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

Percentage Accurate: 99.9% → 99.9%

Time: 5.3s

Alternatives: 14

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

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

5. Applied rewrites 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 65.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-272}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-80}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 34000:\\ \;\;\;\;\mathsf{fma}\left(0.5, -y, x\right)\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 5.6e-272)
   (* (- x y) 0.5)
   (if (<= x 5.5e-80)
     (* y 0.5)
     (if (<= x 34000.0) (fma 0.5 (- y) x) (* 1.5 x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5.6e-272) {
		tmp = (x - y) * 0.5;
	} else if (x <= 5.5e-80) {
		tmp = y * 0.5;
	} else if (x <= 34000.0) {
		tmp = fma(0.5, -y, x);
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5.6e-272)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (x <= 5.5e-80)
		tmp = Float64(y * 0.5);
	elseif (x <= 34000.0)
		tmp = fma(0.5, Float64(-y), x);
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5.6e-272], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 5.5e-80], N[(y * 0.5), $MachinePrecision], If[LessEqual[x, 34000.0], N[(0.5 * (-y) + x), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 5.59999999999999987e-272

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. /-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-/r/ N/A

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

      5. /-rgt-identity N/A

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

      6. metadata-eval N/A

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

      7. fabs-mul N/A

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. fabs-sub N/A

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

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

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

      14. lower-fabs.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. *-lft-identity N/A

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

      18. metadata-eval N/A

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

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

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

      20. *-commutative N/A

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

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. metadata-eval N/A

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

      24. *-lft-identity N/A

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

      25. lower--.f64 49.3

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

   5. Applied rewrites 49.3%

      \[\leadsto \color{blue}{0.5 \cdot \left|x - y\right|} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. fabs-sub N/A

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

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

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

      7. unpow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 17.5

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

   7. Applied rewrites 17.5%

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

      1. lift--.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. unpow2 N/A

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

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

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

      7. fabs-sub N/A

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

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

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

      9. sqrt-unprod N/A

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

      10. rem-square-sqrt N/A

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

      11. lift--.f64 82.2

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

   9. Applied rewrites 82.2%

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

   if 5.59999999999999987e-272 < x < 5.4999999999999997e-80

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. /-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-/r/ N/A

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

      5. /-rgt-identity N/A

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

      6. metadata-eval N/A

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

      7. fabs-mul N/A

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. fabs-sub N/A

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

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

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

      14. lower-fabs.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. *-lft-identity N/A

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

      18. metadata-eval N/A

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

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

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

      20. *-commutative N/A

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

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. metadata-eval N/A

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

      24. *-lft-identity N/A

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

      25. lower--.f64 85.0

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

   5. Applied rewrites 85.0%

      \[\leadsto \color{blue}{0.5 \cdot \left|x - y\right|} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. fabs-sub N/A

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

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

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

      7. unpow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 61.6

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

   7. Applied rewrites 61.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 62.2%

       \[\leadsto y \cdot 0.5 \]

   if 5.4999999999999997e-80 < x < 34000

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|-1 \cdot y\right|, x\right) \]
   7. 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 80.0

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

   8. Applied rewrites 80.0%

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

   9. Taylor expanded in x around 0

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

       1. fabs-sub N/A

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

       2. *-commutative N/A

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

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

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

       4. *-lft-identity N/A

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

       5. metadata-eval N/A

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

       6. cancel-sign-sub N/A

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

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

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

       8. *-commutative N/A

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

       9. fabs-neg N/A

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

       10. fabs-neg N/A

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

       11. *-commutative N/A

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

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

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

   11. Applied rewrites 70.9%

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

   12. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 51.0

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

   14. Applied rewrites 51.0%

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

   if 34000 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

   5. Applied rewrites 99.9%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 85.8

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

   7. Applied rewrites 85.8%

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

   8. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. rem-square-sqrt N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

   10. Applied rewrites 76.1%

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

Alternative 3: 65.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - y\right) \cdot 0.5\\ \mathbf{if}\;x \leq 5.6 \cdot 10^{-272}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-80}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 34000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- x y) 0.5)))
   (if (<= x 5.6e-272)
     t_0
     (if (<= x 5.5e-80) (* y 0.5) (if (<= x 34000.0) t_0 (* 1.5 x))))))

C

double code(double x, double y) {
	double t_0 = (x - y) * 0.5;
	double tmp;
	if (x <= 5.6e-272) {
		tmp = t_0;
	} else if (x <= 5.5e-80) {
		tmp = y * 0.5;
	} else if (x <= 34000.0) {
		tmp = t_0;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - y) * 0.5d0
    if (x <= 5.6d-272) then
        tmp = t_0
    else if (x <= 5.5d-80) then
        tmp = y * 0.5d0
    else if (x <= 34000.0d0) then
        tmp = t_0
    else
        tmp = 1.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) * 0.5;
	double tmp;
	if (x <= 5.6e-272) {
		tmp = t_0;
	} else if (x <= 5.5e-80) {
		tmp = y * 0.5;
	} else if (x <= 34000.0) {
		tmp = t_0;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) * 0.5
	tmp = 0
	if x <= 5.6e-272:
		tmp = t_0
	elif x <= 5.5e-80:
		tmp = y * 0.5
	elif x <= 34000.0:
		tmp = t_0
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) * 0.5)
	tmp = 0.0
	if (x <= 5.6e-272)
		tmp = t_0;
	elseif (x <= 5.5e-80)
		tmp = Float64(y * 0.5);
	elseif (x <= 34000.0)
		tmp = t_0;
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) * 0.5;
	tmp = 0.0;
	if (x <= 5.6e-272)
		tmp = t_0;
	elseif (x <= 5.5e-80)
		tmp = y * 0.5;
	elseif (x <= 34000.0)
		tmp = t_0;
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[x, 5.6e-272], t$95$0, If[LessEqual[x, 5.5e-80], N[(y * 0.5), $MachinePrecision], If[LessEqual[x, 34000.0], t$95$0, N[(1.5 * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.59999999999999987e-272 or 5.4999999999999997e-80 < x < 34000

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. /-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-/r/ N/A

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

      5. /-rgt-identity N/A

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

      6. metadata-eval N/A

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

      7. fabs-mul N/A

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. fabs-sub N/A

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

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

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

      14. lower-fabs.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. *-lft-identity N/A

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

      18. metadata-eval N/A

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

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

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

      20. *-commutative N/A

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

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. metadata-eval N/A

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

      24. *-lft-identity N/A

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

      25. lower--.f64 53.9

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

   5. Applied rewrites 53.9%

      \[\leadsto \color{blue}{0.5 \cdot \left|x - y\right|} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. fabs-sub N/A

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

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

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

      7. unpow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 19.4

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

   7. Applied rewrites 19.4%

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

      1. lift--.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. unpow2 N/A

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

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

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

      7. fabs-sub N/A

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

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

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

      9. sqrt-unprod N/A

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

      10. rem-square-sqrt N/A

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

      11. lift--.f64 77.3

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

   9. Applied rewrites 77.3%

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

   if 5.59999999999999987e-272 < x < 5.4999999999999997e-80

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. /-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-/r/ N/A

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

      5. /-rgt-identity N/A

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

      6. metadata-eval N/A

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

      7. fabs-mul N/A

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. fabs-sub N/A

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

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

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

      14. lower-fabs.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. *-lft-identity N/A

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

      18. metadata-eval N/A

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

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

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

      20. *-commutative N/A

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

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. metadata-eval N/A

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

      24. *-lft-identity N/A

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

      25. lower--.f64 85.0

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

   5. Applied rewrites 85.0%

      \[\leadsto \color{blue}{0.5 \cdot \left|x - y\right|} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. fabs-sub N/A

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

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

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

      7. unpow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 61.6

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

   7. Applied rewrites 61.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 62.2%

       \[\leadsto y \cdot 0.5 \]

   if 34000 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

   5. Applied rewrites 99.9%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 85.8

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

   7. Applied rewrites 85.8%

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

   8. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. rem-square-sqrt N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

   10. Applied rewrites 76.1%

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

Alternative 4: 82.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -195000:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|-y\right|, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x - y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -195000.0)
   (* (- x y) 0.5)
   (if (<= x 2.3e+14) (fma 0.5 (fabs (- y)) x) (fma 0.5 (- x y) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -195000.0) {
		tmp = (x - y) * 0.5;
	} else if (x <= 2.3e+14) {
		tmp = fma(0.5, fabs(-y), x);
	} else {
		tmp = fma(0.5, (x - y), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -195000.0)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (x <= 2.3e+14)
		tmp = fma(0.5, abs(Float64(-y)), x);
	else
		tmp = fma(0.5, Float64(x - y), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -195000.0], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 2.3e+14], N[(0.5 * N[Abs[(-y)], $MachinePrecision] + x), $MachinePrecision], N[(0.5 * N[(x - y), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -195000

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. /-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-/r/ N/A

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

      5. /-rgt-identity N/A

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

      6. metadata-eval N/A

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

      7. fabs-mul N/A

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. fabs-sub N/A

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

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

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

      14. lower-fabs.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. *-lft-identity N/A

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

      18. metadata-eval N/A

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

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

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

      20. *-commutative N/A

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

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. metadata-eval N/A

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

      24. *-lft-identity N/A

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

      25. lower--.f64 15.9

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

   5. Applied rewrites 15.9%

      \[\leadsto \color{blue}{0.5 \cdot \left|x - y\right|} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. fabs-sub N/A

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

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

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

      7. unpow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 5.5

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

   7. Applied rewrites 5.5%

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

      1. lift--.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. unpow2 N/A

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

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

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

      7. fabs-sub N/A

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

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

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

      9. sqrt-unprod N/A

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

      10. rem-square-sqrt N/A

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

      11. lift--.f64 93.8

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

   9. Applied rewrites 93.8%

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

   if -195000 < x < 2.3e14

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|-1 \cdot y\right|, x\right) \]
   7. 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 82.0

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

   8. Applied rewrites 82.0%

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

   if 2.3e14 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|-1 \cdot y\right|, x\right) \]
   7. 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 38.6

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

   8. Applied rewrites 38.6%

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

   9. Taylor expanded in x around 0

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

       1. fabs-sub N/A

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

       2. *-commutative N/A

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

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

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

       4. *-lft-identity N/A

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

       5. metadata-eval N/A

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

       6. cancel-sign-sub N/A

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

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

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

       8. *-commutative N/A

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

       9. fabs-neg N/A

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

       10. fabs-neg N/A

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

       11. *-commutative N/A

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

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

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

   11. Applied rewrites 87.2%

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

Alternative 5: 82.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-141}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+14}:\\ \;\;\;\;0.5 \cdot \left|y - x\right|\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x - y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4e-141)
   (* (- x y) 0.5)
   (if (<= x 2.3e+14) (* 0.5 (fabs (- y x))) (fma 0.5 (- x y) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4e-141) {
		tmp = (x - y) * 0.5;
	} else if (x <= 2.3e+14) {
		tmp = 0.5 * fabs((y - x));
	} else {
		tmp = fma(0.5, (x - y), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4e-141)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (x <= 2.3e+14)
		tmp = Float64(0.5 * abs(Float64(y - x)));
	else
		tmp = fma(0.5, Float64(x - y), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4e-141], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 2.3e+14], N[(0.5 * N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x - y), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.0000000000000002e-141

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. /-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-/r/ N/A

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

      5. /-rgt-identity N/A

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

      6. metadata-eval N/A

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

      7. fabs-mul N/A

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. fabs-sub N/A

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

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

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

      14. lower-fabs.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. *-lft-identity N/A

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

      18. metadata-eval N/A

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

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

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

      20. *-commutative N/A

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

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. metadata-eval N/A

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

      24. *-lft-identity N/A

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

      25. lower--.f64 33.1

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

   5. Applied rewrites 33.1%

      \[\leadsto \color{blue}{0.5 \cdot \left|x - y\right|} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. fabs-sub N/A

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

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

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

      7. unpow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 11.9

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

   7. Applied rewrites 11.9%

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

      1. lift--.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. unpow2 N/A

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

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

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

      7. fabs-sub N/A

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

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

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

      9. sqrt-unprod N/A

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

      10. rem-square-sqrt N/A

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

      11. lift--.f64 87.7

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

   9. Applied rewrites 87.7%

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

   if -4.0000000000000002e-141 < x < 2.3e14

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. /-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-/r/ N/A

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

      5. /-rgt-identity N/A

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

      6. metadata-eval N/A

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

      7. fabs-mul N/A

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. fabs-sub N/A

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

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

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

      14. lower-fabs.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. *-lft-identity N/A

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

      18. metadata-eval N/A

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

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

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

      20. *-commutative N/A

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

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. metadata-eval N/A

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

      24. *-lft-identity N/A

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

      25. lower--.f64 82.2

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

   5. Applied rewrites 82.2%

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

   if 2.3e14 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|-1 \cdot y\right|, x\right) \]
   7. 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 38.6

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

   8. Applied rewrites 38.6%

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

   9. Taylor expanded in x around 0

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

       1. fabs-sub N/A

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

       2. *-commutative N/A

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

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

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

       4. *-lft-identity N/A

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

       5. metadata-eval N/A

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

       6. cancel-sign-sub N/A

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

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

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

       8. *-commutative N/A

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

       9. fabs-neg N/A

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

       10. fabs-neg N/A

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

       11. *-commutative N/A

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

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

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

   11. Applied rewrites 87.2%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-141}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+14}:\\ \;\;\;\;0.5 \cdot \left|y - x\right|\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x - y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-131}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x - y, x\right)\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-137}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|x\right|, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, 0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.1e-131)
   (fma 0.5 (- x y) x)
   (if (<= y 3.7e-137) (fma 0.5 (fabs x) x) (fma (- y x) 0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.1e-131) {
		tmp = fma(0.5, (x - y), x);
	} else if (y <= 3.7e-137) {
		tmp = fma(0.5, fabs(x), x);
	} else {
		tmp = fma((y - x), 0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.1e-131)
		tmp = fma(0.5, Float64(x - y), x);
	elseif (y <= 3.7e-137)
		tmp = fma(0.5, abs(x), x);
	else
		tmp = fma(Float64(y - x), 0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.1e-131], N[(0.5 * N[(x - y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 3.7e-137], N[(0.5 * N[Abs[x], $MachinePrecision] + x), $MachinePrecision], N[(N[(y - x), $MachinePrecision] * 0.5 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.09999999999999997e-131

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|-1 \cdot y\right|, x\right) \]
   7. 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 73.8

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

   8. Applied rewrites 73.8%

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

   9. Taylor expanded in x around 0

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

       1. fabs-sub N/A

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

       2. *-commutative N/A

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

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

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

       4. *-lft-identity N/A

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

       5. metadata-eval N/A

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

       6. cancel-sign-sub N/A

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

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

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

       8. *-commutative N/A

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

       9. fabs-neg N/A

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

       10. fabs-neg N/A

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

       11. *-commutative N/A

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

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

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

   11. Applied rewrites 87.6%

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

   if -2.09999999999999997e-131 < y < 3.7e-137

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 87.0%

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

   if 3.7e-137 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

   5. Applied rewrites 99.9%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. fabs-sub N/A

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

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

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

      7. unpow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift--.f64 81.0

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

   7. Applied rewrites 81.0%

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

Alternative 7: 81.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-141}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+14}:\\ \;\;\;\;0.5 \cdot \left|-y\right|\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x - y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4e-141)
   (* (- x y) 0.5)
   (if (<= x 2.3e+14) (* 0.5 (fabs (- y))) (fma 0.5 (- x y) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4e-141) {
		tmp = (x - y) * 0.5;
	} else if (x <= 2.3e+14) {
		tmp = 0.5 * fabs(-y);
	} else {
		tmp = fma(0.5, (x - y), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4e-141)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (x <= 2.3e+14)
		tmp = Float64(0.5 * abs(Float64(-y)));
	else
		tmp = fma(0.5, Float64(x - y), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4e-141], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 2.3e+14], N[(0.5 * N[Abs[(-y)], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x - y), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.0000000000000002e-141

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. /-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-/r/ N/A

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

      5. /-rgt-identity N/A

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

      6. metadata-eval N/A

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

      7. fabs-mul N/A

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. fabs-sub N/A

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

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

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

      14. lower-fabs.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. *-lft-identity N/A

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

      18. metadata-eval N/A

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

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

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

      20. *-commutative N/A

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

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. metadata-eval N/A

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

      24. *-lft-identity N/A

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

      25. lower--.f64 33.1

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

   5. Applied rewrites 33.1%

      \[\leadsto \color{blue}{0.5 \cdot \left|x - y\right|} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. fabs-sub N/A

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

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

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

      7. unpow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 11.9

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

   7. Applied rewrites 11.9%

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

      1. lift--.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. unpow2 N/A

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

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

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

      7. fabs-sub N/A

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

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

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

      9. sqrt-unprod N/A

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

      10. rem-square-sqrt N/A

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

      11. lift--.f64 87.7

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

   9. Applied rewrites 87.7%

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

   if -4.0000000000000002e-141 < x < 2.3e14

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. /-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-/r/ N/A

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

      5. /-rgt-identity N/A

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

      6. metadata-eval N/A

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

      7. fabs-mul N/A

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. fabs-sub N/A

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

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

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

      14. lower-fabs.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. *-lft-identity N/A

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

      18. metadata-eval N/A

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

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

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

      20. *-commutative N/A

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

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. metadata-eval N/A

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

      24. *-lft-identity N/A

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

      25. lower--.f64 82.2

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

   5. Applied rewrites 82.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{1}{2} \cdot \left|-1 \cdot y\right| \]
   7. 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 80.8

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

   8. Applied rewrites 80.8%

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

   if 2.3e14 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|-1 \cdot y\right|, x\right) \]
   7. 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 38.6

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

   8. Applied rewrites 38.6%

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

   9. Taylor expanded in x around 0

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

       1. fabs-sub N/A

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

       2. *-commutative N/A

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

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

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

       4. *-lft-identity N/A

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

       5. metadata-eval N/A

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

       6. cancel-sign-sub N/A

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

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

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

       8. *-commutative N/A

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

       9. fabs-neg N/A

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

       10. fabs-neg N/A

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

       11. *-commutative N/A

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

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

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

   11. Applied rewrites 87.2%

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

Alternative 8: 78.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-141}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+42}:\\ \;\;\;\;0.5 \cdot \left|-y\right|\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4e-141)
   (* (- x y) 0.5)
   (if (<= x 6e+42) (* 0.5 (fabs (- y))) (* 1.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4e-141) {
		tmp = (x - y) * 0.5;
	} else if (x <= 6e+42) {
		tmp = 0.5 * fabs(-y);
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4d-141)) then
        tmp = (x - y) * 0.5d0
    else if (x <= 6d+42) then
        tmp = 0.5d0 * abs(-y)
    else
        tmp = 1.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4e-141) {
		tmp = (x - y) * 0.5;
	} else if (x <= 6e+42) {
		tmp = 0.5 * Math.abs(-y);
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4e-141:
		tmp = (x - y) * 0.5
	elif x <= 6e+42:
		tmp = 0.5 * math.fabs(-y)
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4e-141)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (x <= 6e+42)
		tmp = Float64(0.5 * abs(Float64(-y)));
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4e-141)
		tmp = (x - y) * 0.5;
	elseif (x <= 6e+42)
		tmp = 0.5 * abs(-y);
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4e-141], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 6e+42], N[(0.5 * N[Abs[(-y)], $MachinePrecision]), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.0000000000000002e-141

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. /-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-/r/ N/A

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

      5. /-rgt-identity N/A

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

      6. metadata-eval N/A

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

      7. fabs-mul N/A

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. fabs-sub N/A

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

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

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

      14. lower-fabs.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. *-lft-identity N/A

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

      18. metadata-eval N/A

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

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

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

      20. *-commutative N/A

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

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. metadata-eval N/A

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

      24. *-lft-identity N/A

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

      25. lower--.f64 33.1

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

   5. Applied rewrites 33.1%

      \[\leadsto \color{blue}{0.5 \cdot \left|x - y\right|} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. fabs-sub N/A

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

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

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

      7. unpow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 11.9

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

   7. Applied rewrites 11.9%

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

      1. lift--.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. unpow2 N/A

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

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

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

      7. fabs-sub N/A

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

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

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

      9. sqrt-unprod N/A

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

      10. rem-square-sqrt N/A

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

      11. lift--.f64 87.7

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

   9. Applied rewrites 87.7%

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

   if -4.0000000000000002e-141 < x < 6.00000000000000058e42

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. /-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-/r/ N/A

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

      5. /-rgt-identity N/A

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

      6. metadata-eval N/A

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

      7. fabs-mul N/A

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. fabs-sub N/A

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

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

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

      14. lower-fabs.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. *-lft-identity N/A

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

      18. metadata-eval N/A

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

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

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

      20. *-commutative N/A

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

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. metadata-eval N/A

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

      24. *-lft-identity N/A

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

      25. lower--.f64 80.6

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

   5. Applied rewrites 80.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{1}{2} \cdot \left|-1 \cdot y\right| \]
   7. 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 79.0

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

   8. Applied rewrites 79.0%

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

   if 6.00000000000000058e42 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

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

   5. Applied rewrites 99.8%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 90.4

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

   7. Applied rewrites 90.4%

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

   8. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. rem-square-sqrt N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

   10. Applied rewrites 81.8%

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

Alternative 9: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-251}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|x\right|, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.5e-251)
   (* (- x y) 0.5)
   (if (<= y 2.3e-24) (fma 0.5 (fabs x) x) (* y 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.5e-251) {
		tmp = (x - y) * 0.5;
	} else if (y <= 2.3e-24) {
		tmp = fma(0.5, fabs(x), x);
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.5e-251)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (y <= 2.3e-24)
		tmp = fma(0.5, abs(x), x);
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.5e-251], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[y, 2.3e-24], N[(0.5 * N[Abs[x], $MachinePrecision] + x), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.5000000000000001e-251

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. /-rgt-identity N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\left|y - x\right|}{\color{blue}{1}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\left|y - x\right|}{\frac{1}{\color{blue}{1}}} \]

      4. associate-/r/ N/A

         \[\leadsto \frac{1}{2} \cdot \left(\frac{\left|y - x\right|}{1} \cdot \color{blue}{1}\right) \]

      5. /-rgt-identity N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left|y - x\right| \cdot 1\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left|y - x\right| \cdot \left|1\right|\right) \]

      7. fabs-mul N/A

         \[\leadsto \frac{1}{2} \cdot \left|\left(y - x\right) \cdot 1\right| \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left|1 \cdot \left(y - x\right)\right| \]

      9. distribute-lft-out-- N/A

         \[\leadsto \frac{1}{2} \cdot \left|1 \cdot y - 1 \cdot x\right| \]

      10. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot y - 1 \cdot x\right| \]

      11. *-lft-identity N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot y - x\right| \]

      12. fabs-sub N/A

          \[\leadsto \frac{1}{2} \cdot \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right| \]

      13. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{1}{2} \cdot \left|x + -1 \cdot y\right| \]

      14. lower-fabs.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left|x + -1 \cdot y\right| \]

      15. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left|x + y \cdot -1\right| \]

      16. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{1}{2} \cdot \left|x - \left(\mathsf{neg}\left(y\right)\right) \cdot -1\right| \]

      17. *-lft-identity N/A

          \[\leadsto \frac{1}{2} \cdot \left|1 \cdot x - \left(\mathsf{neg}\left(y\right)\right) \cdot -1\right| \]

      18. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - \left(\mathsf{neg}\left(y\right)\right) \cdot -1\right| \]

      19. distribute-lft-neg-in N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - \left(\mathsf{neg}\left(y \cdot -1\right)\right)\right| \]

      20. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - \left(\mathsf{neg}\left(-1 \cdot y\right)\right)\right| \]

      21. mul-1-neg N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right| \]

      22. remove-double-neg N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - y\right| \]

      23. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot \left|1 \cdot x - y\right| \]

      24. *-lft-identity N/A

          \[\leadsto \frac{1}{2} \cdot \left|x - y\right| \]

      25. lower--.f64 60.1

          \[\leadsto 0.5 \cdot \left|x - y\right| \]

   5. Applied rewrites 60.1%

      \[\leadsto \color{blue}{0.5 \cdot \left|x - y\right|} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left|x - y\right|} \]

      2. *-commutative N/A

         \[\leadsto \left|x - y\right| \cdot \color{blue}{\frac{1}{2}} \]

      3. lift-fabs.f64 N/A

         \[\leadsto \left|x - y\right| \cdot \frac{1}{2} \]

      4. lift--.f64 N/A

         \[\leadsto \left|x - y\right| \cdot \frac{1}{2} \]

      5. fabs-sub N/A

         \[\leadsto \left|y - x\right| \cdot \frac{1}{2} \]

      6. rem-sqrt-square-rev N/A

         \[\leadsto \sqrt{\left(y - x\right) \cdot \left(y - x\right)} \cdot \frac{1}{2} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{{\left(y - x\right)}^{2}} \cdot \frac{1}{2} \]

      8. sqrt-pow1 N/A

         \[\leadsto {\left(y - x\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{1}{2} \]

      9. metadata-eval N/A

         \[\leadsto {\left(y - x\right)}^{1} \cdot \frac{1}{2} \]

      10. unpow1 N/A

          \[\leadsto \left(y - x\right) \cdot \frac{1}{2} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(y - x\right) \cdot \color{blue}{\frac{1}{2}} \]

      12. lift--.f64 1.2

          \[\leadsto \left(y - x\right) \cdot 0.5 \]

   7. Applied rewrites 1.2%

      \[\leadsto \left(y - x\right) \cdot \color{blue}{0.5} \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(y - x\right) \cdot \frac{1}{2} \]

      2. unpow1 N/A

         \[\leadsto {\left(y - x\right)}^{1} \cdot \frac{1}{2} \]

      3. metadata-eval N/A

         \[\leadsto {\left(y - x\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{1}{2} \]

      4. sqrt-pow1 N/A

         \[\leadsto \sqrt{{\left(y - x\right)}^{2}} \cdot \frac{1}{2} \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\left(y - x\right) \cdot \left(y - x\right)} \cdot \frac{1}{2} \]

      6. rem-sqrt-square-rev N/A

         \[\leadsto \left|y - x\right| \cdot \frac{1}{2} \]

      7. fabs-sub N/A

         \[\leadsto \left|x - y\right| \cdot \frac{1}{2} \]

      8. rem-sqrt-square-rev N/A

         \[\leadsto \sqrt{\left(x - y\right) \cdot \left(x - y\right)} \cdot \frac{1}{2} \]

      9. sqrt-unprod N/A

         \[\leadsto \left(\sqrt{x - y} \cdot \sqrt{x - y}\right) \cdot \frac{1}{2} \]

      10. rem-square-sqrt N/A

          \[\leadsto \left(x - y\right) \cdot \frac{1}{2} \]

      11. lift--.f64 83.6

          \[\leadsto \left(x - y\right) \cdot 0.5 \]

   9. Applied rewrites 83.6%

      \[\leadsto \left(x - y\right) \cdot 0.5 \]

   if -2.5000000000000001e-251 < y < 2.3000000000000001e-24

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]
   4. 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

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left|y - x\right|}, x\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|x - y\right|, x\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x\right|, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 82.5%

      \[\leadsto \mathsf{fma}\left(0.5, \left|x\right|, x\right) \]

   if 2.3000000000000001e-24 < y

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left|y - x\right|} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left|y - x\right|} \]

      2. /-rgt-identity N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\left|y - x\right|}{\color{blue}{1}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\left|y - x\right|}{\frac{1}{\color{blue}{1}}} \]

      4. associate-/r/ N/A

         \[\leadsto \frac{1}{2} \cdot \left(\frac{\left|y - x\right|}{1} \cdot \color{blue}{1}\right) \]

      5. /-rgt-identity N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left|y - x\right| \cdot 1\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left|y - x\right| \cdot \left|1\right|\right) \]

      7. fabs-mul N/A

         \[\leadsto \frac{1}{2} \cdot \left|\left(y - x\right) \cdot 1\right| \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left|1 \cdot \left(y - x\right)\right| \]

      9. distribute-lft-out-- N/A

         \[\leadsto \frac{1}{2} \cdot \left|1 \cdot y - 1 \cdot x\right| \]

      10. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot y - 1 \cdot x\right| \]

      11. *-lft-identity N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot y - x\right| \]

      12. fabs-sub N/A

          \[\leadsto \frac{1}{2} \cdot \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right| \]

      13. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{1}{2} \cdot \left|x + -1 \cdot y\right| \]

      14. lower-fabs.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left|x + -1 \cdot y\right| \]

      15. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left|x + y \cdot -1\right| \]

      16. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{1}{2} \cdot \left|x - \left(\mathsf{neg}\left(y\right)\right) \cdot -1\right| \]

      17. *-lft-identity N/A

          \[\leadsto \frac{1}{2} \cdot \left|1 \cdot x - \left(\mathsf{neg}\left(y\right)\right) \cdot -1\right| \]

      18. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - \left(\mathsf{neg}\left(y\right)\right) \cdot -1\right| \]

      19. distribute-lft-neg-in N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - \left(\mathsf{neg}\left(y \cdot -1\right)\right)\right| \]

      20. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - \left(\mathsf{neg}\left(-1 \cdot y\right)\right)\right| \]

      21. mul-1-neg N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right| \]

      22. remove-double-neg N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - y\right| \]

      23. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot \left|1 \cdot x - y\right| \]

      24. *-lft-identity N/A

          \[\leadsto \frac{1}{2} \cdot \left|x - y\right| \]

      25. lower--.f64 70.6

          \[\leadsto 0.5 \cdot \left|x - y\right| \]

   5. Applied rewrites 70.6%

      \[\leadsto \color{blue}{0.5 \cdot \left|x - y\right|} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left|x - y\right|} \]

      2. *-commutative N/A

         \[\leadsto \left|x - y\right| \cdot \color{blue}{\frac{1}{2}} \]

      3. lift-fabs.f64 N/A

         \[\leadsto \left|x - y\right| \cdot \frac{1}{2} \]

      4. lift--.f64 N/A

         \[\leadsto \left|x - y\right| \cdot \frac{1}{2} \]

      5. fabs-sub N/A

         \[\leadsto \left|y - x\right| \cdot \frac{1}{2} \]

      6. rem-sqrt-square-rev N/A

         \[\leadsto \sqrt{\left(y - x\right) \cdot \left(y - x\right)} \cdot \frac{1}{2} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{{\left(y - x\right)}^{2}} \cdot \frac{1}{2} \]

      8. sqrt-pow1 N/A

         \[\leadsto {\left(y - x\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{1}{2} \]

      9. metadata-eval N/A

         \[\leadsto {\left(y - x\right)}^{1} \cdot \frac{1}{2} \]

      10. unpow1 N/A

          \[\leadsto \left(y - x\right) \cdot \frac{1}{2} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(y - x\right) \cdot \color{blue}{\frac{1}{2}} \]

      12. lift--.f64 66.9

          \[\leadsto \left(y - x\right) \cdot 0.5 \]

   7. Applied rewrites 66.9%

      \[\leadsto \left(y - x\right) \cdot \color{blue}{0.5} \]

   8. Taylor expanded in x around 0

      \[\leadsto y \cdot \frac{1}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 68.2%

       \[\leadsto y \cdot 0.5 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 60.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{-131}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-89}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.3e-131) (* -0.5 y) (if (<= y 1.05e-89) (* x 0.5) (* y 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.3e-131) {
		tmp = -0.5 * y;
	} else if (y <= 1.05e-89) {
		tmp = x * 0.5;
	} 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 (y <= (-2.3d-131)) then
        tmp = (-0.5d0) * y
    else if (y <= 1.05d-89) then
        tmp = x * 0.5d0
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.3e-131) {
		tmp = -0.5 * y;
	} else if (y <= 1.05e-89) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.3e-131:
		tmp = -0.5 * y
	elif y <= 1.05e-89:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.3e-131)
		tmp = Float64(-0.5 * y);
	elseif (y <= 1.05e-89)
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.3e-131)
		tmp = -0.5 * y;
	elseif (y <= 1.05e-89)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.3e-131], N[(-0.5 * y), $MachinePrecision], If[LessEqual[y, 1.05e-89], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.30000000000000022e-131

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]
   4. 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

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left|y - x\right|}, x\right) \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|x - y\right|, x\right)} \]
   6. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      2. rem-sqrt-square-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\left(x - y\right) \cdot \left(x - y\right)}, x\right) \]

      3. sqrt-prod N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{\color{blue}{x - y}}, x\right) \]

      6. lower-sqrt.f64 84.3

         \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \sqrt{x - y}, x\right) \]

   7. Applied rewrites 84.3%

      \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{2} \cdot \color{blue}{y} \]
   9. Step-by-step derivation

      1. lower-*.f64 68.8

         \[\leadsto -0.5 \cdot y \]

   10. Applied rewrites 68.8%

       \[\leadsto -0.5 \cdot \color{blue}{y} \]

   if -2.30000000000000022e-131 < y < 1.05e-89

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left|y - x\right|} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left|y - x\right|} \]

      2. /-rgt-identity N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\left|y - x\right|}{\color{blue}{1}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\left|y - x\right|}{\frac{1}{\color{blue}{1}}} \]

      4. associate-/r/ N/A

         \[\leadsto \frac{1}{2} \cdot \left(\frac{\left|y - x\right|}{1} \cdot \color{blue}{1}\right) \]

      5. /-rgt-identity N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left|y - x\right| \cdot 1\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left|y - x\right| \cdot \left|1\right|\right) \]

      7. fabs-mul N/A

         \[\leadsto \frac{1}{2} \cdot \left|\left(y - x\right) \cdot 1\right| \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left|1 \cdot \left(y - x\right)\right| \]

      9. distribute-lft-out-- N/A

         \[\leadsto \frac{1}{2} \cdot \left|1 \cdot y - 1 \cdot x\right| \]

      10. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot y - 1 \cdot x\right| \]

      11. *-lft-identity N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot y - x\right| \]

      12. fabs-sub N/A

          \[\leadsto \frac{1}{2} \cdot \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right| \]

      13. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{1}{2} \cdot \left|x + -1 \cdot y\right| \]

      14. lower-fabs.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left|x + -1 \cdot y\right| \]

      15. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left|x + y \cdot -1\right| \]

      16. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{1}{2} \cdot \left|x - \left(\mathsf{neg}\left(y\right)\right) \cdot -1\right| \]

      17. *-lft-identity N/A

          \[\leadsto \frac{1}{2} \cdot \left|1 \cdot x - \left(\mathsf{neg}\left(y\right)\right) \cdot -1\right| \]

      18. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - \left(\mathsf{neg}\left(y\right)\right) \cdot -1\right| \]

      19. distribute-lft-neg-in N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - \left(\mathsf{neg}\left(y \cdot -1\right)\right)\right| \]

      20. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - \left(\mathsf{neg}\left(-1 \cdot y\right)\right)\right| \]

      21. mul-1-neg N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right| \]

      22. remove-double-neg N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - y\right| \]

      23. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot \left|1 \cdot x - y\right| \]

      24. *-lft-identity N/A

          \[\leadsto \frac{1}{2} \cdot \left|x - y\right| \]

      25. lower--.f64 20.6

          \[\leadsto 0.5 \cdot \left|x - y\right| \]

   5. Applied rewrites 20.6%

      \[\leadsto \color{blue}{0.5 \cdot \left|x - y\right|} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left|x - y\right|} \]

      2. *-commutative N/A

         \[\leadsto \left|x - y\right| \cdot \color{blue}{\frac{1}{2}} \]

      3. lift-fabs.f64 N/A

         \[\leadsto \left|x - y\right| \cdot \frac{1}{2} \]

      4. lift--.f64 N/A

         \[\leadsto \left|x - y\right| \cdot \frac{1}{2} \]

      5. fabs-sub N/A

         \[\leadsto \left|y - x\right| \cdot \frac{1}{2} \]

      6. rem-sqrt-square-rev N/A

         \[\leadsto \sqrt{\left(y - x\right) \cdot \left(y - x\right)} \cdot \frac{1}{2} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{{\left(y - x\right)}^{2}} \cdot \frac{1}{2} \]

      8. sqrt-pow1 N/A

         \[\leadsto {\left(y - x\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{1}{2} \]

      9. metadata-eval N/A

         \[\leadsto {\left(y - x\right)}^{1} \cdot \frac{1}{2} \]

      10. unpow1 N/A

          \[\leadsto \left(y - x\right) \cdot \frac{1}{2} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(y - x\right) \cdot \color{blue}{\frac{1}{2}} \]

      12. lift--.f64 9.1

          \[\leadsto \left(y - x\right) \cdot 0.5 \]

   7. Applied rewrites 9.1%

      \[\leadsto \left(y - x\right) \cdot \color{blue}{0.5} \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(y - x\right) \cdot \frac{1}{2} \]

      2. unpow1 N/A

         \[\leadsto {\left(y - x\right)}^{1} \cdot \frac{1}{2} \]

      3. metadata-eval N/A

         \[\leadsto {\left(y - x\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{1}{2} \]

      4. sqrt-pow1 N/A

         \[\leadsto \sqrt{{\left(y - x\right)}^{2}} \cdot \frac{1}{2} \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\left(y - x\right) \cdot \left(y - x\right)} \cdot \frac{1}{2} \]

      6. rem-sqrt-square-rev N/A

         \[\leadsto \left|y - x\right| \cdot \frac{1}{2} \]

      7. fabs-sub N/A

         \[\leadsto \left|x - y\right| \cdot \frac{1}{2} \]

      8. rem-sqrt-square-rev N/A

         \[\leadsto \sqrt{\left(x - y\right) \cdot \left(x - y\right)} \cdot \frac{1}{2} \]

      9. sqrt-unprod N/A

         \[\leadsto \left(\sqrt{x - y} \cdot \sqrt{x - y}\right) \cdot \frac{1}{2} \]

      10. rem-square-sqrt N/A

          \[\leadsto \left(x - y\right) \cdot \frac{1}{2} \]

      11. lift--.f64 63.1

          \[\leadsto \left(x - y\right) \cdot 0.5 \]

   9. Applied rewrites 63.1%

      \[\leadsto \left(x - y\right) \cdot 0.5 \]

   10. Taylor expanded in x around inf

       \[\leadsto x \cdot \frac{1}{2} \]
   11. Step-by-step derivation

   12. Applied rewrites 57.5%

       \[\leadsto x \cdot 0.5 \]

   if 1.05e-89 < y

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left|y - x\right|} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left|y - x\right|} \]

      2. /-rgt-identity N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\left|y - x\right|}{\color{blue}{1}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\left|y - x\right|}{\frac{1}{\color{blue}{1}}} \]

      4. associate-/r/ N/A

         \[\leadsto \frac{1}{2} \cdot \left(\frac{\left|y - x\right|}{1} \cdot \color{blue}{1}\right) \]

      5. /-rgt-identity N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left|y - x\right| \cdot 1\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left|y - x\right| \cdot \left|1\right|\right) \]

      7. fabs-mul N/A

         \[\leadsto \frac{1}{2} \cdot \left|\left(y - x\right) \cdot 1\right| \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left|1 \cdot \left(y - x\right)\right| \]

      9. distribute-lft-out-- N/A

         \[\leadsto \frac{1}{2} \cdot \left|1 \cdot y - 1 \cdot x\right| \]

      10. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot y - 1 \cdot x\right| \]

      11. *-lft-identity N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot y - x\right| \]

      12. fabs-sub N/A

          \[\leadsto \frac{1}{2} \cdot \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right| \]

      13. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{1}{2} \cdot \left|x + -1 \cdot y\right| \]

      14. lower-fabs.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left|x + -1 \cdot y\right| \]

      15. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left|x + y \cdot -1\right| \]

      16. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{1}{2} \cdot \left|x - \left(\mathsf{neg}\left(y\right)\right) \cdot -1\right| \]

      17. *-lft-identity N/A

          \[\leadsto \frac{1}{2} \cdot \left|1 \cdot x - \left(\mathsf{neg}\left(y\right)\right) \cdot -1\right| \]

      18. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - \left(\mathsf{neg}\left(y\right)\right) \cdot -1\right| \]

      19. distribute-lft-neg-in N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - \left(\mathsf{neg}\left(y \cdot -1\right)\right)\right| \]

      20. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - \left(\mathsf{neg}\left(-1 \cdot y\right)\right)\right| \]

      21. mul-1-neg N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right| \]

      22. remove-double-neg N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - y\right| \]

      23. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot \left|1 \cdot x - y\right| \]

      24. *-lft-identity N/A

          \[\leadsto \frac{1}{2} \cdot \left|x - y\right| \]

      25. lower--.f64 67.4

          \[\leadsto 0.5 \cdot \left|x - y\right| \]

   5. Applied rewrites 67.4%

      \[\leadsto \color{blue}{0.5 \cdot \left|x - y\right|} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left|x - y\right|} \]

      2. *-commutative N/A

         \[\leadsto \left|x - y\right| \cdot \color{blue}{\frac{1}{2}} \]

      3. lift-fabs.f64 N/A

         \[\leadsto \left|x - y\right| \cdot \frac{1}{2} \]

      4. lift--.f64 N/A

         \[\leadsto \left|x - y\right| \cdot \frac{1}{2} \]

      5. fabs-sub N/A

         \[\leadsto \left|y - x\right| \cdot \frac{1}{2} \]

      6. rem-sqrt-square-rev N/A

         \[\leadsto \sqrt{\left(y - x\right) \cdot \left(y - x\right)} \cdot \frac{1}{2} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{{\left(y - x\right)}^{2}} \cdot \frac{1}{2} \]

      8. sqrt-pow1 N/A

         \[\leadsto {\left(y - x\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{1}{2} \]

      9. metadata-eval N/A

         \[\leadsto {\left(y - x\right)}^{1} \cdot \frac{1}{2} \]

      10. unpow1 N/A

          \[\leadsto \left(y - x\right) \cdot \frac{1}{2} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(y - x\right) \cdot \color{blue}{\frac{1}{2}} \]

      12. lift--.f64 63.7

          \[\leadsto \left(y - x\right) \cdot 0.5 \]

   7. Applied rewrites 63.7%

      \[\leadsto \left(y - x\right) \cdot \color{blue}{0.5} \]

   8. Taylor expanded in x around 0

      \[\leadsto y \cdot \frac{1}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 65.0%

       \[\leadsto y \cdot 0.5 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 57.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.092:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 34000:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.092) (* x 0.5) (if (<= x 34000.0) (* -0.5 y) (* 1.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.092) {
		tmp = x * 0.5;
	} else if (x <= 34000.0) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.092d0)) then
        tmp = x * 0.5d0
    else if (x <= 34000.0d0) then
        tmp = (-0.5d0) * y
    else
        tmp = 1.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.092) {
		tmp = x * 0.5;
	} else if (x <= 34000.0) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.092:
		tmp = x * 0.5
	elif x <= 34000.0:
		tmp = -0.5 * y
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.092)
		tmp = Float64(x * 0.5);
	elseif (x <= 34000.0)
		tmp = Float64(-0.5 * y);
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.092)
		tmp = x * 0.5;
	elseif (x <= 34000.0)
		tmp = -0.5 * y;
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.092], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 34000.0], N[(-0.5 * y), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.091999999999999998

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left|y - x\right|} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left|y - x\right|} \]

      2. /-rgt-identity N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\left|y - x\right|}{\color{blue}{1}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\left|y - x\right|}{\frac{1}{\color{blue}{1}}} \]

      4. associate-/r/ N/A

         \[\leadsto \frac{1}{2} \cdot \left(\frac{\left|y - x\right|}{1} \cdot \color{blue}{1}\right) \]

      5. /-rgt-identity N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left|y - x\right| \cdot 1\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left|y - x\right| \cdot \left|1\right|\right) \]

      7. fabs-mul N/A

         \[\leadsto \frac{1}{2} \cdot \left|\left(y - x\right) \cdot 1\right| \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left|1 \cdot \left(y - x\right)\right| \]

      9. distribute-lft-out-- N/A

         \[\leadsto \frac{1}{2} \cdot \left|1 \cdot y - 1 \cdot x\right| \]

      10. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot y - 1 \cdot x\right| \]

      11. *-lft-identity N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot y - x\right| \]

      12. fabs-sub N/A

          \[\leadsto \frac{1}{2} \cdot \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right| \]

      13. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{1}{2} \cdot \left|x + -1 \cdot y\right| \]

      14. lower-fabs.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left|x + -1 \cdot y\right| \]

      15. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left|x + y \cdot -1\right| \]

      16. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{1}{2} \cdot \left|x - \left(\mathsf{neg}\left(y\right)\right) \cdot -1\right| \]

      17. *-lft-identity N/A

          \[\leadsto \frac{1}{2} \cdot \left|1 \cdot x - \left(\mathsf{neg}\left(y\right)\right) \cdot -1\right| \]

      18. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - \left(\mathsf{neg}\left(y\right)\right) \cdot -1\right| \]

      19. distribute-lft-neg-in N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - \left(\mathsf{neg}\left(y \cdot -1\right)\right)\right| \]

      20. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - \left(\mathsf{neg}\left(-1 \cdot y\right)\right)\right| \]

      21. mul-1-neg N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right| \]

      22. remove-double-neg N/A

          \[\leadsto \frac{1}{2} \cdot \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - y\right| \]

      23. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot \left|1 \cdot x - y\right| \]

      24. *-lft-identity N/A

          \[\leadsto \frac{1}{2} \cdot \left|x - y\right| \]

      25. lower--.f64 17.3

          \[\leadsto 0.5 \cdot \left|x - y\right| \]

   5. Applied rewrites 17.3%

      \[\leadsto \color{blue}{0.5 \cdot \left|x - y\right|} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left|x - y\right|} \]

      2. *-commutative N/A

         \[\leadsto \left|x - y\right| \cdot \color{blue}{\frac{1}{2}} \]

      3. lift-fabs.f64 N/A

         \[\leadsto \left|x - y\right| \cdot \frac{1}{2} \]

      4. lift--.f64 N/A

         \[\leadsto \left|x - y\right| \cdot \frac{1}{2} \]

      5. fabs-sub N/A

         \[\leadsto \left|y - x\right| \cdot \frac{1}{2} \]

      6. rem-sqrt-square-rev N/A

         \[\leadsto \sqrt{\left(y - x\right) \cdot \left(y - x\right)} \cdot \frac{1}{2} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{{\left(y - x\right)}^{2}} \cdot \frac{1}{2} \]

      8. sqrt-pow1 N/A

         \[\leadsto {\left(y - x\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{1}{2} \]

      9. metadata-eval N/A

         \[\leadsto {\left(y - x\right)}^{1} \cdot \frac{1}{2} \]

      10. unpow1 N/A

          \[\leadsto \left(y - x\right) \cdot \frac{1}{2} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(y - x\right) \cdot \color{blue}{\frac{1}{2}} \]

      12. lift--.f64 7.0

          \[\leadsto \left(y - x\right) \cdot 0.5 \]

   7. Applied rewrites 7.0%

      \[\leadsto \left(y - x\right) \cdot \color{blue}{0.5} \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(y - x\right) \cdot \frac{1}{2} \]

      2. unpow1 N/A

         \[\leadsto {\left(y - x\right)}^{1} \cdot \frac{1}{2} \]

      3. metadata-eval N/A

         \[\leadsto {\left(y - x\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{1}{2} \]

      4. sqrt-pow1 N/A

         \[\leadsto \sqrt{{\left(y - x\right)}^{2}} \cdot \frac{1}{2} \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\left(y - x\right) \cdot \left(y - x\right)} \cdot \frac{1}{2} \]

      6. rem-sqrt-square-rev N/A

         \[\leadsto \left|y - x\right| \cdot \frac{1}{2} \]

      7. fabs-sub N/A

         \[\leadsto \left|x - y\right| \cdot \frac{1}{2} \]

      8. rem-sqrt-square-rev N/A

         \[\leadsto \sqrt{\left(x - y\right) \cdot \left(x - y\right)} \cdot \frac{1}{2} \]

      9. sqrt-unprod N/A

         \[\leadsto \left(\sqrt{x - y} \cdot \sqrt{x - y}\right) \cdot \frac{1}{2} \]

      10. rem-square-sqrt N/A

          \[\leadsto \left(x - y\right) \cdot \frac{1}{2} \]

      11. lift--.f64 92.3

          \[\leadsto \left(x - y\right) \cdot 0.5 \]

   9. Applied rewrites 92.3%

      \[\leadsto \left(x - y\right) \cdot 0.5 \]

   10. Taylor expanded in x around inf

       \[\leadsto x \cdot \frac{1}{2} \]
   11. Step-by-step derivation

   12. Applied rewrites 82.1%

       \[\leadsto x \cdot 0.5 \]

   if -0.091999999999999998 < x < 34000

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]
   4. 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

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left|y - x\right|}, x\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|x - y\right|, x\right)} \]
   6. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      2. rem-sqrt-square-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\left(x - y\right) \cdot \left(x - y\right)}, x\right) \]

      3. sqrt-prod N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{\color{blue}{x - y}}, x\right) \]

      6. lower-sqrt.f64 51.9

         \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \sqrt{x - y}, x\right) \]

   7. Applied rewrites 51.9%

      \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{2} \cdot \color{blue}{y} \]
   9. Step-by-step derivation

      1. lower-*.f64 44.4

         \[\leadsto -0.5 \cdot y \]

   10. Applied rewrites 44.4%

       \[\leadsto -0.5 \cdot \color{blue}{y} \]

   if 34000 < x

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]
   4. 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

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left|y - x\right|}, x\right) \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|x - y\right|, x\right)} \]
   6. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      2. rem-sqrt-square-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\left(x - y\right) \cdot \left(x - y\right)}, x\right) \]

      3. sqrt-prod N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{\color{blue}{x - y}}, x\right) \]

      6. lower-sqrt.f64 85.8

         \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \sqrt{x - y}, x\right) \]

   7. Applied rewrites 85.8%

      \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{3}{2} \cdot \color{blue}{x} \]
   9. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left(-1 \cdot \frac{-3}{2}\right) \cdot x \]

      2. metadata-eval N/A

         \[\leadsto \left(-1 \cdot \left(\frac{-1}{2} - 1\right)\right) \cdot x \]

      3. metadata-eval N/A

         \[\leadsto \left(-1 \cdot \left(\frac{1}{2} \cdot -1 - 1\right)\right) \cdot x \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(-1 \cdot \left(\frac{1}{2} \cdot \left(\sqrt{-1} \cdot \sqrt{-1}\right) - 1\right)\right) \cdot x \]

      5. unpow2 N/A

         \[\leadsto \left(-1 \cdot \left(\frac{1}{2} \cdot {\left(\sqrt{-1}\right)}^{2} - 1\right)\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto -1 \cdot \left(\left(\frac{1}{2} \cdot {\left(\sqrt{-1}\right)}^{2} - 1\right) \cdot \color{blue}{x}\right) \]

      7. *-commutative N/A

         \[\leadsto -1 \cdot \left(x \cdot \left(\frac{1}{2} \cdot {\left(\sqrt{-1}\right)}^{2} - \color{blue}{1}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto -1 \cdot \left(\left(\frac{1}{2} \cdot {\left(\sqrt{-1}\right)}^{2} - 1\right) \cdot x\right) \]

      9. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(\frac{1}{2} \cdot {\left(\sqrt{-1}\right)}^{2} - 1\right)\right) \cdot x \]

   10. Applied rewrites 76.1%

       \[\leadsto 1.5 \cdot \color{blue}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 46.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{-131}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -2.15e-131) (* -0.5 y) (* 1.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.15e-131) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.15d-131)) then
        tmp = (-0.5d0) * y
    else
        tmp = 1.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.15e-131) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.15e-131:
		tmp = -0.5 * y
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.15e-131)
		tmp = Float64(-0.5 * y);
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.15e-131)
		tmp = -0.5 * y;
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.15e-131], N[(-0.5 * y), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.15000000000000009e-131

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]
   4. 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

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left|y - x\right|}, x\right) \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|x - y\right|, x\right)} \]
   6. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      2. rem-sqrt-square-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\left(x - y\right) \cdot \left(x - y\right)}, x\right) \]

      3. sqrt-prod N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{\color{blue}{x - y}}, x\right) \]

      6. lower-sqrt.f64 84.3

         \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \sqrt{x - y}, x\right) \]

   7. Applied rewrites 84.3%

      \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{2} \cdot \color{blue}{y} \]
   9. Step-by-step derivation

      1. lower-*.f64 68.8

         \[\leadsto -0.5 \cdot y \]

   10. Applied rewrites 68.8%

       \[\leadsto -0.5 \cdot \color{blue}{y} \]

   if -2.15000000000000009e-131 < y

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]
   4. 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

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left|y - x\right|}, x\right) \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|x - y\right|, x\right)} \]
   6. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      2. rem-sqrt-square-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\left(x - y\right) \cdot \left(x - y\right)}, x\right) \]

      3. sqrt-prod N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{\color{blue}{x - y}}, x\right) \]

      6. lower-sqrt.f64 31.1

         \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \sqrt{x - y}, x\right) \]

   7. Applied rewrites 31.1%

      \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{3}{2} \cdot \color{blue}{x} \]
   9. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left(-1 \cdot \frac{-3}{2}\right) \cdot x \]

      2. metadata-eval N/A

         \[\leadsto \left(-1 \cdot \left(\frac{-1}{2} - 1\right)\right) \cdot x \]

      3. metadata-eval N/A

         \[\leadsto \left(-1 \cdot \left(\frac{1}{2} \cdot -1 - 1\right)\right) \cdot x \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(-1 \cdot \left(\frac{1}{2} \cdot \left(\sqrt{-1} \cdot \sqrt{-1}\right) - 1\right)\right) \cdot x \]

      5. unpow2 N/A

         \[\leadsto \left(-1 \cdot \left(\frac{1}{2} \cdot {\left(\sqrt{-1}\right)}^{2} - 1\right)\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto -1 \cdot \left(\left(\frac{1}{2} \cdot {\left(\sqrt{-1}\right)}^{2} - 1\right) \cdot \color{blue}{x}\right) \]

      7. *-commutative N/A

         \[\leadsto -1 \cdot \left(x \cdot \left(\frac{1}{2} \cdot {\left(\sqrt{-1}\right)}^{2} - \color{blue}{1}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto -1 \cdot \left(\left(\frac{1}{2} \cdot {\left(\sqrt{-1}\right)}^{2} - 1\right) \cdot x\right) \]

      9. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(\frac{1}{2} \cdot {\left(\sqrt{-1}\right)}^{2} - 1\right)\right) \cdot x \]

   10. Applied rewrites 34.8%

       \[\leadsto 1.5 \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 31.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-251}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -2.4e-251) (* -0.5 y) x))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.4e-251) {
		tmp = -0.5 * y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.4d-251)) then
        tmp = (-0.5d0) * y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.4e-251) {
		tmp = -0.5 * y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.4e-251:
		tmp = -0.5 * y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.4e-251)
		tmp = Float64(-0.5 * y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.4e-251)
		tmp = -0.5 * y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.4e-251], N[(-0.5 * y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -2.39999999999999996e-251

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]
   4. 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

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left|y - x\right|}, x\right) \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|x - y\right|, x\right)} \]
   6. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      2. rem-sqrt-square-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\left(x - y\right) \cdot \left(x - y\right)}, x\right) \]

      3. sqrt-prod N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{x - y} \cdot \sqrt{\color{blue}{x - y}}, x\right) \]

      6. lower-sqrt.f64 74.4

         \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \sqrt{x - y}, x\right) \]

   7. Applied rewrites 74.4%

      \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{2} \cdot \color{blue}{y} \]
   9. Step-by-step derivation

      1. lower-*.f64 58.0

         \[\leadsto -0.5 \cdot y \]

   10. Applied rewrites 58.0%

       \[\leadsto -0.5 \cdot \color{blue}{y} \]

   if -2.39999999999999996e-251 < y

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 12.7%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 11.4% accurate, 20.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 x)

C

double code(double x, double y) {
	return 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 = x
end function

Java

public static double code(double x, double y) {
	return x;
}

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

function tmp = code(x, y)
	tmp = x;
end

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 99.9%

   \[x + \frac{\left|y - x\right|}{2} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Applied rewrites 11.4%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025026 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))