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

Percentage Accurate: 99.9% → 99.9%

Time: 4.6s

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: 83.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-39}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.42 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|-y\right|, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7.8e-39)
   (* (- x y) 0.5)
   (if (<= x 1.42e-11) (fma 0.5 (fabs (- y)) x) (fma 1.5 x (* -0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.8e-39) {
		tmp = (x - y) * 0.5;
	} else if (x <= 1.42e-11) {
		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 <= -7.8e-39)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (x <= 1.42e-11)
		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, -7.8e-39], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 1.42e-11], 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 < -7.80000000000000059e-39

   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 88.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, \frac{0.5}{x}, 1\right) \cdot x} \]
   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. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-rgt-identity N/A

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

      17. lower-fma.f64 N/A

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

   6. Applied rewrites 88.5%

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

   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 53.0

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

   9. Applied rewrites 53.0%

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

   if -7.80000000000000059e-39 < x < 1.42e-11

   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 58.5

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

   7. Applied rewrites 58.5%

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

   if 1.42e-11 < 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 49.2

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

   6. Applied rewrites 49.2%

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

   7. Taylor expanded in x around inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. unpow1 N/A

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

      4. inv-pow N/A

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

      5. pow-prod-up N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot {x}^{1} \]

      7. unpow1 N/A

         \[\leadsto \frac{1}{2} \cdot x \]

      8. lift-*.f64 30.6

         \[\leadsto 0.5 \cdot x \]

   9. Applied rewrites 30.6%

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

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lift-neg.f64 N/A

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

       3. sqrt-pow2 N/A

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

       4. metadata-eval N/A

          \[\leadsto \frac{3}{2} \cdot x + \frac{1}{2} \cdot {\left(-y\right)}^{1} \]

       5. unpow1 N/A

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

       6. lift-neg.f64 N/A

          \[\leadsto \frac{3}{2} \cdot x + \frac{1}{2} \cdot \left(\mathsf{neg}\left(y\right)\right) \]

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

          \[\leadsto \frac{3}{2} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2} \cdot y\right)\right) \]

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

          \[\leadsto \frac{3}{2} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot y \]

       9. metadata-eval N/A

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

       10. lower-fma.f64 N/A

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

       11. lower-*.f64 54.2

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

   12. Applied rewrites 54.2%

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

Alternative 3: 83.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-39}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.42 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|-y\right|, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7.8e-39)
   (* (- x y) 0.5)
   (if (<= x 1.42e-11) (fma 0.5 (fabs (- y)) x) (fma (- x y) 0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.8e-39) {
		tmp = (x - y) * 0.5;
	} else if (x <= 1.42e-11) {
		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 <= -7.8e-39)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (x <= 1.42e-11)
		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, -7.8e-39], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 1.42e-11], 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 < -7.80000000000000059e-39

   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 88.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, \frac{0.5}{x}, 1\right) \cdot x} \]
   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. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-rgt-identity N/A

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

      17. lower-fma.f64 N/A

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

   6. Applied rewrites 88.5%

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

   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 53.0

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

   9. Applied rewrites 53.0%

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

   if -7.80000000000000059e-39 < x < 1.42e-11

   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 58.5

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

   7. Applied rewrites 58.5%

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

   if 1.42e-11 < 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 49.2

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

   6. Applied rewrites 49.2%

      \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]
   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 54.1

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

   8. Applied rewrites 54.1%

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

Alternative 4: 80.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-39}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 230000000:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|-y\right|, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7.8e-39)
   (* (- x y) 0.5)
   (if (<= x 230000000.0) (fma 0.5 (fabs (- y)) x) (fma 0.5 x x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.8e-39) {
		tmp = (x - y) * 0.5;
	} else if (x <= 230000000.0) {
		tmp = fma(0.5, fabs(-y), x);
	} else {
		tmp = fma(0.5, x, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7.8e-39)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (x <= 230000000.0)
		tmp = fma(0.5, abs(Float64(-y)), x);
	else
		tmp = fma(0.5, x, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7.8e-39], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 230000000.0], N[(0.5 * N[Abs[(-y)], $MachinePrecision] + x), $MachinePrecision], N[(0.5 * x + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.80000000000000059e-39

   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 88.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, \frac{0.5}{x}, 1\right) \cdot x} \]
   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. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-rgt-identity N/A

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

      17. lower-fma.f64 N/A

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

   6. Applied rewrites 88.5%

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

   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 53.0

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

   9. Applied rewrites 53.0%

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

   if -7.80000000000000059e-39 < x < 2.3e8

   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 58.5

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

   7. Applied rewrites 58.5%

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

   if 2.3e8 < 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 49.2

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

   6. Applied rewrites 49.2%

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

   7. Taylor expanded in x around inf

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

      1. sqrt-unprod N/A

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

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

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

      3. sqrt-pow2 N/A

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

      4. metadata-eval N/A

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

      5. unpow1 N/A

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

      6. inv-pow N/A

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

      7. pow-prod-up N/A

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

      8. metadata-eval N/A

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

      9. unpow1 30.6

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

   9. Applied rewrites 30.6%

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

Alternative 5: 79.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-39}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 230000000:\\ \;\;\;\;0.5 \cdot \left|x - y\right|\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.7e-39)
   (* (- x y) 0.5)
   (if (<= x 230000000.0) (* 0.5 (fabs (- x y))) (fma 0.5 x x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.7e-39) {
		tmp = (x - y) * 0.5;
	} else if (x <= 230000000.0) {
		tmp = 0.5 * fabs((x - y));
	} else {
		tmp = fma(0.5, x, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.7e-39)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (x <= 230000000.0)
		tmp = Float64(0.5 * abs(Float64(x - y)));
	else
		tmp = fma(0.5, x, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.7e-39], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 230000000.0], N[(0.5 * N[Abs[N[(x - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * x + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.7000000000000001e-39

   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 88.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, \frac{0.5}{x}, 1\right) \cdot x} \]
   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. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-rgt-identity N/A

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

      17. lower-fma.f64 N/A

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

   6. Applied rewrites 88.5%

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

   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 53.0

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

   9. Applied rewrites 53.0%

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

   if -2.7000000000000001e-39 < x < 2.3e8

   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 53.5

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

   4. Applied rewrites 53.5%

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

   if 2.3e8 < 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 49.2

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

   6. Applied rewrites 49.2%

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

   7. Taylor expanded in x around inf

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

      1. sqrt-unprod N/A

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

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

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

      3. sqrt-pow2 N/A

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

      4. metadata-eval N/A

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

      5. unpow1 N/A

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

      6. inv-pow N/A

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

      7. pow-prod-up N/A

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

      8. metadata-eval N/A

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

      9. unpow1 30.6

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

   9. Applied rewrites 30.6%

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

Alternative 6: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-39}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 135000000:\\ \;\;\;\;0.5 \cdot \left|-y\right|\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.7e-39)
   (* (- x y) 0.5)
   (if (<= x 135000000.0) (* 0.5 (fabs (- y))) (fma 0.5 x x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.7e-39) {
		tmp = (x - y) * 0.5;
	} else if (x <= 135000000.0) {
		tmp = 0.5 * fabs(-y);
	} else {
		tmp = fma(0.5, x, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.7e-39)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (x <= 135000000.0)
		tmp = Float64(0.5 * abs(Float64(-y)));
	else
		tmp = fma(0.5, x, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.7e-39], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 135000000.0], N[(0.5 * N[Abs[(-y)], $MachinePrecision]), $MachinePrecision], N[(0.5 * x + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.7000000000000001e-39

   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 88.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, \frac{0.5}{x}, 1\right) \cdot x} \]
   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. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-rgt-identity N/A

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

      17. lower-fma.f64 N/A

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

   6. Applied rewrites 88.5%

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

   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 53.0

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

   9. Applied rewrites 53.0%

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

   if -2.7000000000000001e-39 < x < 1.35e8

   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 53.5

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

   4. Applied rewrites 53.5%

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

   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 50.5

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

   7. Applied rewrites 50.5%

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

   if 1.35e8 < 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 49.2

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

   6. Applied rewrites 49.2%

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

   7. Taylor expanded in x around inf

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

      1. sqrt-unprod N/A

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

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

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

      3. sqrt-pow2 N/A

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

      4. metadata-eval N/A

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

      5. unpow1 N/A

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

      6. inv-pow N/A

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

      7. pow-prod-up N/A

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

      8. metadata-eval N/A

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

      9. unpow1 30.6

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

   9. Applied rewrites 30.6%

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

Alternative 7: 74.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \frac{\left|y - x\right|}{2} \leq -5 \cdot 10^{-252}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left|-y\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ x (/ (fabs (- y x)) 2.0)) -5e-252) (* 0.5 x) (* 0.5 (fabs (- y)))))

C

double code(double x, double y) {
	double tmp;
	if ((x + (fabs((y - x)) / 2.0)) <= -5e-252) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * fabs(-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)) <= (-5d-252)) then
        tmp = 0.5d0 * x
    else
        tmp = 0.5d0 * abs(-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)) <= -5e-252) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * Math.abs(-y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x + (math.fabs((y - x)) / 2.0)) <= -5e-252:
		tmp = 0.5 * x
	else:
		tmp = 0.5 * math.fabs(-y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x + Float64(abs(Float64(y - x)) / 2.0)) <= -5e-252)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(0.5 * abs(Float64(-y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x + (abs((y - x)) / 2.0)) <= -5e-252)
		tmp = 0.5 * x;
	else
		tmp = 0.5 * abs(-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], -5e-252], N[(0.5 * x), $MachinePrecision], N[(0.5 * N[Abs[(-y)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (fabs.f64 (-.f64 y x)) #s(literal 2 binary64))) < -5.00000000000000008e-252

   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 49.2

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

   6. Applied rewrites 49.2%

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

   7. Taylor expanded in x around inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. unpow1 N/A

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

      4. inv-pow N/A

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

      5. pow-prod-up N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot {x}^{1} \]

      7. unpow1 N/A

         \[\leadsto \frac{1}{2} \cdot x \]

      8. lift-*.f64 30.6

         \[\leadsto 0.5 \cdot x \]

   9. Applied rewrites 30.6%

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

   if -5.00000000000000008e-252 < (+.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

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left|y - x\right|} \]
   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 53.5

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

   4. Applied rewrites 53.5%

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

   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 50.5

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

   7. Applied rewrites 50.5%

      \[\leadsto 0.5 \cdot \left|-y\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 58.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-49}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-90}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.8e-49) (* 0.5 x) (if (<= x 5e-90) (* y -0.5) (fma 0.5 x x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.8e-49) {
		tmp = 0.5 * x;
	} else if (x <= 5e-90) {
		tmp = y * -0.5;
	} else {
		tmp = fma(0.5, x, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.8e-49)
		tmp = Float64(0.5 * x);
	elseif (x <= 5e-90)
		tmp = Float64(y * -0.5);
	else
		tmp = fma(0.5, x, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.8e-49], N[(0.5 * x), $MachinePrecision], If[LessEqual[x, 5e-90], N[(y * -0.5), $MachinePrecision], N[(0.5 * x + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.79999999999999985e-49

   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 49.2

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

   6. Applied rewrites 49.2%

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

   7. Taylor expanded in x around inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. unpow1 N/A

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

      4. inv-pow N/A

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

      5. pow-prod-up N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot {x}^{1} \]

      7. unpow1 N/A

         \[\leadsto \frac{1}{2} \cdot x \]

      8. lift-*.f64 30.6

         \[\leadsto 0.5 \cdot x \]

   9. Applied rewrites 30.6%

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

   if -1.79999999999999985e-49 < x < 5.00000000000000019e-90

   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 49.2

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

   6. Applied rewrites 49.2%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lift-neg.f64 N/A

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

      3. sqrt-pow2 N/A

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

      4. metadata-eval N/A

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

      5. unpow1 N/A

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

      6. lower-*.f64 25.8

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

   9. Applied rewrites 25.8%

      \[\leadsto \left(-y\right) \cdot \color{blue}{0.5} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-neg.f64 N/A

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

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

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

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

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

       5. metadata-eval N/A

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

       6. lower-*.f64 25.8

          \[\leadsto y \cdot -0.5 \]

   11. Applied rewrites 25.8%

       \[\leadsto y \cdot -0.5 \]

   if 5.00000000000000019e-90 < 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 49.2

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

   6. Applied rewrites 49.2%

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

   7. Taylor expanded in x around inf

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

      1. sqrt-unprod N/A

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

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

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

      3. sqrt-pow2 N/A

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

      4. metadata-eval N/A

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

      5. unpow1 N/A

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

      6. inv-pow N/A

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

      7. pow-prod-up N/A

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

      8. metadata-eval N/A

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

      9. unpow1 30.6

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

   9. Applied rewrites 30.6%

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

Alternative 9: 49.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \frac{\left|y - x\right|}{2} \leq 5 \cdot 10^{-285}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ x (/ (fabs (- y x)) 2.0)) 5e-285) (* 0.5 x) (* y -0.5)))

C

double code(double x, double y) {
	double tmp;
	if ((x + (fabs((y - x)) / 2.0)) <= 5e-285) {
		tmp = 0.5 * x;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x + (abs((y - x)) / 2.0d0)) <= 5d-285) then
        tmp = 0.5d0 * x
    else
        tmp = y * (-0.5d0)
    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)) <= 5e-285) {
		tmp = 0.5 * x;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x + (math.fabs((y - x)) / 2.0)) <= 5e-285:
		tmp = 0.5 * x
	else:
		tmp = y * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x + Float64(abs(Float64(y - x)) / 2.0)) <= 5e-285)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(y * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x + (abs((y - x)) / 2.0)) <= 5e-285)
		tmp = 0.5 * x;
	else
		tmp = y * -0.5;
	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], 5e-285], N[(0.5 * x), $MachinePrecision], N[(y * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (fabs.f64 (-.f64 y x)) #s(literal 2 binary64))) < 5.00000000000000018e-285

   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 49.2

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

   6. Applied rewrites 49.2%

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

   7. Taylor expanded in x around inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. unpow1 N/A

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

      4. inv-pow N/A

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

      5. pow-prod-up N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot {x}^{1} \]

      7. unpow1 N/A

         \[\leadsto \frac{1}{2} \cdot x \]

      8. lift-*.f64 30.6

         \[\leadsto 0.5 \cdot x \]

   9. Applied rewrites 30.6%

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

   if 5.00000000000000018e-285 < (+.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 49.2

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

   6. Applied rewrites 49.2%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lift-neg.f64 N/A

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

      3. sqrt-pow2 N/A

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

      4. metadata-eval N/A

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

      5. unpow1 N/A

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

      6. lower-*.f64 25.8

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

   9. Applied rewrites 25.8%

      \[\leadsto \left(-y\right) \cdot \color{blue}{0.5} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-neg.f64 N/A

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

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

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

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

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

       5. metadata-eval N/A

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

       6. lower-*.f64 25.8

          \[\leadsto y \cdot -0.5 \]

   11. Applied rewrites 25.8%

       \[\leadsto y \cdot -0.5 \]
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 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 49.2

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

6. Applied rewrites 49.2%

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

7. Taylor expanded in x around inf

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

   1. sqrt-pow2 N/A

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

   2. metadata-eval N/A

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

   3. unpow1 N/A

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

   4. inv-pow N/A

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

   5. pow-prod-up N/A

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

   6. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot {x}^{1} \]

   7. unpow1 N/A

      \[\leadsto \frac{1}{2} \cdot x \]

   8. lift-*.f64 30.6

      \[\leadsto 0.5 \cdot x \]

9. Applied rewrites 30.6%

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

Reproduce

herbie shell --seed 2025139 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))