Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3

Percentage Accurate: 99.9% → 99.9%

Time: 6.9s

Alternatives: 10

Speedup: 1.7×

Specification

Math

\[\begin{array}{l} \\ x + \frac{\left|y - x\right|}{2} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (/ (fabs (- y x)) 2.0)))

C

double code(double x, double y) {
	return x + (fabs((y - x)) / 2.0);
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + (abs((y - x)) / 2.0d0)
end function

Java

public static double code(double x, double y) {
	return x + (Math.abs((y - x)) / 2.0);
}

Python

def code(x, y):
	return x + (math.fabs((y - x)) / 2.0)

Julia

function code(x, y)
	return Float64(x + Float64(abs(Float64(y - x)) / 2.0))
end

MATLAB

function tmp = code(x, y)
	tmp = x + (abs((y - x)) / 2.0);
end

Wolfram

code[x_, y_] := N[(x + N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{\left|y - x\right|}{2} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (/ (fabs (- y x)) 2.0)))

C

double code(double x, double y) {
	return x + (fabs((y - x)) / 2.0);
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + (abs((y - x)) / 2.0d0)
end function

Java

public static double code(double x, double y) {
	return x + (Math.abs((y - x)) / 2.0);
}

Python

def code(x, y):
	return x + (math.fabs((y - x)) / 2.0)

Julia

function code(x, y)
	return Float64(x + Float64(abs(Float64(y - x)) / 2.0))
end

MATLAB

function tmp = code(x, y)
	tmp = x + (abs((y - x)) / 2.0);
end

Wolfram

code[x_, y_] := N[(x + N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left|y - x\right|, 0.5, x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma (fabs (- y x)) 0.5 x))

C

double code(double x, double y) {
	return fma(fabs((y - x)), 0.5, x);
}

Julia

function code(x, y)
	return fma(abs(Float64(y - x)), 0.5, x)
end

Wolfram

code[x_, y_] := N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] * 0.5 + x), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x + \frac{\left|y - x\right|}{2} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]

5. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, 0.5, x\right)} \]

6. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(\left|y - x\right|, 0.5, x\right) \]
7. Add Preprocessing

Alternative 2: 61.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-99}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-205}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-59}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.7e-99)
   (* -0.5 y)
   (if (<= y 4.4e-205) (* 1.5 x) (if (<= y 3.5e-59) (* 0.5 x) (* 0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.7e-99) {
		tmp = -0.5 * y;
	} else if (y <= 4.4e-205) {
		tmp = 1.5 * x;
	} else if (y <= 3.5e-59) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * y;
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.7d-99)) then
        tmp = (-0.5d0) * y
    else if (y <= 4.4d-205) then
        tmp = 1.5d0 * x
    else if (y <= 3.5d-59) then
        tmp = 0.5d0 * x
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.7e-99) {
		tmp = -0.5 * y;
	} else if (y <= 4.4e-205) {
		tmp = 1.5 * x;
	} else if (y <= 3.5e-59) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.7e-99:
		tmp = -0.5 * y
	elif y <= 4.4e-205:
		tmp = 1.5 * x
	elif y <= 3.5e-59:
		tmp = 0.5 * x
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.7e-99)
		tmp = Float64(-0.5 * y);
	elseif (y <= 4.4e-205)
		tmp = Float64(1.5 * x);
	elseif (y <= 3.5e-59)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.7e-99)
		tmp = -0.5 * y;
	elseif (y <= 4.4e-205)
		tmp = 1.5 * x;
	elseif (y <= 3.5e-59)
		tmp = 0.5 * x;
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.7e-99], N[(-0.5 * y), $MachinePrecision], If[LessEqual[y, 4.4e-205], N[(1.5 * x), $MachinePrecision], If[LessEqual[y, 3.5e-59], N[(0.5 * x), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.7e-99

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, 0.5, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 78.4%

      \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, 0.5, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{2} \cdot \color{blue}{y} \]
   9. Step-by-step derivation

   10. Applied rewrites 70.0%

       \[\leadsto -0.5 \cdot \color{blue}{y} \]

   if -2.7e-99 < y < 4.40000000000000018e-205

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, 0.5, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 60.5%

      \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, 0.5, x\right) \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 63.8%

       \[\leadsto 1.5 \cdot \color{blue}{x} \]

   if 4.40000000000000018e-205 < y < 3.5000000000000001e-59

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left|y - x\right|}}{2} \]

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

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 72.0

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

   4. Applied rewrites 72.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x} \]
   6. Step-by-step derivation

      1. lower-*.f64 66.1

         \[\leadsto \color{blue}{0.5 \cdot x} \]

   7. Applied rewrites 66.1%

      \[\leadsto \color{blue}{0.5 \cdot x} \]

   if 3.5000000000000001e-59 < y

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left|y - x\right|}}{2} \]

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

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 84.2

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

   4. Applied rewrites 84.2%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot y} \]
   6. Step-by-step derivation

      1. lower-*.f64 78.4

         \[\leadsto \color{blue}{0.5 \cdot y} \]

   7. Applied rewrites 78.4%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-99}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-205}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-59}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3500000000:\\ \;\;\;\;\left(y + x\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-178}:\\ \;\;\;\;\mathsf{fma}\left(\left|-y\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 1.5 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3500000000.0)
   (* (+ y x) 0.5)
   (if (<= x 3.5e-178) (fma (fabs (- y)) 0.5 x) (fma -0.5 y (* 1.5 x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3500000000.0) {
		tmp = (y + x) * 0.5;
	} else if (x <= 3.5e-178) {
		tmp = fma(fabs(-y), 0.5, x);
	} else {
		tmp = fma(-0.5, y, (1.5 * x));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3500000000.0)
		tmp = Float64(Float64(y + x) * 0.5);
	elseif (x <= 3.5e-178)
		tmp = fma(abs(Float64(-y)), 0.5, x);
	else
		tmp = fma(-0.5, y, Float64(1.5 * x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3500000000.0], N[(N[(y + x), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 3.5e-178], N[(N[Abs[(-y)], $MachinePrecision] * 0.5 + x), $MachinePrecision], N[(-0.5 * y + N[(1.5 * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.5e9

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left|y - x\right|}}{2} \]

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

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 92.3

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

   4. Applied rewrites 92.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x + \frac{1}{2} \cdot y} \]
   6. Step-by-step derivation

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 93.1

         \[\leadsto \color{blue}{\left(y + x\right)} \cdot 0.5 \]

   7. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\left(y + x\right) \cdot 0.5} \]

   if -3.5e9 < x < 3.49999999999999983e-178

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, 0.5, x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\left|-1 \cdot y\right|, \frac{1}{2}, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 80.2%

      \[\leadsto \mathsf{fma}\left(\left|-y\right|, 0.5, x\right) \]

   if 3.49999999999999983e-178 < x

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, 0.5, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 82.7%

      \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, 0.5, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{2} \cdot y + \color{blue}{\frac{3}{2} \cdot x} \]
   9. Step-by-step derivation

   10. Applied rewrites 83.2%

       \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{y}, 1.5 \cdot x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3500000000:\\ \;\;\;\;\left(y + x\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-178}:\\ \;\;\;\;\mathsf{fma}\left(\left|-y\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 1.5 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3500000000:\\ \;\;\;\;\left(y + x\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\left|-y\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3500000000.0)
   (* (+ y x) 0.5)
   (if (<= x 5.2e+14) (fma (fabs (- y)) 0.5 x) (* 1.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3500000000.0) {
		tmp = (y + x) * 0.5;
	} else if (x <= 5.2e+14) {
		tmp = fma(fabs(-y), 0.5, x);
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3500000000.0)
		tmp = Float64(Float64(y + x) * 0.5);
	elseif (x <= 5.2e+14)
		tmp = fma(abs(Float64(-y)), 0.5, x);
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3500000000.0], N[(N[(y + x), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 5.2e+14], N[(N[Abs[(-y)], $MachinePrecision] * 0.5 + x), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.5e9

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left|y - x\right|}}{2} \]

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

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 92.3

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

   4. Applied rewrites 92.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x + \frac{1}{2} \cdot y} \]
   6. Step-by-step derivation

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 93.1

         \[\leadsto \color{blue}{\left(y + x\right)} \cdot 0.5 \]

   7. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\left(y + x\right) \cdot 0.5} \]

   if -3.5e9 < x < 5.2e14

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, 0.5, x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\left|-1 \cdot y\right|, \frac{1}{2}, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 76.0%

      \[\leadsto \mathsf{fma}\left(\left|-y\right|, 0.5, x\right) \]

   if 5.2e14 < x

   1. Initial program 99.8%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, 0.5, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 87.8%

      \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, 0.5, x\right) \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 79.1%

       \[\leadsto 1.5 \cdot \color{blue}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3500000000:\\ \;\;\;\;\left(y + x\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\left|-y\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1350000000:\\ \;\;\;\;\left(y + x\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+14}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1350000000.0)
   (* (+ y x) 0.5)
   (if (<= x 5.2e+14) (* (fabs (- y x)) 0.5) (* 1.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1350000000.0) {
		tmp = (y + x) * 0.5;
	} else if (x <= 5.2e+14) {
		tmp = fabs((y - x)) * 0.5;
	} else {
		tmp = 1.5 * x;
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1350000000.0d0)) then
        tmp = (y + x) * 0.5d0
    else if (x <= 5.2d+14) then
        tmp = abs((y - x)) * 0.5d0
    else
        tmp = 1.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1350000000.0) {
		tmp = (y + x) * 0.5;
	} else if (x <= 5.2e+14) {
		tmp = Math.abs((y - x)) * 0.5;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1350000000.0:
		tmp = (y + x) * 0.5
	elif x <= 5.2e+14:
		tmp = math.fabs((y - x)) * 0.5
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1350000000.0)
		tmp = Float64(Float64(y + x) * 0.5);
	elseif (x <= 5.2e+14)
		tmp = Float64(abs(Float64(y - x)) * 0.5);
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1350000000.0)
		tmp = (y + x) * 0.5;
	elseif (x <= 5.2e+14)
		tmp = abs((y - x)) * 0.5;
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1350000000.0], N[(N[(y + x), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 5.2e+14], N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35e9

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left|y - x\right|}}{2} \]

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

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 92.3

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

   4. Applied rewrites 92.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x + \frac{1}{2} \cdot y} \]
   6. Step-by-step derivation

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 93.1

         \[\leadsto \color{blue}{\left(y + x\right)} \cdot 0.5 \]

   7. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\left(y + x\right) \cdot 0.5} \]

   if -1.35e9 < x < 5.2e14

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left|y - x\right|} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 73.5%

      \[\leadsto \color{blue}{\left|x - y\right| \cdot 0.5} \]

   if 5.2e14 < x

   1. Initial program 99.8%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, 0.5, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 87.8%

      \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, 0.5, x\right) \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 79.1%

       \[\leadsto 1.5 \cdot \color{blue}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1350000000:\\ \;\;\;\;\left(y + x\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+14}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1350000000:\\ \;\;\;\;\left(y + x\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+14}:\\ \;\;\;\;\left|-y\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1350000000.0)
   (* (+ y x) 0.5)
   (if (<= x 1.95e+14) (* (fabs (- y)) 0.5) (* 1.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1350000000.0) {
		tmp = (y + x) * 0.5;
	} else if (x <= 1.95e+14) {
		tmp = fabs(-y) * 0.5;
	} else {
		tmp = 1.5 * x;
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1350000000.0d0)) then
        tmp = (y + x) * 0.5d0
    else if (x <= 1.95d+14) then
        tmp = abs(-y) * 0.5d0
    else
        tmp = 1.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1350000000.0) {
		tmp = (y + x) * 0.5;
	} else if (x <= 1.95e+14) {
		tmp = Math.abs(-y) * 0.5;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1350000000.0:
		tmp = (y + x) * 0.5
	elif x <= 1.95e+14:
		tmp = math.fabs(-y) * 0.5
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1350000000.0)
		tmp = Float64(Float64(y + x) * 0.5);
	elseif (x <= 1.95e+14)
		tmp = Float64(abs(Float64(-y)) * 0.5);
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1350000000.0)
		tmp = (y + x) * 0.5;
	elseif (x <= 1.95e+14)
		tmp = abs(-y) * 0.5;
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1350000000.0], N[(N[(y + x), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 1.95e+14], N[(N[Abs[(-y)], $MachinePrecision] * 0.5), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35e9

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left|y - x\right|}}{2} \]

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

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 92.3

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

   4. Applied rewrites 92.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x + \frac{1}{2} \cdot y} \]
   6. Step-by-step derivation

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 93.1

         \[\leadsto \color{blue}{\left(y + x\right)} \cdot 0.5 \]

   7. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\left(y + x\right) \cdot 0.5} \]

   if -1.35e9 < x < 1.95e14

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left|y - x\right|} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 73.5%

      \[\leadsto \color{blue}{\left|x - y\right| \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left|-1 \cdot y\right| \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 71.9%

      \[\leadsto \left|-y\right| \cdot 0.5 \]

   if 1.95e14 < x

   1. Initial program 99.8%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, 0.5, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 87.8%

      \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, 0.5, x\right) \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 79.1%

       \[\leadsto 1.5 \cdot \color{blue}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1350000000:\\ \;\;\;\;\left(y + x\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+14}:\\ \;\;\;\;\left|-y\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-99}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-246}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.7e-99)
   (* -0.5 y)
   (if (<= y 5.6e-246) (* 1.5 x) (* (+ y x) 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.7e-99) {
		tmp = -0.5 * y;
	} else if (y <= 5.6e-246) {
		tmp = 1.5 * x;
	} else {
		tmp = (y + x) * 0.5;
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.7d-99)) then
        tmp = (-0.5d0) * y
    else if (y <= 5.6d-246) then
        tmp = 1.5d0 * x
    else
        tmp = (y + x) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.7e-99) {
		tmp = -0.5 * y;
	} else if (y <= 5.6e-246) {
		tmp = 1.5 * x;
	} else {
		tmp = (y + x) * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.7e-99:
		tmp = -0.5 * y
	elif y <= 5.6e-246:
		tmp = 1.5 * x
	else:
		tmp = (y + x) * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.7e-99)
		tmp = Float64(-0.5 * y);
	elseif (y <= 5.6e-246)
		tmp = Float64(1.5 * x);
	else
		tmp = Float64(Float64(y + x) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.7e-99)
		tmp = -0.5 * y;
	elseif (y <= 5.6e-246)
		tmp = 1.5 * x;
	else
		tmp = (y + x) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.7e-99], N[(-0.5 * y), $MachinePrecision], If[LessEqual[y, 5.6e-246], N[(1.5 * x), $MachinePrecision], N[(N[(y + x), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.7e-99

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, 0.5, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 78.4%

      \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, 0.5, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{2} \cdot \color{blue}{y} \]
   9. Step-by-step derivation

   10. Applied rewrites 70.0%

       \[\leadsto -0.5 \cdot \color{blue}{y} \]

   if -2.7e-99 < y < 5.5999999999999999e-246

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, 0.5, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 64.1%

      \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, 0.5, x\right) \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 66.1%

       \[\leadsto 1.5 \cdot \color{blue}{x} \]

   if 5.5999999999999999e-246 < y

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left|y - x\right|}}{2} \]

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

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 77.0

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

   4. Applied rewrites 77.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x + \frac{1}{2} \cdot y} \]
   6. Step-by-step derivation

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 81.6

         \[\leadsto \color{blue}{\left(y + x\right)} \cdot 0.5 \]

   7. Applied rewrites 81.6%

      \[\leadsto \color{blue}{\left(y + x\right) \cdot 0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-99}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-246}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1350000000:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-147}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1350000000.0) (* 0.5 x) (if (<= x 7.2e-147) (* -0.5 y) (* 1.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1350000000.0) {
		tmp = 0.5 * x;
	} else if (x <= 7.2e-147) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1350000000.0d0)) then
        tmp = 0.5d0 * x
    else if (x <= 7.2d-147) then
        tmp = (-0.5d0) * y
    else
        tmp = 1.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1350000000.0) {
		tmp = 0.5 * x;
	} else if (x <= 7.2e-147) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1350000000.0:
		tmp = 0.5 * x
	elif x <= 7.2e-147:
		tmp = -0.5 * y
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1350000000.0)
		tmp = Float64(0.5 * x);
	elseif (x <= 7.2e-147)
		tmp = Float64(-0.5 * y);
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1350000000.0)
		tmp = 0.5 * x;
	elseif (x <= 7.2e-147)
		tmp = -0.5 * y;
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1350000000.0], N[(0.5 * x), $MachinePrecision], If[LessEqual[x, 7.2e-147], N[(-0.5 * y), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35e9

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left|y - x\right|}}{2} \]

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

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 92.3

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

   4. Applied rewrites 92.3%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x} \]
   6. Step-by-step derivation

      1. lower-*.f64 83.6

         \[\leadsto \color{blue}{0.5 \cdot x} \]

   7. Applied rewrites 83.6%

      \[\leadsto \color{blue}{0.5 \cdot x} \]

   if -1.35e9 < x < 7.20000000000000023e-147

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, 0.5, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 41.1%

      \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, 0.5, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{2} \cdot \color{blue}{y} \]
   9. Step-by-step derivation

   10. Applied rewrites 37.5%

       \[\leadsto -0.5 \cdot \color{blue}{y} \]

   if 7.20000000000000023e-147 < x

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, 0.5, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 82.5%

      \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, 0.5, x\right) \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 67.5%

       \[\leadsto 1.5 \cdot \color{blue}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1350000000:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-147}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-99}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -2.7e-99) (* -0.5 y) (* 1.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.7e-99) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.7d-99)) then
        tmp = (-0.5d0) * y
    else
        tmp = 1.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.7e-99) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.7e-99:
		tmp = -0.5 * y
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.7e-99)
		tmp = Float64(-0.5 * y);
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.7e-99)
		tmp = -0.5 * y;
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.7e-99], N[(-0.5 * y), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.7e-99

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, 0.5, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 78.4%

      \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, 0.5, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{2} \cdot \color{blue}{y} \]
   9. Step-by-step derivation

   10. Applied rewrites 70.0%

       \[\leadsto -0.5 \cdot \color{blue}{y} \]

   if -2.7e-99 < y

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, 0.5, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 35.9%

      \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, 0.5, x\right) \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 40.4%

       \[\leadsto 1.5 \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 26.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* -0.5 y))

C

double code(double x, double y) {
	return -0.5 * y;
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (-0.5d0) * y
end function

Java

public static double code(double x, double y) {
	return -0.5 * y;
}

Python

def code(x, y):
	return -0.5 * y

Julia

function code(x, y)
	return Float64(-0.5 * y)
end

MATLAB

function tmp = code(x, y)
	tmp = -0.5 * y;
end

Wolfram

code[x_, y_] := N[(-0.5 * y), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x + \frac{\left|y - x\right|}{2} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]

5. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, 0.5, x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 48.5%

   \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, 0.5, x\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{-1}{2} \cdot \color{blue}{y} \]
9. Step-by-step derivation

10. Applied rewrites 23.2%

    \[\leadsto -0.5 \cdot \color{blue}{y} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025017 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))