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

Percentage Accurate: 71.1% → 92.9%

Time: 6.6s

Alternatives: 11

Speedup: 1.2×

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: 71.1% 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: 92.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.06 \cdot 10^{+27}:\\ \;\;\;\;2 \cdot e^{\mathsf{fma}\left(\log \left(\frac{-1}{x}\right), -1, \log \left(-\left(z + y\right)\right)\right) \cdot 0.5}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+45}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{{z}^{-3}}{y + x}}, y \cdot x, \sqrt{\frac{y + x}{z}} \cdot 2\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.06e+27)
   (* 2.0 (exp (* (fma (log (/ -1.0 x)) -1.0 (log (- (+ z y)))) 0.5)))
   (if (<= y 3.2e+45)
     (* 2.0 (sqrt (fma y x (* (+ y x) z))))
     (*
      (fma
       (sqrt (/ (pow z -3.0) (+ y x)))
       (* y x)
       (* (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 <= -1.06e+27) {
		tmp = 2.0 * exp((fma(log((-1.0 / x)), -1.0, log(-(z + y))) * 0.5));
	} else if (y <= 3.2e+45) {
		tmp = 2.0 * sqrt(fma(y, x, ((y + x) * z)));
	} else {
		tmp = fma(sqrt((pow(z, -3.0) / (y + x))), (y * x), (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 <= -1.06e+27)
		tmp = Float64(2.0 * exp(Float64(fma(log(Float64(-1.0 / x)), -1.0, log(Float64(-Float64(z + y)))) * 0.5)));
	elseif (y <= 3.2e+45)
		tmp = Float64(2.0 * sqrt(fma(y, x, Float64(Float64(y + x) * z))));
	else
		tmp = Float64(fma(sqrt(Float64((z ^ -3.0) / Float64(y + x))), Float64(y * x), 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, -1.06e+27], N[(2.0 * N[Exp[N[(N[(N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision] * -1.0 + N[Log[(-N[(z + y), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.2e+45], N[(2.0 * N[Sqrt[N[(y * x + N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[Power[z, -3.0], $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(y * x), $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 < -1.05999999999999994e27

   1. Initial program 50.3%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   2. Add Preprocessing
   3. 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

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

      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

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

      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

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

   4. Applied rewrites 47.0%

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

   5. Taylor expanded in x around -inf

      \[\leadsto 2 \cdot e^{\color{blue}{\left(\log \left(-1 \cdot y + -1 \cdot z\right) + -1 \cdot \log \left(\frac{-1}{x}\right)\right)} \cdot \frac{1}{2}} \]
   6. 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 45.2

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

   7. Applied rewrites 45.2%

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

   if -1.05999999999999994e27 < y < 3.2000000000000003e45

   1. Initial program 83.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+l+ N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 83.2

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

   4. Applied rewrites 83.2%

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

   if 3.2000000000000003e45 < y

   1. Initial program 65.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(2 \cdot \sqrt{\frac{x + y}{z}} + \left(x \cdot y\right) \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right)} \]
   4. 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

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

   5. Applied rewrites 34.6%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.06 \cdot 10^{+27}:\\ \;\;\;\;2 \cdot e^{\mathsf{fma}\left(\log \left(\frac{-1}{x}\right), -1, \log \left(-\left(z + y\right)\right)\right) \cdot 0.5}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+45}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{{z}^{-3}}{y + x}}, y \cdot x, \sqrt{\frac{y + x}{z}} \cdot 2\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.18 \cdot 10^{+27}:\\ \;\;\;\;2 \cdot e^{\mathsf{fma}\left(\log \left(\frac{-1}{y}\right), -1, \log \left(-x\right)\right) \cdot 0.5}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+45}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{{z}^{-3}}{y + x}}, y \cdot x, \sqrt{\frac{y + x}{z}} \cdot 2\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.18e+27)
   (* 2.0 (exp (* (fma (log (/ -1.0 y)) -1.0 (log (- x))) 0.5)))
   (if (<= y 3.2e+45)
     (* 2.0 (sqrt (fma y x (* (+ y x) z))))
     (*
      (fma
       (sqrt (/ (pow z -3.0) (+ y x)))
       (* y x)
       (* (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 <= -1.18e+27) {
		tmp = 2.0 * exp((fma(log((-1.0 / y)), -1.0, log(-x)) * 0.5));
	} else if (y <= 3.2e+45) {
		tmp = 2.0 * sqrt(fma(y, x, ((y + x) * z)));
	} else {
		tmp = fma(sqrt((pow(z, -3.0) / (y + x))), (y * x), (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 <= -1.18e+27)
		tmp = Float64(2.0 * exp(Float64(fma(log(Float64(-1.0 / y)), -1.0, log(Float64(-x))) * 0.5)));
	elseif (y <= 3.2e+45)
		tmp = Float64(2.0 * sqrt(fma(y, x, Float64(Float64(y + x) * z))));
	else
		tmp = Float64(fma(sqrt(Float64((z ^ -3.0) / Float64(y + x))), Float64(y * x), 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, -1.18e+27], N[(2.0 * N[Exp[N[(N[(N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision] * -1.0 + N[Log[(-x)], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.2e+45], N[(2.0 * N[Sqrt[N[(y * x + N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[Power[z, -3.0], $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(y * x), $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 < -1.18000000000000006e27

   1. Initial program 50.3%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   2. Add Preprocessing
   3. 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

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

      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

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

      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

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

   4. Applied rewrites 47.0%

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

   5. Taylor expanded in z around 0

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

      1. lower-log.f64 N/A

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

      2. lower-*.f64 24.1

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

   7. Applied rewrites 24.1%

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

   8. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 43.2

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

   10. Applied rewrites 43.2%

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

   if -1.18000000000000006e27 < y < 3.2000000000000003e45

   1. Initial program 83.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+l+ N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 83.2

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

   4. Applied rewrites 83.2%

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

   if 3.2000000000000003e45 < y

   1. Initial program 65.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(2 \cdot \sqrt{\frac{x + y}{z}} + \left(x \cdot y\right) \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right)} \]
   4. 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

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

   5. Applied rewrites 34.6%

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

Alternative 3: 82.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(y + x, z, y \cdot x\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{{z}^{-3}}{y + x}}, y \cdot x, \sqrt{\frac{y + x}{z}} \cdot 2\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 3.2e+45)
   (* (sqrt (fma (+ y x) z (* y x))) 2.0)
   (*
    (fma (sqrt (/ (pow z -3.0) (+ y x))) (* y x) (* (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 <= 3.2e+45) {
		tmp = sqrt(fma((y + x), z, (y * x))) * 2.0;
	} else {
		tmp = fma(sqrt((pow(z, -3.0) / (y + x))), (y * x), (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 <= 3.2e+45)
		tmp = Float64(sqrt(fma(Float64(y + x), z, Float64(y * x))) * 2.0);
	else
		tmp = Float64(fma(sqrt(Float64((z ^ -3.0) / Float64(y + x))), Float64(y * x), 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, 3.2e+45], N[(N[Sqrt[N[(N[(y + x), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[Sqrt[N[(N[Power[z, -3.0], $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(y * x), $MachinePrecision] + N[(N[Sqrt[N[(N[(y + x), $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.2000000000000003e45

   1. Initial program 75.3%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   2. Add Preprocessing
   3. 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

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

      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

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

      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

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

   4. Applied rewrites 75.4%

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

   if 3.2000000000000003e45 < y

   1. Initial program 65.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(2 \cdot \sqrt{\frac{x + y}{z}} + \left(x \cdot y\right) \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right)} \]
   4. 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

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

   5. Applied rewrites 34.6%

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

Alternative 4: 80.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.36 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(y + x, z, y \cdot x\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{{y}^{-3}}{z + x}}, z \cdot x, \sqrt{\frac{z + x}{y}} \cdot 2\right) \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.36e+45)
   (* (sqrt (fma (+ y x) z (* y x))) 2.0)
   (*
    (fma (sqrt (/ (pow y -3.0) (+ z x))) (* z x) (* (sqrt (/ (+ z x) y)) 2.0))
    y)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.36e+45) {
		tmp = sqrt(fma((y + x), z, (y * x))) * 2.0;
	} else {
		tmp = fma(sqrt((pow(y, -3.0) / (z + x))), (z * x), (sqrt(((z + x) / y)) * 2.0)) * y;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1.36e+45)
		tmp = Float64(sqrt(fma(Float64(y + x), z, Float64(y * x))) * 2.0);
	else
		tmp = Float64(fma(sqrt(Float64((y ^ -3.0) / Float64(z + x))), Float64(z * x), Float64(sqrt(Float64(Float64(z + x) / y)) * 2.0)) * y);
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 1.36e+45], N[(N[Sqrt[N[(N[(y + x), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[Sqrt[N[(N[Power[y, -3.0], $MachinePrecision] / N[(z + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(z * x), $MachinePrecision] + N[(N[Sqrt[N[(N[(z + x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.36e45

   1. Initial program 75.3%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   2. Add Preprocessing
   3. 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

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

      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

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

      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

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

   4. Applied rewrites 75.4%

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

   if 1.36e45 < y

   1. Initial program 65.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 75.9%

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

Alternative 5: 71.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, z, y \cdot x\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{\left(y + x\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-288)
   (* (sqrt (fma x z (* y x))) 2.0)
   (* 2.0 (sqrt (* (+ y x) z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-288) {
		tmp = sqrt(fma(x, z, (y * x))) * 2.0;
	} else {
		tmp = 2.0 * sqrt(((y + x) * z));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -5e-288)
		tmp = Float64(sqrt(fma(x, z, Float64(y * x))) * 2.0);
	else
		tmp = Float64(2.0 * sqrt(Float64(Float64(y + x) * 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, -5e-288], N[(N[Sqrt[N[(x * z + N[(y * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $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 < -5.00000000000000011e-288

   1. Initial program 68.3%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   2. Add Preprocessing
   3. 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

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

      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

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

      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

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

   4. Applied rewrites 68.5%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 41.4%

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

   if -5.00000000000000011e-288 < y

   1. Initial program 77.4%

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

   3. Taylor expanded in z around inf

      \[\leadsto 2 \cdot \sqrt{\color{blue}{z \cdot \left(x + y\right)}} \]
   4. 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 53.4

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

   5. Applied rewrites 53.4%

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

Alternative 6: 71.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-288}:\\ \;\;\;\;2 \cdot \sqrt{\left(z + y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{\left(y + x\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-288) (* 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 <= -5e-288) {
		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 <= (-5d-288)) 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 <= -5e-288) {
		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 <= -5e-288:
		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 <= -5e-288)
		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 <= -5e-288)
		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, -5e-288], 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 < -5.00000000000000011e-288

   1. Initial program 68.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)}} \]
   4. 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 41.4

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

   5. Applied rewrites 41.4%

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

   if -5.00000000000000011e-288 < y

   1. Initial program 77.4%

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

   3. Taylor expanded in z around inf

      \[\leadsto 2 \cdot \sqrt{\color{blue}{z \cdot \left(x + y\right)}} \]
   4. 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 53.4

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

   5. Applied rewrites 53.4%

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

Alternative 7: 69.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-266}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{\left(y + x\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.1e-266) (* 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 <= -1.1e-266) {
		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 <= (-1.1d-266)) 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 <= -1.1e-266) {
		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 <= -1.1e-266:
		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 <= -1.1e-266)
		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 <= -1.1e-266)
		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, -1.1e-266], 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 < -1.1e-266

   1. Initial program 68.8%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 24.4

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

   5. Applied rewrites 24.4%

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

   if -1.1e-266 < y

   1. Initial program 76.6%

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

   3. Taylor expanded in z around inf

      \[\leadsto 2 \cdot \sqrt{\color{blue}{z \cdot \left(x + y\right)}} \]
   4. 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 53.4

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

   5. Applied rewrites 53.4%

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

Alternative 8: 71.2% 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 73.0%

   \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
2. Add Preprocessing
3. 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

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

   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

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

   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

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

4. Applied rewrites 73.1%

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

Alternative 9: 71.2% accurate, 1.2× speedup

Math

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

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 (fma y x (* (+ y x) z)))))

C

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

Julia

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

Derivation

1. Initial program 73.0%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-+l+ N/A

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

   7. distribute-rgt-in N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 73.1

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

4. Applied rewrites 73.1%

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

Alternative 10: 69.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-295}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2.1e-295) (* 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 <= -2.1e-295) {
		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 <= (-2.1d-295)) 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 <= -2.1e-295) {
		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 <= -2.1e-295:
		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 <= -2.1e-295)
		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 <= -2.1e-295)
		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, -2.1e-295], 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 < -2.09999999999999993e-295

   1. Initial program 68.6%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 23.4

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

   5. Applied rewrites 23.4%

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

   if -2.09999999999999993e-295 < y

   1. Initial program 77.2%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 29.6

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

   5. Applied rewrites 29.6%

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

Alternative 11: 36.4% accurate, 1.8× 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 73.0%

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

3. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 25.3

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

5. Applied rewrites 25.3%

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

Developer Target 1: 83.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.25 \cdot \left(\left({y}^{-0.75} \cdot \left({z}^{-0.75} \cdot x\right)\right) \cdot \left(y + z\right)\right) + {z}^{0.25} \cdot {y}^{0.25}\\ \mathbf{if}\;z < 7.636950090573675 \cdot 10^{+176}:\\ \;\;\;\;2 \cdot \sqrt{\left(x + y\right) \cdot z + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot t\_0\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* 0.25 (* (* (pow y -0.75) (* (pow z -0.75) x)) (+ y z)))
          (* (pow z 0.25) (pow y 0.25)))))
   (if (< z 7.636950090573675e+176)
     (* 2.0 (sqrt (+ (* (+ x y) z) (* x y))))
     (* (* t_0 t_0) 2.0))))

C

double code(double x, double y, double z) {
	double t_0 = (0.25 * ((pow(y, -0.75) * (pow(z, -0.75) * x)) * (y + z))) + (pow(z, 0.25) * pow(y, 0.25));
	double tmp;
	if (z < 7.636950090573675e+176) {
		tmp = 2.0 * sqrt((((x + y) * z) + (x * y)));
	} else {
		tmp = (t_0 * t_0) * 2.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.25d0 * (((y ** (-0.75d0)) * ((z ** (-0.75d0)) * x)) * (y + z))) + ((z ** 0.25d0) * (y ** 0.25d0))
    if (z < 7.636950090573675d+176) then
        tmp = 2.0d0 * sqrt((((x + y) * z) + (x * y)))
    else
        tmp = (t_0 * t_0) * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (0.25 * ((Math.pow(y, -0.75) * (Math.pow(z, -0.75) * x)) * (y + z))) + (Math.pow(z, 0.25) * Math.pow(y, 0.25));
	double tmp;
	if (z < 7.636950090573675e+176) {
		tmp = 2.0 * Math.sqrt((((x + y) * z) + (x * y)));
	} else {
		tmp = (t_0 * t_0) * 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (0.25 * ((math.pow(y, -0.75) * (math.pow(z, -0.75) * x)) * (y + z))) + (math.pow(z, 0.25) * math.pow(y, 0.25))
	tmp = 0
	if z < 7.636950090573675e+176:
		tmp = 2.0 * math.sqrt((((x + y) * z) + (x * y)))
	else:
		tmp = (t_0 * t_0) * 2.0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(0.25 * Float64(Float64((y ^ -0.75) * Float64((z ^ -0.75) * x)) * Float64(y + z))) + Float64((z ^ 0.25) * (y ^ 0.25)))
	tmp = 0.0
	if (z < 7.636950090573675e+176)
		tmp = Float64(2.0 * sqrt(Float64(Float64(Float64(x + y) * z) + Float64(x * y))));
	else
		tmp = Float64(Float64(t_0 * t_0) * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (0.25 * (((y ^ -0.75) * ((z ^ -0.75) * x)) * (y + z))) + ((z ^ 0.25) * (y ^ 0.25));
	tmp = 0.0;
	if (z < 7.636950090573675e+176)
		tmp = 2.0 * sqrt((((x + y) * z) + (x * y)));
	else
		tmp = (t_0 * t_0) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.25 * N[(N[(N[Power[y, -0.75], $MachinePrecision] * N[(N[Power[z, -0.75], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[z, 0.25], $MachinePrecision] * N[Power[y, 0.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[z, 7.636950090573675e+176], N[(2.0 * N[Sqrt[N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 2.0), $MachinePrecision]]]

Reproduce

herbie shell --seed 2025051 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:descartes from diagrams-contrib-1.3.0.5"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z 763695009057367500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* 2 (sqrt (+ (* (+ x y) z) (* x y)))) (* (* (+ (* 1/4 (* (* (pow y -3/4) (* (pow z -3/4) x)) (+ y z))) (* (pow z 1/4) (pow y 1/4))) (+ (* 1/4 (* (* (pow y -3/4) (* (pow z -3/4) x)) (+ y z))) (* (pow z 1/4) (pow y 1/4)))) 2)))

  (* 2.0 (sqrt (+ (+ (* x y) (* x z)) (* y z)))))