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

Percentage Accurate: 99.9% → 99.9%

Time: 5.5s

Alternatives: 9

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: 48.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \frac{\left|y - x\right|}{2} \leq 2 \cdot 10^{-160}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ x (/ (fabs (- y x)) 2.0)) 2e-160) (* 0.5 x) (* 0.5 y)))

C

double code(double x, double y) {
	double tmp;
	if ((x + (fabs((y - x)) / 2.0)) <= 2e-160) {
		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 ((x + (abs((y - x)) / 2.0d0)) <= 2d-160) 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 ((x + (Math.abs((y - x)) / 2.0)) <= 2e-160) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x + (math.fabs((y - x)) / 2.0)) <= 2e-160:
		tmp = 0.5 * x
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x + Float64(abs(Float64(y - x)) / 2.0)) <= 2e-160)
		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 ((x + (abs((y - x)) / 2.0)) <= 2e-160)
		tmp = 0.5 * x;
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x + N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2e-160], N[(0.5 * x), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (fabs.f64 (-.f64 y x)) #s(literal 2 binary64))) < 2e-160

   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. rem-square-sqrt 87.3

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. difference-of-squares N/A

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

      8. lift--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 87.3

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

   4. Applied rewrites 87.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 87.3

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

   7. Applied rewrites 87.3%

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

   if 2e-160 < (+.f64 x (/.f64 (fabs.f64 (-.f64 y x)) #s(literal 2 binary64)))

   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. rem-square-sqrt 40.5

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. difference-of-squares N/A

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

      8. lift--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 40.5

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

   4. Applied rewrites 40.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 34.2

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

   7. Applied rewrites 34.2%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 56.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-84}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-232}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-214}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.2e-84)
   (* 0.5 x)
   (if (<= x -7.5e-232) (* -0.5 y) (if (<= x 3.4e-214) (* 0.5 y) (* 1.5 x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.2e-84) {
		tmp = 0.5 * x;
	} else if (x <= -7.5e-232) {
		tmp = -0.5 * y;
	} else if (x <= 3.4e-214) {
		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 <= (-1.2d-84)) then
        tmp = 0.5d0 * x
    else if (x <= (-7.5d-232)) then
        tmp = (-0.5d0) * y
    else if (x <= 3.4d-214) 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 <= -1.2e-84) {
		tmp = 0.5 * x;
	} else if (x <= -7.5e-232) {
		tmp = -0.5 * y;
	} else if (x <= 3.4e-214) {
		tmp = 0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.2e-84:
		tmp = 0.5 * x
	elif x <= -7.5e-232:
		tmp = -0.5 * y
	elif x <= 3.4e-214:
		tmp = 0.5 * y
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.2e-84)
		tmp = Float64(0.5 * x);
	elseif (x <= -7.5e-232)
		tmp = Float64(-0.5 * y);
	elseif (x <= 3.4e-214)
		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 <= -1.2e-84)
		tmp = 0.5 * x;
	elseif (x <= -7.5e-232)
		tmp = -0.5 * y;
	elseif (x <= 3.4e-214)
		tmp = 0.5 * y;
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.2e-84], N[(0.5 * x), $MachinePrecision], If[LessEqual[x, -7.5e-232], N[(-0.5 * y), $MachinePrecision], If[LessEqual[x, 3.4e-214], N[(0.5 * y), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.20000000000000009e-84

   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. rem-square-sqrt 90.8

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. difference-of-squares N/A

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

      8. lift--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 90.8

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

   4. Applied rewrites 90.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 75.1

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

   7. Applied rewrites 75.1%

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

   if -1.20000000000000009e-84 < x < -7.5000000000000006e-232

   1. Initial program 99.9%

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

   4. Applied rewrites 55.8%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

         \[\leadsto \color{blue}{\frac{-1}{2}} \cdot y \]

      4. lower-*.f64 57.2

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

   7. Applied rewrites 57.2%

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

   if -7.5000000000000006e-232 < x < 3.3999999999999999e-214

   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. rem-square-sqrt 64.4

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. difference-of-squares N/A

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

      8. lift--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 64.4

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

   4. Applied rewrites 64.4%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 64.2

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

   7. Applied rewrites 64.2%

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

   if 3.3999999999999999e-214 < x

   1. Initial program 99.8%

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

         \[\leadsto x \cdot \color{blue}{\frac{3}{2}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{3}{2} \cdot x} \]

      7. lower-*.f64 63.9

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

   7. Applied rewrites 63.9%

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

Alternative 4: 82.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-84}:\\ \;\;\;\;\left(y + x\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-195}:\\ \;\;\;\;\mathsf{fma}\left(\left|-y\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, -0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.2e-84)
   (* (+ y x) 0.5)
   (if (<= x 1.4e-195) (fma (fabs (- y)) 0.5 x) (fma (- y x) -0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.2e-84) {
		tmp = (y + x) * 0.5;
	} else if (x <= 1.4e-195) {
		tmp = fma(fabs(-y), 0.5, x);
	} else {
		tmp = fma((y - x), -0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.2e-84)
		tmp = Float64(Float64(y + x) * 0.5);
	elseif (x <= 1.4e-195)
		tmp = fma(abs(Float64(-y)), 0.5, x);
	else
		tmp = fma(Float64(y - x), -0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.2e-84], N[(N[(y + x), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 1.4e-195], N[(N[Abs[(-y)], $MachinePrecision] * 0.5 + x), $MachinePrecision], N[(N[(y - x), $MachinePrecision] * -0.5 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.1999999999999999e-84

   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. rem-square-sqrt 90.8

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. difference-of-squares N/A

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

      8. lift--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 90.8

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

   4. Applied rewrites 90.8%

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

   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 90.8

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

   7. Applied rewrites 90.8%

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

   if -3.1999999999999999e-84 < x < 1.40000000000000002e-195

   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 91.9%

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

   if 1.40000000000000002e-195 < x

   1. Initial program 99.8%

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

   4. Applied rewrites 85.7%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

         \[\leadsto \color{blue}{\frac{-1}{2}} \cdot y \]

      4. lower-*.f64 22.1

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

   7. Applied rewrites 22.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 86.2%

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

Alternative 5: 82.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-84}:\\ \;\;\;\;\left(y + x\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-195}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, -0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.2e-84)
   (* (+ y x) 0.5)
   (if (<= x 1.4e-195) (* (fabs (- y x)) 0.5) (fma (- y x) -0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.2e-84) {
		tmp = (y + x) * 0.5;
	} else if (x <= 1.4e-195) {
		tmp = fabs((y - x)) * 0.5;
	} else {
		tmp = fma((y - x), -0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.2e-84)
		tmp = Float64(Float64(y + x) * 0.5);
	elseif (x <= 1.4e-195)
		tmp = Float64(abs(Float64(y - x)) * 0.5);
	else
		tmp = fma(Float64(y - x), -0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.2e-84], N[(N[(y + x), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 1.4e-195], N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(y - x), $MachinePrecision] * -0.5 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.20000000000000009e-84

   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. rem-square-sqrt 90.8

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. difference-of-squares N/A

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

      8. lift--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 90.8

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

   4. Applied rewrites 90.8%

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

   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 90.8

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

   7. Applied rewrites 90.8%

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

   if -1.20000000000000009e-84 < x < 1.40000000000000002e-195

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left|y - x\right|} \]
   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 91.1%

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

   if 1.40000000000000002e-195 < x

   1. Initial program 99.8%

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

   4. Applied rewrites 85.7%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

         \[\leadsto \color{blue}{\frac{-1}{2}} \cdot y \]

      4. lower-*.f64 22.1

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

   7. Applied rewrites 22.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 86.2%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-84}:\\ \;\;\;\;\left(y + x\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-195}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, -0.5, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-5}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-189}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9e-5) (* -0.5 y) (if (<= y 6e-189) (* 1.5 x) (* (+ y x) 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9e-5) {
		tmp = -0.5 * y;
	} else if (y <= 6e-189) {
		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 <= (-9d-5)) then
        tmp = (-0.5d0) * y
    else if (y <= 6d-189) 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 <= -9e-5) {
		tmp = -0.5 * y;
	} else if (y <= 6e-189) {
		tmp = 1.5 * x;
	} else {
		tmp = (y + x) * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9e-5:
		tmp = -0.5 * y
	elif y <= 6e-189:
		tmp = 1.5 * x
	else:
		tmp = (y + x) * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9e-5)
		tmp = Float64(-0.5 * y);
	elseif (y <= 6e-189)
		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 <= -9e-5)
		tmp = -0.5 * y;
	elseif (y <= 6e-189)
		tmp = 1.5 * x;
	else
		tmp = (y + x) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9e-5], N[(-0.5 * y), $MachinePrecision], If[LessEqual[y, 6e-189], N[(1.5 * x), $MachinePrecision], N[(N[(y + x), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.00000000000000057e-5

   1. Initial program 100.0%

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

   4. Applied rewrites 89.1%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

         \[\leadsto \color{blue}{\frac{-1}{2}} \cdot y \]

      4. lower-*.f64 77.3

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

   7. Applied rewrites 77.3%

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

   if -9.00000000000000057e-5 < y < 6e-189

   1. Initial program 99.9%

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

   4. Applied rewrites 64.2%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

         \[\leadsto x \cdot \color{blue}{\frac{3}{2}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{3}{2} \cdot x} \]

      7. lower-*.f64 54.8

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

   7. Applied rewrites 54.8%

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

   if 6e-189 < 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. rem-square-sqrt 84.2

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. difference-of-squares N/A

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

      8. lift--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 84.2

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

   4. Applied rewrites 84.2%

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

   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 84.2

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

   7. Applied rewrites 84.2%

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

Alternative 7: 60.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-5}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-117}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.5e-5) (* -0.5 y) (if (<= y 2.4e-117) (* 0.5 x) (* 0.5 y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.5e-5) {
		tmp = -0.5 * y;
	} else if (y <= 2.4e-117) {
		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 <= (-8.5d-5)) then
        tmp = (-0.5d0) * y
    else if (y <= 2.4d-117) 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 <= -8.5e-5) {
		tmp = -0.5 * y;
	} else if (y <= 2.4e-117) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.5e-5:
		tmp = -0.5 * y
	elif y <= 2.4e-117:
		tmp = 0.5 * x
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.5e-5)
		tmp = Float64(-0.5 * y);
	elseif (y <= 2.4e-117)
		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 <= -8.5e-5)
		tmp = -0.5 * y;
	elseif (y <= 2.4e-117)
		tmp = 0.5 * x;
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.5e-5], N[(-0.5 * y), $MachinePrecision], If[LessEqual[y, 2.4e-117], N[(0.5 * x), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.500000000000001e-5

   1. Initial program 100.0%

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

   4. Applied rewrites 89.1%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

         \[\leadsto \color{blue}{\frac{-1}{2}} \cdot y \]

      4. lower-*.f64 77.3

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

   7. Applied rewrites 77.3%

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

   if -8.500000000000001e-5 < y < 2.40000000000000014e-117

   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. rem-square-sqrt 49.1

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. difference-of-squares N/A

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

      8. lift--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 49.1

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

   4. Applied rewrites 49.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 46.9

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

   7. Applied rewrites 46.9%

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

   if 2.40000000000000014e-117 < 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. rem-square-sqrt 86.6

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. difference-of-squares N/A

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

      8. lift--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 86.6

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

   4. Applied rewrites 86.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 69.0

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

   7. Applied rewrites 69.0%

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

Alternative 8: 78.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-189}:\\ \;\;\;\;\mathsf{fma}\left(y - x, -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6e-189) (fma (- y x) -0.5 x) (* (+ y x) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6e-189) {
		tmp = fma((y - x), -0.5, x);
	} else {
		tmp = (y + x) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6e-189)
		tmp = fma(Float64(y - x), -0.5, x);
	else
		tmp = Float64(Float64(y + x) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6e-189], N[(N[(y - x), $MachinePrecision] * -0.5 + x), $MachinePrecision], N[(N[(y + x), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6e-189

   1. Initial program 99.9%

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

   4. Applied rewrites 73.9%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

         \[\leadsto \color{blue}{\frac{-1}{2}} \cdot y \]

      4. lower-*.f64 40.5

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

   7. Applied rewrites 40.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 78.8%

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

   if 6e-189 < 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. rem-square-sqrt 84.2

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. difference-of-squares N/A

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

      8. lift--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 84.2

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

   4. Applied rewrites 84.2%

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

   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 84.2

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

   7. Applied rewrites 84.2%

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

Alternative 9: 30.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot x \end{array} \]

FPCore

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

C

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

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 * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. rem-square-sqrt 53.3

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

   5. lift--.f64 N/A

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

   6. flip-- N/A

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

   7. difference-of-squares N/A

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

   8. lift--.f64 N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-+.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-+.f64 53.3

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

4. Applied rewrites 53.3%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 30.1

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

7. Applied rewrites 30.1%

   \[\leadsto \color{blue}{0.5 \cdot x} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024364 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))