Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.5% → 95.8%

Time: 6.0s

Alternatives: 13

Speedup: 1.2×

• 31.3% 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: 59.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.8% accurate, 1.2× 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 62.9%

   \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
2. 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 95.7%

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

Alternative 2: 76.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.94:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 6.8 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1, \cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 0.94)
   (* (hypot (* (cos (* 0.5 phi1)) (- lambda1 lambda2)) phi1) R)
   (if (<= phi2 6.8e+38)
     (* (hypot phi1 (* (cos (* 0.5 phi2)) (- lambda1 lambda2))) R)
     (* R (- phi2 phi1)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 0.94) {
		tmp = hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R;
	} else if (phi2 <= 6.8e+38) {
		tmp = hypot(phi1, (cos((0.5 * phi2)) * (lambda1 - lambda2))) * R;
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 0.94) {
		tmp = Math.hypot((Math.cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R;
	} else if (phi2 <= 6.8e+38) {
		tmp = Math.hypot(phi1, (Math.cos((0.5 * phi2)) * (lambda1 - lambda2))) * R;
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 0.94:
		tmp = math.hypot((math.cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R
	elif phi2 <= 6.8e+38:
		tmp = math.hypot(phi1, (math.cos((0.5 * phi2)) * (lambda1 - lambda2))) * R
	else:
		tmp = R * (phi2 - phi1)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 0.94)
		tmp = Float64(hypot(Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2)), phi1) * R);
	elseif (phi2 <= 6.8e+38)
		tmp = Float64(hypot(phi1, Float64(cos(Float64(0.5 * phi2)) * Float64(lambda1 - lambda2))) * R);
	else
		tmp = Float64(R * Float64(phi2 - phi1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 0.94)
		tmp = hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R;
	elseif (phi2 <= 6.8e+38)
		tmp = hypot(phi1, (cos((0.5 * phi2)) * (lambda1 - lambda2))) * R;
	else
		tmp = R * (phi2 - phi1);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 0.94], N[(N[Sqrt[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi2, 6.8e+38], N[(N[Sqrt[phi1 ^ 2 + N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 0.93999999999999995

   1. Initial program 64.8%

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

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

   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. unpow2 N/A

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

      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. lift-cos.f64 N/A

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

      6. lift-*.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. lift-*.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. lift--.f64 82.5

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

   7. Applied rewrites 82.5%

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

   if 0.93999999999999995 < phi2 < 6.79999999999999992e38

   1. Initial program 74.9%

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

      1. 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 89.1%

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

   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 88.7

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

   7. Applied rewrites 88.7%

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

   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 64.5%

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

   if 6.79999999999999992e38 < phi2

   1. Initial program 53.1%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 61.5

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

   5. Applied rewrites 61.5%

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

   6. Taylor expanded in phi1 around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 72.2

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

   8. Applied rewrites 72.2%

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

   9. Taylor expanded in R around 0

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

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

       3. lower-*.f64 70.3

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

   11. Applied rewrites 70.3%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.94:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 6.8 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1, \cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 7.2 \cdot 10^{-47}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, \cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 7.2e-47)
   (* (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 <= 7.2e-47) {
		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 <= 7.2e-47) {
		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 <= 7.2e-47:
		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 <= 7.2e-47)
		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 <= 7.2e-47)
		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, 7.2e-47], 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 < 7.19999999999999982e-47

   1. Initial program 63.7%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 96.3%

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

   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. unpow2 N/A

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

      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. lift-cos.f64 N/A

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

      6. lift-*.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. lift-*.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. lift--.f64 83.9

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

   7. Applied rewrites 83.9%

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

   if 7.19999999999999982e-47 < phi2

   1. Initial program 61.1%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 94.2%

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

   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 93.1

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

   7. Applied rewrites 93.1%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 7.2 \cdot 10^{-47}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, \cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.1 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(-\phi_2, \cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.1e-46)
   (* (hypot (* (cos (* 0.5 phi1)) (- lambda1 lambda2)) phi1) R)
   (* (hypot (- 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 <= 1.1e-46) {
		tmp = hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R;
	} else {
		tmp = hypot(-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 <= 1.1e-46) {
		tmp = Math.hypot((Math.cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R;
	} else {
		tmp = Math.hypot(-phi2, (Math.cos((0.5 * phi2)) * (lambda1 - lambda2))) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.1e-46:
		tmp = math.hypot((math.cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R
	else:
		tmp = math.hypot(-phi2, (math.cos((0.5 * phi2)) * (lambda1 - lambda2))) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.1e-46)
		tmp = Float64(hypot(Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2)), phi1) * R);
	else
		tmp = Float64(hypot(Float64(-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 <= 1.1e-46)
		tmp = hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R;
	else
		tmp = hypot(-phi2, (cos((0.5 * phi2)) * (lambda1 - lambda2))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.1e-46], 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[(-phi2) ^ 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 < 1.1e-46

   1. Initial program 63.9%

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

      1. 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 96.3%

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

   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. unpow2 N/A

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

      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. lift-cos.f64 N/A

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

      6. lift-*.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. lift-*.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. lift--.f64 84.0

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

   7. Applied rewrites 84.0%

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

   if 1.1e-46 < phi2

   1. Initial program 60.6%

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

      1. 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 94.2%

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

   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 93.0

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

   7. Applied rewrites 93.0%

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

   8. Taylor expanded in phi1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 80.3

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

   10. Applied rewrites 80.3%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.1 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(-\phi_2, \cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 8.6 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 8.6e+36)
   (* (hypot (* (cos (* 0.5 phi1)) (- lambda1 lambda2)) phi1) R)
   (* R (- phi2 phi1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 8.6e+36) {
		tmp = hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R;
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 8.6e+36) {
		tmp = Math.hypot((Math.cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R;
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 8.6e+36:
		tmp = math.hypot((math.cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R
	else:
		tmp = R * (phi2 - phi1)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 8.6e+36)
		tmp = Float64(hypot(Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2)), phi1) * R);
	else
		tmp = Float64(R * Float64(phi2 - phi1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 8.6e+36)
		tmp = hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R;
	else
		tmp = R * (phi2 - phi1);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 8.6e+36], 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[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 8.6000000000000001e36

   1. Initial program 65.1%

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

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

   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. unpow2 N/A

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

      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. lift-cos.f64 N/A

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

      6. lift-*.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. lift-*.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. lift--.f64 80.7

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

   7. Applied rewrites 80.7%

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

   if 8.6000000000000001e36 < phi2

   1. Initial program 54.8%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 61.2

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

   5. Applied rewrites 61.2%

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

   6. Taylor expanded in phi1 around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 71.5

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

   8. Applied rewrites 71.5%

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

   9. Taylor expanded in R around 0

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

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

       3. lower-*.f64 69.7

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

   11. Applied rewrites 69.7%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 8.6 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 41.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;R \leq 5 \cdot 10^{+67}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= R 5e+67)
   (* R (- phi2 phi1))
   (*
    R
    (sqrt
     (+
      (* (- lambda1 lambda2) (- lambda1 lambda2))
      (* (- phi1 phi2) (- phi1 phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (R <= 5e+67) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * sqrt((((lambda1 - lambda2) * (lambda1 - lambda2)) + ((phi1 - phi2) * (phi1 - 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 (r <= 5d+67) then
        tmp = r * (phi2 - phi1)
    else
        tmp = r * sqrt((((lambda1 - lambda2) * (lambda1 - lambda2)) + ((phi1 - phi2) * (phi1 - 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 (R <= 5e+67) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * Math.sqrt((((lambda1 - lambda2) * (lambda1 - lambda2)) + ((phi1 - phi2) * (phi1 - phi2))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if R <= 5e+67:
		tmp = R * (phi2 - phi1)
	else:
		tmp = R * math.sqrt((((lambda1 - lambda2) * (lambda1 - lambda2)) + ((phi1 - phi2) * (phi1 - phi2))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (R <= 5e+67)
		tmp = Float64(R * Float64(phi2 - phi1));
	else
		tmp = Float64(R * sqrt(Float64(Float64(Float64(lambda1 - lambda2) * Float64(lambda1 - lambda2)) + Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (R <= 5e+67)
		tmp = R * (phi2 - phi1);
	else
		tmp = R * sqrt((((lambda1 - lambda2) * (lambda1 - lambda2)) + ((phi1 - phi2) * (phi1 - phi2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[R, 5e+67], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] + N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if R < 4.99999999999999976e67

   1. Initial program 55.6%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 28.7

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

   5. Applied rewrites 28.7%

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

   6. Taylor expanded in phi1 around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 30.6

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

   8. Applied rewrites 30.6%

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

   9. Taylor expanded in R around 0

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

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

       3. lower-*.f64 30.6

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

   11. Applied rewrites 30.6%

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

   if 4.99999999999999976e67 < R

   1. Initial program 94.4%

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

   3. Taylor expanded in phi2 around 0

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

      1. pow-prod-down N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 92.7

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

   5. Applied rewrites 92.7%

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

   6. Taylor expanded in phi1 around 0

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

      1. lift--.f64 92.7

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

   8. Applied rewrites 92.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 92.7

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

   10. Applied rewrites 92.7%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;R \leq 5 \cdot 10^{+67}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 30.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.3 \cdot 10^{+107}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \left(\phi_2 \cdot \mathsf{fma}\left(-1, \frac{R}{\phi_1}, \frac{R}{\phi_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -2.3e+107)
   (* (- phi1) (* phi2 (fma -1.0 (/ R phi1) (/ R phi2))))
   (* R (- phi2 phi1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -2.3e+107) {
		tmp = -phi1 * (phi2 * fma(-1.0, (R / phi1), (R / phi2)));
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -2.3e+107)
		tmp = Float64(Float64(-phi1) * Float64(phi2 * fma(-1.0, Float64(R / phi1), Float64(R / phi2))));
	else
		tmp = Float64(R * Float64(phi2 - phi1));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -2.3e+107], N[((-phi1) * N[(phi2 * N[(-1.0 * N[(R / phi1), $MachinePrecision] + N[(R / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -2.3e107

   1. Initial program 67.7%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 24.5

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

   5. Applied rewrites 24.5%

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

   6. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 24.8

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

   8. Applied rewrites 24.8%

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

   if -2.3e107 < lambda1

   1. Initial program 62.1%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 32.9

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

   5. Applied rewrites 32.9%

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

   6. Taylor expanded in phi1 around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 35.6

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

   8. Applied rewrites 35.6%

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

   9. Taylor expanded in R around 0

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

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

       3. lower-*.f64 35.6

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

   11. Applied rewrites 35.6%

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

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.3 \cdot 10^{+107}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \left(\phi_2 \cdot \mathsf{fma}\left(-1, \frac{R}{\phi_1}, \frac{R}{\phi_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 31.8% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.9 \cdot 10^{-152}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \left(\phi_2 \cdot \frac{R}{\phi_2}\right)\\ \mathbf{elif}\;\phi_2 \leq 9.5 \cdot 10^{+36}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \mathsf{fma}\left(\frac{\phi_2 \cdot R}{\phi_1}, -1, R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.9e-152)
   (* (- phi1) (* phi2 (/ R phi2)))
   (if (<= phi2 9.5e+36)
     (* (- phi1) (fma (/ (* phi2 R) phi1) -1.0 R))
     (* R (- phi2 phi1)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.9e-152) {
		tmp = -phi1 * (phi2 * (R / phi2));
	} else if (phi2 <= 9.5e+36) {
		tmp = -phi1 * fma(((phi2 * R) / phi1), -1.0, R);
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.9e-152)
		tmp = Float64(Float64(-phi1) * Float64(phi2 * Float64(R / phi2)));
	elseif (phi2 <= 9.5e+36)
		tmp = Float64(Float64(-phi1) * fma(Float64(Float64(phi2 * R) / phi1), -1.0, R));
	else
		tmp = Float64(R * Float64(phi2 - phi1));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.9e-152], N[((-phi1) * N[(phi2 * N[(R / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 9.5e+36], N[((-phi1) * N[(N[(N[(phi2 * R), $MachinePrecision] / phi1), $MachinePrecision] * -1.0 + R), $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 1.90000000000000006e-152

   1. Initial program 62.2%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 18.6

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

   5. Applied rewrites 18.6%

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

   6. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 19.7

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

   8. Applied rewrites 19.7%

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

   9. Taylor expanded in phi1 around inf

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

       1. lift-/.f64 21.6

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

   11. Applied rewrites 21.6%

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

   if 1.90000000000000006e-152 < phi2 < 9.49999999999999974e36

   1. Initial program 74.0%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 39.1

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

   5. Applied rewrites 39.1%

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

   if 9.49999999999999974e36 < phi2

   1. Initial program 54.8%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 61.2

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

   5. Applied rewrites 61.2%

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

   6. Taylor expanded in phi1 around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 71.5

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

   8. Applied rewrites 71.5%

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

   9. Taylor expanded in R around 0

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

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

       3. lower-*.f64 69.7

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

   11. Applied rewrites 69.7%

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

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.9 \cdot 10^{-152}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \left(\phi_2 \cdot \frac{R}{\phi_2}\right)\\ \mathbf{elif}\;\phi_2 \leq 9.5 \cdot 10^{+36}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \mathsf{fma}\left(\frac{\phi_2 \cdot R}{\phi_1}, -1, R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 31.7% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.9 \cdot 10^{-152}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \left(\phi_2 \cdot \frac{R}{\phi_2}\right)\\ \mathbf{elif}\;\phi_2 \leq 8.6 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(R, \frac{\phi_2}{\phi_1}, -R\right) \cdot \phi_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.9e-152)
   (* (- phi1) (* phi2 (/ R phi2)))
   (if (<= phi2 8.6e+36)
     (* (fma R (/ phi2 phi1) (- R)) phi1)
     (* R (- phi2 phi1)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.9e-152) {
		tmp = -phi1 * (phi2 * (R / phi2));
	} else if (phi2 <= 8.6e+36) {
		tmp = fma(R, (phi2 / phi1), -R) * phi1;
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.9e-152)
		tmp = Float64(Float64(-phi1) * Float64(phi2 * Float64(R / phi2)));
	elseif (phi2 <= 8.6e+36)
		tmp = Float64(fma(R, Float64(phi2 / phi1), Float64(-R)) * phi1);
	else
		tmp = Float64(R * Float64(phi2 - phi1));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.9e-152], N[((-phi1) * N[(phi2 * N[(R / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 8.6e+36], N[(N[(R * N[(phi2 / phi1), $MachinePrecision] + (-R)), $MachinePrecision] * phi1), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 1.90000000000000006e-152

   1. Initial program 62.2%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 18.6

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

   5. Applied rewrites 18.6%

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

   6. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 19.7

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

   8. Applied rewrites 19.7%

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

   9. Taylor expanded in phi1 around inf

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

       1. lift-/.f64 21.6

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

   11. Applied rewrites 21.6%

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

   if 1.90000000000000006e-152 < phi2 < 8.6000000000000001e36

   1. Initial program 74.0%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 39.1

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

   5. Applied rewrites 39.1%

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

   6. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 37.3

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

   8. Applied rewrites 37.3%

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

   9. Taylor expanded in phi1 around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. associate-/l* N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. mul-1-neg N/A

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

       8. lower-neg.f64 39.1

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

   11. Applied rewrites 39.1%

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

   if 8.6000000000000001e36 < phi2

   1. Initial program 54.8%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 61.2

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

   5. Applied rewrites 61.2%

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

   6. Taylor expanded in phi1 around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 71.5

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

   8. Applied rewrites 71.5%

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

   9. Taylor expanded in R around 0

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

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

       3. lower-*.f64 69.7

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

   11. Applied rewrites 69.7%

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

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.9 \cdot 10^{-152}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \left(\phi_2 \cdot \frac{R}{\phi_2}\right)\\ \mathbf{elif}\;\phi_2 \leq 8.6 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(R, \frac{\phi_2}{\phi_1}, -R\right) \cdot \phi_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 31.3% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 9 \cdot 10^{-80}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \left(\phi_2 \cdot \frac{R}{\phi_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 9e-80) (* (- phi1) (* phi2 (/ R phi2))) (* R (- phi2 phi1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 9e-80) {
		tmp = -phi1 * (phi2 * (R / phi2));
	} else {
		tmp = R * (phi2 - phi1);
	}
	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 <= 9d-80) then
        tmp = -phi1 * (phi2 * (r / phi2))
    else
        tmp = r * (phi2 - phi1)
    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 <= 9e-80) {
		tmp = -phi1 * (phi2 * (R / phi2));
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 9e-80:
		tmp = -phi1 * (phi2 * (R / phi2))
	else:
		tmp = R * (phi2 - phi1)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 9e-80)
		tmp = Float64(Float64(-phi1) * Float64(phi2 * Float64(R / phi2)));
	else
		tmp = Float64(R * Float64(phi2 - phi1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 9e-80)
		tmp = -phi1 * (phi2 * (R / phi2));
	else
		tmp = R * (phi2 - phi1);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 9e-80], N[((-phi1) * N[(phi2 * N[(R / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 9.0000000000000006e-80

   1. Initial program 64.1%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 19.5

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

   5. Applied rewrites 19.5%

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

   6. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 20.5

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

   8. Applied rewrites 20.5%

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

   9. Taylor expanded in phi1 around inf

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

       1. lift-/.f64 22.2

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

   11. Applied rewrites 22.2%

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

   if 9.0000000000000006e-80 < phi2

   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

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 54.8

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

   5. Applied rewrites 54.8%

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

   6. Taylor expanded in phi1 around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 58.1

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

   8. Applied rewrites 58.1%

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

   9. Taylor expanded in R around 0

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

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

       3. lower-*.f64 57.0

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

   11. Applied rewrites 57.0%

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

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 9 \cdot 10^{-80}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \left(\phi_2 \cdot \frac{R}{\phi_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 28.5% accurate, 19.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.1 \cdot 10^{-46}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.1e-46) (* R (- phi1)) (* R phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.1e-46) {
		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 <= 1.1d-46) 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 <= 1.1e-46) {
		tmp = R * -phi1;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.1e-46:
		tmp = R * -phi1
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.1e-46)
		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 <= 1.1e-46)
		tmp = R * -phi1;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.1e-46], N[(R * (-phi1)), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.1e-46

   1. Initial program 63.9%

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

   3. Taylor expanded in 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 21.6

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

   5. Applied rewrites 21.6%

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

   if 1.1e-46 < phi2

   1. Initial program 60.6%

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

   3. Taylor expanded in phi2 around inf

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

   5. Applied rewrites 56.0%

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

Alternative 12: 29.8% 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 62.9%

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

3. Taylor expanded in phi1 around -inf

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 31.7

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

5. Applied rewrites 31.7%

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

6. Taylor expanded in phi1 around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-*.f64 33.3

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

8. Applied rewrites 33.3%

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

9. Taylor expanded in R around 0

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

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

    3. lower-*.f64 33.3

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

11. Applied rewrites 33.3%

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

12. Final simplification 33.3%

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

Alternative 13: 17.3% 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 62.9%

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

3. Taylor expanded in phi2 around inf

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

5. Applied rewrites 19.7%

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

Reproduce

herbie shell --seed 2025085 
(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))))))