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

Percentage Accurate: 70.0% → 94.7%

Time: 9.5s

Alternatives: 10

Speedup: 1.1×

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: 70.0% 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: 94.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+45}:\\ \;\;\;\;\left(-x\right) \cdot \mathsf{fma}\left(\sqrt{\frac{z + y}{x}}, 2, \left(z \cdot y\right) \cdot \sqrt{-1 \cdot \frac{-1}{\left(x \cdot x\right) \cdot \left(\left(z + y\right) \cdot x\right)}}\right)\\ \mathbf{elif}\;y \leq 1100000000000:\\ \;\;\;\;2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{y}{z}} \cdot 2\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.25e+45)
   (*
    (- x)
    (fma
     (sqrt (/ (+ z y) x))
     2.0
     (* (* z y) (sqrt (* -1.0 (/ -1.0 (* (* x x) (* (+ z y) x))))))))
   (if (<= y 1100000000000.0)
     (* 2.0 (sqrt (+ (+ (* x y) (* x z)) (* y z))))
     (* (* (sqrt (/ y z)) 2.0) z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.25e+45) {
		tmp = -x * fma(sqrt(((z + y) / x)), 2.0, ((z * y) * sqrt((-1.0 * (-1.0 / ((x * x) * ((z + y) * x)))))));
	} else if (y <= 1100000000000.0) {
		tmp = 2.0 * sqrt((((x * y) + (x * z)) + (y * z)));
	} else {
		tmp = (sqrt((y / z)) * 2.0) * z;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1.25e+45)
		tmp = Float64(Float64(-x) * fma(sqrt(Float64(Float64(z + y) / x)), 2.0, Float64(Float64(z * y) * sqrt(Float64(-1.0 * Float64(-1.0 / Float64(Float64(x * x) * Float64(Float64(z + y) * x))))))));
	elseif (y <= 1100000000000.0)
		tmp = Float64(2.0 * sqrt(Float64(Float64(Float64(x * y) + Float64(x * z)) + Float64(y * z))));
	else
		tmp = Float64(Float64(sqrt(Float64(y / z)) * 2.0) * 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, -1.25e+45], N[((-x) * N[(N[Sqrt[N[(N[(z + y), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] * 2.0 + N[(N[(z * y), $MachinePrecision] * N[Sqrt[N[(-1.0 * N[(-1.0 / N[(N[(x * x), $MachinePrecision] * N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1100000000000.0], N[(2.0 * N[Sqrt[N[(N[(N[(x * y), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(y / z), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.25e45

   1. Initial program 33.0%

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

   2. Taylor expanded in x around -inf

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

   4. Applied rewrites 92.1%

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

   if -1.25e45 < y < 1.1e12

   1. Initial program 94.0%

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

   if 1.1e12 < y

   1. Initial program 42.1%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 92.4%

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

   5. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 98.3

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

   7. Applied rewrites 98.3%

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

Alternative 2: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1100000000000:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(y + x, z, y \cdot x\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{y}{z}} \cdot 2\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1100000000000.0)
   (* (sqrt (fma (+ y x) z (* y x))) 2.0)
   (* (* (sqrt (/ y z)) 2.0) z)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1100000000000.0) {
		tmp = sqrt(fma((y + x), z, (y * x))) * 2.0;
	} else {
		tmp = (sqrt((y / z)) * 2.0) * z;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1100000000000.0)
		tmp = Float64(sqrt(fma(Float64(y + x), z, Float64(y * x))) * 2.0);
	else
		tmp = Float64(Float64(sqrt(Float64(y / z)) * 2.0) * 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, 1100000000000.0], N[(N[Sqrt[N[(N[(y + x), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[Sqrt[N[(y / z), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.1e12

   1. Initial program 78.6%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 78.8%

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

   if 1.1e12 < y

   1. Initial program 42.1%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 92.4%

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

   5. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 98.3

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

   7. Applied rewrites 98.3%

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

Alternative 3: 83.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-273}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, z, y \cdot x\right)} \cdot 2\\ \mathbf{elif}\;y \leq 1100000000000:\\ \;\;\;\;2 \cdot \sqrt{\left(y + x\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{y}{z}} \cdot 2\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.1e-273)
   (* (sqrt (fma x z (* y x))) 2.0)
   (if (<= y 1100000000000.0)
     (* 2.0 (sqrt (* (+ y x) z)))
     (* (* (sqrt (/ y z)) 2.0) z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.1e-273) {
		tmp = sqrt(fma(x, z, (y * x))) * 2.0;
	} else if (y <= 1100000000000.0) {
		tmp = 2.0 * sqrt(((y + x) * z));
	} else {
		tmp = (sqrt((y / z)) * 2.0) * z;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1.1e-273)
		tmp = Float64(sqrt(fma(x, z, Float64(y * x))) * 2.0);
	elseif (y <= 1100000000000.0)
		tmp = Float64(2.0 * sqrt(Float64(Float64(y + x) * z)));
	else
		tmp = Float64(Float64(sqrt(Float64(y / z)) * 2.0) * 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, -1.1e-273], N[(N[Sqrt[N[(x * z + N[(y * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[y, 1100000000000.0], N[(2.0 * N[Sqrt[N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(y / z), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.0999999999999999e-273

   1. Initial program 70.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 70.5%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 70.5%

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

   if -1.0999999999999999e-273 < y < 1.1e12

   1. Initial program 93.5%

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

   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. *-lft-identity N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-lft-out N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. distribute-lft-out N/A

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

      10. distribute-lft-neg-in N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 92.8

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

   4. Applied rewrites 92.8%

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

   if 1.1e12 < y

   1. Initial program 42.1%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 92.4%

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

   5. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 98.3

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

   7. Applied rewrites 98.3%

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

Alternative 4: 83.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-273}:\\ \;\;\;\;2 \cdot \sqrt{\left(z + y\right) \cdot x}\\ \mathbf{elif}\;y \leq 1100000000000:\\ \;\;\;\;2 \cdot \sqrt{\left(y + x\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{y}{z}} \cdot 2\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.1e-273)
   (* 2.0 (sqrt (* (+ z y) x)))
   (if (<= y 1100000000000.0)
     (* 2.0 (sqrt (* (+ y x) z)))
     (* (* (sqrt (/ y z)) 2.0) z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.1e-273) {
		tmp = 2.0 * sqrt(((z + y) * x));
	} else if (y <= 1100000000000.0) {
		tmp = 2.0 * sqrt(((y + x) * z));
	} else {
		tmp = (sqrt((y / z)) * 2.0) * 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 <= (-1.1d-273)) then
        tmp = 2.0d0 * sqrt(((z + y) * x))
    else if (y <= 1100000000000.0d0) then
        tmp = 2.0d0 * sqrt(((y + x) * z))
    else
        tmp = (sqrt((y / z)) * 2.0d0) * 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 <= -1.1e-273) {
		tmp = 2.0 * Math.sqrt(((z + y) * x));
	} else if (y <= 1100000000000.0) {
		tmp = 2.0 * Math.sqrt(((y + x) * z));
	} else {
		tmp = (Math.sqrt((y / z)) * 2.0) * z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -1.1e-273:
		tmp = 2.0 * math.sqrt(((z + y) * x))
	elif y <= 1100000000000.0:
		tmp = 2.0 * math.sqrt(((y + x) * z))
	else:
		tmp = (math.sqrt((y / z)) * 2.0) * z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1.1e-273)
		tmp = Float64(2.0 * sqrt(Float64(Float64(z + y) * x)));
	elseif (y <= 1100000000000.0)
		tmp = Float64(2.0 * sqrt(Float64(Float64(y + x) * z)));
	else
		tmp = Float64(Float64(sqrt(Float64(y / z)) * 2.0) * 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 <= -1.1e-273)
		tmp = 2.0 * sqrt(((z + y) * x));
	elseif (y <= 1100000000000.0)
		tmp = 2.0 * sqrt(((y + x) * z));
	else
		tmp = (sqrt((y / z)) * 2.0) * 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, -1.1e-273], N[(2.0 * N[Sqrt[N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1100000000000.0], N[(2.0 * N[Sqrt[N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(y / z), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.0999999999999999e-273

   1. Initial program 70.3%

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

   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 70.5

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

   4. Applied rewrites 70.5%

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

   if -1.0999999999999999e-273 < y < 1.1e12

   1. Initial program 93.5%

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

   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. *-lft-identity N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-lft-out N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. distribute-lft-out N/A

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

      10. distribute-lft-neg-in N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 92.8

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

   4. Applied rewrites 92.8%

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

   if 1.1e12 < y

   1. Initial program 42.1%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 92.4%

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

   5. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 98.3

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

   7. Applied rewrites 98.3%

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

Alternative 5: 83.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 8.8 \cdot 10^{-298}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, z, y \cdot x\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{y}}{\sqrt{z}} \cdot 2\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 8.8e-298)
   (* (sqrt (fma x z (* y x))) 2.0)
   (* (* (/ (sqrt y) (sqrt z)) 2.0) z)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.8e-298) {
		tmp = sqrt(fma(x, z, (y * x))) * 2.0;
	} else {
		tmp = ((sqrt(y) / sqrt(z)) * 2.0) * z;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 8.8e-298)
		tmp = Float64(sqrt(fma(x, z, Float64(y * x))) * 2.0);
	else
		tmp = Float64(Float64(Float64(sqrt(y) / sqrt(z)) * 2.0) * 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, 8.8e-298], N[(N[Sqrt[N[(x * z + N[(y * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[Sqrt[y], $MachinePrecision] / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.799999999999999e-298

   1. Initial program 70.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 70.3%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 70.2%

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

   if 8.799999999999999e-298 < y

   1. Initial program 70.0%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 78.2%

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

   5. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 87.5

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

   7. Applied rewrites 87.5%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 96.8

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

   9. Applied rewrites 96.8%

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

Alternative 6: 70.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-273}:\\ \;\;\;\;2 \cdot \sqrt{\left(z + y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{\left(y + x\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.1e-273)
   (* 2.0 (sqrt (* (+ z 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 <= -1.1e-273) {
		tmp = 2.0 * sqrt(((z + 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 <= (-1.1d-273)) then
        tmp = 2.0d0 * sqrt(((z + 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 <= -1.1e-273) {
		tmp = 2.0 * Math.sqrt(((z + 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 <= -1.1e-273:
		tmp = 2.0 * math.sqrt(((z + 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 <= -1.1e-273)
		tmp = Float64(2.0 * sqrt(Float64(Float64(z + 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 <= -1.1e-273)
		tmp = 2.0 * sqrt(((z + 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, -1.1e-273], N[(2.0 * N[Sqrt[N[(N[(z + y), $MachinePrecision] * 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 < -1.0999999999999999e-273

   1. Initial program 70.3%

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

   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 70.5

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

   4. Applied rewrites 70.5%

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

   if -1.0999999999999999e-273 < y

   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 inf

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

      1. *-lft-identity N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-lft-out N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. distribute-lft-out N/A

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

      10. distribute-lft-neg-in N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 69.6

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

   4. Applied rewrites 69.6%

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

Alternative 7: 69.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{-272}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{\left(y + x\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2.9e-272) (* 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 <= -2.9e-272) {
		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 <= (-2.9d-272)) 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 <= -2.9e-272) {
		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 <= -2.9e-272:
		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 <= -2.9e-272)
		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 <= -2.9e-272)
		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, -2.9e-272], 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 < -2.89999999999999995e-272

   1. Initial program 70.3%

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

   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 68.5

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

   4. Applied rewrites 68.5%

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

   if -2.89999999999999995e-272 < y

   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 inf

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

      1. *-lft-identity N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-lft-out N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. distribute-lft-out N/A

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

      10. distribute-lft-neg-in N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 69.6

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

   4. Applied rewrites 69.6%

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

Alternative 8: 68.2% 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 70.0%

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

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 68.2

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

4. Applied rewrites 68.2%

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

Alternative 9: 68.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-273}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.1e-273) (* 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 <= -1.1e-273) {
		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 <= (-1.1d-273)) 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 <= -1.1e-273) {
		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 <= -1.1e-273:
		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 <= -1.1e-273)
		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 <= -1.1e-273)
		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, -1.1e-273], 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 < -1.0999999999999999e-273

   1. Initial program 70.3%

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

   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 68.5

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

   4. Applied rewrites 68.5%

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

   if -1.0999999999999999e-273 < y

   1. Initial program 69.8%

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

   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 67.6

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

   4. Applied rewrites 67.6%

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

Alternative 10: 35.1% 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 70.0%

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

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 35.1

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

4. Applied rewrites 35.1%

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

Reproduce

herbie shell --seed 2025130 
(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)))))