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

Percentage Accurate: 70.5% → 96.8%

Time: 4.6s

Alternatives: 15

Speedup: 0.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 70.5% accurate, 1.0× speedup

Math

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

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: 96.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ \mathbf{if}\;t\_2 \leq -1.5 \cdot 10^{+14}:\\ \;\;\;\;-2 \cdot \left(t\_0 \cdot \sqrt{-1 \cdot \frac{\mathsf{fma}\left(-1, t\_2, -1 \cdot t\_3\right)}{t\_0}}\right)\\ \mathbf{elif}\;t\_2 \leq 6.4 \cdot 10^{-12}:\\ \;\;\;\;2 \cdot \sqrt{\left(t\_0 \cdot t\_2 + t\_0 \cdot t\_3\right) + t\_2 \cdot t\_3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_3 \cdot \sqrt{\frac{1}{t\_3} \cdot \left(t\_2 + t\_0\right)}\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fmin (fmin x y) z))
       (t_1 (fmax (fmin x y) z))
       (t_2 (fmin (fmax x y) t_1))
       (t_3 (fmax (fmax x y) t_1)))
  (if (<= t_2 -1.5e+14)
    (*
     -2.0
     (* t_0 (sqrt (* -1.0 (/ (fma -1.0 t_2 (* -1.0 t_3)) t_0)))))
    (if (<= t_2 6.4e-12)
      (* 2.0 (sqrt (+ (+ (* t_0 t_2) (* t_0 t_3)) (* t_2 t_3))))
      (* 2.0 (* t_3 (sqrt (* (/ 1.0 t_3) (+ t_2 t_0)))))))))

C

double code(double x, double y, double z) {
	double t_0 = fmin(fmin(x, y), z);
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmin(fmax(x, y), t_1);
	double t_3 = fmax(fmax(x, y), t_1);
	double tmp;
	if (t_2 <= -1.5e+14) {
		tmp = -2.0 * (t_0 * sqrt((-1.0 * (fma(-1.0, t_2, (-1.0 * t_3)) / t_0))));
	} else if (t_2 <= 6.4e-12) {
		tmp = 2.0 * sqrt((((t_0 * t_2) + (t_0 * t_3)) + (t_2 * t_3)));
	} else {
		tmp = 2.0 * (t_3 * sqrt(((1.0 / t_3) * (t_2 + t_0))));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fmin(fmin(x, y), z)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmin(fmax(x, y), t_1)
	t_3 = fmax(fmax(x, y), t_1)
	tmp = 0.0
	if (t_2 <= -1.5e+14)
		tmp = Float64(-2.0 * Float64(t_0 * sqrt(Float64(-1.0 * Float64(fma(-1.0, t_2, Float64(-1.0 * t_3)) / t_0)))));
	elseif (t_2 <= 6.4e-12)
		tmp = Float64(2.0 * sqrt(Float64(Float64(Float64(t_0 * t_2) + Float64(t_0 * t_3)) + Float64(t_2 * t_3))));
	else
		tmp = Float64(2.0 * Float64(t_3 * sqrt(Float64(Float64(1.0 / t_3) * Float64(t_2 + t_0)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, If[LessEqual[t$95$2, -1.5e+14], N[(-2.0 * N[(t$95$0 * N[Sqrt[N[(-1.0 * N[(N[(-1.0 * t$95$2 + N[(-1.0 * t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 6.4e-12], N[(2.0 * N[Sqrt[N[(N[(N[(t$95$0 * t$95$2), $MachinePrecision] + N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$3 * N[Sqrt[N[(N[(1.0 / t$95$3), $MachinePrecision] * N[(t$95$2 + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.5e14

   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 -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 29.5%

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

   4. Applied rewrites 29.5%

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

   if -1.5e14 < y < 6.4000000000000002e-12

   1. Initial program 70.5%

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

   if 6.4000000000000002e-12 < 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

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 31.2%

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

   4. Applied rewrites 31.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 31.2%

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 31.2%

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

   6. Applied rewrites 31.2%

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

Alternative 2: 96.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ \mathbf{if}\;t\_2 \leq -1.5 \cdot 10^{+14}:\\ \;\;\;\;2 \cdot \left(-1 \cdot \left(t\_0 \cdot \sqrt{\frac{t\_2}{t\_0}}\right)\right)\\ \mathbf{elif}\;t\_2 \leq 6.4 \cdot 10^{-12}:\\ \;\;\;\;2 \cdot \sqrt{\left(t\_0 \cdot t\_2 + t\_0 \cdot t\_3\right) + t\_2 \cdot t\_3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_3 \cdot \sqrt{\frac{1}{t\_3} \cdot \left(t\_2 + t\_0\right)}\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fmin (fmin x y) z))
       (t_1 (fmax (fmin x y) z))
       (t_2 (fmin (fmax x y) t_1))
       (t_3 (fmax (fmax x y) t_1)))
  (if (<= t_2 -1.5e+14)
    (* 2.0 (* -1.0 (* t_0 (sqrt (/ t_2 t_0)))))
    (if (<= t_2 6.4e-12)
      (* 2.0 (sqrt (+ (+ (* t_0 t_2) (* t_0 t_3)) (* t_2 t_3))))
      (* 2.0 (* t_3 (sqrt (* (/ 1.0 t_3) (+ t_2 t_0)))))))))

C

double code(double x, double y, double z) {
	double t_0 = fmin(fmin(x, y), z);
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmin(fmax(x, y), t_1);
	double t_3 = fmax(fmax(x, y), t_1);
	double tmp;
	if (t_2 <= -1.5e+14) {
		tmp = 2.0 * (-1.0 * (t_0 * sqrt((t_2 / t_0))));
	} else if (t_2 <= 6.4e-12) {
		tmp = 2.0 * sqrt((((t_0 * t_2) + (t_0 * t_3)) + (t_2 * t_3)));
	} else {
		tmp = 2.0 * (t_3 * sqrt(((1.0 / t_3) * (t_2 + t_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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = fmin(fmin(x, y), z)
    t_1 = fmax(fmin(x, y), z)
    t_2 = fmin(fmax(x, y), t_1)
    t_3 = fmax(fmax(x, y), t_1)
    if (t_2 <= (-1.5d+14)) then
        tmp = 2.0d0 * ((-1.0d0) * (t_0 * sqrt((t_2 / t_0))))
    else if (t_2 <= 6.4d-12) then
        tmp = 2.0d0 * sqrt((((t_0 * t_2) + (t_0 * t_3)) + (t_2 * t_3)))
    else
        tmp = 2.0d0 * (t_3 * sqrt(((1.0d0 / t_3) * (t_2 + t_0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = fmin(fmin(x, y), z);
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmin(fmax(x, y), t_1);
	double t_3 = fmax(fmax(x, y), t_1);
	double tmp;
	if (t_2 <= -1.5e+14) {
		tmp = 2.0 * (-1.0 * (t_0 * Math.sqrt((t_2 / t_0))));
	} else if (t_2 <= 6.4e-12) {
		tmp = 2.0 * Math.sqrt((((t_0 * t_2) + (t_0 * t_3)) + (t_2 * t_3)));
	} else {
		tmp = 2.0 * (t_3 * Math.sqrt(((1.0 / t_3) * (t_2 + t_0))));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = fmin(fmin(x, y), z)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmin(fmax(x, y), t_1)
	t_3 = fmax(fmax(x, y), t_1)
	tmp = 0
	if t_2 <= -1.5e+14:
		tmp = 2.0 * (-1.0 * (t_0 * math.sqrt((t_2 / t_0))))
	elif t_2 <= 6.4e-12:
		tmp = 2.0 * math.sqrt((((t_0 * t_2) + (t_0 * t_3)) + (t_2 * t_3)))
	else:
		tmp = 2.0 * (t_3 * math.sqrt(((1.0 / t_3) * (t_2 + t_0))))
	return tmp

Julia

function code(x, y, z)
	t_0 = fmin(fmin(x, y), z)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmin(fmax(x, y), t_1)
	t_3 = fmax(fmax(x, y), t_1)
	tmp = 0.0
	if (t_2 <= -1.5e+14)
		tmp = Float64(2.0 * Float64(-1.0 * Float64(t_0 * sqrt(Float64(t_2 / t_0)))));
	elseif (t_2 <= 6.4e-12)
		tmp = Float64(2.0 * sqrt(Float64(Float64(Float64(t_0 * t_2) + Float64(t_0 * t_3)) + Float64(t_2 * t_3))));
	else
		tmp = Float64(2.0 * Float64(t_3 * sqrt(Float64(Float64(1.0 / t_3) * Float64(t_2 + t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = min(min(x, y), z);
	t_1 = max(min(x, y), z);
	t_2 = min(max(x, y), t_1);
	t_3 = max(max(x, y), t_1);
	tmp = 0.0;
	if (t_2 <= -1.5e+14)
		tmp = 2.0 * (-1.0 * (t_0 * sqrt((t_2 / t_0))));
	elseif (t_2 <= 6.4e-12)
		tmp = 2.0 * sqrt((((t_0 * t_2) + (t_0 * t_3)) + (t_2 * t_3)));
	else
		tmp = 2.0 * (t_3 * sqrt(((1.0 / t_3) * (t_2 + t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, If[LessEqual[t$95$2, -1.5e+14], N[(2.0 * N[(-1.0 * N[(t$95$0 * N[Sqrt[N[(t$95$2 / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 6.4e-12], N[(2.0 * N[Sqrt[N[(N[(N[(t$95$0 * t$95$2), $MachinePrecision] + N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$3 * N[Sqrt[N[(N[(1.0 / t$95$3), $MachinePrecision] * N[(t$95$2 + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.5e14

   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 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 24.1%

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

   4. Applied rewrites 24.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 15.5%

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

   7. Applied rewrites 15.5%

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

   8. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 15.4%

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

   10. Applied rewrites 15.4%

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

   if -1.5e14 < y < 6.4000000000000002e-12

   1. Initial program 70.5%

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

   if 6.4000000000000002e-12 < 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

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 31.2%

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

   4. Applied rewrites 31.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 31.2%

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 31.2%

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

   6. Applied rewrites 31.2%

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

Alternative 3: 96.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ \mathbf{if}\;t\_2 \leq -1.5 \cdot 10^{+14}:\\ \;\;\;\;2 \cdot \left(-1 \cdot \left(t\_0 \cdot \sqrt{\frac{t\_2}{t\_0}}\right)\right)\\ \mathbf{elif}\;t\_2 \leq 6.4 \cdot 10^{-12}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(t\_3, t\_0, \left(t\_3 + t\_0\right) \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_3 \cdot \sqrt{\frac{1}{t\_3} \cdot \left(t\_2 + t\_0\right)}\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fmin (fmin x y) z))
       (t_1 (fmax (fmin x y) z))
       (t_2 (fmin (fmax x y) t_1))
       (t_3 (fmax (fmax x y) t_1)))
  (if (<= t_2 -1.5e+14)
    (* 2.0 (* -1.0 (* t_0 (sqrt (/ t_2 t_0)))))
    (if (<= t_2 6.4e-12)
      (* 2.0 (sqrt (fma t_3 t_0 (* (+ t_3 t_0) t_2))))
      (* 2.0 (* t_3 (sqrt (* (/ 1.0 t_3) (+ t_2 t_0)))))))))

C

double code(double x, double y, double z) {
	double t_0 = fmin(fmin(x, y), z);
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmin(fmax(x, y), t_1);
	double t_3 = fmax(fmax(x, y), t_1);
	double tmp;
	if (t_2 <= -1.5e+14) {
		tmp = 2.0 * (-1.0 * (t_0 * sqrt((t_2 / t_0))));
	} else if (t_2 <= 6.4e-12) {
		tmp = 2.0 * sqrt(fma(t_3, t_0, ((t_3 + t_0) * t_2)));
	} else {
		tmp = 2.0 * (t_3 * sqrt(((1.0 / t_3) * (t_2 + t_0))));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fmin(fmin(x, y), z)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmin(fmax(x, y), t_1)
	t_3 = fmax(fmax(x, y), t_1)
	tmp = 0.0
	if (t_2 <= -1.5e+14)
		tmp = Float64(2.0 * Float64(-1.0 * Float64(t_0 * sqrt(Float64(t_2 / t_0)))));
	elseif (t_2 <= 6.4e-12)
		tmp = Float64(2.0 * sqrt(fma(t_3, t_0, Float64(Float64(t_3 + t_0) * t_2))));
	else
		tmp = Float64(2.0 * Float64(t_3 * sqrt(Float64(Float64(1.0 / t_3) * Float64(t_2 + t_0)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, If[LessEqual[t$95$2, -1.5e+14], N[(2.0 * N[(-1.0 * N[(t$95$0 * N[Sqrt[N[(t$95$2 / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 6.4e-12], N[(2.0 * N[Sqrt[N[(t$95$3 * t$95$0 + N[(N[(t$95$3 + t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$3 * N[Sqrt[N[(N[(1.0 / t$95$3), $MachinePrecision] * N[(t$95$2 + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.5e14

   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 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 24.1%

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

   4. Applied rewrites 24.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 15.5%

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

   7. Applied rewrites 15.5%

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

   8. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 15.4%

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

   10. Applied rewrites 15.4%

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

   if -1.5e14 < y < 6.4000000000000002e-12

   1. Initial program 70.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-lft-neg-out N/A

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

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

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

      16. distribute-lft-out-- N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. add-flip-rev N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 70.7%

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

   3. Applied rewrites 70.7%

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

   if 6.4000000000000002e-12 < 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

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 31.2%

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

   4. Applied rewrites 31.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 31.2%

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 31.2%

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

   6. Applied rewrites 31.2%

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

Alternative 4: 96.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ \mathbf{if}\;t\_2 \leq -1.5 \cdot 10^{+14}:\\ \;\;\;\;2 \cdot \left(-1 \cdot \left(t\_0 \cdot \sqrt{\frac{t\_2}{t\_0}}\right)\right)\\ \mathbf{elif}\;t\_2 \leq 6.4 \cdot 10^{-12}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(t\_3, t\_0, \left(t\_3 + t\_0\right) \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_3 \cdot \sqrt{\frac{t\_0 + t\_2}{t\_3}}\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fmin (fmin x y) z))
       (t_1 (fmax (fmin x y) z))
       (t_2 (fmin (fmax x y) t_1))
       (t_3 (fmax (fmax x y) t_1)))
  (if (<= t_2 -1.5e+14)
    (* 2.0 (* -1.0 (* t_0 (sqrt (/ t_2 t_0)))))
    (if (<= t_2 6.4e-12)
      (* 2.0 (sqrt (fma t_3 t_0 (* (+ t_3 t_0) t_2))))
      (* 2.0 (* t_3 (sqrt (/ (+ t_0 t_2) t_3))))))))

C

double code(double x, double y, double z) {
	double t_0 = fmin(fmin(x, y), z);
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmin(fmax(x, y), t_1);
	double t_3 = fmax(fmax(x, y), t_1);
	double tmp;
	if (t_2 <= -1.5e+14) {
		tmp = 2.0 * (-1.0 * (t_0 * sqrt((t_2 / t_0))));
	} else if (t_2 <= 6.4e-12) {
		tmp = 2.0 * sqrt(fma(t_3, t_0, ((t_3 + t_0) * t_2)));
	} else {
		tmp = 2.0 * (t_3 * sqrt(((t_0 + t_2) / t_3)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fmin(fmin(x, y), z)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmin(fmax(x, y), t_1)
	t_3 = fmax(fmax(x, y), t_1)
	tmp = 0.0
	if (t_2 <= -1.5e+14)
		tmp = Float64(2.0 * Float64(-1.0 * Float64(t_0 * sqrt(Float64(t_2 / t_0)))));
	elseif (t_2 <= 6.4e-12)
		tmp = Float64(2.0 * sqrt(fma(t_3, t_0, Float64(Float64(t_3 + t_0) * t_2))));
	else
		tmp = Float64(2.0 * Float64(t_3 * sqrt(Float64(Float64(t_0 + t_2) / t_3))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, If[LessEqual[t$95$2, -1.5e+14], N[(2.0 * N[(-1.0 * N[(t$95$0 * N[Sqrt[N[(t$95$2 / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 6.4e-12], N[(2.0 * N[Sqrt[N[(t$95$3 * t$95$0 + N[(N[(t$95$3 + t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$3 * N[Sqrt[N[(N[(t$95$0 + t$95$2), $MachinePrecision] / t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.5e14

   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 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 24.1%

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

   4. Applied rewrites 24.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 15.5%

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

   7. Applied rewrites 15.5%

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

   8. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 15.4%

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

   10. Applied rewrites 15.4%

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

   if -1.5e14 < y < 6.4000000000000002e-12

   1. Initial program 70.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-lft-neg-out N/A

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

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

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

      16. distribute-lft-out-- N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. add-flip-rev N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 70.7%

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

   3. Applied rewrites 70.7%

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

   if 6.4000000000000002e-12 < 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

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 31.2%

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

   4. Applied rewrites 31.2%

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

Alternative 5: 96.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := t\_0 + t\_2\\ t_4 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ \mathbf{if}\;t\_2 \leq -1.35 \cdot 10^{+16}:\\ \;\;\;\;2 \cdot \left(-1 \cdot \left(t\_0 \cdot \sqrt{\frac{t\_2}{t\_0}}\right)\right)\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{-304}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(t\_4, t\_0, t\_2 \cdot t\_0\right)}\\ \mathbf{elif}\;t\_2 \leq 2.15 \cdot 10^{-11}:\\ \;\;\;\;2 \cdot \sqrt{t\_4 \cdot t\_3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_4 \cdot \sqrt{\frac{t\_3}{t\_4}}\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fmin (fmin x y) z))
       (t_1 (fmax (fmin x y) z))
       (t_2 (fmin (fmax x y) t_1))
       (t_3 (+ t_0 t_2))
       (t_4 (fmax (fmax x y) t_1)))
  (if (<= t_2 -1.35e+16)
    (* 2.0 (* -1.0 (* t_0 (sqrt (/ t_2 t_0)))))
    (if (<= t_2 -4e-304)
      (* 2.0 (sqrt (fma t_4 t_0 (* t_2 t_0))))
      (if (<= t_2 2.15e-11)
        (* 2.0 (sqrt (* t_4 t_3)))
        (* 2.0 (* t_4 (sqrt (/ t_3 t_4)))))))))

C

double code(double x, double y, double z) {
	double t_0 = fmin(fmin(x, y), z);
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmin(fmax(x, y), t_1);
	double t_3 = t_0 + t_2;
	double t_4 = fmax(fmax(x, y), t_1);
	double tmp;
	if (t_2 <= -1.35e+16) {
		tmp = 2.0 * (-1.0 * (t_0 * sqrt((t_2 / t_0))));
	} else if (t_2 <= -4e-304) {
		tmp = 2.0 * sqrt(fma(t_4, t_0, (t_2 * t_0)));
	} else if (t_2 <= 2.15e-11) {
		tmp = 2.0 * sqrt((t_4 * t_3));
	} else {
		tmp = 2.0 * (t_4 * sqrt((t_3 / t_4)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fmin(fmin(x, y), z)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmin(fmax(x, y), t_1)
	t_3 = Float64(t_0 + t_2)
	t_4 = fmax(fmax(x, y), t_1)
	tmp = 0.0
	if (t_2 <= -1.35e+16)
		tmp = Float64(2.0 * Float64(-1.0 * Float64(t_0 * sqrt(Float64(t_2 / t_0)))));
	elseif (t_2 <= -4e-304)
		tmp = Float64(2.0 * sqrt(fma(t_4, t_0, Float64(t_2 * t_0))));
	elseif (t_2 <= 2.15e-11)
		tmp = Float64(2.0 * sqrt(Float64(t_4 * t_3)));
	else
		tmp = Float64(2.0 * Float64(t_4 * sqrt(Float64(t_3 / t_4))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, If[LessEqual[t$95$2, -1.35e+16], N[(2.0 * N[(-1.0 * N[(t$95$0 * N[Sqrt[N[(t$95$2 / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -4e-304], N[(2.0 * N[Sqrt[N[(t$95$4 * t$95$0 + N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.15e-11], N[(2.0 * N[Sqrt[N[(t$95$4 * t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$4 * N[Sqrt[N[(t$95$3 / t$95$4), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.35e16

   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 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 24.1%

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

   4. Applied rewrites 24.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 15.5%

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

   7. Applied rewrites 15.5%

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

   8. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 15.4%

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

   10. Applied rewrites 15.4%

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

   if -1.35e16 < y < -3.9999999999999999e-304

   1. Initial program 70.5%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 46.8%

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 47.6%

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

   7. Applied rewrites 47.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. +-commutative N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. *-commutative N/A

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

      8. remove-double-neg N/A

         \[\leadsto 2 \cdot \sqrt{z \cdot x + x \cdot y} \]

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 47.6%

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

   9. Applied rewrites 47.6%

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

   if -3.9999999999999999e-304 < y < 2.15e-11

   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. lower-*.f64 N/A

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

      2. lower-+.f64 48.5%

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

   4. Applied rewrites 48.5%

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

   if 2.15e-11 < 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

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 31.2%

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

   4. Applied rewrites 31.2%

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

Alternative 6: 96.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_1 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_0\right)\\ t_2 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_0\right)\\ t_3 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_4 := t\_3 + t\_2\\ \mathbf{if}\;t\_2 \leq -1.45 \cdot 10^{+19}:\\ \;\;\;\;-2 \cdot \left(t\_2 \cdot \sqrt{\frac{t\_3 + t\_1}{t\_2}}\right)\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{-304}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(t\_1, t\_3, t\_2 \cdot t\_3\right)}\\ \mathbf{elif}\;t\_2 \leq 2.15 \cdot 10^{-11}:\\ \;\;\;\;2 \cdot \sqrt{t\_1 \cdot t\_4}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_1 \cdot \sqrt{\frac{t\_4}{t\_1}}\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fmax (fmin x y) z))
       (t_1 (fmax (fmax x y) t_0))
       (t_2 (fmin (fmax x y) t_0))
       (t_3 (fmin (fmin x y) z))
       (t_4 (+ t_3 t_2)))
  (if (<= t_2 -1.45e+19)
    (* -2.0 (* t_2 (sqrt (/ (+ t_3 t_1) t_2))))
    (if (<= t_2 -4e-304)
      (* 2.0 (sqrt (fma t_1 t_3 (* t_2 t_3))))
      (if (<= t_2 2.15e-11)
        (* 2.0 (sqrt (* t_1 t_4)))
        (* 2.0 (* t_1 (sqrt (/ t_4 t_1)))))))))

C

double code(double x, double y, double z) {
	double t_0 = fmax(fmin(x, y), z);
	double t_1 = fmax(fmax(x, y), t_0);
	double t_2 = fmin(fmax(x, y), t_0);
	double t_3 = fmin(fmin(x, y), z);
	double t_4 = t_3 + t_2;
	double tmp;
	if (t_2 <= -1.45e+19) {
		tmp = -2.0 * (t_2 * sqrt(((t_3 + t_1) / t_2)));
	} else if (t_2 <= -4e-304) {
		tmp = 2.0 * sqrt(fma(t_1, t_3, (t_2 * t_3)));
	} else if (t_2 <= 2.15e-11) {
		tmp = 2.0 * sqrt((t_1 * t_4));
	} else {
		tmp = 2.0 * (t_1 * sqrt((t_4 / t_1)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fmax(fmin(x, y), z)
	t_1 = fmax(fmax(x, y), t_0)
	t_2 = fmin(fmax(x, y), t_0)
	t_3 = fmin(fmin(x, y), z)
	t_4 = Float64(t_3 + t_2)
	tmp = 0.0
	if (t_2 <= -1.45e+19)
		tmp = Float64(-2.0 * Float64(t_2 * sqrt(Float64(Float64(t_3 + t_1) / t_2))));
	elseif (t_2 <= -4e-304)
		tmp = Float64(2.0 * sqrt(fma(t_1, t_3, Float64(t_2 * t_3))));
	elseif (t_2 <= 2.15e-11)
		tmp = Float64(2.0 * sqrt(Float64(t_1 * t_4)));
	else
		tmp = Float64(2.0 * Float64(t_1 * sqrt(Float64(t_4 / t_1))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Max[x, y], $MachinePrecision], t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Max[x, y], $MachinePrecision], t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 + t$95$2), $MachinePrecision]}, If[LessEqual[t$95$2, -1.45e+19], N[(-2.0 * N[(t$95$2 * N[Sqrt[N[(N[(t$95$3 + t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -4e-304], N[(2.0 * N[Sqrt[N[(t$95$1 * t$95$3 + N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.15e-11], N[(2.0 * N[Sqrt[N[(t$95$1 * t$95$4), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$1 * N[Sqrt[N[(t$95$4 / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.45e19

   1. Initial program 70.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-lft-neg-out N/A

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

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

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

      16. distribute-lft-out-- N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. add-flip-rev N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 70.7%

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

   3. Applied rewrites 70.7%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 48.5%

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

   7. Taylor expanded in y around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 29.4%

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

   9. Applied rewrites 29.4%

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

   if -1.45e19 < y < -3.9999999999999999e-304

   1. Initial program 70.5%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 46.8%

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 47.6%

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

   7. Applied rewrites 47.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. +-commutative N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. *-commutative N/A

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

      8. remove-double-neg N/A

         \[\leadsto 2 \cdot \sqrt{z \cdot x + x \cdot y} \]

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 47.6%

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

   9. Applied rewrites 47.6%

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

   if -3.9999999999999999e-304 < y < 2.15e-11

   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. lower-*.f64 N/A

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

      2. lower-+.f64 48.5%

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

   4. Applied rewrites 48.5%

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

   if 2.15e-11 < 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

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 31.2%

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

   4. Applied rewrites 31.2%

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

Alternative 7: 96.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ \mathbf{if}\;t\_3 \leq -1.45 \cdot 10^{+19}:\\ \;\;\;\;-2 \cdot \left(t\_3 \cdot \sqrt{\frac{t\_0 + t\_2}{t\_3}}\right)\\ \mathbf{elif}\;t\_3 \leq -4 \cdot 10^{-304}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(t\_2, t\_0, t\_3 \cdot t\_0\right)}\\ \mathbf{elif}\;t\_3 \leq 21000000000000:\\ \;\;\;\;2 \cdot \sqrt{t\_2 \cdot \left(t\_0 + t\_3\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_2 \cdot \sqrt{\frac{t\_3}{t\_2}}\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fmin (fmin x y) z))
       (t_1 (fmax (fmin x y) z))
       (t_2 (fmax (fmax x y) t_1))
       (t_3 (fmin (fmax x y) t_1)))
  (if (<= t_3 -1.45e+19)
    (* -2.0 (* t_3 (sqrt (/ (+ t_0 t_2) t_3))))
    (if (<= t_3 -4e-304)
      (* 2.0 (sqrt (fma t_2 t_0 (* t_3 t_0))))
      (if (<= t_3 21000000000000.0)
        (* 2.0 (sqrt (* t_2 (+ t_0 t_3))))
        (* 2.0 (* t_2 (sqrt (/ t_3 t_2)))))))))

C

double code(double x, double y, double z) {
	double t_0 = fmin(fmin(x, y), z);
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmax(x, y), t_1);
	double tmp;
	if (t_3 <= -1.45e+19) {
		tmp = -2.0 * (t_3 * sqrt(((t_0 + t_2) / t_3)));
	} else if (t_3 <= -4e-304) {
		tmp = 2.0 * sqrt(fma(t_2, t_0, (t_3 * t_0)));
	} else if (t_3 <= 21000000000000.0) {
		tmp = 2.0 * sqrt((t_2 * (t_0 + t_3)));
	} else {
		tmp = 2.0 * (t_2 * sqrt((t_3 / t_2)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fmin(fmin(x, y), z)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmax(x, y), t_1)
	tmp = 0.0
	if (t_3 <= -1.45e+19)
		tmp = Float64(-2.0 * Float64(t_3 * sqrt(Float64(Float64(t_0 + t_2) / t_3))));
	elseif (t_3 <= -4e-304)
		tmp = Float64(2.0 * sqrt(fma(t_2, t_0, Float64(t_3 * t_0))));
	elseif (t_3 <= 21000000000000.0)
		tmp = Float64(2.0 * sqrt(Float64(t_2 * Float64(t_0 + t_3))));
	else
		tmp = Float64(2.0 * Float64(t_2 * sqrt(Float64(t_3 / t_2))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, If[LessEqual[t$95$3, -1.45e+19], N[(-2.0 * N[(t$95$3 * N[Sqrt[N[(N[(t$95$0 + t$95$2), $MachinePrecision] / t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -4e-304], N[(2.0 * N[Sqrt[N[(t$95$2 * t$95$0 + N[(t$95$3 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 21000000000000.0], N[(2.0 * N[Sqrt[N[(t$95$2 * N[(t$95$0 + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$2 * N[Sqrt[N[(t$95$3 / t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.45e19

   1. Initial program 70.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-lft-neg-out N/A

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

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

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

      16. distribute-lft-out-- N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. add-flip-rev N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 70.7%

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

   3. Applied rewrites 70.7%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 48.5%

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

   7. Taylor expanded in y around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 29.4%

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

   9. Applied rewrites 29.4%

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

   if -1.45e19 < y < -3.9999999999999999e-304

   1. Initial program 70.5%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 46.8%

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 47.6%

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

   7. Applied rewrites 47.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. +-commutative N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. *-commutative N/A

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

      8. remove-double-neg N/A

         \[\leadsto 2 \cdot \sqrt{z \cdot x + x \cdot y} \]

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 47.6%

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

   9. Applied rewrites 47.6%

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

   if -3.9999999999999999e-304 < y < 2.1e13

   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. lower-*.f64 N/A

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

      2. lower-+.f64 48.5%

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

   4. Applied rewrites 48.5%

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

   if 2.1e13 < 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

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 31.2%

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

   4. Applied rewrites 31.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 16.0%

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

Alternative 8: 83.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ \mathbf{if}\;t\_3 \leq -4 \cdot 10^{-304}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(t\_2, t\_0, t\_3 \cdot t\_0\right)}\\ \mathbf{elif}\;t\_3 \leq 21000000000000:\\ \;\;\;\;2 \cdot \sqrt{t\_2 \cdot \left(t\_0 + t\_3\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_2 \cdot \sqrt{\frac{t\_3}{t\_2}}\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fmin (fmin x y) z))
       (t_1 (fmax (fmin x y) z))
       (t_2 (fmax (fmax x y) t_1))
       (t_3 (fmin (fmax x y) t_1)))
  (if (<= t_3 -4e-304)
    (* 2.0 (sqrt (fma t_2 t_0 (* t_3 t_0))))
    (if (<= t_3 21000000000000.0)
      (* 2.0 (sqrt (* t_2 (+ t_0 t_3))))
      (* 2.0 (* t_2 (sqrt (/ t_3 t_2))))))))

C

double code(double x, double y, double z) {
	double t_0 = fmin(fmin(x, y), z);
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmax(x, y), t_1);
	double tmp;
	if (t_3 <= -4e-304) {
		tmp = 2.0 * sqrt(fma(t_2, t_0, (t_3 * t_0)));
	} else if (t_3 <= 21000000000000.0) {
		tmp = 2.0 * sqrt((t_2 * (t_0 + t_3)));
	} else {
		tmp = 2.0 * (t_2 * sqrt((t_3 / t_2)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fmin(fmin(x, y), z)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmax(x, y), t_1)
	tmp = 0.0
	if (t_3 <= -4e-304)
		tmp = Float64(2.0 * sqrt(fma(t_2, t_0, Float64(t_3 * t_0))));
	elseif (t_3 <= 21000000000000.0)
		tmp = Float64(2.0 * sqrt(Float64(t_2 * Float64(t_0 + t_3))));
	else
		tmp = Float64(2.0 * Float64(t_2 * sqrt(Float64(t_3 / t_2))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, If[LessEqual[t$95$3, -4e-304], N[(2.0 * N[Sqrt[N[(t$95$2 * t$95$0 + N[(t$95$3 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 21000000000000.0], N[(2.0 * N[Sqrt[N[(t$95$2 * N[(t$95$0 + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$2 * N[Sqrt[N[(t$95$3 / t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.9999999999999999e-304

   1. Initial program 70.5%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 46.8%

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 47.6%

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

   7. Applied rewrites 47.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. +-commutative N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. *-commutative N/A

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

      8. remove-double-neg N/A

         \[\leadsto 2 \cdot \sqrt{z \cdot x + x \cdot y} \]

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 47.6%

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

   9. Applied rewrites 47.6%

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

   if -3.9999999999999999e-304 < y < 2.1e13

   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. lower-*.f64 N/A

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

      2. lower-+.f64 48.5%

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

   4. Applied rewrites 48.5%

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

   if 2.1e13 < 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

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 31.2%

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

   4. Applied rewrites 31.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 16.0%

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

Alternative 9: 83.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ \mathbf{if}\;t\_3 \leq -4 \cdot 10^{-304}:\\ \;\;\;\;2 \cdot \sqrt{t\_0 \cdot \left(t\_3 + t\_2\right)}\\ \mathbf{elif}\;t\_3 \leq 21000000000000:\\ \;\;\;\;2 \cdot \sqrt{t\_2 \cdot \left(t\_0 + t\_3\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_2 \cdot \sqrt{\frac{t\_3}{t\_2}}\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fmin (fmin x y) z))
       (t_1 (fmax (fmin x y) z))
       (t_2 (fmax (fmax x y) t_1))
       (t_3 (fmin (fmax x y) t_1)))
  (if (<= t_3 -4e-304)
    (* 2.0 (sqrt (* t_0 (+ t_3 t_2))))
    (if (<= t_3 21000000000000.0)
      (* 2.0 (sqrt (* t_2 (+ t_0 t_3))))
      (* 2.0 (* t_2 (sqrt (/ t_3 t_2))))))))

C

double code(double x, double y, double z) {
	double t_0 = fmin(fmin(x, y), z);
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmax(x, y), t_1);
	double tmp;
	if (t_3 <= -4e-304) {
		tmp = 2.0 * sqrt((t_0 * (t_3 + t_2)));
	} else if (t_3 <= 21000000000000.0) {
		tmp = 2.0 * sqrt((t_2 * (t_0 + t_3)));
	} else {
		tmp = 2.0 * (t_2 * sqrt((t_3 / t_2)));
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = fmin(fmin(x, y), z)
    t_1 = fmax(fmin(x, y), z)
    t_2 = fmax(fmax(x, y), t_1)
    t_3 = fmin(fmax(x, y), t_1)
    if (t_3 <= (-4d-304)) then
        tmp = 2.0d0 * sqrt((t_0 * (t_3 + t_2)))
    else if (t_3 <= 21000000000000.0d0) then
        tmp = 2.0d0 * sqrt((t_2 * (t_0 + t_3)))
    else
        tmp = 2.0d0 * (t_2 * sqrt((t_3 / t_2)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = fmin(fmin(x, y), z);
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmax(x, y), t_1);
	double tmp;
	if (t_3 <= -4e-304) {
		tmp = 2.0 * Math.sqrt((t_0 * (t_3 + t_2)));
	} else if (t_3 <= 21000000000000.0) {
		tmp = 2.0 * Math.sqrt((t_2 * (t_0 + t_3)));
	} else {
		tmp = 2.0 * (t_2 * Math.sqrt((t_3 / t_2)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = fmin(fmin(x, y), z)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmax(x, y), t_1)
	tmp = 0
	if t_3 <= -4e-304:
		tmp = 2.0 * math.sqrt((t_0 * (t_3 + t_2)))
	elif t_3 <= 21000000000000.0:
		tmp = 2.0 * math.sqrt((t_2 * (t_0 + t_3)))
	else:
		tmp = 2.0 * (t_2 * math.sqrt((t_3 / t_2)))
	return tmp

Julia

function code(x, y, z)
	t_0 = fmin(fmin(x, y), z)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmax(x, y), t_1)
	tmp = 0.0
	if (t_3 <= -4e-304)
		tmp = Float64(2.0 * sqrt(Float64(t_0 * Float64(t_3 + t_2))));
	elseif (t_3 <= 21000000000000.0)
		tmp = Float64(2.0 * sqrt(Float64(t_2 * Float64(t_0 + t_3))));
	else
		tmp = Float64(2.0 * Float64(t_2 * sqrt(Float64(t_3 / t_2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = min(min(x, y), z);
	t_1 = max(min(x, y), z);
	t_2 = max(max(x, y), t_1);
	t_3 = min(max(x, y), t_1);
	tmp = 0.0;
	if (t_3 <= -4e-304)
		tmp = 2.0 * sqrt((t_0 * (t_3 + t_2)));
	elseif (t_3 <= 21000000000000.0)
		tmp = 2.0 * sqrt((t_2 * (t_0 + t_3)));
	else
		tmp = 2.0 * (t_2 * sqrt((t_3 / t_2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, If[LessEqual[t$95$3, -4e-304], N[(2.0 * N[Sqrt[N[(t$95$0 * N[(t$95$3 + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 21000000000000.0], N[(2.0 * N[Sqrt[N[(t$95$2 * N[(t$95$0 + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$2 * N[Sqrt[N[(t$95$3 / t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.9999999999999999e-304

   1. Initial program 70.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-lft-neg-out N/A

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

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

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

      16. distribute-lft-out-- N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. add-flip-rev N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 70.7%

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

   3. Applied rewrites 70.7%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 47.6%

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

   6. Applied rewrites 47.6%

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

   if -3.9999999999999999e-304 < y < 2.1e13

   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. lower-*.f64 N/A

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

      2. lower-+.f64 48.5%

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

   4. Applied rewrites 48.5%

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

   if 2.1e13 < 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

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 31.2%

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

   4. Applied rewrites 31.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 16.0%

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

Alternative 10: 70.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ \mathbf{if}\;t\_3 \leq -4 \cdot 10^{-304}:\\ \;\;\;\;2 \cdot \sqrt{t\_0 \cdot \left(t\_3 + t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{t\_2 \cdot \left(t\_0 + t\_3\right)}\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fmin (fmin x y) z))
       (t_1 (fmax (fmin x y) z))
       (t_2 (fmax (fmax x y) t_1))
       (t_3 (fmin (fmax x y) t_1)))
  (if (<= t_3 -4e-304)
    (* 2.0 (sqrt (* t_0 (+ t_3 t_2))))
    (* 2.0 (sqrt (* t_2 (+ t_0 t_3)))))))

C

double code(double x, double y, double z) {
	double t_0 = fmin(fmin(x, y), z);
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmax(x, y), t_1);
	double tmp;
	if (t_3 <= -4e-304) {
		tmp = 2.0 * sqrt((t_0 * (t_3 + t_2)));
	} else {
		tmp = 2.0 * sqrt((t_2 * (t_0 + t_3)));
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = fmin(fmin(x, y), z)
    t_1 = fmax(fmin(x, y), z)
    t_2 = fmax(fmax(x, y), t_1)
    t_3 = fmin(fmax(x, y), t_1)
    if (t_3 <= (-4d-304)) then
        tmp = 2.0d0 * sqrt((t_0 * (t_3 + t_2)))
    else
        tmp = 2.0d0 * sqrt((t_2 * (t_0 + t_3)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = fmin(fmin(x, y), z);
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmax(x, y), t_1);
	double tmp;
	if (t_3 <= -4e-304) {
		tmp = 2.0 * Math.sqrt((t_0 * (t_3 + t_2)));
	} else {
		tmp = 2.0 * Math.sqrt((t_2 * (t_0 + t_3)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = fmin(fmin(x, y), z)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmax(x, y), t_1)
	tmp = 0
	if t_3 <= -4e-304:
		tmp = 2.0 * math.sqrt((t_0 * (t_3 + t_2)))
	else:
		tmp = 2.0 * math.sqrt((t_2 * (t_0 + t_3)))
	return tmp

Julia

function code(x, y, z)
	t_0 = fmin(fmin(x, y), z)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmax(x, y), t_1)
	tmp = 0.0
	if (t_3 <= -4e-304)
		tmp = Float64(2.0 * sqrt(Float64(t_0 * Float64(t_3 + t_2))));
	else
		tmp = Float64(2.0 * sqrt(Float64(t_2 * Float64(t_0 + t_3))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = min(min(x, y), z);
	t_1 = max(min(x, y), z);
	t_2 = max(max(x, y), t_1);
	t_3 = min(max(x, y), t_1);
	tmp = 0.0;
	if (t_3 <= -4e-304)
		tmp = 2.0 * sqrt((t_0 * (t_3 + t_2)));
	else
		tmp = 2.0 * sqrt((t_2 * (t_0 + t_3)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, If[LessEqual[t$95$3, -4e-304], N[(2.0 * N[Sqrt[N[(t$95$0 * N[(t$95$3 + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(t$95$2 * N[(t$95$0 + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.9999999999999999e-304

   1. Initial program 70.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-lft-neg-out N/A

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

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

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

      16. distribute-lft-out-- N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. add-flip-rev N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 70.7%

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

   3. Applied rewrites 70.7%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 47.6%

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

   6. Applied rewrites 47.6%

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

   if -3.9999999999999999e-304 < 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. lower-*.f64 N/A

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

      2. lower-+.f64 48.5%

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

   4. Applied rewrites 48.5%

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

Alternative 11: 69.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_1 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_0\right)\\ t_2 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_0\right)\\ \mathbf{if}\;t\_2 \leq -4.1 \cdot 10^{-305}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right) \cdot \left(t\_2 + t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{\left|t\_2 \cdot t\_1\right|}\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fmax (fmin x y) z))
       (t_1 (fmax (fmax x y) t_0))
       (t_2 (fmin (fmax x y) t_0)))
  (if (<= t_2 -4.1e-305)
    (* 2.0 (sqrt (* (fmin (fmin x y) z) (+ t_2 t_1))))
    (* 2.0 (sqrt (fabs (* t_2 t_1)))))))

C

double code(double x, double y, double z) {
	double t_0 = fmax(fmin(x, y), z);
	double t_1 = fmax(fmax(x, y), t_0);
	double t_2 = fmin(fmax(x, y), t_0);
	double tmp;
	if (t_2 <= -4.1e-305) {
		tmp = 2.0 * sqrt((fmin(fmin(x, y), z) * (t_2 + t_1)));
	} else {
		tmp = 2.0 * sqrt(fabs((t_2 * t_1)));
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = fmax(fmin(x, y), z)
    t_1 = fmax(fmax(x, y), t_0)
    t_2 = fmin(fmax(x, y), t_0)
    if (t_2 <= (-4.1d-305)) then
        tmp = 2.0d0 * sqrt((fmin(fmin(x, y), z) * (t_2 + t_1)))
    else
        tmp = 2.0d0 * sqrt(abs((t_2 * t_1)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = fmax(fmin(x, y), z);
	double t_1 = fmax(fmax(x, y), t_0);
	double t_2 = fmin(fmax(x, y), t_0);
	double tmp;
	if (t_2 <= -4.1e-305) {
		tmp = 2.0 * Math.sqrt((fmin(fmin(x, y), z) * (t_2 + t_1)));
	} else {
		tmp = 2.0 * Math.sqrt(Math.abs((t_2 * t_1)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = fmax(fmin(x, y), z)
	t_1 = fmax(fmax(x, y), t_0)
	t_2 = fmin(fmax(x, y), t_0)
	tmp = 0
	if t_2 <= -4.1e-305:
		tmp = 2.0 * math.sqrt((fmin(fmin(x, y), z) * (t_2 + t_1)))
	else:
		tmp = 2.0 * math.sqrt(math.fabs((t_2 * t_1)))
	return tmp

Julia

function code(x, y, z)
	t_0 = fmax(fmin(x, y), z)
	t_1 = fmax(fmax(x, y), t_0)
	t_2 = fmin(fmax(x, y), t_0)
	tmp = 0.0
	if (t_2 <= -4.1e-305)
		tmp = Float64(2.0 * sqrt(Float64(fmin(fmin(x, y), z) * Float64(t_2 + t_1))));
	else
		tmp = Float64(2.0 * sqrt(abs(Float64(t_2 * t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = max(min(x, y), z);
	t_1 = max(max(x, y), t_0);
	t_2 = min(max(x, y), t_0);
	tmp = 0.0;
	if (t_2 <= -4.1e-305)
		tmp = 2.0 * sqrt((min(min(x, y), z) * (t_2 + t_1)));
	else
		tmp = 2.0 * sqrt(abs((t_2 * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Max[x, y], $MachinePrecision], t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Max[x, y], $MachinePrecision], t$95$0], $MachinePrecision]}, If[LessEqual[t$95$2, -4.1e-305], N[(2.0 * N[Sqrt[N[(N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision] * N[(t$95$2 + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[Abs[N[(t$95$2 * t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.1000000000000002e-305

   1. Initial program 70.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-lft-neg-out N/A

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

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

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

      16. distribute-lft-out-- N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. add-flip-rev N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 70.7%

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

   3. Applied rewrites 70.7%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 47.6%

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

   6. Applied rewrites 47.6%

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

   if -4.1000000000000002e-305 < 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. lower-*.f64 N/A

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

      2. lower-+.f64 48.5%

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

   4. Applied rewrites 48.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 25.0%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. rem-sqrt-square-rev N/A

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

      4. lower-fabs.f64 26.8%

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 26.8%

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

   9. Applied rewrites 26.8%

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

Alternative 12: 68.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ 2 \cdot \sqrt{\mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_0\right) \cdot \left(\mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right) + \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_0\right)\right)} \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fmax (fmin x y) z)))
  (*
   2.0
   (sqrt
    (*
     (fmin (fmax x y) t_0)
     (+ (fmin (fmin x y) z) (fmax (fmax x y) t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = fmax(fmin(x, y), z);
	return 2.0 * sqrt((fmin(fmax(x, y), t_0) * (fmin(fmin(x, y), z) + fmax(fmax(x, y), t_0))));
}

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
    t_0 = fmax(fmin(x, y), z)
    code = 2.0d0 * sqrt((fmin(fmax(x, y), t_0) * (fmin(fmin(x, y), z) + fmax(fmax(x, y), t_0))))
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = fmax(fmin(x, y), z);
	return 2.0 * Math.sqrt((fmin(fmax(x, y), t_0) * (fmin(fmin(x, y), z) + fmax(fmax(x, y), t_0))));
}

Python

def code(x, y, z):
	t_0 = fmax(fmin(x, y), z)
	return 2.0 * math.sqrt((fmin(fmax(x, y), t_0) * (fmin(fmin(x, y), z) + fmax(fmax(x, y), t_0))))

Julia

function code(x, y, z)
	t_0 = fmax(fmin(x, y), z)
	return Float64(2.0 * sqrt(Float64(fmin(fmax(x, y), t_0) * Float64(fmin(fmin(x, y), z) + fmax(fmax(x, y), t_0)))))
end

MATLAB

function tmp = code(x, y, z)
	t_0 = max(min(x, y), z);
	tmp = 2.0 * sqrt((min(max(x, y), t_0) * (min(min(x, y), z) + max(max(x, y), t_0))));
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, N[(2.0 * N[Sqrt[N[(N[Min[N[Max[x, y], $MachinePrecision], t$95$0], $MachinePrecision] * N[(N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision] + N[Max[N[Max[x, y], $MachinePrecision], t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 70.5%

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

2. Taylor expanded in y around inf

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 46.8%

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

4. Applied rewrites 46.8%

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

Alternative 13: 68.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_1 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_0\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{t\_1 \cdot \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_0\right)}\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fmax (fmin x y) z)) (t_1 (fmin (fmax x y) t_0)))
  (if (<= t_1 -5e-310)
    (* 2.0 (sqrt (* (fmin (fmin x y) z) t_1)))
    (* 2.0 (sqrt (* t_1 (fmax (fmax x y) t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = fmax(fmin(x, y), z);
	double t_1 = fmin(fmax(x, y), t_0);
	double tmp;
	if (t_1 <= -5e-310) {
		tmp = 2.0 * sqrt((fmin(fmin(x, y), z) * t_1));
	} else {
		tmp = 2.0 * sqrt((t_1 * fmax(fmax(x, y), t_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) :: t_1
    real(8) :: tmp
    t_0 = fmax(fmin(x, y), z)
    t_1 = fmin(fmax(x, y), t_0)
    if (t_1 <= (-5d-310)) then
        tmp = 2.0d0 * sqrt((fmin(fmin(x, y), z) * t_1))
    else
        tmp = 2.0d0 * sqrt((t_1 * fmax(fmax(x, y), t_0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = fmax(fmin(x, y), z);
	double t_1 = fmin(fmax(x, y), t_0);
	double tmp;
	if (t_1 <= -5e-310) {
		tmp = 2.0 * Math.sqrt((fmin(fmin(x, y), z) * t_1));
	} else {
		tmp = 2.0 * Math.sqrt((t_1 * fmax(fmax(x, y), t_0)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = fmax(fmin(x, y), z)
	t_1 = fmin(fmax(x, y), t_0)
	tmp = 0
	if t_1 <= -5e-310:
		tmp = 2.0 * math.sqrt((fmin(fmin(x, y), z) * t_1))
	else:
		tmp = 2.0 * math.sqrt((t_1 * fmax(fmax(x, y), t_0)))
	return tmp

Julia

function code(x, y, z)
	t_0 = fmax(fmin(x, y), z)
	t_1 = fmin(fmax(x, y), t_0)
	tmp = 0.0
	if (t_1 <= -5e-310)
		tmp = Float64(2.0 * sqrt(Float64(fmin(fmin(x, y), z) * t_1)));
	else
		tmp = Float64(2.0 * sqrt(Float64(t_1 * fmax(fmax(x, y), t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = max(min(x, y), z);
	t_1 = min(max(x, y), t_0);
	tmp = 0.0;
	if (t_1 <= -5e-310)
		tmp = 2.0 * sqrt((min(min(x, y), z) * t_1));
	else
		tmp = 2.0 * sqrt((t_1 * max(max(x, y), t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Max[x, y], $MachinePrecision], t$95$0], $MachinePrecision]}, If[LessEqual[t$95$1, -5e-310], N[(2.0 * N[Sqrt[N[(N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(t$95$1 * N[Max[N[Max[x, y], $MachinePrecision], t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.9999999999999847e-310

   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 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 24.1%

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

   4. Applied rewrites 24.1%

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

   if -4.9999999999999847e-310 < 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. lower-*.f64 25.0%

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

   4. Applied rewrites 25.0%

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

Alternative 14: 68.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_1 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_0\right)\\ \mathbf{if}\;t\_1 \leq -1.62 \cdot 10^{-304}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{\left|t\_1 \cdot \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_0\right)\right|}\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fmax (fmin x y) z)) (t_1 (fmin (fmax x y) t_0)))
  (if (<= t_1 -1.62e-304)
    (* 2.0 (sqrt (* (fmin (fmin x y) z) t_1)))
    (* 2.0 (sqrt (fabs (* t_1 (fmax (fmax x y) t_0))))))))

C

double code(double x, double y, double z) {
	double t_0 = fmax(fmin(x, y), z);
	double t_1 = fmin(fmax(x, y), t_0);
	double tmp;
	if (t_1 <= -1.62e-304) {
		tmp = 2.0 * sqrt((fmin(fmin(x, y), z) * t_1));
	} else {
		tmp = 2.0 * sqrt(fabs((t_1 * fmax(fmax(x, y), t_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) :: t_1
    real(8) :: tmp
    t_0 = fmax(fmin(x, y), z)
    t_1 = fmin(fmax(x, y), t_0)
    if (t_1 <= (-1.62d-304)) then
        tmp = 2.0d0 * sqrt((fmin(fmin(x, y), z) * t_1))
    else
        tmp = 2.0d0 * sqrt(abs((t_1 * fmax(fmax(x, y), t_0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = fmax(fmin(x, y), z);
	double t_1 = fmin(fmax(x, y), t_0);
	double tmp;
	if (t_1 <= -1.62e-304) {
		tmp = 2.0 * Math.sqrt((fmin(fmin(x, y), z) * t_1));
	} else {
		tmp = 2.0 * Math.sqrt(Math.abs((t_1 * fmax(fmax(x, y), t_0))));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = fmax(fmin(x, y), z)
	t_1 = fmin(fmax(x, y), t_0)
	tmp = 0
	if t_1 <= -1.62e-304:
		tmp = 2.0 * math.sqrt((fmin(fmin(x, y), z) * t_1))
	else:
		tmp = 2.0 * math.sqrt(math.fabs((t_1 * fmax(fmax(x, y), t_0))))
	return tmp

Julia

function code(x, y, z)
	t_0 = fmax(fmin(x, y), z)
	t_1 = fmin(fmax(x, y), t_0)
	tmp = 0.0
	if (t_1 <= -1.62e-304)
		tmp = Float64(2.0 * sqrt(Float64(fmin(fmin(x, y), z) * t_1)));
	else
		tmp = Float64(2.0 * sqrt(abs(Float64(t_1 * fmax(fmax(x, y), t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = max(min(x, y), z);
	t_1 = min(max(x, y), t_0);
	tmp = 0.0;
	if (t_1 <= -1.62e-304)
		tmp = 2.0 * sqrt((min(min(x, y), z) * t_1));
	else
		tmp = 2.0 * sqrt(abs((t_1 * max(max(x, y), t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Max[x, y], $MachinePrecision], t$95$0], $MachinePrecision]}, If[LessEqual[t$95$1, -1.62e-304], N[(2.0 * N[Sqrt[N[(N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[Abs[N[(t$95$1 * N[Max[N[Max[x, y], $MachinePrecision], t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6199999999999999e-304

   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 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 24.1%

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

   4. Applied rewrites 24.1%

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

   if -1.6199999999999999e-304 < 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. lower-*.f64 N/A

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

      2. lower-+.f64 48.5%

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

   4. Applied rewrites 48.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 25.0%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. rem-sqrt-square-rev N/A

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

      4. lower-fabs.f64 26.8%

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 26.8%

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

   9. Applied rewrites 26.8%

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

Alternative 15: 35.4% accurate, 1.1× speedup

Math

\[2 \cdot \sqrt{\mathsf{min}\left(x, z\right) \cdot \mathsf{min}\left(y, \mathsf{max}\left(x, z\right)\right)} \]

FPCore

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

C

double code(double x, double y, double z) {
	return 2.0 * sqrt((fmin(x, z) * fmin(y, fmax(x, 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((fmin(x, z) * fmin(y, fmax(x, z))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 0

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

   1. lower-sqrt.f64 N/A

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

   2. lower-*.f64 24.1%

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

4. Applied rewrites 24.1%

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

Reproduce

herbie shell --seed 2025212 
(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)))))