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

Percentage Accurate: 99.9% → 99.9%

Time: 2.0s

Alternatives: 10

Speedup: 1.1×

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, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. fabs-sub N/A

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

   4. *-lft-identity N/A

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

   5. metadata-eval N/A

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

   6. fabs-sub N/A

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

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

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

   8. remove-double-neg N/A

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

   9. mul-1-neg N/A

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

   10. mul-1-neg N/A

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

   11. distribute-neg-in N/A

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

   12. +-commutative N/A

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

   13. fabs-neg N/A

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

   14. lower-fabs.f64 N/A

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

   15. *-commutative N/A

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

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

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

   17. distribute-lft-neg-in N/A

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

   18. distribute-rgt-neg-out N/A

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

   19. metadata-eval N/A

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

   20. *-rgt-identity N/A

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

   21. lower--.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 82.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+20}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|-y\right|, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.4e+20)
   (* (- x y) 0.5)
   (if (<= x 9.5e-80) (fma 0.5 (fabs (- y)) x) (fma 1.5 x (* -0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.4e+20) {
		tmp = (x - y) * 0.5;
	} else if (x <= 9.5e-80) {
		tmp = fma(0.5, fabs(-y), x);
	} else {
		tmp = fma(1.5, x, (-0.5 * y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.4e+20)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (x <= 9.5e-80)
		tmp = fma(0.5, abs(Float64(-y)), x);
	else
		tmp = fma(1.5, x, Float64(-0.5 * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.4e+20], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 9.5e-80], N[(0.5 * N[Abs[(-y)], $MachinePrecision] + x), $MachinePrecision], N[(1.5 * x + N[(-0.5 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.4e20

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 88.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, \frac{0.5}{x}, 1\right) \cdot x} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. lift--.f64 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-*r/ N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. +-commutative N/A

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

      12. distribute-lft-in N/A

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

      13. *-commutative N/A

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

      14. associate-*r/ N/A

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

      15. associate-*l/ N/A

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

      16. associate-*l* N/A

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

      17. *-rgt-identity N/A

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

   6. Applied rewrites 65.0%

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

   7. Taylor expanded in x around 0

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

      1. fabs-sub N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

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

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

      5. sqrt-unprod N/A

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

      6. rem-square-sqrt N/A

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

      7. lift--.f64 53.9

         \[\leadsto \left(x - y\right) \cdot 0.5 \]

   9. Applied rewrites 53.9%

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

   if -3.4e20 < x < 9.5000000000000003e-80

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. fabs-sub N/A

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

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

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

      8. remove-double-neg N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-in N/A

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

      12. +-commutative N/A

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

      13. fabs-neg N/A

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

      14. lower-fabs.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. distribute-lft-neg-in N/A

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

      18. distribute-rgt-neg-out N/A

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

      19. metadata-eval N/A

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

      20. *-rgt-identity N/A

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

      21. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(y\right)\right|, x\right) \]

      2. lower-neg.f64 58.7

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

   7. Applied rewrites 58.7%

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

   if 9.5000000000000003e-80 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. fabs-sub N/A

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

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

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

      8. remove-double-neg N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-in N/A

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

      12. +-commutative N/A

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

      13. fabs-neg N/A

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

      14. lower-fabs.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. distribute-lft-neg-in N/A

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

      18. distribute-rgt-neg-out N/A

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

      19. metadata-eval N/A

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

      20. *-rgt-identity N/A

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

      21. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower-fabs.f64 N/A

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

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

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

      7. sqrt-prod N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lift--.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

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

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. lift--.f64 49.7

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

   6. Applied rewrites 49.7%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{3}{2} \cdot x + \frac{-1}{2} \cdot \color{blue}{y} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{3}{2}, x, \frac{-1}{2} \cdot y\right) \]

      3. lower-*.f64 54.8

         \[\leadsto \mathsf{fma}\left(1.5, x, -0.5 \cdot y\right) \]

   9. Applied rewrites 54.8%

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

Alternative 3: 82.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+20}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|-y\right|, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.4e+20)
   (* (- x y) 0.5)
   (if (<= x 9.5e-80) (fma 0.5 (fabs (- y)) x) (fma (- x y) 0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.4e+20) {
		tmp = (x - y) * 0.5;
	} else if (x <= 9.5e-80) {
		tmp = fma(0.5, fabs(-y), x);
	} else {
		tmp = fma((x - y), 0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.4e+20)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (x <= 9.5e-80)
		tmp = fma(0.5, abs(Float64(-y)), x);
	else
		tmp = fma(Float64(x - y), 0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.4e+20], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 9.5e-80], N[(0.5 * N[Abs[(-y)], $MachinePrecision] + x), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * 0.5 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.4e20

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 88.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, \frac{0.5}{x}, 1\right) \cdot x} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. lift--.f64 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-*r/ N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. +-commutative N/A

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

      12. distribute-lft-in N/A

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

      13. *-commutative N/A

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

      14. associate-*r/ N/A

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

      15. associate-*l/ N/A

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

      16. associate-*l* N/A

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

      17. *-rgt-identity N/A

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

   6. Applied rewrites 65.0%

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

   7. Taylor expanded in x around 0

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

      1. fabs-sub N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

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

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

      5. sqrt-unprod N/A

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

      6. rem-square-sqrt N/A

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

      7. lift--.f64 53.9

         \[\leadsto \left(x - y\right) \cdot 0.5 \]

   9. Applied rewrites 53.9%

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

   if -3.4e20 < x < 9.5000000000000003e-80

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. fabs-sub N/A

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

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

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

      8. remove-double-neg N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-in N/A

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

      12. +-commutative N/A

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

      13. fabs-neg N/A

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

      14. lower-fabs.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. distribute-lft-neg-in N/A

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

      18. distribute-rgt-neg-out N/A

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

      19. metadata-eval N/A

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

      20. *-rgt-identity N/A

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

      21. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(y\right)\right|, x\right) \]

      2. lower-neg.f64 58.7

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

   7. Applied rewrites 58.7%

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

   if 9.5000000000000003e-80 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. fabs-sub N/A

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

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

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

      8. remove-double-neg N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-in N/A

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

      12. +-commutative N/A

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

      13. fabs-neg N/A

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

      14. lower-fabs.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. distribute-lft-neg-in N/A

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

      18. distribute-rgt-neg-out N/A

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

      19. metadata-eval N/A

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

      20. *-rgt-identity N/A

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

      21. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower-fabs.f64 N/A

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

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

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

      7. sqrt-prod N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lift--.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

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

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. lift--.f64 49.7

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

   6. Applied rewrites 49.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. sqrt-unprod N/A

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

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

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

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

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

      12. sqrt-unprod N/A

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

      13. rem-square-sqrt N/A

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

      14. lift--.f64 54.7

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

   8. Applied rewrites 54.7%

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

Alternative 4: 79.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+20}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|-y\right|, x\right)\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.4e+20)
   (* (- x y) 0.5)
   (if (<= x 4e-22) (fma 0.5 (fabs (- y)) x) (* 1.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.4e+20) {
		tmp = (x - y) * 0.5;
	} else if (x <= 4e-22) {
		tmp = fma(0.5, fabs(-y), x);
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.4e+20)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (x <= 4e-22)
		tmp = fma(0.5, abs(Float64(-y)), x);
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.4e+20], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 4e-22], N[(0.5 * N[Abs[(-y)], $MachinePrecision] + x), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.4e20

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 88.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, \frac{0.5}{x}, 1\right) \cdot x} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. lift--.f64 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-*r/ N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. +-commutative N/A

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

      12. distribute-lft-in N/A

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

      13. *-commutative N/A

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

      14. associate-*r/ N/A

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

      15. associate-*l/ N/A

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

      16. associate-*l* N/A

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

      17. *-rgt-identity N/A

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

   6. Applied rewrites 65.0%

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

   7. Taylor expanded in x around 0

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

      1. fabs-sub N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

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

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

      5. sqrt-unprod N/A

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

      6. rem-square-sqrt N/A

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

      7. lift--.f64 53.9

         \[\leadsto \left(x - y\right) \cdot 0.5 \]

   9. Applied rewrites 53.9%

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

   if -3.4e20 < x < 4.0000000000000002e-22

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. fabs-sub N/A

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

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

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

      8. remove-double-neg N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-in N/A

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

      12. +-commutative N/A

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

      13. fabs-neg N/A

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

      14. lower-fabs.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. distribute-lft-neg-in N/A

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

      18. distribute-rgt-neg-out N/A

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

      19. metadata-eval N/A

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

      20. *-rgt-identity N/A

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

      21. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(y\right)\right|, x\right) \]

      2. lower-neg.f64 58.7

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

   7. Applied rewrites 58.7%

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

   if 4.0000000000000002e-22 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. fabs-sub N/A

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

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

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

      8. remove-double-neg N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-in N/A

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

      12. +-commutative N/A

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

      13. fabs-neg N/A

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

      14. lower-fabs.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. distribute-lft-neg-in N/A

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

      18. distribute-rgt-neg-out N/A

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

      19. metadata-eval N/A

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

      20. *-rgt-identity N/A

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

      21. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower-fabs.f64 N/A

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

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

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

      7. sqrt-prod N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lift--.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

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

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. lift--.f64 49.7

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

   6. Applied rewrites 49.7%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 30.4

         \[\leadsto 1.5 \cdot x \]

   9. Applied rewrites 30.4%

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

Alternative 5: 78.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-124}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-22}:\\ \;\;\;\;0.5 \cdot \left|x - y\right|\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.05e-124)
   (* (- x y) 0.5)
   (if (<= x 4e-22) (* 0.5 (fabs (- x y))) (* 1.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.05e-124) {
		tmp = (x - y) * 0.5;
	} else if (x <= 4e-22) {
		tmp = 0.5 * fabs((x - 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 <= (-1.05d-124)) then
        tmp = (x - y) * 0.5d0
    else if (x <= 4d-22) then
        tmp = 0.5d0 * abs((x - 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 <= -1.05e-124) {
		tmp = (x - y) * 0.5;
	} else if (x <= 4e-22) {
		tmp = 0.5 * Math.abs((x - y));
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.05e-124:
		tmp = (x - y) * 0.5
	elif x <= 4e-22:
		tmp = 0.5 * math.fabs((x - y))
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.05e-124)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (x <= 4e-22)
		tmp = Float64(0.5 * abs(Float64(x - y)));
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.05e-124)
		tmp = (x - y) * 0.5;
	elseif (x <= 4e-22)
		tmp = 0.5 * abs((x - y));
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.05e-124], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 4e-22], N[(0.5 * N[Abs[N[(x - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05e-124

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 88.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, \frac{0.5}{x}, 1\right) \cdot x} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. lift--.f64 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-*r/ N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. +-commutative N/A

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

      12. distribute-lft-in N/A

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

      13. *-commutative N/A

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

      14. associate-*r/ N/A

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

      15. associate-*l/ N/A

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

      16. associate-*l* N/A

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

      17. *-rgt-identity N/A

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

   6. Applied rewrites 65.0%

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

   7. Taylor expanded in x around 0

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

      1. fabs-sub N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

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

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

      5. sqrt-unprod N/A

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

      6. rem-square-sqrt N/A

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

      7. lift--.f64 53.9

         \[\leadsto \left(x - y\right) \cdot 0.5 \]

   9. Applied rewrites 53.9%

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

   if -1.05e-124 < x < 4.0000000000000002e-22

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. fabs-sub N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fabs-sub N/A

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

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

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

      7. remove-double-neg N/A

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

      8. mul-1-neg N/A

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

      9. mul-1-neg N/A

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

      10. distribute-neg-in N/A

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

      11. +-commutative N/A

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

      12. fabs-neg N/A

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

      13. lower-fabs.f64 N/A

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

      14. *-commutative N/A

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

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

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

      16. distribute-lft-neg-in N/A

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

      17. distribute-rgt-neg-out N/A

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

      18. metadata-eval N/A

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

      19. *-rgt-identity N/A

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

      20. lower--.f64 53.6

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

   4. Applied rewrites 53.6%

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

   if 4.0000000000000002e-22 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. fabs-sub N/A

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

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

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

      8. remove-double-neg N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-in N/A

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

      12. +-commutative N/A

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

      13. fabs-neg N/A

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

      14. lower-fabs.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. distribute-lft-neg-in N/A

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

      18. distribute-rgt-neg-out N/A

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

      19. metadata-eval N/A

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

      20. *-rgt-identity N/A

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

      21. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower-fabs.f64 N/A

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

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

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

      7. sqrt-prod N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lift--.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

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

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. lift--.f64 49.7

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

   6. Applied rewrites 49.7%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 30.4

         \[\leadsto 1.5 \cdot x \]

   9. Applied rewrites 30.4%

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

Alternative 6: 78.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-124}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-22}:\\ \;\;\;\;0.5 \cdot \left|-y\right|\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.05e-124)
   (* (- x y) 0.5)
   (if (<= x 1.45e-22) (* 0.5 (fabs (- y))) (* 1.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.05e-124) {
		tmp = (x - y) * 0.5;
	} else if (x <= 1.45e-22) {
		tmp = 0.5 * fabs(-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 <= (-1.05d-124)) then
        tmp = (x - y) * 0.5d0
    else if (x <= 1.45d-22) then
        tmp = 0.5d0 * abs(-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 <= -1.05e-124) {
		tmp = (x - y) * 0.5;
	} else if (x <= 1.45e-22) {
		tmp = 0.5 * Math.abs(-y);
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.05e-124:
		tmp = (x - y) * 0.5
	elif x <= 1.45e-22:
		tmp = 0.5 * math.fabs(-y)
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.05e-124)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (x <= 1.45e-22)
		tmp = Float64(0.5 * abs(Float64(-y)));
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.05e-124)
		tmp = (x - y) * 0.5;
	elseif (x <= 1.45e-22)
		tmp = 0.5 * abs(-y);
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.05e-124], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 1.45e-22], N[(0.5 * N[Abs[(-y)], $MachinePrecision]), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05e-124

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 88.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, \frac{0.5}{x}, 1\right) \cdot x} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. lift--.f64 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-*r/ N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. +-commutative N/A

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

      12. distribute-lft-in N/A

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

      13. *-commutative N/A

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

      14. associate-*r/ N/A

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

      15. associate-*l/ N/A

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

      16. associate-*l* N/A

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

      17. *-rgt-identity N/A

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

   6. Applied rewrites 65.0%

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

   7. Taylor expanded in x around 0

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

      1. fabs-sub N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

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

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

      5. sqrt-unprod N/A

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

      6. rem-square-sqrt N/A

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

      7. lift--.f64 53.9

         \[\leadsto \left(x - y\right) \cdot 0.5 \]

   9. Applied rewrites 53.9%

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

   if -1.05e-124 < x < 1.4500000000000001e-22

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. fabs-sub N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fabs-sub N/A

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

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

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

      7. remove-double-neg N/A

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

      8. mul-1-neg N/A

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

      9. mul-1-neg N/A

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

      10. distribute-neg-in N/A

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

      11. +-commutative N/A

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

      12. fabs-neg N/A

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

      13. lower-fabs.f64 N/A

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

      14. *-commutative N/A

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

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

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

      16. distribute-lft-neg-in N/A

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

      17. distribute-rgt-neg-out N/A

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

      18. metadata-eval N/A

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

      19. *-rgt-identity N/A

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

      20. lower--.f64 53.6

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

   4. Applied rewrites 53.6%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left|\mathsf{neg}\left(y\right)\right| \]

      2. lower-neg.f64 50.6

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

   7. Applied rewrites 50.6%

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

   if 1.4500000000000001e-22 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. fabs-sub N/A

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

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

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

      8. remove-double-neg N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-in N/A

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

      12. +-commutative N/A

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

      13. fabs-neg N/A

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

      14. lower-fabs.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. distribute-lft-neg-in N/A

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

      18. distribute-rgt-neg-out N/A

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

      19. metadata-eval N/A

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

      20. *-rgt-identity N/A

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

      21. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower-fabs.f64 N/A

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

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

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

      7. sqrt-prod N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lift--.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

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

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. lift--.f64 49.7

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

   6. Applied rewrites 49.7%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 30.4

         \[\leadsto 1.5 \cdot x \]

   9. Applied rewrites 30.4%

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

Alternative 7: 73.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+84}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-22}:\\ \;\;\;\;0.5 \cdot \left|-y\right|\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.8e+84)
   (* 0.5 x)
   (if (<= x 1.45e-22) (* 0.5 (fabs (- y))) (* 1.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.8e+84) {
		tmp = 0.5 * x;
	} else if (x <= 1.45e-22) {
		tmp = 0.5 * fabs(-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 <= (-3.8d+84)) then
        tmp = 0.5d0 * x
    else if (x <= 1.45d-22) then
        tmp = 0.5d0 * abs(-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 <= -3.8e+84) {
		tmp = 0.5 * x;
	} else if (x <= 1.45e-22) {
		tmp = 0.5 * Math.abs(-y);
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.8e+84:
		tmp = 0.5 * x
	elif x <= 1.45e-22:
		tmp = 0.5 * math.fabs(-y)
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.8e+84)
		tmp = Float64(0.5 * x);
	elseif (x <= 1.45e-22)
		tmp = Float64(0.5 * abs(Float64(-y)));
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.8e+84)
		tmp = 0.5 * x;
	elseif (x <= 1.45e-22)
		tmp = 0.5 * abs(-y);
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.8e+84], N[(0.5 * x), $MachinePrecision], If[LessEqual[x, 1.45e-22], N[(0.5 * N[Abs[(-y)], $MachinePrecision]), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.8000000000000001e84

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 88.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, \frac{0.5}{x}, 1\right) \cdot x} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. lift--.f64 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-*r/ N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. +-commutative N/A

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

      12. distribute-lft-in N/A

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

      13. *-commutative N/A

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

      14. associate-*r/ N/A

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

      15. associate-*l/ N/A

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

      16. associate-*l* N/A

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

      17. *-rgt-identity N/A

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

   6. Applied rewrites 65.0%

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

   7. Taylor expanded in x around 0

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

      1. fabs-sub N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

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

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

      5. sqrt-unprod N/A

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

      6. rem-square-sqrt N/A

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

      7. lift--.f64 53.9

         \[\leadsto \left(x - y\right) \cdot 0.5 \]

   9. Applied rewrites 53.9%

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

   10. Taylor expanded in x around inf

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

       1. lower-*.f64 30.6

          \[\leadsto 0.5 \cdot x \]

   12. Applied rewrites 30.6%

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

   if -3.8000000000000001e84 < x < 1.4500000000000001e-22

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. fabs-sub N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fabs-sub N/A

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

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

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

      7. remove-double-neg N/A

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

      8. mul-1-neg N/A

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

      9. mul-1-neg N/A

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

      10. distribute-neg-in N/A

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

      11. +-commutative N/A

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

      12. fabs-neg N/A

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

      13. lower-fabs.f64 N/A

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

      14. *-commutative N/A

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

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

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

      16. distribute-lft-neg-in N/A

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

      17. distribute-rgt-neg-out N/A

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

      18. metadata-eval N/A

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

      19. *-rgt-identity N/A

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

      20. lower--.f64 53.6

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

   4. Applied rewrites 53.6%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left|\mathsf{neg}\left(y\right)\right| \]

      2. lower-neg.f64 50.6

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

   7. Applied rewrites 50.6%

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

   if 1.4500000000000001e-22 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. fabs-sub N/A

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

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

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

      8. remove-double-neg N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-in N/A

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

      12. +-commutative N/A

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

      13. fabs-neg N/A

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

      14. lower-fabs.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. distribute-lft-neg-in N/A

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

      18. distribute-rgt-neg-out N/A

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

      19. metadata-eval N/A

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

      20. *-rgt-identity N/A

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

      21. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower-fabs.f64 N/A

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

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

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

      7. sqrt-prod N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lift--.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

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

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. lift--.f64 49.7

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

   6. Applied rewrites 49.7%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 30.4

         \[\leadsto 1.5 \cdot x \]

   9. Applied rewrites 30.4%

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

Alternative 8: 58.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-74}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-99}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.9e-74) (* 0.5 x) (if (<= x 2.2e-99) (* -0.5 y) (* 1.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.9e-74) {
		tmp = 0.5 * x;
	} else if (x <= 2.2e-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 (x <= (-2.9d-74)) then
        tmp = 0.5d0 * x
    else if (x <= 2.2d-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 (x <= -2.9e-74) {
		tmp = 0.5 * x;
	} else if (x <= 2.2e-99) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.9e-74:
		tmp = 0.5 * x
	elif x <= 2.2e-99:
		tmp = -0.5 * y
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.9e-74)
		tmp = Float64(0.5 * x);
	elseif (x <= 2.2e-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 (x <= -2.9e-74)
		tmp = 0.5 * x;
	elseif (x <= 2.2e-99)
		tmp = -0.5 * y;
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.9e-74], N[(0.5 * x), $MachinePrecision], If[LessEqual[x, 2.2e-99], N[(-0.5 * y), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.9e-74

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 88.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, \frac{0.5}{x}, 1\right) \cdot x} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. lift--.f64 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-*r/ N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. +-commutative N/A

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

      12. distribute-lft-in N/A

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

      13. *-commutative N/A

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

      14. associate-*r/ N/A

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

      15. associate-*l/ N/A

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

      16. associate-*l* N/A

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

      17. *-rgt-identity N/A

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

   6. Applied rewrites 65.0%

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

   7. Taylor expanded in x around 0

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

      1. fabs-sub N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

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

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

      5. sqrt-unprod N/A

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

      6. rem-square-sqrt N/A

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

      7. lift--.f64 53.9

         \[\leadsto \left(x - y\right) \cdot 0.5 \]

   9. Applied rewrites 53.9%

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

   10. Taylor expanded in x around inf

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

       1. lower-*.f64 30.6

          \[\leadsto 0.5 \cdot x \]

   12. Applied rewrites 30.6%

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

   if -2.9e-74 < x < 2.20000000000000004e-99

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. fabs-sub N/A

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

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

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

      8. remove-double-neg N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-in N/A

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

      12. +-commutative N/A

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

      13. fabs-neg N/A

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

      14. lower-fabs.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. distribute-lft-neg-in N/A

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

      18. distribute-rgt-neg-out N/A

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

      19. metadata-eval N/A

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

      20. *-rgt-identity N/A

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

      21. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower-fabs.f64 N/A

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

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

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

      7. sqrt-prod N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lift--.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

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

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. lift--.f64 49.7

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

   6. Applied rewrites 49.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 26.7

         \[\leadsto -0.5 \cdot y \]

   9. Applied rewrites 26.7%

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

   if 2.20000000000000004e-99 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|y - \left(\mathsf{neg}\left(-1\right)\right) \cdot x\right|, x\right) \]

      7. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|y + -1 \cdot x\right|, x\right) \]

      8. remove-double-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + -1 \cdot x\right|, x\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\left(\mathsf{neg}\left(-1 \cdot y\right)\right) + -1 \cdot x\right|, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\left(\mathsf{neg}\left(-1 \cdot y\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right|, x\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\left(-1 \cdot y + x\right)\right)\right|, x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\left(x + -1 \cdot y\right)\right)\right|, x\right) \]

      13. fabs-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      14. lower-fabs.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + y \cdot -1\right|, x\right) \]

      16. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(y\right)\right) \cdot -1\right|, x\right) \]

      17. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(y \cdot -1\right)\right)\right|, x\right) \]

      18. distribute-rgt-neg-out N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y \cdot \left(\mathsf{neg}\left(-1\right)\right)\right|, x\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y \cdot 1\right|, x\right) \]

      20. *-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      21. lower--.f64 99.9

          \[\leadsto \mathsf{fma}\left(0.5, \left|x - y\right|, x\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|x - y\right|, x\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      5. lower-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      6. rem-sqrt-square-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\left(x + -1 \cdot y\right) \cdot \left(x + -1 \cdot y\right)}, x\right) \]

      7. sqrt-prod N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x + -1 \cdot y} \cdot \color{blue}{\sqrt{x + -1 \cdot y}}, x\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x + -1 \cdot y} \cdot \color{blue}{\sqrt{x + -1 \cdot y}}, x\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x + -1 \cdot y} \cdot \sqrt{\color{blue}{x + -1 \cdot y}}, x\right) \]

      10. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y} \cdot \sqrt{\color{blue}{x} + -1 \cdot y}, x\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - 1 \cdot y} \cdot \sqrt{x + -1 \cdot y}, x\right) \]

      12. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{x + -1 \cdot y}, x\right) \]

      13. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{\color{blue}{x} + -1 \cdot y}, x\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{x + -1 \cdot y}, x\right) \]

      15. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y}, x\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{x - 1 \cdot y}, x\right) \]

      17. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{x - y}, x\right) \]

      18. lift--.f64 49.7

          \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \sqrt{x - y}, x\right) \]

   6. Applied rewrites 49.7%

      \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

   7. Taylor expanded in x around inf

      \[\leadsto \frac{3}{2} \cdot \color{blue}{x} \]
   8. Step-by-step derivation

      1. lower-*.f64 30.4

         \[\leadsto 1.5 \cdot x \]

   9. Applied rewrites 30.4%

      \[\leadsto 1.5 \cdot \color{blue}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 50.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \frac{\left|y - x\right|}{2} \leq 10^{-300}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ x (/ (fabs (- y x)) 2.0)) 1e-300) (* 0.5 x) (* -0.5 y)))

C

double code(double x, double y) {
	double tmp;
	if ((x + (fabs((y - x)) / 2.0)) <= 1e-300) {
		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)) <= 1d-300) 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)) <= 1e-300) {
		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)) <= 1e-300:
		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)) <= 1e-300)
		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)) <= 1e-300)
		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], 1e-300], 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))) < 1.00000000000000003e-300

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot \frac{\left|y - x\right|}{x}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + \frac{1}{2} \cdot \frac{\left|y - x\right|}{x}\right) \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{1}{2} \cdot \frac{\left|y - x\right|}{x}\right) \cdot \color{blue}{x} \]

   4. Applied rewrites 88.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, \frac{0.5}{x}, 1\right) \cdot x} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left|x - y\right|, \frac{\frac{1}{2}}{x}, 1\right) \cdot \color{blue}{x} \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left|x - y\right|, \frac{\frac{1}{2}}{x}, 1\right) \cdot x \]

      3. lift-fma.f64 N/A

         \[\leadsto \left(\left|x - y\right| \cdot \frac{\frac{1}{2}}{x} + 1\right) \cdot x \]

      4. lift-fabs.f64 N/A

         \[\leadsto \left(\left|x - y\right| \cdot \frac{\frac{1}{2}}{x} + 1\right) \cdot x \]

      5. lift--.f64 N/A

         \[\leadsto \left(\left|x - y\right| \cdot \frac{\frac{1}{2}}{x} + 1\right) \cdot x \]

      6. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left|x - y\right| \cdot \frac{\frac{1}{2}}{x} + 1\right)} \]

      7. +-commutative N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{\left|x - y\right| \cdot \frac{\frac{1}{2}}{x}}\right) \]

      8. associate-*r/ N/A

         \[\leadsto x \cdot \left(1 + \frac{\left|x - y\right| \cdot \frac{1}{2}}{\color{blue}{x}}\right) \]

      9. *-commutative N/A

         \[\leadsto x \cdot \left(1 + \frac{\frac{1}{2} \cdot \left|x - y\right|}{x}\right) \]

      10. associate-*r/ N/A

          \[\leadsto x \cdot \left(1 + \frac{1}{2} \cdot \color{blue}{\frac{\left|x - y\right|}{x}}\right) \]

      11. +-commutative N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \frac{\left|x - y\right|}{x} + \color{blue}{1}\right) \]

      12. distribute-lft-in N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \frac{\left|x - y\right|}{x}\right) + \color{blue}{x \cdot 1} \]

      13. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} \cdot \frac{\left|x - y\right|}{x}\right) \cdot x + \color{blue}{x} \cdot 1 \]

      14. associate-*r/ N/A

          \[\leadsto \frac{\frac{1}{2} \cdot \left|x - y\right|}{x} \cdot x + x \cdot 1 \]

      15. associate-*l/ N/A

          \[\leadsto \left(\frac{\frac{1}{2}}{x} \cdot \left|x - y\right|\right) \cdot x + x \cdot 1 \]

      16. associate-*l* N/A

          \[\leadsto \frac{\frac{1}{2}}{x} \cdot \left(\left|x - y\right| \cdot x\right) + \color{blue}{x} \cdot 1 \]

      17. *-rgt-identity N/A

          \[\leadsto \frac{\frac{1}{2}}{x} \cdot \left(\left|x - y\right| \cdot x\right) + x \]

   6. Applied rewrites 65.0%

      \[\leadsto \mathsf{fma}\left(\frac{0.5}{x}, \color{blue}{\left|x - y\right| \cdot x}, x\right) \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left|y - x\right|} \]
   8. Step-by-step derivation

      1. fabs-sub N/A

         \[\leadsto \frac{1}{2} \cdot \left|x - y\right| \]

      2. *-commutative N/A

         \[\leadsto \left|x - y\right| \cdot \color{blue}{\frac{1}{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left|x - y\right| \cdot \color{blue}{\frac{1}{2}} \]

      4. rem-sqrt-square-rev N/A

         \[\leadsto \sqrt{\left(x - y\right) \cdot \left(x - y\right)} \cdot \frac{1}{2} \]

      5. sqrt-unprod N/A

         \[\leadsto \left(\sqrt{x - y} \cdot \sqrt{x - y}\right) \cdot \frac{1}{2} \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(x - y\right) \cdot \frac{1}{2} \]

      7. lift--.f64 53.9

         \[\leadsto \left(x - y\right) \cdot 0.5 \]

   9. Applied rewrites 53.9%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot 0.5} \]

   10. Taylor expanded in x around inf

       \[\leadsto \frac{1}{2} \cdot \color{blue}{x} \]
   11. Step-by-step derivation

       1. lower-*.f64 30.6

          \[\leadsto 0.5 \cdot x \]

   12. Applied rewrites 30.6%

       \[\leadsto 0.5 \cdot \color{blue}{x} \]

   if 1.00000000000000003e-300 < (+.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. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left|y - x\right| + \color{blue}{x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left|y - x\right|}, x\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|1 \cdot x - y\right|, x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - y\right|, x\right) \]

      6. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|y - \left(\mathsf{neg}\left(-1\right)\right) \cdot x\right|, x\right) \]

      7. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|y + -1 \cdot x\right|, x\right) \]

      8. remove-double-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + -1 \cdot x\right|, x\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\left(\mathsf{neg}\left(-1 \cdot y\right)\right) + -1 \cdot x\right|, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\left(\mathsf{neg}\left(-1 \cdot y\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right|, x\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\left(-1 \cdot y + x\right)\right)\right|, x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\left(x + -1 \cdot y\right)\right)\right|, x\right) \]

      13. fabs-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      14. lower-fabs.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + y \cdot -1\right|, x\right) \]

      16. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(y\right)\right) \cdot -1\right|, x\right) \]

      17. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(y \cdot -1\right)\right)\right|, x\right) \]

      18. distribute-rgt-neg-out N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y \cdot \left(\mathsf{neg}\left(-1\right)\right)\right|, x\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y \cdot 1\right|, x\right) \]

      20. *-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      21. lower--.f64 99.9

          \[\leadsto \mathsf{fma}\left(0.5, \left|x - y\right|, x\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|x - y\right|, x\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      5. lower-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      6. rem-sqrt-square-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\left(x + -1 \cdot y\right) \cdot \left(x + -1 \cdot y\right)}, x\right) \]

      7. sqrt-prod N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x + -1 \cdot y} \cdot \color{blue}{\sqrt{x + -1 \cdot y}}, x\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x + -1 \cdot y} \cdot \color{blue}{\sqrt{x + -1 \cdot y}}, x\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x + -1 \cdot y} \cdot \sqrt{\color{blue}{x + -1 \cdot y}}, x\right) \]

      10. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y} \cdot \sqrt{\color{blue}{x} + -1 \cdot y}, x\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - 1 \cdot y} \cdot \sqrt{x + -1 \cdot y}, x\right) \]

      12. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{x + -1 \cdot y}, x\right) \]

      13. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{\color{blue}{x} + -1 \cdot y}, x\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{x + -1 \cdot y}, x\right) \]

      15. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y}, x\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{x - 1 \cdot y}, x\right) \]

      17. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{x - y}, x\right) \]

      18. lift--.f64 49.7

          \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \sqrt{x - y}, x\right) \]

   6. Applied rewrites 49.7%

      \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{2} \cdot \color{blue}{y} \]
   8. Step-by-step derivation

      1. lower-*.f64 26.7

         \[\leadsto -0.5 \cdot y \]

   9. Applied rewrites 26.7%

      \[\leadsto -0.5 \cdot \color{blue}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 30.6% accurate, 2.8× 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. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot \frac{\left|y - x\right|}{x}\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(1 + \frac{1}{2} \cdot \frac{\left|y - x\right|}{x}\right) \cdot \color{blue}{x} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(1 + \frac{1}{2} \cdot \frac{\left|y - x\right|}{x}\right) \cdot \color{blue}{x} \]

4. Applied rewrites 88.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, \frac{0.5}{x}, 1\right) \cdot x} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left|x - y\right|, \frac{\frac{1}{2}}{x}, 1\right) \cdot \color{blue}{x} \]

   2. lift-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left|x - y\right|, \frac{\frac{1}{2}}{x}, 1\right) \cdot x \]

   3. lift-fma.f64 N/A

      \[\leadsto \left(\left|x - y\right| \cdot \frac{\frac{1}{2}}{x} + 1\right) \cdot x \]

   4. lift-fabs.f64 N/A

      \[\leadsto \left(\left|x - y\right| \cdot \frac{\frac{1}{2}}{x} + 1\right) \cdot x \]

   5. lift--.f64 N/A

      \[\leadsto \left(\left|x - y\right| \cdot \frac{\frac{1}{2}}{x} + 1\right) \cdot x \]

   6. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\left|x - y\right| \cdot \frac{\frac{1}{2}}{x} + 1\right)} \]

   7. +-commutative N/A

      \[\leadsto x \cdot \left(1 + \color{blue}{\left|x - y\right| \cdot \frac{\frac{1}{2}}{x}}\right) \]

   8. associate-*r/ N/A

      \[\leadsto x \cdot \left(1 + \frac{\left|x - y\right| \cdot \frac{1}{2}}{\color{blue}{x}}\right) \]

   9. *-commutative N/A

      \[\leadsto x \cdot \left(1 + \frac{\frac{1}{2} \cdot \left|x - y\right|}{x}\right) \]

   10. associate-*r/ N/A

       \[\leadsto x \cdot \left(1 + \frac{1}{2} \cdot \color{blue}{\frac{\left|x - y\right|}{x}}\right) \]

   11. +-commutative N/A

       \[\leadsto x \cdot \left(\frac{1}{2} \cdot \frac{\left|x - y\right|}{x} + \color{blue}{1}\right) \]

   12. distribute-lft-in N/A

       \[\leadsto x \cdot \left(\frac{1}{2} \cdot \frac{\left|x - y\right|}{x}\right) + \color{blue}{x \cdot 1} \]

   13. *-commutative N/A

       \[\leadsto \left(\frac{1}{2} \cdot \frac{\left|x - y\right|}{x}\right) \cdot x + \color{blue}{x} \cdot 1 \]

   14. associate-*r/ N/A

       \[\leadsto \frac{\frac{1}{2} \cdot \left|x - y\right|}{x} \cdot x + x \cdot 1 \]

   15. associate-*l/ N/A

       \[\leadsto \left(\frac{\frac{1}{2}}{x} \cdot \left|x - y\right|\right) \cdot x + x \cdot 1 \]

   16. associate-*l* N/A

       \[\leadsto \frac{\frac{1}{2}}{x} \cdot \left(\left|x - y\right| \cdot x\right) + \color{blue}{x} \cdot 1 \]

   17. *-rgt-identity N/A

       \[\leadsto \frac{\frac{1}{2}}{x} \cdot \left(\left|x - y\right| \cdot x\right) + x \]

6. Applied rewrites 65.0%

   \[\leadsto \mathsf{fma}\left(\frac{0.5}{x}, \color{blue}{\left|x - y\right| \cdot x}, x\right) \]

7. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left|y - x\right|} \]
8. Step-by-step derivation

   1. fabs-sub N/A

      \[\leadsto \frac{1}{2} \cdot \left|x - y\right| \]

   2. *-commutative N/A

      \[\leadsto \left|x - y\right| \cdot \color{blue}{\frac{1}{2}} \]

   3. lower-*.f64 N/A

      \[\leadsto \left|x - y\right| \cdot \color{blue}{\frac{1}{2}} \]

   4. rem-sqrt-square-rev N/A

      \[\leadsto \sqrt{\left(x - y\right) \cdot \left(x - y\right)} \cdot \frac{1}{2} \]

   5. sqrt-unprod N/A

      \[\leadsto \left(\sqrt{x - y} \cdot \sqrt{x - y}\right) \cdot \frac{1}{2} \]

   6. rem-square-sqrt N/A

      \[\leadsto \left(x - y\right) \cdot \frac{1}{2} \]

   7. lift--.f64 53.9

      \[\leadsto \left(x - y\right) \cdot 0.5 \]

9. Applied rewrites 53.9%

   \[\leadsto \color{blue}{\left(x - y\right) \cdot 0.5} \]

10. Taylor expanded in x around inf

    \[\leadsto \frac{1}{2} \cdot \color{blue}{x} \]
11. Step-by-step derivation

    1. lower-*.f64 30.6

       \[\leadsto 0.5 \cdot x \]

12. Applied rewrites 30.6%

    \[\leadsto 0.5 \cdot \color{blue}{x} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025131 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))