Diagrams.TwoD.Apollonian:descartes from diagrams-contrib-1.3.0.5

Percentage Accurate: 70.6% → 97.5%

Time: 5.2s

Alternatives: 13

Speedup: 1.3×

Specification

Math

\[\begin{array}{l} \\ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (* 2.0 (sqrt (+ (+ (* x y) (* x z)) (* y z)))))

C

double code(double x, double y, double z) {
	return 2.0 * sqrt((((x * y) + (x * z)) + (y * z)));
}

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(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 2.0d0 * sqrt((((x * y) + (x * z)) + (y * z)))
end function

Java

public static double code(double x, double y, double z) {
	return 2.0 * Math.sqrt((((x * y) + (x * z)) + (y * z)));
}

Python

def code(x, y, z):
	return 2.0 * math.sqrt((((x * y) + (x * z)) + (y * z)))

Julia

function code(x, y, z)
	return Float64(2.0 * sqrt(Float64(Float64(Float64(x * y) + Float64(x * z)) + Float64(y * z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 2.0 * sqrt((((x * y) + (x * z)) + (y * z)));
end

Wolfram

code[x_, y_, z_] := N[(2.0 * N[Sqrt[N[(N[(N[(x * y), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 70.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (* 2.0 (sqrt (+ (+ (* x y) (* x z)) (* y z)))))

C

double code(double x, double y, double z) {
	return 2.0 * sqrt((((x * y) + (x * z)) + (y * z)));
}

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(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 2.0d0 * sqrt((((x * y) + (x * z)) + (y * z)))
end function

Java

public static double code(double x, double y, double z) {
	return 2.0 * Math.sqrt((((x * y) + (x * z)) + (y * z)));
}

Python

def code(x, y, z):
	return 2.0 * math.sqrt((((x * y) + (x * z)) + (y * z)))

Julia

function code(x, y, z)
	return Float64(2.0 * sqrt(Float64(Float64(Float64(x * y) + Float64(x * z)) + Float64(y * z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 2.0 * sqrt((((x * y) + (x * z)) + (y * z)));
end

Wolfram

code[x_, y_, z_] := N[(2.0 * N[Sqrt[N[(N[(N[(x * y), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -63000000000000:\\ \;\;\;\;\left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x\\ \mathbf{elif}\;y \leq -4.4 \cdot 10^{-239}:\\ \;\;\;\;2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-303}:\\ \;\;\;\;\left(2 \cdot \left(\sqrt{\frac{z \cdot \left(1 + \frac{y}{z}\right)}{x}} \cdot -1\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\sqrt{x + y}}{\sqrt{z}}\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -63000000000000.0)
   (* (* -2.0 (sqrt (/ y x))) x)
   (if (<= y -4.4e-239)
     (* 2.0 (sqrt (+ (+ (* x y) (* x z)) (* y z))))
     (if (<= y 3.6e-303)
       (* (* 2.0 (* (sqrt (/ (* z (+ 1.0 (/ y z))) x)) -1.0)) x)
       (* (* 2.0 (/ (sqrt (+ x y)) (sqrt z))) z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -63000000000000.0) {
		tmp = (-2.0 * sqrt((y / x))) * x;
	} else if (y <= -4.4e-239) {
		tmp = 2.0 * sqrt((((x * y) + (x * z)) + (y * z)));
	} else if (y <= 3.6e-303) {
		tmp = (2.0 * (sqrt(((z * (1.0 + (y / z))) / x)) * -1.0)) * x;
	} else {
		tmp = (2.0 * (sqrt((x + y)) / sqrt(z))) * z;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-63000000000000.0d0)) then
        tmp = ((-2.0d0) * sqrt((y / x))) * x
    else if (y <= (-4.4d-239)) then
        tmp = 2.0d0 * sqrt((((x * y) + (x * z)) + (y * z)))
    else if (y <= 3.6d-303) then
        tmp = (2.0d0 * (sqrt(((z * (1.0d0 + (y / z))) / x)) * (-1.0d0))) * x
    else
        tmp = (2.0d0 * (sqrt((x + y)) / sqrt(z))) * z
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -63000000000000.0) {
		tmp = (-2.0 * Math.sqrt((y / x))) * x;
	} else if (y <= -4.4e-239) {
		tmp = 2.0 * Math.sqrt((((x * y) + (x * z)) + (y * z)));
	} else if (y <= 3.6e-303) {
		tmp = (2.0 * (Math.sqrt(((z * (1.0 + (y / z))) / x)) * -1.0)) * x;
	} else {
		tmp = (2.0 * (Math.sqrt((x + y)) / Math.sqrt(z))) * z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -63000000000000.0:
		tmp = (-2.0 * math.sqrt((y / x))) * x
	elif y <= -4.4e-239:
		tmp = 2.0 * math.sqrt((((x * y) + (x * z)) + (y * z)))
	elif y <= 3.6e-303:
		tmp = (2.0 * (math.sqrt(((z * (1.0 + (y / z))) / x)) * -1.0)) * x
	else:
		tmp = (2.0 * (math.sqrt((x + y)) / math.sqrt(z))) * z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -63000000000000.0)
		tmp = Float64(Float64(-2.0 * sqrt(Float64(y / x))) * x);
	elseif (y <= -4.4e-239)
		tmp = Float64(2.0 * sqrt(Float64(Float64(Float64(x * y) + Float64(x * z)) + Float64(y * z))));
	elseif (y <= 3.6e-303)
		tmp = Float64(Float64(2.0 * Float64(sqrt(Float64(Float64(z * Float64(1.0 + Float64(y / z))) / x)) * -1.0)) * x);
	else
		tmp = Float64(Float64(2.0 * Float64(sqrt(Float64(x + y)) / sqrt(z))) * z);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -63000000000000.0)
		tmp = (-2.0 * sqrt((y / x))) * x;
	elseif (y <= -4.4e-239)
		tmp = 2.0 * sqrt((((x * y) + (x * z)) + (y * z)));
	elseif (y <= 3.6e-303)
		tmp = (2.0 * (sqrt(((z * (1.0 + (y / z))) / x)) * -1.0)) * x;
	else
		tmp = (2.0 * (sqrt((x + y)) / sqrt(z))) * z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -63000000000000.0], N[(N[(-2.0 * N[Sqrt[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, -4.4e-239], N[(2.0 * N[Sqrt[N[(N[(N[(x * y), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.6e-303], N[(N[(2.0 * N[(N[Sqrt[N[(N[(z * N[(1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(2.0 * N[(N[Sqrt[N[(x + y), $MachinePrecision]], $MachinePrecision] / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -6.3e13

   1. Initial program 41.9%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right) \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right) \cdot \color{blue}{x} \]

   4. Applied rewrites 0.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y, \sqrt{\frac{1}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(z + y\right)}}, \sqrt{\frac{z + y}{x}} \cdot 2\right) \cdot x} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot x \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot x \]

      2. sqrt-pow2 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {-1}^{\left(\frac{2}{2}\right)}\right)\right) \cdot x \]

      3. metadata-eval N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {-1}^{1}\right)\right) \cdot x \]

      4. metadata-eval N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      5. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      7. lower-/.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      8. lower-+.f64 99.5

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

   7. Applied rewrites 99.5%

      \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

   8. Taylor expanded in y around inf

      \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]

      3. lower-/.f64 98.0

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]

   10. Applied rewrites 98.0%

       \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]

   if -6.3e13 < y < -4.39999999999999965e-239

   1. Initial program 96.5%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   if -4.39999999999999965e-239 < y < 3.5999999999999998e-303

   1. Initial program 65.2%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right) \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right) \cdot \color{blue}{x} \]

   4. Applied rewrites 1.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y, \sqrt{\frac{1}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(z + y\right)}}, \sqrt{\frac{z + y}{x}} \cdot 2\right) \cdot x} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot x \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot x \]

      2. sqrt-pow2 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {-1}^{\left(\frac{2}{2}\right)}\right)\right) \cdot x \]

      3. metadata-eval N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {-1}^{1}\right)\right) \cdot x \]

      4. metadata-eval N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      5. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      7. lower-/.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      8. lower-+.f64 58.6

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

   7. Applied rewrites 58.6%

      \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

   8. Taylor expanded in z around inf

      \[\leadsto \left(2 \cdot \left(\sqrt{\frac{z \cdot \left(1 + \frac{y}{z}\right)}{x}} \cdot -1\right)\right) \cdot x \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{z \cdot \left(1 + \frac{y}{z}\right)}{x}} \cdot -1\right)\right) \cdot x \]

      2. lower-+.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{z \cdot \left(1 + \frac{y}{z}\right)}{x}} \cdot -1\right)\right) \cdot x \]

      3. lift-/.f64 58.6

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{z \cdot \left(1 + \frac{y}{z}\right)}{x}} \cdot -1\right)\right) \cdot x \]

   10. Applied rewrites 58.6%

       \[\leadsto \left(2 \cdot \left(\sqrt{\frac{z \cdot \left(1 + \frac{y}{z}\right)}{x}} \cdot -1\right)\right) \cdot x \]

   if 3.5999999999999998e-303 < y

   1. Initial program 71.6%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(2 \cdot \sqrt{\frac{x + y}{z}} + \left(x \cdot y\right) \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}} + \left(x \cdot y\right) \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right) \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}} + \left(x \cdot y\right) \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right) \cdot \color{blue}{z} \]

   4. Applied rewrites 77.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot x, \sqrt{\frac{1}{\left(y + x\right) \cdot \left(\left(z \cdot z\right) \cdot z\right)}}, \sqrt{\frac{y + x}{z}} \cdot 2\right) \cdot z} \]

   5. Taylor expanded in z around inf

      \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]

      3. lower-/.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]

      4. lower-+.f64 87.6

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]

   7. Applied rewrites 87.6%

      \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]

      2. lift-+.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]

      3. lift-/.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]

      4. sqrt-div N/A

         \[\leadsto \left(2 \cdot \frac{\sqrt{x + y}}{\sqrt{z}}\right) \cdot z \]

      5. lower-/.f64 N/A

         \[\leadsto \left(2 \cdot \frac{\sqrt{x + y}}{\sqrt{z}}\right) \cdot z \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \frac{\sqrt{x + y}}{\sqrt{z}}\right) \cdot z \]

      7. lift-+.f64 N/A

         \[\leadsto \left(2 \cdot \frac{\sqrt{x + y}}{\sqrt{z}}\right) \cdot z \]

      8. lift-sqrt.f64 99.3

         \[\leadsto \left(2 \cdot \frac{\sqrt{x + y}}{\sqrt{z}}\right) \cdot z \]

   9. Applied rewrites 99.3%

      \[\leadsto \left(2 \cdot \frac{\sqrt{x + y}}{\sqrt{z}}\right) \cdot z \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -63000000000000:\\ \;\;\;\;\left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x\\ \mathbf{elif}\;y \leq -4.4 \cdot 10^{-239}:\\ \;\;\;\;2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-303}:\\ \;\;\;\;\left(-2 \cdot \sqrt{\frac{y + z}{x}}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\sqrt{x + y}}{\sqrt{z}}\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -63000000000000.0)
   (* (* -2.0 (sqrt (/ y x))) x)
   (if (<= y -4.4e-239)
     (* 2.0 (sqrt (+ (+ (* x y) (* x z)) (* y z))))
     (if (<= y 3.6e-303)
       (* (* -2.0 (sqrt (/ (+ y z) x))) x)
       (* (* 2.0 (/ (sqrt (+ x y)) (sqrt z))) z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -63000000000000.0) {
		tmp = (-2.0 * sqrt((y / x))) * x;
	} else if (y <= -4.4e-239) {
		tmp = 2.0 * sqrt((((x * y) + (x * z)) + (y * z)));
	} else if (y <= 3.6e-303) {
		tmp = (-2.0 * sqrt(((y + z) / x))) * x;
	} else {
		tmp = (2.0 * (sqrt((x + y)) / sqrt(z))) * z;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-63000000000000.0d0)) then
        tmp = ((-2.0d0) * sqrt((y / x))) * x
    else if (y <= (-4.4d-239)) then
        tmp = 2.0d0 * sqrt((((x * y) + (x * z)) + (y * z)))
    else if (y <= 3.6d-303) then
        tmp = ((-2.0d0) * sqrt(((y + z) / x))) * x
    else
        tmp = (2.0d0 * (sqrt((x + y)) / sqrt(z))) * z
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -63000000000000.0) {
		tmp = (-2.0 * Math.sqrt((y / x))) * x;
	} else if (y <= -4.4e-239) {
		tmp = 2.0 * Math.sqrt((((x * y) + (x * z)) + (y * z)));
	} else if (y <= 3.6e-303) {
		tmp = (-2.0 * Math.sqrt(((y + z) / x))) * x;
	} else {
		tmp = (2.0 * (Math.sqrt((x + y)) / Math.sqrt(z))) * z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -63000000000000.0:
		tmp = (-2.0 * math.sqrt((y / x))) * x
	elif y <= -4.4e-239:
		tmp = 2.0 * math.sqrt((((x * y) + (x * z)) + (y * z)))
	elif y <= 3.6e-303:
		tmp = (-2.0 * math.sqrt(((y + z) / x))) * x
	else:
		tmp = (2.0 * (math.sqrt((x + y)) / math.sqrt(z))) * z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -63000000000000.0)
		tmp = Float64(Float64(-2.0 * sqrt(Float64(y / x))) * x);
	elseif (y <= -4.4e-239)
		tmp = Float64(2.0 * sqrt(Float64(Float64(Float64(x * y) + Float64(x * z)) + Float64(y * z))));
	elseif (y <= 3.6e-303)
		tmp = Float64(Float64(-2.0 * sqrt(Float64(Float64(y + z) / x))) * x);
	else
		tmp = Float64(Float64(2.0 * Float64(sqrt(Float64(x + y)) / sqrt(z))) * z);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -63000000000000.0)
		tmp = (-2.0 * sqrt((y / x))) * x;
	elseif (y <= -4.4e-239)
		tmp = 2.0 * sqrt((((x * y) + (x * z)) + (y * z)));
	elseif (y <= 3.6e-303)
		tmp = (-2.0 * sqrt(((y + z) / x))) * x;
	else
		tmp = (2.0 * (sqrt((x + y)) / sqrt(z))) * z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -63000000000000.0], N[(N[(-2.0 * N[Sqrt[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, -4.4e-239], N[(2.0 * N[Sqrt[N[(N[(N[(x * y), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.6e-303], N[(N[(-2.0 * N[Sqrt[N[(N[(y + z), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(2.0 * N[(N[Sqrt[N[(x + y), $MachinePrecision]], $MachinePrecision] / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -6.3e13

   1. Initial program 41.9%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right) \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right) \cdot \color{blue}{x} \]

   4. Applied rewrites 0.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y, \sqrt{\frac{1}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(z + y\right)}}, \sqrt{\frac{z + y}{x}} \cdot 2\right) \cdot x} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot x \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot x \]

      2. sqrt-pow2 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {-1}^{\left(\frac{2}{2}\right)}\right)\right) \cdot x \]

      3. metadata-eval N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {-1}^{1}\right)\right) \cdot x \]

      4. metadata-eval N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      5. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      7. lower-/.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      8. lower-+.f64 99.5

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

   7. Applied rewrites 99.5%

      \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

   8. Taylor expanded in y around inf

      \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]

      3. lower-/.f64 98.0

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]

   10. Applied rewrites 98.0%

       \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]

   if -6.3e13 < y < -4.39999999999999965e-239

   1. Initial program 96.5%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   if -4.39999999999999965e-239 < y < 3.5999999999999998e-303

   1. Initial program 65.2%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right) \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right) \cdot \color{blue}{x} \]

   4. Applied rewrites 1.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y, \sqrt{\frac{1}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(z + y\right)}}, \sqrt{\frac{z + y}{x}} \cdot 2\right) \cdot x} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot x \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot x \]

      2. sqrt-pow2 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {-1}^{\left(\frac{2}{2}\right)}\right)\right) \cdot x \]

      3. metadata-eval N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {-1}^{1}\right)\right) \cdot x \]

      4. metadata-eval N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      5. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      7. lower-/.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      8. lower-+.f64 58.6

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

   7. Applied rewrites 58.6%

      \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

   8. Taylor expanded in x around 0

      \[\leadsto \left(-2 \cdot \sqrt{\frac{y + z}{x}}\right) \cdot x \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y + z}{x}}\right) \cdot x \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y + z}{x}}\right) \cdot x \]

      3. lift-+.f64 N/A

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y + z}{x}}\right) \cdot x \]

      4. lift-sqrt.f64 58.6

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y + z}{x}}\right) \cdot x \]

   10. Applied rewrites 58.6%

       \[\leadsto \left(-2 \cdot \sqrt{\frac{y + z}{x}}\right) \cdot x \]

   if 3.5999999999999998e-303 < y

   1. Initial program 71.6%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(2 \cdot \sqrt{\frac{x + y}{z}} + \left(x \cdot y\right) \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}} + \left(x \cdot y\right) \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right) \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}} + \left(x \cdot y\right) \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right) \cdot \color{blue}{z} \]

   4. Applied rewrites 77.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot x, \sqrt{\frac{1}{\left(y + x\right) \cdot \left(\left(z \cdot z\right) \cdot z\right)}}, \sqrt{\frac{y + x}{z}} \cdot 2\right) \cdot z} \]

   5. Taylor expanded in z around inf

      \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]

      3. lower-/.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]

      4. lower-+.f64 87.6

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]

   7. Applied rewrites 87.6%

      \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]

      2. lift-+.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]

      3. lift-/.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]

      4. sqrt-div N/A

         \[\leadsto \left(2 \cdot \frac{\sqrt{x + y}}{\sqrt{z}}\right) \cdot z \]

      5. lower-/.f64 N/A

         \[\leadsto \left(2 \cdot \frac{\sqrt{x + y}}{\sqrt{z}}\right) \cdot z \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \frac{\sqrt{x + y}}{\sqrt{z}}\right) \cdot z \]

      7. lift-+.f64 N/A

         \[\leadsto \left(2 \cdot \frac{\sqrt{x + y}}{\sqrt{z}}\right) \cdot z \]

      8. lift-sqrt.f64 99.3

         \[\leadsto \left(2 \cdot \frac{\sqrt{x + y}}{\sqrt{z}}\right) \cdot z \]

   9. Applied rewrites 99.3%

      \[\leadsto \left(2 \cdot \frac{\sqrt{x + y}}{\sqrt{z}}\right) \cdot z \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -63000000000000:\\ \;\;\;\;\left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x\\ \mathbf{elif}\;y \leq -4.4 \cdot 10^{-239}:\\ \;\;\;\;2 \cdot \sqrt{\left(z + y\right) \cdot x}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-303}:\\ \;\;\;\;\left(-2 \cdot \sqrt{\frac{y + z}{x}}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\sqrt{x + y}}{\sqrt{z}}\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -63000000000000.0)
   (* (* -2.0 (sqrt (/ y x))) x)
   (if (<= y -4.4e-239)
     (* 2.0 (sqrt (* (+ z y) x)))
     (if (<= y 3.6e-303)
       (* (* -2.0 (sqrt (/ (+ y z) x))) x)
       (* (* 2.0 (/ (sqrt (+ x y)) (sqrt z))) z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -63000000000000.0) {
		tmp = (-2.0 * sqrt((y / x))) * x;
	} else if (y <= -4.4e-239) {
		tmp = 2.0 * sqrt(((z + y) * x));
	} else if (y <= 3.6e-303) {
		tmp = (-2.0 * sqrt(((y + z) / x))) * x;
	} else {
		tmp = (2.0 * (sqrt((x + y)) / sqrt(z))) * z;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-63000000000000.0d0)) then
        tmp = ((-2.0d0) * sqrt((y / x))) * x
    else if (y <= (-4.4d-239)) then
        tmp = 2.0d0 * sqrt(((z + y) * x))
    else if (y <= 3.6d-303) then
        tmp = ((-2.0d0) * sqrt(((y + z) / x))) * x
    else
        tmp = (2.0d0 * (sqrt((x + y)) / sqrt(z))) * z
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -63000000000000.0) {
		tmp = (-2.0 * Math.sqrt((y / x))) * x;
	} else if (y <= -4.4e-239) {
		tmp = 2.0 * Math.sqrt(((z + y) * x));
	} else if (y <= 3.6e-303) {
		tmp = (-2.0 * Math.sqrt(((y + z) / x))) * x;
	} else {
		tmp = (2.0 * (Math.sqrt((x + y)) / Math.sqrt(z))) * z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -63000000000000.0:
		tmp = (-2.0 * math.sqrt((y / x))) * x
	elif y <= -4.4e-239:
		tmp = 2.0 * math.sqrt(((z + y) * x))
	elif y <= 3.6e-303:
		tmp = (-2.0 * math.sqrt(((y + z) / x))) * x
	else:
		tmp = (2.0 * (math.sqrt((x + y)) / math.sqrt(z))) * z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -63000000000000.0)
		tmp = Float64(Float64(-2.0 * sqrt(Float64(y / x))) * x);
	elseif (y <= -4.4e-239)
		tmp = Float64(2.0 * sqrt(Float64(Float64(z + y) * x)));
	elseif (y <= 3.6e-303)
		tmp = Float64(Float64(-2.0 * sqrt(Float64(Float64(y + z) / x))) * x);
	else
		tmp = Float64(Float64(2.0 * Float64(sqrt(Float64(x + y)) / sqrt(z))) * z);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -63000000000000.0)
		tmp = (-2.0 * sqrt((y / x))) * x;
	elseif (y <= -4.4e-239)
		tmp = 2.0 * sqrt(((z + y) * x));
	elseif (y <= 3.6e-303)
		tmp = (-2.0 * sqrt(((y + z) / x))) * x;
	else
		tmp = (2.0 * (sqrt((x + y)) / sqrt(z))) * z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -63000000000000.0], N[(N[(-2.0 * N[Sqrt[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, -4.4e-239], N[(2.0 * N[Sqrt[N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.6e-303], N[(N[(-2.0 * N[Sqrt[N[(N[(y + z), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(2.0 * N[(N[Sqrt[N[(x + y), $MachinePrecision]], $MachinePrecision] / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -6.3e13

   1. Initial program 41.9%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right) \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right) \cdot \color{blue}{x} \]

   4. Applied rewrites 0.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y, \sqrt{\frac{1}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(z + y\right)}}, \sqrt{\frac{z + y}{x}} \cdot 2\right) \cdot x} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot x \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot x \]

      2. sqrt-pow2 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {-1}^{\left(\frac{2}{2}\right)}\right)\right) \cdot x \]

      3. metadata-eval N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {-1}^{1}\right)\right) \cdot x \]

      4. metadata-eval N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      5. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      7. lower-/.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      8. lower-+.f64 99.5

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

   7. Applied rewrites 99.5%

      \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

   8. Taylor expanded in y around inf

      \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]

      3. lower-/.f64 98.0

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]

   10. Applied rewrites 98.0%

       \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]

   if -6.3e13 < y < -4.39999999999999965e-239

   1. Initial program 96.5%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Taylor expanded in x around inf

      \[\leadsto 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 2 \cdot \sqrt{\left(y + z\right) \cdot \color{blue}{x}} \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\left(y + z\right) \cdot \color{blue}{x}} \]

      3. +-commutative N/A

         \[\leadsto 2 \cdot \sqrt{\left(z + y\right) \cdot x} \]

      4. lower-+.f64 96.5

         \[\leadsto 2 \cdot \sqrt{\left(z + y\right) \cdot x} \]

   4. Applied rewrites 96.5%

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(z + y\right) \cdot x}} \]

   if -4.39999999999999965e-239 < y < 3.5999999999999998e-303

   1. Initial program 65.2%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right) \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right) \cdot \color{blue}{x} \]

   4. Applied rewrites 1.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y, \sqrt{\frac{1}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(z + y\right)}}, \sqrt{\frac{z + y}{x}} \cdot 2\right) \cdot x} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot x \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot x \]

      2. sqrt-pow2 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {-1}^{\left(\frac{2}{2}\right)}\right)\right) \cdot x \]

      3. metadata-eval N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {-1}^{1}\right)\right) \cdot x \]

      4. metadata-eval N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      5. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      7. lower-/.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      8. lower-+.f64 58.6

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

   7. Applied rewrites 58.6%

      \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

   8. Taylor expanded in x around 0

      \[\leadsto \left(-2 \cdot \sqrt{\frac{y + z}{x}}\right) \cdot x \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y + z}{x}}\right) \cdot x \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y + z}{x}}\right) \cdot x \]

      3. lift-+.f64 N/A

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y + z}{x}}\right) \cdot x \]

      4. lift-sqrt.f64 58.6

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y + z}{x}}\right) \cdot x \]

   10. Applied rewrites 58.6%

       \[\leadsto \left(-2 \cdot \sqrt{\frac{y + z}{x}}\right) \cdot x \]

   if 3.5999999999999998e-303 < y

   1. Initial program 71.6%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(2 \cdot \sqrt{\frac{x + y}{z}} + \left(x \cdot y\right) \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}} + \left(x \cdot y\right) \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right) \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}} + \left(x \cdot y\right) \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right) \cdot \color{blue}{z} \]

   4. Applied rewrites 77.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot x, \sqrt{\frac{1}{\left(y + x\right) \cdot \left(\left(z \cdot z\right) \cdot z\right)}}, \sqrt{\frac{y + x}{z}} \cdot 2\right) \cdot z} \]

   5. Taylor expanded in z around inf

      \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]

      3. lower-/.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]

      4. lower-+.f64 87.6

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]

   7. Applied rewrites 87.6%

      \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]

      2. lift-+.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]

      3. lift-/.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]

      4. sqrt-div N/A

         \[\leadsto \left(2 \cdot \frac{\sqrt{x + y}}{\sqrt{z}}\right) \cdot z \]

      5. lower-/.f64 N/A

         \[\leadsto \left(2 \cdot \frac{\sqrt{x + y}}{\sqrt{z}}\right) \cdot z \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \frac{\sqrt{x + y}}{\sqrt{z}}\right) \cdot z \]

      7. lift-+.f64 N/A

         \[\leadsto \left(2 \cdot \frac{\sqrt{x + y}}{\sqrt{z}}\right) \cdot z \]

      8. lift-sqrt.f64 99.3

         \[\leadsto \left(2 \cdot \frac{\sqrt{x + y}}{\sqrt{z}}\right) \cdot z \]

   9. Applied rewrites 99.3%

      \[\leadsto \left(2 \cdot \frac{\sqrt{x + y}}{\sqrt{z}}\right) \cdot z \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 96.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-59}:\\ \;\;\;\;\left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x\\ \mathbf{elif}\;y \leq -1.5 \cdot 10^{-289}:\\ \;\;\;\;2 \cdot e^{\left(\log \left(\mathsf{fma}\left(-1, x, -1 \cdot z\right)\right) + -1 \cdot \log \left(\frac{-1}{y}\right)\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\sqrt{x + y}}{\sqrt{z}}\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.5e-59)
   (* (* -2.0 (sqrt (/ y x))) x)
   (if (<= y -1.5e-289)
     (*
      2.0
      (exp
       (* (+ (log (fma -1.0 x (* -1.0 z))) (* -1.0 (log (/ -1.0 y)))) 0.5)))
     (* (* 2.0 (/ (sqrt (+ x y)) (sqrt z))) z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.5e-59) {
		tmp = (-2.0 * sqrt((y / x))) * x;
	} else if (y <= -1.5e-289) {
		tmp = 2.0 * exp(((log(fma(-1.0, x, (-1.0 * z))) + (-1.0 * log((-1.0 / y)))) * 0.5));
	} else {
		tmp = (2.0 * (sqrt((x + y)) / sqrt(z))) * z;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1.5e-59)
		tmp = Float64(Float64(-2.0 * sqrt(Float64(y / x))) * x);
	elseif (y <= -1.5e-289)
		tmp = Float64(2.0 * exp(Float64(Float64(log(fma(-1.0, x, Float64(-1.0 * z))) + Float64(-1.0 * log(Float64(-1.0 / y)))) * 0.5)));
	else
		tmp = Float64(Float64(2.0 * Float64(sqrt(Float64(x + y)) / sqrt(z))) * z);
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -1.5e-59], N[(N[(-2.0 * N[Sqrt[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, -1.5e-289], N[(2.0 * N[Exp[N[(N[(N[Log[N[(-1.0 * x + N[(-1.0 * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-1.0 * N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[Sqrt[N[(x + y), $MachinePrecision]], $MachinePrecision] / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.5e-59

   1. Initial program 57.2%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right) \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right) \cdot \color{blue}{x} \]

   4. Applied rewrites 0.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y, \sqrt{\frac{1}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(z + y\right)}}, \sqrt{\frac{z + y}{x}} \cdot 2\right) \cdot x} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot x \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot x \]

      2. sqrt-pow2 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {-1}^{\left(\frac{2}{2}\right)}\right)\right) \cdot x \]

      3. metadata-eval N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {-1}^{1}\right)\right) \cdot x \]

      4. metadata-eval N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      5. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      7. lower-/.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      8. lower-+.f64 98.3

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

   7. Applied rewrites 98.3%

      \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

   8. Taylor expanded in y around inf

      \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]

      3. lower-/.f64 96.7

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]

   10. Applied rewrites 96.7%

       \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]

   if -1.5e-59 < y < -1.4999999999999999e-289

   1. Initial program 92.1%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}} \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + \color{blue}{y \cdot z}} \]

      3. lift-+.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(x \cdot y + x \cdot z\right) + y \cdot z}} \]

      4. lift-*.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\left(\color{blue}{x \cdot y} + x \cdot z\right) + y \cdot z} \]

      5. lift-*.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\left(x \cdot y + \color{blue}{x \cdot z}\right) + y \cdot z} \]

      6. lift-+.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(x \cdot y + x \cdot z\right)} + y \cdot z} \]

      7. pow1/2 N/A

         \[\leadsto 2 \cdot \color{blue}{{\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)}^{\frac{1}{2}}} \]

      8. pow-to-exp N/A

         \[\leadsto 2 \cdot \color{blue}{e^{\log \left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right) \cdot \frac{1}{2}}} \]

      9. lower-exp.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{e^{\log \left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right) \cdot \frac{1}{2}}} \]

      10. lower-*.f64 N/A

          \[\leadsto 2 \cdot e^{\color{blue}{\log \left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right) \cdot \frac{1}{2}}} \]

   3. Applied rewrites 86.1%

      \[\leadsto 2 \cdot \color{blue}{e^{\log \left(\mathsf{fma}\left(y + x, z, y \cdot x\right)\right) \cdot 0.5}} \]

   4. Taylor expanded in y around -inf

      \[\leadsto 2 \cdot e^{\color{blue}{\left(\log \left(-1 \cdot x + -1 \cdot z\right) + -1 \cdot \log \left(\frac{-1}{y}\right)\right)} \cdot \frac{1}{2}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 2 \cdot e^{\left(\log \left(-1 \cdot x + -1 \cdot z\right) + \color{blue}{-1 \cdot \log \left(\frac{-1}{y}\right)}\right) \cdot \frac{1}{2}} \]

      2. lower-log.f64 N/A

         \[\leadsto 2 \cdot e^{\left(\log \left(-1 \cdot x + -1 \cdot z\right) + \color{blue}{-1} \cdot \log \left(\frac{-1}{y}\right)\right) \cdot \frac{1}{2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto 2 \cdot e^{\left(\log \left(\mathsf{fma}\left(-1, x, -1 \cdot z\right)\right) + -1 \cdot \log \left(\frac{-1}{y}\right)\right) \cdot \frac{1}{2}} \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot e^{\left(\log \left(\mathsf{fma}\left(-1, x, -1 \cdot z\right)\right) + -1 \cdot \log \left(\frac{-1}{y}\right)\right) \cdot \frac{1}{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto 2 \cdot e^{\left(\log \left(\mathsf{fma}\left(-1, x, -1 \cdot z\right)\right) + -1 \cdot \color{blue}{\log \left(\frac{-1}{y}\right)}\right) \cdot \frac{1}{2}} \]

      6. lower-log.f64 N/A

         \[\leadsto 2 \cdot e^{\left(\log \left(\mathsf{fma}\left(-1, x, -1 \cdot z\right)\right) + -1 \cdot \log \left(\frac{-1}{y}\right)\right) \cdot \frac{1}{2}} \]

      7. lower-/.f64 87.2

         \[\leadsto 2 \cdot e^{\left(\log \left(\mathsf{fma}\left(-1, x, -1 \cdot z\right)\right) + -1 \cdot \log \left(\frac{-1}{y}\right)\right) \cdot 0.5} \]

   6. Applied rewrites 87.2%

      \[\leadsto 2 \cdot e^{\color{blue}{\left(\log \left(\mathsf{fma}\left(-1, x, -1 \cdot z\right)\right) + -1 \cdot \log \left(\frac{-1}{y}\right)\right)} \cdot 0.5} \]

   if -1.4999999999999999e-289 < y

   1. Initial program 71.5%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(2 \cdot \sqrt{\frac{x + y}{z}} + \left(x \cdot y\right) \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}} + \left(x \cdot y\right) \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right) \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}} + \left(x \cdot y\right) \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right) \cdot \color{blue}{z} \]

   4. Applied rewrites 77.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot x, \sqrt{\frac{1}{\left(y + x\right) \cdot \left(\left(z \cdot z\right) \cdot z\right)}}, \sqrt{\frac{y + x}{z}} \cdot 2\right) \cdot z} \]

   5. Taylor expanded in z around inf

      \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]

      3. lower-/.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]

      4. lower-+.f64 87.3

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]

   7. Applied rewrites 87.3%

      \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]

      2. lift-+.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]

      3. lift-/.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]

      4. sqrt-div N/A

         \[\leadsto \left(2 \cdot \frac{\sqrt{x + y}}{\sqrt{z}}\right) \cdot z \]

      5. lower-/.f64 N/A

         \[\leadsto \left(2 \cdot \frac{\sqrt{x + y}}{\sqrt{z}}\right) \cdot z \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \frac{\sqrt{x + y}}{\sqrt{z}}\right) \cdot z \]

      7. lift-+.f64 N/A

         \[\leadsto \left(2 \cdot \frac{\sqrt{x + y}}{\sqrt{z}}\right) \cdot z \]

      8. lift-sqrt.f64 99.1

         \[\leadsto \left(2 \cdot \frac{\sqrt{x + y}}{\sqrt{z}}\right) \cdot z \]

   9. Applied rewrites 99.1%

      \[\leadsto \left(2 \cdot \frac{\sqrt{x + y}}{\sqrt{z}}\right) \cdot z \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 96.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -63000000000000:\\ \;\;\;\;\left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x\\ \mathbf{elif}\;y \leq -4.4 \cdot 10^{-239}:\\ \;\;\;\;2 \cdot \sqrt{\left(z + y\right) \cdot x}\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-295}:\\ \;\;\;\;\left(-2 \cdot \sqrt{\frac{y + z}{x}}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\sqrt{y}}{\sqrt{z}}\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -63000000000000.0)
   (* (* -2.0 (sqrt (/ y x))) x)
   (if (<= y -4.4e-239)
     (* 2.0 (sqrt (* (+ z y) x)))
     (if (<= y 5.4e-295)
       (* (* -2.0 (sqrt (/ (+ y z) x))) x)
       (* (* 2.0 (/ (sqrt y) (sqrt z))) z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -63000000000000.0) {
		tmp = (-2.0 * sqrt((y / x))) * x;
	} else if (y <= -4.4e-239) {
		tmp = 2.0 * sqrt(((z + y) * x));
	} else if (y <= 5.4e-295) {
		tmp = (-2.0 * sqrt(((y + z) / x))) * x;
	} else {
		tmp = (2.0 * (sqrt(y) / sqrt(z))) * z;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-63000000000000.0d0)) then
        tmp = ((-2.0d0) * sqrt((y / x))) * x
    else if (y <= (-4.4d-239)) then
        tmp = 2.0d0 * sqrt(((z + y) * x))
    else if (y <= 5.4d-295) then
        tmp = ((-2.0d0) * sqrt(((y + z) / x))) * x
    else
        tmp = (2.0d0 * (sqrt(y) / sqrt(z))) * z
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -63000000000000.0) {
		tmp = (-2.0 * Math.sqrt((y / x))) * x;
	} else if (y <= -4.4e-239) {
		tmp = 2.0 * Math.sqrt(((z + y) * x));
	} else if (y <= 5.4e-295) {
		tmp = (-2.0 * Math.sqrt(((y + z) / x))) * x;
	} else {
		tmp = (2.0 * (Math.sqrt(y) / Math.sqrt(z))) * z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -63000000000000.0:
		tmp = (-2.0 * math.sqrt((y / x))) * x
	elif y <= -4.4e-239:
		tmp = 2.0 * math.sqrt(((z + y) * x))
	elif y <= 5.4e-295:
		tmp = (-2.0 * math.sqrt(((y + z) / x))) * x
	else:
		tmp = (2.0 * (math.sqrt(y) / math.sqrt(z))) * z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -63000000000000.0)
		tmp = Float64(Float64(-2.0 * sqrt(Float64(y / x))) * x);
	elseif (y <= -4.4e-239)
		tmp = Float64(2.0 * sqrt(Float64(Float64(z + y) * x)));
	elseif (y <= 5.4e-295)
		tmp = Float64(Float64(-2.0 * sqrt(Float64(Float64(y + z) / x))) * x);
	else
		tmp = Float64(Float64(2.0 * Float64(sqrt(y) / sqrt(z))) * z);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -63000000000000.0)
		tmp = (-2.0 * sqrt((y / x))) * x;
	elseif (y <= -4.4e-239)
		tmp = 2.0 * sqrt(((z + y) * x));
	elseif (y <= 5.4e-295)
		tmp = (-2.0 * sqrt(((y + z) / x))) * x;
	else
		tmp = (2.0 * (sqrt(y) / sqrt(z))) * z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -63000000000000.0], N[(N[(-2.0 * N[Sqrt[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, -4.4e-239], N[(2.0 * N[Sqrt[N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.4e-295], N[(N[(-2.0 * N[Sqrt[N[(N[(y + z), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(2.0 * N[(N[Sqrt[y], $MachinePrecision] / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -6.3e13

   1. Initial program 41.9%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right) \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right) \cdot \color{blue}{x} \]

   4. Applied rewrites 0.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y, \sqrt{\frac{1}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(z + y\right)}}, \sqrt{\frac{z + y}{x}} \cdot 2\right) \cdot x} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot x \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot x \]

      2. sqrt-pow2 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {-1}^{\left(\frac{2}{2}\right)}\right)\right) \cdot x \]

      3. metadata-eval N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {-1}^{1}\right)\right) \cdot x \]

      4. metadata-eval N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      5. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      7. lower-/.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      8. lower-+.f64 99.5

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

   7. Applied rewrites 99.5%

      \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

   8. Taylor expanded in y around inf

      \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]

      3. lower-/.f64 98.0

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]

   10. Applied rewrites 98.0%

       \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]

   if -6.3e13 < y < -4.39999999999999965e-239

   1. Initial program 96.5%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Taylor expanded in x around inf

      \[\leadsto 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 2 \cdot \sqrt{\left(y + z\right) \cdot \color{blue}{x}} \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\left(y + z\right) \cdot \color{blue}{x}} \]

      3. +-commutative N/A

         \[\leadsto 2 \cdot \sqrt{\left(z + y\right) \cdot x} \]

      4. lower-+.f64 96.5

         \[\leadsto 2 \cdot \sqrt{\left(z + y\right) \cdot x} \]

   4. Applied rewrites 96.5%

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(z + y\right) \cdot x}} \]

   if -4.39999999999999965e-239 < y < 5.4000000000000002e-295

   1. Initial program 65.3%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right) \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right) \cdot \color{blue}{x} \]

   4. Applied rewrites 1.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y, \sqrt{\frac{1}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(z + y\right)}}, \sqrt{\frac{z + y}{x}} \cdot 2\right) \cdot x} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot x \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot x \]

      2. sqrt-pow2 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {-1}^{\left(\frac{2}{2}\right)}\right)\right) \cdot x \]

      3. metadata-eval N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {-1}^{1}\right)\right) \cdot x \]

      4. metadata-eval N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      5. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      7. lower-/.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      8. lower-+.f64 57.9

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

   7. Applied rewrites 57.9%

      \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

   8. Taylor expanded in x around 0

      \[\leadsto \left(-2 \cdot \sqrt{\frac{y + z}{x}}\right) \cdot x \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y + z}{x}}\right) \cdot x \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y + z}{x}}\right) \cdot x \]

      3. lift-+.f64 N/A

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y + z}{x}}\right) \cdot x \]

      4. lift-sqrt.f64 57.9

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y + z}{x}}\right) \cdot x \]

   10. Applied rewrites 57.9%

       \[\leadsto \left(-2 \cdot \sqrt{\frac{y + z}{x}}\right) \cdot x \]

   if 5.4000000000000002e-295 < y

   1. Initial program 71.6%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(2 \cdot \sqrt{\frac{x + y}{z}} + \left(x \cdot y\right) \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}} + \left(x \cdot y\right) \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right) \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}} + \left(x \cdot y\right) \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right) \cdot \color{blue}{z} \]

   4. Applied rewrites 78.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot x, \sqrt{\frac{1}{\left(y + x\right) \cdot \left(\left(z \cdot z\right) \cdot z\right)}}, \sqrt{\frac{y + x}{z}} \cdot 2\right) \cdot z} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(2 \cdot \sqrt{\frac{y}{z}}\right) \cdot z \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y}{z}}\right) \cdot z \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y}{z}}\right) \cdot z \]

      3. lower-/.f64 85.5

         \[\leadsto \left(2 \cdot \sqrt{\frac{y}{z}}\right) \cdot z \]

   7. Applied rewrites 85.5%

      \[\leadsto \left(2 \cdot \sqrt{\frac{y}{z}}\right) \cdot z \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y}{z}}\right) \cdot z \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y}{z}}\right) \cdot z \]

      3. sqrt-div N/A

         \[\leadsto \left(2 \cdot \frac{\sqrt{y}}{\sqrt{z}}\right) \cdot z \]

      4. lower-/.f64 N/A

         \[\leadsto \left(2 \cdot \frac{\sqrt{y}}{\sqrt{z}}\right) \cdot z \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \frac{\sqrt{y}}{\sqrt{z}}\right) \cdot z \]

      6. lift-sqrt.f64 96.4

         \[\leadsto \left(2 \cdot \frac{\sqrt{y}}{\sqrt{z}}\right) \cdot z \]

   9. Applied rewrites 96.4%

      \[\leadsto \left(2 \cdot \frac{\sqrt{y}}{\sqrt{z}}\right) \cdot z \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 96.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -63000000000000:\\ \;\;\;\;\left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-19}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -63000000000000.0)
   (* (* -2.0 (sqrt (/ y x))) x)
   (if (<= y 1.7e-19)
     (* 2.0 (sqrt (fma y x (* (+ y x) z))))
     (* (* 2.0 (sqrt (/ (+ x y) z))) z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -63000000000000.0) {
		tmp = (-2.0 * sqrt((y / x))) * x;
	} else if (y <= 1.7e-19) {
		tmp = 2.0 * sqrt(fma(y, x, ((y + x) * z)));
	} else {
		tmp = (2.0 * sqrt(((x + y) / z))) * z;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -63000000000000.0)
		tmp = Float64(Float64(-2.0 * sqrt(Float64(y / x))) * x);
	elseif (y <= 1.7e-19)
		tmp = Float64(2.0 * sqrt(fma(y, x, Float64(Float64(y + x) * z))));
	else
		tmp = Float64(Float64(2.0 * sqrt(Float64(Float64(x + y) / z))) * z);
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -63000000000000.0], N[(N[(-2.0 * N[Sqrt[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 1.7e-19], N[(2.0 * N[Sqrt[N[(y * x + N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sqrt[N[(N[(x + y), $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.3e13

   1. Initial program 41.9%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right) \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right) \cdot \color{blue}{x} \]

   4. Applied rewrites 0.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y, \sqrt{\frac{1}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(z + y\right)}}, \sqrt{\frac{z + y}{x}} \cdot 2\right) \cdot x} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot x \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot x \]

      2. sqrt-pow2 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {-1}^{\left(\frac{2}{2}\right)}\right)\right) \cdot x \]

      3. metadata-eval N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {-1}^{1}\right)\right) \cdot x \]

      4. metadata-eval N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      5. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      7. lower-/.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      8. lower-+.f64 99.5

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

   7. Applied rewrites 99.5%

      \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

   8. Taylor expanded in y around inf

      \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]

      3. lower-/.f64 98.0

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]

   10. Applied rewrites 98.0%

       \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]

   if -6.3e13 < y < 1.7000000000000001e-19

   1. Initial program 94.1%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + \color{blue}{y \cdot z}} \]

      2. lift-+.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(x \cdot y + x \cdot z\right) + y \cdot z}} \]

      3. lift-*.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\left(\color{blue}{x \cdot y} + x \cdot z\right) + y \cdot z} \]

      4. lift-*.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\left(x \cdot y + \color{blue}{x \cdot z}\right) + y \cdot z} \]

      5. lift-+.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(x \cdot y + x \cdot z\right)} + y \cdot z} \]

      6. associate-+l+ N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{x \cdot y + \left(x \cdot z + y \cdot z\right)}} \]

      7. distribute-rgt-in N/A

         \[\leadsto 2 \cdot \sqrt{x \cdot y + \color{blue}{z \cdot \left(x + y\right)}} \]

      8. *-commutative N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{y \cdot x} + z \cdot \left(x + y\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(y, x, z \cdot \left(x + y\right)\right)}} \]

      10. *-commutative N/A

          \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(y, x, \color{blue}{\left(x + y\right) \cdot z}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(y, x, \color{blue}{\left(x + y\right) \cdot z}\right)} \]

      12. +-commutative N/A

          \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(y, x, \color{blue}{\left(y + x\right)} \cdot z\right)} \]

      13. lower-+.f64 94.1

          \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(y, x, \color{blue}{\left(y + x\right)} \cdot z\right)} \]

   3. Applied rewrites 94.1%

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)}} \]

   if 1.7000000000000001e-19 < y

   1. Initial program 52.4%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(2 \cdot \sqrt{\frac{x + y}{z}} + \left(x \cdot y\right) \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}} + \left(x \cdot y\right) \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right) \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}} + \left(x \cdot y\right) \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right) \cdot \color{blue}{z} \]

   4. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot x, \sqrt{\frac{1}{\left(y + x\right) \cdot \left(\left(z \cdot z\right) \cdot z\right)}}, \sqrt{\frac{y + x}{z}} \cdot 2\right) \cdot z} \]

   5. Taylor expanded in z around inf

      \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]

      3. lower-/.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]

      4. lower-+.f64 99.5

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]

   7. Applied rewrites 99.5%

      \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 96.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -63000000000000:\\ \;\;\;\;\left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-295}:\\ \;\;\;\;2 \cdot \sqrt{\left(z + y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\sqrt{y}}{\sqrt{z}}\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -63000000000000.0)
   (* (* -2.0 (sqrt (/ y x))) x)
   (if (<= y 5.5e-295)
     (* 2.0 (sqrt (* (+ z y) x)))
     (* (* 2.0 (/ (sqrt y) (sqrt z))) z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -63000000000000.0) {
		tmp = (-2.0 * sqrt((y / x))) * x;
	} else if (y <= 5.5e-295) {
		tmp = 2.0 * sqrt(((z + y) * x));
	} else {
		tmp = (2.0 * (sqrt(y) / sqrt(z))) * z;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-63000000000000.0d0)) then
        tmp = ((-2.0d0) * sqrt((y / x))) * x
    else if (y <= 5.5d-295) then
        tmp = 2.0d0 * sqrt(((z + y) * x))
    else
        tmp = (2.0d0 * (sqrt(y) / sqrt(z))) * z
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -63000000000000.0) {
		tmp = (-2.0 * Math.sqrt((y / x))) * x;
	} else if (y <= 5.5e-295) {
		tmp = 2.0 * Math.sqrt(((z + y) * x));
	} else {
		tmp = (2.0 * (Math.sqrt(y) / Math.sqrt(z))) * z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -63000000000000.0:
		tmp = (-2.0 * math.sqrt((y / x))) * x
	elif y <= 5.5e-295:
		tmp = 2.0 * math.sqrt(((z + y) * x))
	else:
		tmp = (2.0 * (math.sqrt(y) / math.sqrt(z))) * z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -63000000000000.0)
		tmp = Float64(Float64(-2.0 * sqrt(Float64(y / x))) * x);
	elseif (y <= 5.5e-295)
		tmp = Float64(2.0 * sqrt(Float64(Float64(z + y) * x)));
	else
		tmp = Float64(Float64(2.0 * Float64(sqrt(y) / sqrt(z))) * z);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -63000000000000.0)
		tmp = (-2.0 * sqrt((y / x))) * x;
	elseif (y <= 5.5e-295)
		tmp = 2.0 * sqrt(((z + y) * x));
	else
		tmp = (2.0 * (sqrt(y) / sqrt(z))) * z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -63000000000000.0], N[(N[(-2.0 * N[Sqrt[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 5.5e-295], N[(2.0 * N[Sqrt[N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[Sqrt[y], $MachinePrecision] / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.3e13

   1. Initial program 41.9%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right) \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right) \cdot \color{blue}{x} \]

   4. Applied rewrites 0.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y, \sqrt{\frac{1}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(z + y\right)}}, \sqrt{\frac{z + y}{x}} \cdot 2\right) \cdot x} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot x \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot x \]

      2. sqrt-pow2 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {-1}^{\left(\frac{2}{2}\right)}\right)\right) \cdot x \]

      3. metadata-eval N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {-1}^{1}\right)\right) \cdot x \]

      4. metadata-eval N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      5. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      7. lower-/.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      8. lower-+.f64 99.5

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

   7. Applied rewrites 99.5%

      \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

   8. Taylor expanded in y around inf

      \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]

      3. lower-/.f64 98.0

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]

   10. Applied rewrites 98.0%

       \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]

   if -6.3e13 < y < 5.5e-295

   1. Initial program 94.2%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Taylor expanded in x around inf

      \[\leadsto 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 2 \cdot \sqrt{\left(y + z\right) \cdot \color{blue}{x}} \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\left(y + z\right) \cdot \color{blue}{x}} \]

      3. +-commutative N/A

         \[\leadsto 2 \cdot \sqrt{\left(z + y\right) \cdot x} \]

      4. lower-+.f64 94.0

         \[\leadsto 2 \cdot \sqrt{\left(z + y\right) \cdot x} \]

   4. Applied rewrites 94.0%

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(z + y\right) \cdot x}} \]

   if 5.5e-295 < y

   1. Initial program 71.6%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(2 \cdot \sqrt{\frac{x + y}{z}} + \left(x \cdot y\right) \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}} + \left(x \cdot y\right) \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right) \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}} + \left(x \cdot y\right) \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right) \cdot \color{blue}{z} \]

   4. Applied rewrites 78.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot x, \sqrt{\frac{1}{\left(y + x\right) \cdot \left(\left(z \cdot z\right) \cdot z\right)}}, \sqrt{\frac{y + x}{z}} \cdot 2\right) \cdot z} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(2 \cdot \sqrt{\frac{y}{z}}\right) \cdot z \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y}{z}}\right) \cdot z \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y}{z}}\right) \cdot z \]

      3. lower-/.f64 85.5

         \[\leadsto \left(2 \cdot \sqrt{\frac{y}{z}}\right) \cdot z \]

   7. Applied rewrites 85.5%

      \[\leadsto \left(2 \cdot \sqrt{\frac{y}{z}}\right) \cdot z \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y}{z}}\right) \cdot z \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y}{z}}\right) \cdot z \]

      3. sqrt-div N/A

         \[\leadsto \left(2 \cdot \frac{\sqrt{y}}{\sqrt{z}}\right) \cdot z \]

      4. lower-/.f64 N/A

         \[\leadsto \left(2 \cdot \frac{\sqrt{y}}{\sqrt{z}}\right) \cdot z \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \frac{\sqrt{y}}{\sqrt{z}}\right) \cdot z \]

      6. lift-sqrt.f64 96.4

         \[\leadsto \left(2 \cdot \frac{\sqrt{y}}{\sqrt{z}}\right) \cdot z \]

   9. Applied rewrites 96.4%

      \[\leadsto \left(2 \cdot \frac{\sqrt{y}}{\sqrt{z}}\right) \cdot z \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 94.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -620000000000:\\ \;\;\;\;\left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x\\ \mathbf{elif}\;y \leq 3200000:\\ \;\;\;\;2 \cdot \sqrt{\left(z + x\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{\frac{y}{z}}\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -620000000000.0)
   (* (* -2.0 (sqrt (/ y x))) x)
   (if (<= y 3200000.0)
     (* 2.0 (sqrt (* (+ z x) y)))
     (* (* 2.0 (sqrt (/ y z))) z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -620000000000.0) {
		tmp = (-2.0 * sqrt((y / x))) * x;
	} else if (y <= 3200000.0) {
		tmp = 2.0 * sqrt(((z + x) * y));
	} else {
		tmp = (2.0 * sqrt((y / z))) * z;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-620000000000.0d0)) then
        tmp = ((-2.0d0) * sqrt((y / x))) * x
    else if (y <= 3200000.0d0) then
        tmp = 2.0d0 * sqrt(((z + x) * y))
    else
        tmp = (2.0d0 * sqrt((y / z))) * z
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -620000000000.0) {
		tmp = (-2.0 * Math.sqrt((y / x))) * x;
	} else if (y <= 3200000.0) {
		tmp = 2.0 * Math.sqrt(((z + x) * y));
	} else {
		tmp = (2.0 * Math.sqrt((y / z))) * z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -620000000000.0:
		tmp = (-2.0 * math.sqrt((y / x))) * x
	elif y <= 3200000.0:
		tmp = 2.0 * math.sqrt(((z + x) * y))
	else:
		tmp = (2.0 * math.sqrt((y / z))) * z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -620000000000.0)
		tmp = Float64(Float64(-2.0 * sqrt(Float64(y / x))) * x);
	elseif (y <= 3200000.0)
		tmp = Float64(2.0 * sqrt(Float64(Float64(z + x) * y)));
	else
		tmp = Float64(Float64(2.0 * sqrt(Float64(y / z))) * z);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -620000000000.0)
		tmp = (-2.0 * sqrt((y / x))) * x;
	elseif (y <= 3200000.0)
		tmp = 2.0 * sqrt(((z + x) * y));
	else
		tmp = (2.0 * sqrt((y / z))) * z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -620000000000.0], N[(N[(-2.0 * N[Sqrt[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 3200000.0], N[(2.0 * N[Sqrt[N[(N[(z + x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sqrt[N[(y / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.2e11

   1. Initial program 42.5%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right) \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y + z}{x}} + \sqrt{\frac{1}{{x}^{3} \cdot \left(y + z\right)}} \cdot \left(y \cdot z\right)\right) \cdot \color{blue}{x} \]

   4. Applied rewrites 0.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y, \sqrt{\frac{1}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(z + y\right)}}, \sqrt{\frac{z + y}{x}} \cdot 2\right) \cdot x} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot x \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot x \]

      2. sqrt-pow2 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {-1}^{\left(\frac{2}{2}\right)}\right)\right) \cdot x \]

      3. metadata-eval N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot {-1}^{1}\right)\right) \cdot x \]

      4. metadata-eval N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      5. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      7. lower-/.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

      8. lower-+.f64 99.5

         \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

   7. Applied rewrites 99.5%

      \[\leadsto \left(2 \cdot \left(\sqrt{\frac{y + z}{x}} \cdot -1\right)\right) \cdot x \]

   8. Taylor expanded in y around inf

      \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]

      3. lower-/.f64 97.9

         \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]

   10. Applied rewrites 97.9%

       \[\leadsto \left(-2 \cdot \sqrt{\frac{y}{x}}\right) \cdot x \]

   if -6.2e11 < y < 3.2e6

   1. Initial program 94.5%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Taylor expanded in y around inf

      \[\leadsto 2 \cdot \sqrt{\color{blue}{y \cdot \left(x + z\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 2 \cdot \sqrt{\left(x + z\right) \cdot \color{blue}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\left(x + z\right) \cdot \color{blue}{y}} \]

      3. +-commutative N/A

         \[\leadsto 2 \cdot \sqrt{\left(z + x\right) \cdot y} \]

      4. lower-+.f64 90.9

         \[\leadsto 2 \cdot \sqrt{\left(z + x\right) \cdot y} \]

   4. Applied rewrites 90.9%

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(z + x\right) \cdot y}} \]

   if 3.2e6 < y

   1. Initial program 46.2%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(2 \cdot \sqrt{\frac{x + y}{z}} + \left(x \cdot y\right) \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}} + \left(x \cdot y\right) \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right) \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}} + \left(x \cdot y\right) \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right) \cdot \color{blue}{z} \]

   4. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot x, \sqrt{\frac{1}{\left(y + x\right) \cdot \left(\left(z \cdot z\right) \cdot z\right)}}, \sqrt{\frac{y + x}{z}} \cdot 2\right) \cdot z} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(2 \cdot \sqrt{\frac{y}{z}}\right) \cdot z \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y}{z}}\right) \cdot z \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y}{z}}\right) \cdot z \]

      3. lower-/.f64 98.2

         \[\leadsto \left(2 \cdot \sqrt{\frac{y}{z}}\right) \cdot z \]

   7. Applied rewrites 98.2%

      \[\leadsto \left(2 \cdot \sqrt{\frac{y}{z}}\right) \cdot z \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 80.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3200000:\\ \;\;\;\;2 \cdot \sqrt{\left(z + x\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{\frac{y}{z}}\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 3200000.0)
   (* 2.0 (sqrt (* (+ z x) y)))
   (* (* 2.0 (sqrt (/ y z))) z)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 3200000.0) {
		tmp = 2.0 * sqrt(((z + x) * y));
	} else {
		tmp = (2.0 * sqrt((y / z))) * z;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 3200000.0d0) then
        tmp = 2.0d0 * sqrt(((z + x) * y))
    else
        tmp = (2.0d0 * sqrt((y / z))) * z
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3200000.0) {
		tmp = 2.0 * Math.sqrt(((z + x) * y));
	} else {
		tmp = (2.0 * Math.sqrt((y / z))) * z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 3200000.0:
		tmp = 2.0 * math.sqrt(((z + x) * y))
	else:
		tmp = (2.0 * math.sqrt((y / z))) * z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 3200000.0)
		tmp = Float64(2.0 * sqrt(Float64(Float64(z + x) * y)));
	else
		tmp = Float64(Float64(2.0 * sqrt(Float64(y / z))) * z);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3200000.0)
		tmp = 2.0 * sqrt(((z + x) * y));
	else
		tmp = (2.0 * sqrt((y / z))) * z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 3200000.0], N[(2.0 * N[Sqrt[N[(N[(z + x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sqrt[N[(y / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.2e6

   1. Initial program 78.2%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Taylor expanded in y around inf

      \[\leadsto 2 \cdot \sqrt{\color{blue}{y \cdot \left(x + z\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 2 \cdot \sqrt{\left(x + z\right) \cdot \color{blue}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\left(x + z\right) \cdot \color{blue}{y}} \]

      3. +-commutative N/A

         \[\leadsto 2 \cdot \sqrt{\left(z + x\right) \cdot y} \]

      4. lower-+.f64 75.5

         \[\leadsto 2 \cdot \sqrt{\left(z + x\right) \cdot y} \]

   4. Applied rewrites 75.5%

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(z + x\right) \cdot y}} \]

   if 3.2e6 < y

   1. Initial program 46.2%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(2 \cdot \sqrt{\frac{x + y}{z}} + \left(x \cdot y\right) \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}} + \left(x \cdot y\right) \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right) \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{x + y}{z}} + \left(x \cdot y\right) \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right) \cdot \color{blue}{z} \]

   4. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot x, \sqrt{\frac{1}{\left(y + x\right) \cdot \left(\left(z \cdot z\right) \cdot z\right)}}, \sqrt{\frac{y + x}{z}} \cdot 2\right) \cdot z} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(2 \cdot \sqrt{\frac{y}{z}}\right) \cdot z \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y}{z}}\right) \cdot z \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y}{z}}\right) \cdot z \]

      3. lower-/.f64 98.2

         \[\leadsto \left(2 \cdot \sqrt{\frac{y}{z}}\right) \cdot z \]

   7. Applied rewrites 98.2%

      \[\leadsto \left(2 \cdot \sqrt{\frac{y}{z}}\right) \cdot z \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 69.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-286}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{\left(y + x\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2.1e-286) (* 2.0 (sqrt (* y x))) (* 2.0 (sqrt (* (+ y x) z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.1e-286) {
		tmp = 2.0 * sqrt((y * x));
	} else {
		tmp = 2.0 * sqrt(((y + x) * z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-2.1d-286)) then
        tmp = 2.0d0 * sqrt((y * x))
    else
        tmp = 2.0d0 * sqrt(((y + x) * z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.1e-286) {
		tmp = 2.0 * Math.sqrt((y * x));
	} else {
		tmp = 2.0 * Math.sqrt(((y + x) * z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -2.1e-286:
		tmp = 2.0 * math.sqrt((y * x))
	else:
		tmp = 2.0 * math.sqrt(((y + x) * z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -2.1e-286)
		tmp = Float64(2.0 * sqrt(Float64(y * x)));
	else
		tmp = Float64(2.0 * sqrt(Float64(Float64(y + x) * z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.1e-286)
		tmp = 2.0 * sqrt((y * x));
	else
		tmp = 2.0 * sqrt(((y + x) * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -2.1e-286], N[(2.0 * N[Sqrt[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.09999999999999988e-286

   1. Initial program 69.8%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Taylor expanded in z around 0

      \[\leadsto 2 \cdot \sqrt{\color{blue}{x \cdot y}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 2 \cdot \sqrt{y \cdot \color{blue}{x}} \]

      2. lower-*.f64 67.8

         \[\leadsto 2 \cdot \sqrt{y \cdot \color{blue}{x}} \]

   4. Applied rewrites 67.8%

      \[\leadsto 2 \cdot \sqrt{\color{blue}{y \cdot x}} \]

   if -2.09999999999999988e-286 < y

   1. Initial program 71.5%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Taylor expanded in z around inf

      \[\leadsto 2 \cdot \sqrt{\color{blue}{z \cdot \left(x + y\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 2 \cdot \sqrt{\left(x + y\right) \cdot \color{blue}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\left(x + y\right) \cdot \color{blue}{z}} \]

      3. +-commutative N/A

         \[\leadsto 2 \cdot \sqrt{\left(y + x\right) \cdot z} \]

      4. lower-+.f64 71.5

         \[\leadsto 2 \cdot \sqrt{\left(y + x\right) \cdot z} \]

   4. Applied rewrites 71.5%

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(y + x\right) \cdot z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 68.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ 2 \cdot \sqrt{\left(z + x\right) \cdot y} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (* 2.0 (sqrt (* (+ z x) y))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return 2.0 * sqrt(((z + x) * y));
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 2.0d0 * sqrt(((z + x) * y))
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	return 2.0 * Math.sqrt(((z + x) * y));
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return 2.0 * math.sqrt(((z + x) * y))

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(2.0 * sqrt(Float64(Float64(z + x) * y)))
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = 2.0 * sqrt(((z + x) * y));
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(2.0 * N[Sqrt[N[(N[(z + x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.6%

   \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

2. Taylor expanded in y around inf

   \[\leadsto 2 \cdot \sqrt{\color{blue}{y \cdot \left(x + z\right)}} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto 2 \cdot \sqrt{\left(x + z\right) \cdot \color{blue}{y}} \]

   2. lower-*.f64 N/A

      \[\leadsto 2 \cdot \sqrt{\left(x + z\right) \cdot \color{blue}{y}} \]

   3. +-commutative N/A

      \[\leadsto 2 \cdot \sqrt{\left(z + x\right) \cdot y} \]

   4. lower-+.f64 68.6

      \[\leadsto 2 \cdot \sqrt{\left(z + x\right) \cdot y} \]

4. Applied rewrites 68.6%

   \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(z + x\right) \cdot y}} \]
5. Add Preprocessing

Alternative 12: 68.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.56 \cdot 10^{-289}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.56e-289) (* 2.0 (sqrt (* y x))) (* 2.0 (sqrt (* z y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.56e-289) {
		tmp = 2.0 * sqrt((y * x));
	} else {
		tmp = 2.0 * sqrt((z * y));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.56d-289)) then
        tmp = 2.0d0 * sqrt((y * x))
    else
        tmp = 2.0d0 * sqrt((z * y))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.56e-289) {
		tmp = 2.0 * Math.sqrt((y * x));
	} else {
		tmp = 2.0 * Math.sqrt((z * y));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -1.56e-289:
		tmp = 2.0 * math.sqrt((y * x))
	else:
		tmp = 2.0 * math.sqrt((z * y))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1.56e-289)
		tmp = Float64(2.0 * sqrt(Float64(y * x)));
	else
		tmp = Float64(2.0 * sqrt(Float64(z * y)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.56e-289)
		tmp = 2.0 * sqrt((y * x));
	else
		tmp = 2.0 * sqrt((z * y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -1.56e-289], N[(2.0 * N[Sqrt[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.55999999999999989e-289

   1. Initial program 69.7%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Taylor expanded in z around 0

      \[\leadsto 2 \cdot \sqrt{\color{blue}{x \cdot y}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 2 \cdot \sqrt{y \cdot \color{blue}{x}} \]

      2. lower-*.f64 67.8

         \[\leadsto 2 \cdot \sqrt{y \cdot \color{blue}{x}} \]

   4. Applied rewrites 67.8%

      \[\leadsto 2 \cdot \sqrt{\color{blue}{y \cdot x}} \]

   if -1.55999999999999989e-289 < y

   1. Initial program 71.5%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Taylor expanded in x around 0

      \[\leadsto 2 \cdot \sqrt{\color{blue}{y \cdot z}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 2 \cdot \sqrt{z \cdot \color{blue}{y}} \]

      2. lower-*.f64 69.3

         \[\leadsto 2 \cdot \sqrt{z \cdot \color{blue}{y}} \]

   4. Applied rewrites 69.3%

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

Alternative 13: 35.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ 2 \cdot \sqrt{y \cdot x} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (* 2.0 (sqrt (* y x))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return 2.0 * sqrt((y * x));
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 2.0d0 * sqrt((y * x))
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	return 2.0 * Math.sqrt((y * x));
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return 2.0 * math.sqrt((y * x))

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(2.0 * sqrt(Float64(y * x)))
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = 2.0 * sqrt((y * x));
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(2.0 * N[Sqrt[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.6%

   \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

2. Taylor expanded in z around 0

   \[\leadsto 2 \cdot \sqrt{\color{blue}{x \cdot y}} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto 2 \cdot \sqrt{y \cdot \color{blue}{x}} \]

   2. lower-*.f64 35.6

      \[\leadsto 2 \cdot \sqrt{y \cdot \color{blue}{x}} \]

4. Applied rewrites 35.6%

   \[\leadsto 2 \cdot \sqrt{\color{blue}{y \cdot x}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025117 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:descartes from diagrams-contrib-1.3.0.5"
  :precision binary64
  (* 2.0 (sqrt (+ (+ (* x y) (* x z)) (* y z)))))