Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.3% → 95.9%

Time: 6.5s

Alternatives: 10

Speedup: 1.2×

• 30.2% 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: 59.3% 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 57.0%

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

   1. lift-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-+.f64 N/A

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

4. Applied rewrites 94.7%

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

Alternative 2: 82.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.5 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, \cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 4.5e-45)
   (* (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 <= 4.5e-45) {
		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 <= 4.5e-45) {
		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 <= 4.5e-45:
		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 <= 4.5e-45)
		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 <= 4.5e-45)
		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, 4.5e-45], 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 < 4.4999999999999999e-45

   1. Initial program 57.4%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

   4. Applied rewrites 95.6%

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

   5. Taylor expanded in phi2 around 0

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

   7. Applied rewrites 79.8%

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

   if 4.4999999999999999e-45 < phi2

   1. Initial program 56.0%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

   4. Applied rewrites 92.2%

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

   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 90.9

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

   7. Applied rewrites 90.9%

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

Alternative 3: 79.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.3 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, \cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_1\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.3e-25)
   (* (hypot (* (cos (* 0.5 phi1)) (- lambda1 lambda2)) phi1) R)
   (* (hypot (- phi1 phi2) (* (cos (* 0.5 phi2)) lambda1)) R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.3e-25) {
		tmp = hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R;
	} else {
		tmp = hypot((phi1 - phi2), (cos((0.5 * phi2)) * lambda1)) * R;
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.3e-25) {
		tmp = Math.hypot((Math.cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R;
	} else {
		tmp = Math.hypot((phi1 - phi2), (Math.cos((0.5 * phi2)) * lambda1)) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.3e-25:
		tmp = math.hypot((math.cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R
	else:
		tmp = math.hypot((phi1 - phi2), (math.cos((0.5 * phi2)) * lambda1)) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.3e-25)
		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)) * lambda1)) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 1.3e-25)
		tmp = hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R;
	else
		tmp = hypot((phi1 - phi2), (cos((0.5 * phi2)) * lambda1)) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.3e-25], 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] * lambda1), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.3e-25

   1. Initial program 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

   4. Applied rewrites 95.6%

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

   5. Taylor expanded in phi2 around 0

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

   7. Applied rewrites 79.9%

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

   if 1.3e-25 < phi2

   1. Initial program 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

   4. Applied rewrites 92.1%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 90.8

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

   7. Applied rewrites 90.8%

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

   8. Taylor expanded in lambda1 around inf

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

   10. Applied rewrites 84.2%

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

Alternative 4: 75.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 5.6 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(-1, \phi_1, \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 5.6e+48)
   (* (hypot (* (cos (* 0.5 phi1)) (- lambda1 lambda2)) phi1) R)
   (* R (fma -1.0 phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 5.6e+48) {
		tmp = hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R;
	} else {
		tmp = R * fma(-1.0, phi1, phi2);
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 5.6e+48)
		tmp = Float64(hypot(Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2)), phi1) * R);
	else
		tmp = Float64(R * fma(-1.0, phi1, phi2));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 5.6e+48], N[(N[Sqrt[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(R * N[(-1.0 * phi1 + phi2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 5.60000000000000025e48

   1. Initial program 56.8%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

   4. Applied rewrites 95.2%

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

   5. Taylor expanded in phi2 around 0

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

   7. Applied rewrites 79.6%

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

   if 5.60000000000000025e48 < phi2

   1. Initial program 57.9%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 70.6

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

   5. Applied rewrites 70.6%

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

   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. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 70.6

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

   8. Applied rewrites 70.6%

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

Alternative 5: 31.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := R \cdot \mathsf{fma}\left(-1, \phi_1, \phi_2\right)\\ \mathbf{if}\;\phi_2 \leq -2.25 \cdot 10^{-246}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 3.8 \cdot 10^{-223}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\phi_2 \leq 5.6 \cdot 10^{-27}:\\ \;\;\;\;\left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* R (fma -1.0 phi1 phi2))))
   (if (<= phi2 -2.25e-246)
     t_0
     (if (<= phi2 3.8e-223)
       (* R (sqrt (* (- lambda1 lambda2) (- lambda1 lambda2))))
       (if (<= phi2 5.6e-27)
         (* (* R lambda2) (cos (* 0.5 (+ phi2 phi1))))
         t_0)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R * fma(-1.0, phi1, phi2);
	double tmp;
	if (phi2 <= -2.25e-246) {
		tmp = t_0;
	} else if (phi2 <= 3.8e-223) {
		tmp = R * sqrt(((lambda1 - lambda2) * (lambda1 - lambda2)));
	} else if (phi2 <= 5.6e-27) {
		tmp = (R * lambda2) * cos((0.5 * (phi2 + phi1)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(R * fma(-1.0, phi1, phi2))
	tmp = 0.0
	if (phi2 <= -2.25e-246)
		tmp = t_0;
	elseif (phi2 <= 3.8e-223)
		tmp = Float64(R * sqrt(Float64(Float64(lambda1 - lambda2) * Float64(lambda1 - lambda2))));
	elseif (phi2 <= 5.6e-27)
		tmp = Float64(Float64(R * lambda2) * cos(Float64(0.5 * Float64(phi2 + phi1))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(R * N[(-1.0 * phi1 + phi2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -2.25e-246], t$95$0, If[LessEqual[phi2, 3.8e-223], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 5.6e-27], N[(N[(R * lambda2), $MachinePrecision] * N[Cos[N[(0.5 * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -2.25e-246 or 5.5999999999999999e-27 < phi2

   1. Initial program 54.0%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 32.3

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

   5. Applied rewrites 32.3%

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

   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. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 33.3

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

   8. Applied rewrites 33.3%

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

   if -2.25e-246 < phi2 < 3.80000000000000012e-223

   1. Initial program 65.2%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 65.2

         \[\leadsto R \cdot \sqrt{\mathsf{fma}\left(\phi_1, \phi_1, {\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)} \]

   5. Applied rewrites 65.2%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \sqrt{{\left(\lambda_1 - \lambda_2\right)}^{\color{blue}{2}}} \]
   7. Step-by-step derivation

      1. lower-pow.f64 N/A

         \[\leadsto R \cdot \sqrt{{\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      2. lift--.f64 54.0

         \[\leadsto R \cdot \sqrt{{\left(\lambda_1 - \lambda_2\right)}^{2}} \]

   8. Applied rewrites 54.0%

      \[\leadsto R \cdot \sqrt{{\left(\lambda_1 - \lambda_2\right)}^{\color{blue}{2}}} \]
   9. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto R \cdot \sqrt{{\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      2. lift-pow.f64 N/A

         \[\leadsto R \cdot \sqrt{{\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      3. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

      5. lift--.f64 N/A

         \[\leadsto R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)} \]

      6. lift--.f64 54.0

         \[\leadsto R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)} \]

   10. Applied rewrites 54.0%

       \[\leadsto R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

   if 3.80000000000000012e-223 < phi2 < 5.5999999999999999e-27

   1. Initial program 63.8%

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

   3. Taylor expanded in lambda2 around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 27.1

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

   5. Applied rewrites 27.1%

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

Alternative 6: 35.5% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.25 \cdot 10^{-246} \lor \neg \left(\phi_2 \leq 2.75 \cdot 10^{-5}\right):\\ \;\;\;\;R \cdot \mathsf{fma}\left(-1, \phi_1, \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi2 -2.25e-246) (not (<= phi2 2.75e-5)))
   (* R (fma -1.0 phi1 phi2))
   (* R (sqrt (* (- lambda1 lambda2) (- lambda1 lambda2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi2 <= -2.25e-246) || !(phi2 <= 2.75e-5)) {
		tmp = R * fma(-1.0, phi1, phi2);
	} else {
		tmp = R * sqrt(((lambda1 - lambda2) * (lambda1 - lambda2)));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi2 <= -2.25e-246) || !(phi2 <= 2.75e-5))
		tmp = Float64(R * fma(-1.0, phi1, phi2));
	else
		tmp = Float64(R * sqrt(Float64(Float64(lambda1 - lambda2) * Float64(lambda1 - lambda2))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi2, -2.25e-246], N[Not[LessEqual[phi2, 2.75e-5]], $MachinePrecision]], N[(R * N[(-1.0 * phi1 + phi2), $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -2.25e-246 or 2.7500000000000001e-5 < phi2

   1. Initial program 54.3%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 32.8

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

   5. Applied rewrites 32.8%

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

   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. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 33.9

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

   8. Applied rewrites 33.9%

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

   if -2.25e-246 < phi2 < 2.7500000000000001e-5

   1. Initial program 63.4%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 63.4

         \[\leadsto R \cdot \sqrt{\mathsf{fma}\left(\phi_1, \phi_1, {\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)} \]

   5. Applied rewrites 63.4%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \sqrt{{\left(\lambda_1 - \lambda_2\right)}^{\color{blue}{2}}} \]
   7. Step-by-step derivation

      1. lower-pow.f64 N/A

         \[\leadsto R \cdot \sqrt{{\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      2. lift--.f64 48.7

         \[\leadsto R \cdot \sqrt{{\left(\lambda_1 - \lambda_2\right)}^{2}} \]

   8. Applied rewrites 48.7%

      \[\leadsto R \cdot \sqrt{{\left(\lambda_1 - \lambda_2\right)}^{\color{blue}{2}}} \]
   9. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto R \cdot \sqrt{{\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      2. lift-pow.f64 N/A

         \[\leadsto R \cdot \sqrt{{\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      3. unpow2 N/A

         \[\leadsto R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \color{blue}{\lambda_2}\right)} \]

      5. lift--.f64 N/A

         \[\leadsto R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)} \]

      6. lift--.f64 48.7

         \[\leadsto R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)} \]

   10. Applied rewrites 48.7%

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

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.25 \cdot 10^{-246} \lor \neg \left(\phi_2 \leq 2.75 \cdot 10^{-5}\right):\\ \;\;\;\;R \cdot \mathsf{fma}\left(-1, \phi_1, \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 31.7% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3.7 \cdot 10^{+192}:\\ \;\;\;\;R \cdot \sqrt{\lambda_1 \cdot \lambda_1}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(-1, \phi_1, \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -3.7e+192)
   (* R (sqrt (* lambda1 lambda1)))
   (* R (fma -1.0 phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -3.7e+192) {
		tmp = R * sqrt((lambda1 * lambda1));
	} else {
		tmp = R * fma(-1.0, phi1, phi2);
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -3.7e+192)
		tmp = Float64(R * sqrt(Float64(lambda1 * lambda1)));
	else
		tmp = Float64(R * fma(-1.0, phi1, phi2));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -3.7e+192], N[(R * N[Sqrt[N[(lambda1 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[(-1.0 * phi1 + phi2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -3.7000000000000001e192

   1. Initial program 47.3%

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

   3. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 47.3

         \[\leadsto R \cdot \sqrt{\mathsf{fma}\left(\phi_1, \phi_1, {\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)} \]

   5. Applied rewrites 47.3%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \sqrt{{\left(\lambda_1 - \lambda_2\right)}^{\color{blue}{2}}} \]
   7. Step-by-step derivation

      1. lower-pow.f64 N/A

         \[\leadsto R \cdot \sqrt{{\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      2. lift--.f64 47.3

         \[\leadsto R \cdot \sqrt{{\left(\lambda_1 - \lambda_2\right)}^{2}} \]

   8. Applied rewrites 47.3%

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

   9. Taylor expanded in lambda1 around inf

      \[\leadsto R \cdot \sqrt{{\lambda_1}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto R \cdot \sqrt{\lambda_1 \cdot \lambda_1} \]

       2. lower-*.f64 47.3

          \[\leadsto R \cdot \sqrt{\lambda_1 \cdot \lambda_1} \]

   11. Applied rewrites 47.3%

       \[\leadsto R \cdot \sqrt{\lambda_1 \cdot \lambda_1} \]

   if -3.7000000000000001e192 < lambda1

   1. Initial program 57.7%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 25.9

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

   5. Applied rewrites 25.9%

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

   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. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 28.8

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

   8. Applied rewrites 28.8%

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

Alternative 8: 28.4% accurate, 19.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 7.2 \cdot 10^{+44}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 7.2e+44) (* R (- phi1)) (* R phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 7.2e+44) {
		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 (phi2 <= 7.2d+44) 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 (phi2 <= 7.2e+44) {
		tmp = R * -phi1;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 7.2e+44:
		tmp = R * -phi1
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 7.2e+44)
		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 (phi2 <= 7.2e+44)
		tmp = R * -phi1;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 7.2e+44], N[(R * (-phi1)), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 7.2e44

   1. Initial program 56.5%

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

   3. Taylor expanded in phi1 around -inf

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \phi_1\right)} \]
   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 18.1

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

   5. Applied rewrites 18.1%

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

   if 7.2e44 < phi2

   1. Initial program 58.7%

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

   3. Taylor expanded in phi2 around inf

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

   5. Applied rewrites 69.5%

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

Alternative 9: 29.5% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ R \cdot \mathsf{fma}\left(-1, \phi_1, \phi_2\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R (fma -1.0 phi1 phi2)))

C

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

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * fma(-1.0, phi1, phi2))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[(-1.0 * phi1 + phi2), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.0%

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

3. Taylor expanded in phi2 around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 25.0

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

5. Applied rewrites 25.0%

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

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. mul-1-neg N/A

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. lower-fma.f64 27.7

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

8. Applied rewrites 27.7%

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

Alternative 10: 16.6% 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 57.0%

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

3. Taylor expanded in phi2 around inf

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

5. Applied rewrites 16.5%

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

Reproduce

herbie shell --seed 2025052 
(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))))))