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

Percentage Accurate: 71.3% → 93.5%

Time: 5.4s

Alternatives: 10

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: 71.3% 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: 93.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+46}:\\ \;\;\;\;2 \cdot e^{\left(\left(-\log \left(\frac{-1}{x}\right)\right) + \log \left(-\left(z + y\right)\right)\right) \cdot 0.5}\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+17}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(y + x, z, y \cdot x\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, \sqrt{\frac{1}{\left(y + x\right) \cdot \left(\left(z \cdot z\right) \cdot z\right)}}, \sqrt{\frac{y + x}{z}} \cdot 2\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -9.5e+46)
   (* 2.0 (exp (* (+ (- (log (/ -1.0 x))) (log (- (+ z y)))) 0.5)))
   (if (<= y 4.7e+17)
     (* (sqrt (fma (+ y x) z (* y x))) 2.0)
     (*
      (fma
       (* y x)
       (sqrt (/ 1.0 (* (+ y x) (* (* z z) z))))
       (* (sqrt (/ (+ y x) z)) 2.0))
      z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -9.5e+46) {
		tmp = 2.0 * exp(((-log((-1.0 / x)) + log(-(z + y))) * 0.5));
	} else if (y <= 4.7e+17) {
		tmp = sqrt(fma((y + x), z, (y * x))) * 2.0;
	} else {
		tmp = fma((y * x), sqrt((1.0 / ((y + x) * ((z * z) * z)))), (sqrt(((y + x) / z)) * 2.0)) * z;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -9.5e+46)
		tmp = Float64(2.0 * exp(Float64(Float64(Float64(-log(Float64(-1.0 / x))) + log(Float64(-Float64(z + y)))) * 0.5)));
	elseif (y <= 4.7e+17)
		tmp = Float64(sqrt(fma(Float64(y + x), z, Float64(y * x))) * 2.0);
	else
		tmp = Float64(fma(Float64(y * x), sqrt(Float64(1.0 / Float64(Float64(y + x) * Float64(Float64(z * z) * z)))), Float64(sqrt(Float64(Float64(y + x) / 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, -9.5e+46], N[(2.0 * N[Exp[N[(N[((-N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]) + N[Log[(-N[(z + y), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.7e+17], N[(N[Sqrt[N[(N[(y + x), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(y * x), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(y + x), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(N[(y + x), $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.5000000000000008e46

   1. Initial program 34.3%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 32.2%

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

   4. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 89.7

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

   6. Applied rewrites 89.7%

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

   if -9.5000000000000008e46 < y < 4.7e17

   1. Initial program 94.5%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 94.5%

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

   if 4.7e17 < y

   1. Initial program 42.3%

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

   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 93.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot x, \sqrt{\frac{1}{\left(y + x\right) \cdot \left(\left(z \cdot z\right) \cdot z\right)}}, \sqrt{\frac{y + x}{z}} \cdot 2\right) \cdot z} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 91.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+66}:\\ \;\;\;\;2 \cdot e^{\left(\left(-\log \left(\frac{-1}{x}\right)\right) + \log \left(-y\right)\right) \cdot 0.5}\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-283}:\\ \;\;\;\;\sqrt{\left(\frac{z}{y} \cdot x + x\right) \cdot y} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot e^{\left(\left(-\left(-\log z\right)\right) + \log \left(x + y\right)\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2.3e+66)
   (* 2.0 (exp (* (+ (- (log (/ -1.0 x))) (log (- y))) 0.5)))
   (if (<= y -5.5e-283)
     (* (sqrt (* (+ (* (/ z y) x) x) y)) 2.0)
     (* 2.0 (exp (* (+ (- (- (log z))) (log (+ x y))) 0.5))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.3e+66) {
		tmp = 2.0 * exp(((-log((-1.0 / x)) + log(-y)) * 0.5));
	} else if (y <= -5.5e-283) {
		tmp = sqrt(((((z / y) * x) + x) * y)) * 2.0;
	} else {
		tmp = 2.0 * exp(((-(-log(z)) + log((x + y))) * 0.5));
	}
	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 <= (-2.3d+66)) then
        tmp = 2.0d0 * exp(((-log(((-1.0d0) / x)) + log(-y)) * 0.5d0))
    else if (y <= (-5.5d-283)) then
        tmp = sqrt(((((z / y) * x) + x) * y)) * 2.0d0
    else
        tmp = 2.0d0 * exp(((-(-log(z)) + log((x + y))) * 0.5d0))
    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 <= -2.3e+66) {
		tmp = 2.0 * Math.exp(((-Math.log((-1.0 / x)) + Math.log(-y)) * 0.5));
	} else if (y <= -5.5e-283) {
		tmp = Math.sqrt(((((z / y) * x) + x) * y)) * 2.0;
	} else {
		tmp = 2.0 * Math.exp(((-(-Math.log(z)) + Math.log((x + y))) * 0.5));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -2.3e+66:
		tmp = 2.0 * math.exp(((-math.log((-1.0 / x)) + math.log(-y)) * 0.5))
	elif y <= -5.5e-283:
		tmp = math.sqrt(((((z / y) * x) + x) * y)) * 2.0
	else:
		tmp = 2.0 * math.exp(((-(-math.log(z)) + math.log((x + y))) * 0.5))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -2.3e+66)
		tmp = Float64(2.0 * exp(Float64(Float64(Float64(-log(Float64(-1.0 / x))) + log(Float64(-y))) * 0.5)));
	elseif (y <= -5.5e-283)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(z / y) * x) + x) * y)) * 2.0);
	else
		tmp = Float64(2.0 * exp(Float64(Float64(Float64(-Float64(-log(z))) + log(Float64(x + y))) * 0.5)));
	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 <= -2.3e+66)
		tmp = 2.0 * exp(((-log((-1.0 / x)) + log(-y)) * 0.5));
	elseif (y <= -5.5e-283)
		tmp = sqrt(((((z / y) * x) + x) * y)) * 2.0;
	else
		tmp = 2.0 * exp(((-(-log(z)) + log((x + y))) * 0.5));
	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, -2.3e+66], N[(2.0 * N[Exp[N[(N[((-N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]) + N[Log[(-y)], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -5.5e-283], N[(N[Sqrt[N[(N[(N[(N[(z / y), $MachinePrecision] * x), $MachinePrecision] + x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(2.0 * N[Exp[N[(N[((-(-N[Log[z], $MachinePrecision])) + N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.3e66

   1. Initial program 28.5%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 26.9%

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

   4. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 89.6

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

   6. Applied rewrites 89.6%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 88.5%

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

   if -2.3e66 < y < -5.49999999999999953e-283

   1. Initial program 94.0%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 94.0%

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

   4. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 94.0

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

   6. Applied rewrites 94.0%

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

   7. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 93.9

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

   9. Applied rewrites 93.9%

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

   if -5.49999999999999953e-283 < y

   1. Initial program 70.6%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 65.8%

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

   4. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. log-rec N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-+.f64 90.2

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

   6. Applied rewrites 90.2%

      \[\leadsto 2 \cdot e^{\color{blue}{\left(\left(-\left(-\log z\right)\right) + \log \left(x + y\right)\right)} \cdot 0.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 90.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-301}:\\ \;\;\;\;2 \cdot e^{\left(\left(-\log \left(\frac{-1}{x}\right)\right) + \log \left(-\left(z + y\right)\right)\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot e^{\left(\left(-\left(-\log z\right)\right) + \log \left(x + y\right)\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 3.1e-301)
   (* 2.0 (exp (* (+ (- (log (/ -1.0 x))) (log (- (+ z y)))) 0.5)))
   (* 2.0 (exp (* (+ (- (- (log z))) (log (+ x y))) 0.5)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.1e-301) {
		tmp = 2.0 * exp(((-log((-1.0 / x)) + log(-(z + y))) * 0.5));
	} else {
		tmp = 2.0 * exp(((-(-log(z)) + log((x + y))) * 0.5));
	}
	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 <= 3.1d-301) then
        tmp = 2.0d0 * exp(((-log(((-1.0d0) / x)) + log(-(z + y))) * 0.5d0))
    else
        tmp = 2.0d0 * exp(((-(-log(z)) + log((x + y))) * 0.5d0))
    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 <= 3.1e-301) {
		tmp = 2.0 * Math.exp(((-Math.log((-1.0 / x)) + Math.log(-(z + y))) * 0.5));
	} else {
		tmp = 2.0 * Math.exp(((-(-Math.log(z)) + Math.log((x + y))) * 0.5));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 3.1e-301:
		tmp = 2.0 * math.exp(((-math.log((-1.0 / x)) + math.log(-(z + y))) * 0.5))
	else:
		tmp = 2.0 * math.exp(((-(-math.log(z)) + math.log((x + y))) * 0.5))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 3.1e-301)
		tmp = Float64(2.0 * exp(Float64(Float64(Float64(-log(Float64(-1.0 / x))) + log(Float64(-Float64(z + y)))) * 0.5)));
	else
		tmp = Float64(2.0 * exp(Float64(Float64(Float64(-Float64(-log(z))) + log(Float64(x + y))) * 0.5)));
	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 <= 3.1e-301)
		tmp = 2.0 * exp(((-log((-1.0 / x)) + log(-(z + y))) * 0.5));
	else
		tmp = 2.0 * exp(((-(-log(z)) + log((x + y))) * 0.5));
	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, 3.1e-301], N[(2.0 * N[Exp[N[(N[((-N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]) + N[Log[(-N[(z + y), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Exp[N[(N[((-(-N[Log[z], $MachinePrecision])) + N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.10000000000000014e-301

   1. Initial program 71.8%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 67.0%

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

   4. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 90.7

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

   6. Applied rewrites 90.7%

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

   if 3.10000000000000014e-301 < y

   1. Initial program 70.7%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 65.9%

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

   4. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. log-rec N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-+.f64 90.7

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

   6. Applied rewrites 90.7%

      \[\leadsto 2 \cdot e^{\color{blue}{\left(\left(-\left(-\log z\right)\right) + \log \left(x + y\right)\right)} \cdot 0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 82.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := 2 \cdot e^{\log \left(\sqrt{z} \cdot \sqrt{y}\right)}\\ t_1 := 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-157}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(y + x, z, y \cdot x\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 2.0 (exp (log (* (sqrt z) (sqrt y))))))
        (t_1 (* 2.0 (sqrt (+ (+ (* x y) (* x z)) (* y z))))))
   (if (<= t_1 5e-157)
     t_0
     (if (<= t_1 2e+153) (* (sqrt (fma (+ y x) z (* y x))) 2.0) t_0))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = 2.0 * exp(log((sqrt(z) * sqrt(y))));
	double t_1 = 2.0 * sqrt((((x * y) + (x * z)) + (y * z)));
	double tmp;
	if (t_1 <= 5e-157) {
		tmp = t_0;
	} else if (t_1 <= 2e+153) {
		tmp = sqrt(fma((y + x), z, (y * x))) * 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(2.0 * exp(log(Float64(sqrt(z) * sqrt(y)))))
	t_1 = Float64(2.0 * sqrt(Float64(Float64(Float64(x * y) + Float64(x * z)) + Float64(y * z))))
	tmp = 0.0
	if (t_1 <= 5e-157)
		tmp = t_0;
	elseif (t_1 <= 2e+153)
		tmp = Float64(sqrt(fma(Float64(y + x), z, Float64(y * x))) * 2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(2.0 * N[Exp[N[Log[N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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[t$95$1, 5e-157], t$95$0, If[LessEqual[t$95$1, 2e+153], N[(N[Sqrt[N[(N[(y + x), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (sqrt.f64 (+.f64 (+.f64 (*.f64 x y) (*.f64 x z)) (*.f64 y z)))) < 5.0000000000000002e-157 or 2e153 < (*.f64 #s(literal 2 binary64) (sqrt.f64 (+.f64 (+.f64 (*.f64 x y) (*.f64 x z)) (*.f64 y z))))

   1. Initial program 6.7%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 7.1%

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

   4. Taylor expanded in x around 0

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

      1. log-pow-rev N/A

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

      2. pow1/2 N/A

         \[\leadsto 2 \cdot e^{\log \left(\sqrt{y \cdot z}\right)} \]

      3. lower-log.f64 N/A

         \[\leadsto 2 \cdot e^{\log \left(\sqrt{y \cdot z}\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto 2 \cdot e^{\log \left(\sqrt{y \cdot z}\right)} \]

      5. *-commutative N/A

         \[\leadsto 2 \cdot e^{\log \left(\sqrt{z \cdot y}\right)} \]

      6. lower-*.f64 5.0

         \[\leadsto 2 \cdot e^{\log \left(\sqrt{z \cdot y}\right)} \]

   6. Applied rewrites 5.0%

      \[\leadsto 2 \cdot e^{\color{blue}{\log \left(\sqrt{z \cdot y}\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot e^{\log \left(\sqrt{z \cdot y}\right)} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto 2 \cdot e^{\log \left(\sqrt{z \cdot y}\right)} \]

      3. sqrt-prod N/A

         \[\leadsto 2 \cdot e^{\log \left(\sqrt{z} \cdot \sqrt{y}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot e^{\log \left(\sqrt{z} \cdot \sqrt{y}\right)} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto 2 \cdot e^{\log \left(\sqrt{z} \cdot \sqrt{y}\right)} \]

      6. lower-sqrt.f64 44.7

         \[\leadsto 2 \cdot e^{\log \left(\sqrt{z} \cdot \sqrt{y}\right)} \]

   8. Applied rewrites 44.7%

      \[\leadsto 2 \cdot e^{\log \left(\sqrt{z} \cdot \sqrt{y}\right)} \]

   if 5.0000000000000002e-157 < (*.f64 #s(literal 2 binary64) (sqrt.f64 (+.f64 (+.f64 (*.f64 x y) (*.f64 x z)) (*.f64 y z)))) < 2e153

   1. Initial program 99.8%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 99.8%

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

Alternative 5: 81.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-283}:\\ \;\;\;\;\sqrt{\left(\frac{z}{y} \cdot x + x\right) \cdot y} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot e^{\left(\left(-\left(-\log z\right)\right) + \log \left(x + y\right)\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -5.5e-283)
   (* (sqrt (* (+ (* (/ z y) x) x) y)) 2.0)
   (* 2.0 (exp (* (+ (- (- (log z))) (log (+ x y))) 0.5)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.5e-283) {
		tmp = sqrt(((((z / y) * x) + x) * y)) * 2.0;
	} else {
		tmp = 2.0 * exp(((-(-log(z)) + log((x + y))) * 0.5));
	}
	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 <= (-5.5d-283)) then
        tmp = sqrt(((((z / y) * x) + x) * y)) * 2.0d0
    else
        tmp = 2.0d0 * exp(((-(-log(z)) + log((x + y))) * 0.5d0))
    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 <= -5.5e-283) {
		tmp = Math.sqrt(((((z / y) * x) + x) * y)) * 2.0;
	} else {
		tmp = 2.0 * Math.exp(((-(-Math.log(z)) + Math.log((x + y))) * 0.5));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -5.5e-283:
		tmp = math.sqrt(((((z / y) * x) + x) * y)) * 2.0
	else:
		tmp = 2.0 * math.exp(((-(-math.log(z)) + math.log((x + y))) * 0.5))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -5.5e-283)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(z / y) * x) + x) * y)) * 2.0);
	else
		tmp = Float64(2.0 * exp(Float64(Float64(Float64(-Float64(-log(z))) + log(Float64(x + y))) * 0.5)));
	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 <= -5.5e-283)
		tmp = sqrt(((((z / y) * x) + x) * y)) * 2.0;
	else
		tmp = 2.0 * exp(((-(-log(z)) + log((x + y))) * 0.5));
	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, -5.5e-283], N[(N[Sqrt[N[(N[(N[(N[(z / y), $MachinePrecision] * x), $MachinePrecision] + x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(2.0 * N[Exp[N[(N[((-(-N[Log[z], $MachinePrecision])) + N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.49999999999999953e-283

   1. Initial program 72.0%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 72.2%

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

   4. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 72.2

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

   6. Applied rewrites 72.2%

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

   7. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 72.1

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

   9. Applied rewrites 72.1%

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

   if -5.49999999999999953e-283 < y

   1. Initial program 70.6%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 65.8%

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

   4. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. log-rec N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-+.f64 90.2

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

   6. Applied rewrites 90.2%

      \[\leadsto 2 \cdot e^{\color{blue}{\left(\left(-\left(-\log z\right)\right) + \log \left(x + y\right)\right)} \cdot 0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 71.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-301}:\\ \;\;\;\;2 \cdot \sqrt{\left(z + y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{\left(y + x\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 6.5e-301)
   (* 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 <= 6.5e-301) {
		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 <= 6.5d-301) 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 <= 6.5e-301) {
		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 <= 6.5e-301:
		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 <= 6.5e-301)
		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 <= 6.5e-301)
		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, 6.5e-301], 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 < 6.49999999999999991e-301

   1. Initial program 71.8%

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

   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 72.0

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

   4. Applied rewrites 72.0%

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

   if 6.49999999999999991e-301 < y

   1. Initial program 70.7%

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

   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 70.8

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

   4. Applied rewrites 70.8%

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

Alternative 7: 71.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.3%

   \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
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 71.4%

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

Alternative 8: 70.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-268}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{\left(y + x\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -4.5e-268) (* 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 <= -4.5e-268) {
		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 <= (-4.5d-268)) 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 <= -4.5e-268) {
		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 <= -4.5e-268:
		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 <= -4.5e-268)
		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 <= -4.5e-268)
		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, -4.5e-268], 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 < -4.5000000000000001e-268

   1. Initial program 72.0%

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

   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 70.0

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

   4. Applied rewrites 70.0%

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

   if -4.5000000000000001e-268 < y

   1. Initial program 70.5%

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

   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 70.1

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

   4. Applied rewrites 70.1%

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

Alternative 9: 69.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{-268}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -4.6e-268) (* 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 <= -4.6e-268) {
		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 <= (-4.6d-268)) 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 <= -4.6e-268) {
		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 <= -4.6e-268:
		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 <= -4.6e-268)
		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 <= -4.6e-268)
		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, -4.6e-268], 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 < -4.60000000000000021e-268

   1. Initial program 72.0%

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

   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 70.0

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

   4. Applied rewrites 70.0%

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

   if -4.60000000000000021e-268 < y

   1. Initial program 70.5%

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

   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 67.9

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

   4. Applied rewrites 67.9%

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

Alternative 10: 36.5% 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 71.3%

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

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 36.5

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

4. Applied rewrites 36.5%

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

Reproduce

herbie shell --seed 2025111 
(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)))))