2) * (lambda1 - lambda2)) * (0.5 + (0.5 * math.cos((2.0 * ((phi2 + phi1) / 2.0)))))) + t_0))
	else:
		tmp = math.hypot((phi1 - phi2), (lambda1 - lambda2)) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2))
	t_1 = Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0)))
	tmp = 0.0
	if (Float64(R * sqrt(Float64(Float64(t_1 * t_1) + t_0))) <= 5e+297)
		tmp = Float64(R * sqrt(Float64(Float64(Float64(Float64(lambda1 - lambda2) * Float64(lambda1 - lambda2)) * Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(Float64(phi2 + phi1) / 2.0)))))) + t_0)));
	else
		tmp = Float64(hypot(Float64(phi1 - phi2), Float64(lambda1 - lambda2)) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (phi1 - phi2) * (phi1 - phi2);
	t_1 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	tmp = 0.0;
	if ((R * sqrt(((t_1 * t_1) + t_0))) <= 5e+297)
		tmp = R * sqrt(((((lambda1 - lambda2) * (lambda1 - lambda2)) * (0.5 + (0.5 * cos((2.0 * ((phi2 + phi1) / 2.0)))))) + t_0));
	else
		tmp = hypot((phi1 - phi2), (lambda1 - lambda2)) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(R * N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e+297], N[(R * N[Sqrt[N[(N[(N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(N[(phi2 + phi1), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(phi1 - phi2), $MachinePrecision] ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 R (sqrt.f64 (+.f64 (*.f64 (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) #s(literal 2 binary64)))) (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) #s(literal 2 binary64))))) (*.f64 (-.f64 phi1 phi2) (-.f64 phi1 phi2))))) < 4.9999999999999998e297

   1. Initial program 70.9%

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

      1. swap-sqr N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower--.f64 N/A

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

      6. sqr-cos-a N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 70.9

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

   4. Applied rewrites 70.9%

      \[\leadsto R \cdot \sqrt{\color{blue}{{\left(\lambda_1 - \lambda_2\right)}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \frac{\phi_2 + \phi_1}{2}\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   5. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 70.9

         \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \frac{\phi_2 + \phi_1}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   6. Applied rewrites 70.9%

      \[\leadsto R \cdot \sqrt{\color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \frac{\phi_2 + \phi_1}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   if 4.9999999999999998e297 < (*.f64 R (sqrt.f64 (+.f64 (*.f64 (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) #s(literal 2 binary64)))) (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) #s(literal 2 binary64))))) (*.f64 (-.f64 phi1 phi2) (-.f64 phi1 phi2)))))

   1. Initial program 48.9%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-hypot.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 98.2

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in phi2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 83.0

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

   7. Applied rewrites 83.0%

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

   8. Taylor expanded in phi1 around 0

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

      1. lower--.f64 83.5

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

   10. Applied rewrites 83.5%

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

Alternative 3: 79.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.25 \cdot 10^{-79}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 4.5 \cdot 10^{+17} \lor \neg \left(\phi_2 \leq 7.5 \cdot 10^{+116}\right):\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \lambda_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1, \cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.25e-79)
   (* (hypot (* (cos (* 0.5 phi1)) (- lambda1 lambda2)) phi1) R)
   (if (or (<= phi2 4.5e+17) (not (<= phi2 7.5e+116)))
     (* (hypot (- phi1 phi2) (- lambda1 lambda2)) R)
     (* (hypot phi1 (* (cos (* 0.5 phi2)) (- lambda1 lambda2))) R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.25e-79) {
		tmp = hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R;
	} else if ((phi2 <= 4.5e+17) || !(phi2 <= 7.5e+116)) {
		tmp = hypot((phi1 - phi2), (lambda1 - lambda2)) * R;
	} else {
		tmp = hypot(phi1, (cos((0.5 * phi2)) * (lambda1 - lambda2))) * R;
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.25e-79) {
		tmp = Math.hypot((Math.cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R;
	} else if ((phi2 <= 4.5e+17) || !(phi2 <= 7.5e+116)) {
		tmp = Math.hypot((phi1 - phi2), (lambda1 - lambda2)) * R;
	} else {
		tmp = Math.hypot(phi1, (Math.cos((0.5 * phi2)) * (lambda1 - lambda2))) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.25e-79:
		tmp = math.hypot((math.cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R
	elif (phi2 <= 4.5e+17) or not (phi2 <= 7.5e+116):
		tmp = math.hypot((phi1 - phi2), (lambda1 - lambda2)) * R
	else:
		tmp = math.hypot(phi1, (math.cos((0.5 * phi2)) * (lambda1 - lambda2))) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.25e-79)
		tmp = Float64(hypot(Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2)), phi1) * R);
	elseif ((phi2 <= 4.5e+17) || !(phi2 <= 7.5e+116))
		tmp = Float64(hypot(Float64(phi1 - phi2), Float64(lambda1 - lambda2)) * R);
	else
		tmp = Float64(hypot(phi1, Float64(cos(Float64(0.5 * phi2)) * Float64(lambda1 - lambda2))) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 1.25e-79)
		tmp = hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R;
	elseif ((phi2 <= 4.5e+17) || ~((phi2 <= 7.5e+116)))
		tmp = hypot((phi1 - phi2), (lambda1 - lambda2)) * R;
	else
		tmp = hypot(phi1, (cos((0.5 * phi2)) * (lambda1 - lambda2))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.25e-79], N[(N[Sqrt[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision] * R), $MachinePrecision], If[Or[LessEqual[phi2, 4.5e+17], N[Not[LessEqual[phi2, 7.5e+116]], $MachinePrecision]], N[(N[Sqrt[N[(phi1 - phi2), $MachinePrecision] ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[phi1 ^ 2 + N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 1.25e-79

   1. Initial program 61.9%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-hypot.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 97.8

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

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in phi2 around 0

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

      1. unpow-prod-down N/A

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

      2. unpow2 N/A

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

      3. pow2 N/A

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

      4. lower-hypot.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 75.2

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

   7. Applied rewrites 75.2%

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

   if 1.25e-79 < phi2 < 4.5e17 or 7.5e116 < phi2

   1. Initial program 70.8%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-hypot.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 97.7

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

   4. Applied rewrites 97.7%

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

   5. Taylor expanded in phi2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 85.5

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

   7. Applied rewrites 85.5%

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

   8. Taylor expanded in phi1 around 0

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

      1. lower--.f64 87.9

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

   10. Applied rewrites 87.9%

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

   if 4.5e17 < phi2 < 7.5e116

   1. Initial program 56.1%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-hypot.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 81.2

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

   4. Applied rewrites 81.2%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 81.4

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

   7. Applied rewrites 81.4%

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

   8. Taylor expanded in phi1 around inf

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

   10. Applied rewrites 70.7%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.25 \cdot 10^{-79}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 4.5 \cdot 10^{+17} \lor \neg \left(\phi_2 \leq 7.5 \cdot 10^{+116}\right):\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \lambda_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1, \cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.25 \cdot 10^{-79}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, \cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.25e-79)
   (* (hypot (* (cos (* 0.5 phi1)) (- lambda1 lambda2)) phi1) R)
   (* (hypot (- phi1 phi2) (* (cos (* 0.5 phi2)) (- lambda1 lambda2))) R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.25e-79) {
		tmp = hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R;
	} else {
		tmp = hypot((phi1 - phi2), (cos((0.5 * phi2)) * (lambda1 - lambda2))) * R;
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.25e-79) {
		tmp = Math.hypot((Math.cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R;
	} else {
		tmp = Math.hypot((phi1 - phi2), (Math.cos((0.5 * phi2)) * (lambda1 - lambda2))) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.25e-79:
		tmp = math.hypot((math.cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R
	else:
		tmp = math.hypot((phi1 - phi2), (math.cos((0.5 * phi2)) * (lambda1 - lambda2))) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.25e-79)
		tmp = Float64(hypot(Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2)), phi1) * R);
	else
		tmp = Float64(hypot(Float64(phi1 - phi2), Float64(cos(Float64(0.5 * phi2)) * Float64(lambda1 - lambda2))) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 1.25e-79)
		tmp = hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R;
	else
		tmp = hypot((phi1 - phi2), (cos((0.5 * phi2)) * (lambda1 - lambda2))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.25e-79], N[(N[Sqrt[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(phi1 - phi2), $MachinePrecision] ^ 2 + N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.25e-79

   1. Initial program 61.9%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-hypot.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 97.8

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

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in phi2 around 0

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

      1. unpow-prod-down N/A

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

      2. unpow2 N/A

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

      3. pow2 N/A

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

      4. lower-hypot.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 75.2

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

   7. Applied rewrites 75.2%

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

   if 1.25e-79 < phi2

   1. Initial program 66.8%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-hypot.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 93.2

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

   4. Applied rewrites 93.2%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 89.2

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

   7. Applied rewrites 89.2%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.25 \cdot 10^{-79}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, \cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.25 \cdot 10^{-79}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \lambda_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.25e-79)
   (* (hypot (* (cos (* 0.5 phi1)) (- lambda1 lambda2)) phi1) R)
   (* (hypot (- phi1 phi2) (- lambda1 lambda2)) R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.25e-79) {
		tmp = hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R;
	} else {
		tmp = hypot((phi1 - phi2), (lambda1 - lambda2)) * R;
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.25e-79) {
		tmp = Math.hypot((Math.cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R;
	} else {
		tmp = Math.hypot((phi1 - phi2), (lambda1 - lambda2)) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.25e-79:
		tmp = math.hypot((math.cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R
	else:
		tmp = math.hypot((phi1 - phi2), (lambda1 - lambda2)) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.25e-79)
		tmp = Float64(hypot(Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2)), phi1) * R);
	else
		tmp = Float64(hypot(Float64(phi1 - phi2), Float64(lambda1 - lambda2)) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 1.25e-79)
		tmp = hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R;
	else
		tmp = hypot((phi1 - phi2), (lambda1 - lambda2)) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.25e-79], N[(N[Sqrt[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(phi1 - phi2), $MachinePrecision] ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.25e-79

   1. Initial program 61.9%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-hypot.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 97.8

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

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in phi2 around 0

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

      1. unpow-prod-down N/A

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

      2. unpow2 N/A

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

      3. pow2 N/A

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

      4. lower-hypot.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 75.2

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

   7. Applied rewrites 75.2%

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

   if 1.25e-79 < phi2

   1. Initial program 66.8%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-hypot.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 93.2

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

   4. Applied rewrites 93.2%

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

   5. Taylor expanded in phi2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 71.4

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

   7. Applied rewrites 71.4%

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

   8. Taylor expanded in phi1 around 0

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

      1. lower--.f64 75.9

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

   10. Applied rewrites 75.9%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.25 \cdot 10^{-79}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \lambda_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.2 \cdot 10^{+215}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \lambda_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \sqrt{0.5 + 0.5 \cdot \cos \left(\phi_1 + \phi_2\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 1.2e+215)
   (* (hypot (- phi1 phi2) (- lambda1 lambda2)) R)
   (* R (* lambda2 (sqrt (+ 0.5 (* 0.5 (cos (+ phi1 phi2)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1.2e+215) {
		tmp = hypot((phi1 - phi2), (lambda1 - lambda2)) * R;
	} else {
		tmp = R * (lambda2 * sqrt((0.5 + (0.5 * cos((phi1 + phi2))))));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1.2e+215) {
		tmp = Math.hypot((phi1 - phi2), (lambda1 - lambda2)) * R;
	} else {
		tmp = R * (lambda2 * Math.sqrt((0.5 + (0.5 * Math.cos((phi1 + phi2))))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 1.2e+215:
		tmp = math.hypot((phi1 - phi2), (lambda1 - lambda2)) * R
	else:
		tmp = R * (lambda2 * math.sqrt((0.5 + (0.5 * math.cos((phi1 + phi2))))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 1.2e+215)
		tmp = Float64(hypot(Float64(phi1 - phi2), Float64(lambda1 - lambda2)) * R);
	else
		tmp = Float64(R * Float64(lambda2 * sqrt(Float64(0.5 + Float64(0.5 * cos(Float64(phi1 + phi2)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 1.2e+215)
		tmp = hypot((phi1 - phi2), (lambda1 - lambda2)) * R;
	else
		tmp = R * (lambda2 * sqrt((0.5 + (0.5 * cos((phi1 + phi2))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 1.2e+215], N[(N[Sqrt[N[(phi1 - phi2), $MachinePrecision] ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(R * N[(lambda2 * N[Sqrt[N[(0.5 + N[(0.5 * N[Cos[N[(phi1 + phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 1.2e215

   1. Initial program 64.0%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-hypot.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 96.7

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

   4. Applied rewrites 96.7%

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

   5. Taylor expanded in phi2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 83.2

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

   7. Applied rewrites 83.2%

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

   8. Taylor expanded in phi1 around 0

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

      1. lower--.f64 83.6

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

   10. Applied rewrites 83.6%

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

   if 1.2e215 < lambda2

   1. Initial program 52.6%

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

      1. swap-sqr N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower--.f64 N/A

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

      6. sqr-cos-a N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 52.6

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

   4. Applied rewrites 52.6%

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

   5. Taylor expanded in lambda2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-+.f64 77.7

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

   7. Applied rewrites 77.7%

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

Alternative 7: 84.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \lambda_2\right) \cdot R \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* (hypot (- phi1 phi2) (- lambda1 lambda2)) R))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return hypot((phi1 - phi2), (lambda1 - lambda2)) * R;
}

Java

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

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return math.hypot((phi1 - phi2), (lambda1 - lambda2)) * R

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(hypot(Float64(phi1 - phi2), Float64(lambda1 - lambda2)) * R)
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = hypot((phi1 - phi2), (lambda1 - lambda2)) * R;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[Sqrt[N[(phi1 - phi2), $MachinePrecision] ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 63.1%

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-hypot.f64 N/A

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

   5. lower--.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lower--.f64 96.7

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

4. Applied rewrites 96.7%

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

5. Taylor expanded in phi2 around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower--.f64 82.8

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

7. Applied rewrites 82.8%

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

8. Taylor expanded in phi1 around 0

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

   1. lower--.f64 82.8

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

10. Applied rewrites 82.8%

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

Alternative 8: 27.3% accurate, 19.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.2 \cdot 10^{-55}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -3.2e-55) (* R (- phi1)) (* R phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3.2e-55) {
		tmp = R * -phi1;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

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) :: tmp
    if (phi1 <= (-3.2d-55)) then
        tmp = r * -phi1
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3.2e-55) {
		tmp = R * -phi1;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -3.2e-55:
		tmp = R * -phi1
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -3.2e-55)
		tmp = Float64(R * Float64(-phi1));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -3.2e-55)
		tmp = R * -phi1;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -3.2e-55], N[(R * (-phi1)), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.2000000000000001e-55

   1. Initial program 60.7%

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

   3. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 50.8

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

   5. Applied rewrites 50.8%

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

   if -3.2000000000000001e-55 < phi1

   1. Initial program 64.2%

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

   3. Taylor expanded in phi2 around inf

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

   5. Applied rewrites 14.7%

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

Alternative 9: 28.5% accurate, 31.0× speedup

Math

\[\begin{array}{l} \\ R \cdot \left(\phi_2 - \phi_1\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2) :precision binary64 (* R (- phi2 phi1)))

C

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

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
    code = r * (phi2 - phi1)
end function

Java

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

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * (phi2 - phi1)

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * Float64(phi2 - phi1))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * (phi2 - phi1);
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.1%

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

3. Taylor expanded in phi1 around -inf

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

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 24.8

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

5. Applied rewrites 24.8%

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

6. Taylor expanded in phi1 around 0

   \[\leadsto R \cdot \left(\phi_2 + \color{blue}{-1 \cdot \phi_1}\right) \]
7. 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 26.0

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

8. Applied rewrites 26.0%

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

9. Final simplification 26.0%

   \[\leadsto R \cdot \left(\phi_2 - \phi_1\right) \]
10. Add Preprocessing

Alternative 10: 16.8% accurate, 46.5× speedup

Math

\[\begin{array}{l} \\ R \cdot \phi_2 \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2) :precision binary64 (* R phi2))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * 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
    code = r * phi2
end function

Java

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

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * phi2

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * phi2)
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * phi2;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * phi2), $MachinePrecision]

Derivation

1. Initial program 63.1%

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

3. Taylor expanded in phi2 around inf

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

5. Applied rewrites 12.6%

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

Reproduce

herbie shell --seed 2025044 
(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))))))