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

Percentage Accurate: 99.9% → 99.9%

Time: 2.9s

Alternatives: 12

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

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

   3. fabs-sub N/A

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

   4. *-lft-identity N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. *-lft-identity N/A

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

   8. *-lft-identity N/A

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

   9. metadata-eval N/A

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

   10. cancel-sign-sub N/A

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

   11. lower-fabs.f64 N/A

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

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

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

   13. metadata-eval N/A

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

   14. *-lft-identity N/A

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

   15. lower--.f64 99.9

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

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: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-151}:\\ \;\;\;\;\mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-171}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|x\right|, x\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.6e-151)
   (fma 1.5 x (* -0.5 y))
   (if (<= y 2.5e-171) (fma 0.5 (fabs x) x) (* 0.5 (+ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.6e-151) {
		tmp = fma(1.5, x, (-0.5 * y));
	} else if (y <= 2.5e-171) {
		tmp = fma(0.5, fabs(x), x);
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.6e-151)
		tmp = fma(1.5, x, Float64(-0.5 * y));
	elseif (y <= 2.5e-171)
		tmp = fma(0.5, abs(x), x);
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.6e-151], N[(1.5 * x + N[(-0.5 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.5e-171], N[(0.5 * N[Abs[x], $MachinePrecision] + x), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.60000000000000011e-151

   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

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub N/A

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

      11. lower-fabs.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. lower--.f64 100.0

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

   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-fma.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. fabs-sub N/A

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

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

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

      8. unpow2 N/A

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

      9. sqrt-pow1 N/A

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

      10. metadata-eval N/A

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

      11. unpow1 N/A

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

      12. lift--.f64 22.5

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

   7. Applied rewrites 22.5%

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

      1. lift--.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. unpow2 N/A

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

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

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

      7. fabs-sub N/A

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

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

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

      9. sqrt-prod N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower--.f64 81.3

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

   9. Applied rewrites 81.3%

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

   10. Taylor expanded in x around 0

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

       1. rem-square-sqrt N/A

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

       2. *-commutative N/A

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

       3. rem-square-sqrt N/A

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

       4. sqrt-unprod N/A

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

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

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

       6. +-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-*.f64 85.0

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

   12. Applied rewrites 85.0%

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

   if -1.60000000000000011e-151 < y < 2.49999999999999996e-171

   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

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub N/A

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

      11. lower-fabs.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. lower--.f64 99.9

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

   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 90.2%

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

   if 2.49999999999999996e-171 < y

   1. Initial program 99.9%

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

   4. Applied rewrites 59.8%

      \[\leadsto \color{blue}{\frac{\frac{{\left(y - x\right)}^{2}}{4} - x \cdot x}{\frac{\left|x - y\right|}{2} - x}} \]

   5. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. fabs-sub N/A

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

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

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

      8. unpow2 N/A

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

      9. sqrt-pow1 N/A

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

      10. metadata-eval N/A

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

      11. unpow1 N/A

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

      12. lift--.f64 46.8

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

   7. Applied rewrites 46.8%

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

   8. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 89.6

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

   10. Applied rewrites 89.6%

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

Alternative 3: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-151}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-171}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|x\right|, x\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.6e-151)
   (fma (- x y) 0.5 x)
   (if (<= y 2.5e-171) (fma 0.5 (fabs x) x) (* 0.5 (+ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.6e-151) {
		tmp = fma((x - y), 0.5, x);
	} else if (y <= 2.5e-171) {
		tmp = fma(0.5, fabs(x), x);
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.6e-151)
		tmp = fma(Float64(x - y), 0.5, x);
	elseif (y <= 2.5e-171)
		tmp = fma(0.5, abs(x), x);
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.6e-151], N[(N[(x - y), $MachinePrecision] * 0.5 + x), $MachinePrecision], If[LessEqual[y, 2.5e-171], N[(0.5 * N[Abs[x], $MachinePrecision] + x), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.60000000000000011e-151

   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

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub N/A

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

      11. lower-fabs.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. lower--.f64 100.0

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

   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-fma.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. fabs-sub N/A

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

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

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

      8. unpow2 N/A

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

      9. sqrt-pow1 N/A

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

      10. metadata-eval N/A

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

      11. unpow1 N/A

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

      12. lift--.f64 22.5

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

   7. Applied rewrites 22.5%

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

      1. lift--.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. unpow2 N/A

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

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

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

      7. fabs-sub N/A

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

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

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

      9. sqrt-prod N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower--.f64 81.3

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

   9. Applied rewrites 81.3%

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

       1. lift-*.f64 N/A

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

       2. lift--.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. lift--.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. rem-square-sqrt N/A

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

       7. lift--.f64 85.0

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

   11. Applied rewrites 85.0%

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

   if -1.60000000000000011e-151 < y < 2.49999999999999996e-171

   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

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub N/A

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

      11. lower-fabs.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. lower--.f64 99.9

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

   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 90.2%

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

   if 2.49999999999999996e-171 < y

   1. Initial program 99.9%

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

   4. Applied rewrites 59.8%

      \[\leadsto \color{blue}{\frac{\frac{{\left(y - x\right)}^{2}}{4} - x \cdot x}{\frac{\left|x - y\right|}{2} - x}} \]

   5. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. fabs-sub N/A

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

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

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

      8. unpow2 N/A

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

      9. sqrt-pow1 N/A

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

      10. metadata-eval N/A

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

      11. unpow1 N/A

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

      12. lift--.f64 46.8

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

   7. Applied rewrites 46.8%

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

   8. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 89.6

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

   10. Applied rewrites 89.6%

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

Alternative 4: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-114}:\\ \;\;\;\;0.5 \cdot \left(x - y\right)\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-171}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|x\right|, x\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.45e-114)
   (* 0.5 (- x y))
   (if (<= y 2.5e-171) (fma 0.5 (fabs x) x) (* 0.5 (+ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.45e-114) {
		tmp = 0.5 * (x - y);
	} else if (y <= 2.5e-171) {
		tmp = fma(0.5, fabs(x), x);
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.45e-114)
		tmp = Float64(0.5 * Float64(x - y));
	elseif (y <= 2.5e-171)
		tmp = fma(0.5, abs(x), x);
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.45e-114], N[(0.5 * N[(x - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.5e-171], N[(0.5 * N[Abs[x], $MachinePrecision] + x), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.44999999999999998e-114

   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. fabs-sub N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. cancel-sign-sub N/A

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

      10. lower-fabs.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. lower--.f64 67.3

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

   5. Applied rewrites 67.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. fabs-sub N/A

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

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

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

      8. unpow2 N/A

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

      9. sqrt-pow1 N/A

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

      10. metadata-eval N/A

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

      11. unpow1 N/A

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

      12. lift--.f64 0.8

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

   7. Applied rewrites 0.8%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 2.3

         \[\leadsto -0.5 \cdot x \]

   10. Applied rewrites 2.3%

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

   11. Taylor expanded in x around 0

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

       1. fabs-sub N/A

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

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

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

       3. sqrt-unprod N/A

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

       4. rem-square-sqrt N/A

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

       5. lower-*.f64 N/A

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

       6. lift--.f64 84.1

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

   13. Applied rewrites 84.1%

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

   if -1.44999999999999998e-114 < y < 2.49999999999999996e-171

   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

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub N/A

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

      11. lower-fabs.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. lower--.f64 99.9

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

   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 87.3%

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

   if 2.49999999999999996e-171 < y

   1. Initial program 99.9%

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

   4. Applied rewrites 59.8%

      \[\leadsto \color{blue}{\frac{\frac{{\left(y - x\right)}^{2}}{4} - x \cdot x}{\frac{\left|x - y\right|}{2} - x}} \]

   5. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. fabs-sub N/A

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

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

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

      8. unpow2 N/A

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

      9. sqrt-pow1 N/A

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

      10. metadata-eval N/A

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

      11. unpow1 N/A

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

      12. lift--.f64 46.8

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

   7. Applied rewrites 46.8%

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

   8. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 89.6

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

   10. Applied rewrites 89.6%

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

Alternative 5: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-114}:\\ \;\;\;\;0.5 \cdot \left(x - y\right)\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|x\right|, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.45e-114)
   (* 0.5 (- x y))
   (if (<= y 1.15e-46) (fma 0.5 (fabs x) x) (fma y 0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.45e-114) {
		tmp = 0.5 * (x - y);
	} else if (y <= 1.15e-46) {
		tmp = fma(0.5, fabs(x), x);
	} else {
		tmp = fma(y, 0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.45e-114)
		tmp = Float64(0.5 * Float64(x - y));
	elseif (y <= 1.15e-46)
		tmp = fma(0.5, abs(x), x);
	else
		tmp = fma(y, 0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.45e-114], N[(0.5 * N[(x - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.15e-46], N[(0.5 * N[Abs[x], $MachinePrecision] + x), $MachinePrecision], N[(y * 0.5 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.44999999999999998e-114

   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. fabs-sub N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. cancel-sign-sub N/A

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

      10. lower-fabs.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. lower--.f64 67.3

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

   5. Applied rewrites 67.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. fabs-sub N/A

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

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

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

      8. unpow2 N/A

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

      9. sqrt-pow1 N/A

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

      10. metadata-eval N/A

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

      11. unpow1 N/A

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

      12. lift--.f64 0.8

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

   7. Applied rewrites 0.8%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 2.3

         \[\leadsto -0.5 \cdot x \]

   10. Applied rewrites 2.3%

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

   11. Taylor expanded in x around 0

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

       1. fabs-sub N/A

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

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

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

       3. sqrt-unprod N/A

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

       4. rem-square-sqrt N/A

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

       5. lower-*.f64 N/A

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

       6. lift--.f64 84.1

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

   13. Applied rewrites 84.1%

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

   if -1.44999999999999998e-114 < y < 1.15e-46

   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

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub N/A

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

      11. lower-fabs.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. lower--.f64 99.9

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

   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 81.4%

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

   if 1.15e-46 < y

   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

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub N/A

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

      11. lower-fabs.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. lower--.f64 100.0

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

   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-fma.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. fabs-sub N/A

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

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

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

      8. unpow2 N/A

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

      9. sqrt-pow1 N/A

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

      10. metadata-eval N/A

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

      11. unpow1 N/A

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

      12. lift--.f64 92.9

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

   7. Applied rewrites 92.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 78.7%

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

Alternative 6: 62.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-114}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-166}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.7e-114) (* -0.5 y) (if (<= y 5.1e-166) (* 0.5 x) (fma y 0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.7e-114) {
		tmp = -0.5 * y;
	} else if (y <= 5.1e-166) {
		tmp = 0.5 * x;
	} else {
		tmp = fma(y, 0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.7e-114)
		tmp = Float64(-0.5 * y);
	elseif (y <= 5.1e-166)
		tmp = Float64(0.5 * x);
	else
		tmp = fma(y, 0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.7e-114], N[(-0.5 * y), $MachinePrecision], If[LessEqual[y, 5.1e-166], N[(0.5 * x), $MachinePrecision], N[(y * 0.5 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.69999999999999991e-114

   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

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub N/A

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

      11. lower-fabs.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. lower--.f64 100.0

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

   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-fma.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. fabs-sub N/A

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

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

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

      8. unpow2 N/A

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

      9. sqrt-pow1 N/A

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

      10. metadata-eval N/A

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

      11. unpow1 N/A

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

      12. lift--.f64 20.5

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

   7. Applied rewrites 20.5%

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

      1. lift--.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. unpow2 N/A

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

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

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

      7. fabs-sub N/A

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

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

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

      9. sqrt-prod N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower--.f64 82.5

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

   9. Applied rewrites 82.5%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 65.2%

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

   if -1.69999999999999991e-114 < y < 5.1000000000000002e-166

   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

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub N/A

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

      11. lower-fabs.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. lower--.f64 99.9

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

   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. lift--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. fabs-sub N/A

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

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

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

      8. unpow2 N/A

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

      9. sqrt-pow1 N/A

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

      10. metadata-eval N/A

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

      11. unpow1 N/A

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

      12. lift--.f64 59.4

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

   7. Applied rewrites 59.4%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 55.0

         \[\leadsto 0.5 \cdot x \]

   10. Applied rewrites 55.0%

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

   if 5.1000000000000002e-166 < 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

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub N/A

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

      11. lower-fabs.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. lower--.f64 99.9

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

   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. lift--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. fabs-sub N/A

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

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

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

      8. unpow2 N/A

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

      9. sqrt-pow1 N/A

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

      10. metadata-eval N/A

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

      11. unpow1 N/A

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

      12. lift--.f64 89.5

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

   7. Applied rewrites 89.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 73.0%

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

Alternative 7: 60.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-114}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-164}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.7e-114) (* -0.5 y) (if (<= y 3e-164) (* 0.5 x) (* 0.5 y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.7e-114) {
		tmp = -0.5 * y;
	} else if (y <= 3e-164) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.7d-114)) then
        tmp = (-0.5d0) * y
    else if (y <= 3d-164) then
        tmp = 0.5d0 * x
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.7e-114) {
		tmp = -0.5 * y;
	} else if (y <= 3e-164) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.7e-114:
		tmp = -0.5 * y
	elif y <= 3e-164:
		tmp = 0.5 * x
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.7e-114)
		tmp = Float64(-0.5 * y);
	elseif (y <= 3e-164)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.7e-114)
		tmp = -0.5 * y;
	elseif (y <= 3e-164)
		tmp = 0.5 * x;
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.7e-114], N[(-0.5 * y), $MachinePrecision], If[LessEqual[y, 3e-164], N[(0.5 * x), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.69999999999999991e-114

   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

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub N/A

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

      11. lower-fabs.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. lower--.f64 100.0

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

   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-fma.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. fabs-sub N/A

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

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

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

      8. unpow2 N/A

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

      9. sqrt-pow1 N/A

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

      10. metadata-eval N/A

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

      11. unpow1 N/A

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

      12. lift--.f64 20.5

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

   7. Applied rewrites 20.5%

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

      1. lift--.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. unpow2 N/A

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

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

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

      7. fabs-sub N/A

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

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

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

      9. sqrt-prod N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower--.f64 82.5

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

   9. Applied rewrites 82.5%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 65.2%

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

   if -1.69999999999999991e-114 < y < 3.0000000000000001e-164

   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

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub N/A

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

      11. lower-fabs.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. lower--.f64 99.9

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

   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. lift--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. fabs-sub N/A

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

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

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

      8. unpow2 N/A

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

      9. sqrt-pow1 N/A

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

      10. metadata-eval N/A

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

      11. unpow1 N/A

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

      12. lift--.f64 59.4

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

   7. Applied rewrites 59.4%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 55.0

         \[\leadsto 0.5 \cdot x \]

   10. Applied rewrites 55.0%

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

   if 3.0000000000000001e-164 < y

   1. Initial program 99.9%

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

   4. Applied rewrites 59.5%

      \[\leadsto \color{blue}{\frac{\frac{{\left(y - x\right)}^{2}}{4} - x \cdot x}{\frac{\left|x - y\right|}{2} - x}} \]

   5. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. fabs-sub N/A

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

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

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

      8. unpow2 N/A

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

      9. sqrt-pow1 N/A

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

      10. metadata-eval N/A

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

      11. unpow1 N/A

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

      12. lift--.f64 47.2

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

   7. Applied rewrites 47.2%

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

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 67.5

         \[\leadsto 0.5 \cdot y \]

   10. Applied rewrites 67.5%

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

Alternative 8: 57.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.04 \cdot 10^{-100}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-109}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.04e-100) (* 0.5 x) (if (<= x 8e-109) (* -0.5 y) (* 1.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.04e-100) {
		tmp = 0.5 * x;
	} else if (x <= 8e-109) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.04d-100)) then
        tmp = 0.5d0 * x
    else if (x <= 8d-109) then
        tmp = (-0.5d0) * y
    else
        tmp = 1.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.04e-100) {
		tmp = 0.5 * x;
	} else if (x <= 8e-109) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.04e-100:
		tmp = 0.5 * x
	elif x <= 8e-109:
		tmp = -0.5 * y
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.04e-100)
		tmp = Float64(0.5 * x);
	elseif (x <= 8e-109)
		tmp = Float64(-0.5 * y);
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.04e-100)
		tmp = 0.5 * x;
	elseif (x <= 8e-109)
		tmp = -0.5 * y;
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.04e-100], N[(0.5 * x), $MachinePrecision], If[LessEqual[x, 8e-109], N[(-0.5 * y), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.04e-100

   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

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub N/A

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

      11. lower-fabs.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. lower--.f64 100.0

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

   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-fma.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. fabs-sub N/A

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

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

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

      8. unpow2 N/A

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

      9. sqrt-pow1 N/A

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

      10. metadata-eval N/A

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

      11. unpow1 N/A

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

      12. lift--.f64 92.6

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

   7. Applied rewrites 92.6%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 71.4

         \[\leadsto 0.5 \cdot x \]

   10. Applied rewrites 71.4%

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

   if -1.04e-100 < x < 7.9999999999999999e-109

   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

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub N/A

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

      11. lower-fabs.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. lower--.f64 100.0

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

   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-fma.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. fabs-sub N/A

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

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

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

      8. unpow2 N/A

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

      9. sqrt-pow1 N/A

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

      10. metadata-eval N/A

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

      11. unpow1 N/A

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

      12. lift--.f64 55.0

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

   7. Applied rewrites 55.0%

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

      1. lift--.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. unpow2 N/A

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

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

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

      7. fabs-sub N/A

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

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

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

      9. sqrt-prod N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower--.f64 46.5

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

   9. Applied rewrites 46.5%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 41.6%

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

   if 7.9999999999999999e-109 < 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

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub N/A

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

      11. lower-fabs.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. lower--.f64 99.8

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

   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-fma.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. fabs-sub N/A

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

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

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

      8. unpow2 N/A

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

      9. sqrt-pow1 N/A

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

      10. metadata-eval N/A

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

      11. unpow1 N/A

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

      12. lift--.f64 32.3

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

   7. Applied rewrites 32.3%

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

      1. lift--.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. unpow2 N/A

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

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

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

      7. fabs-sub N/A

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

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

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

      9. sqrt-prod N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower--.f64 78.7

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

   9. Applied rewrites 78.7%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 65.2%

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

Alternative 9: 70.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.1 \cdot 10^{-166}:\\ \;\;\;\;0.5 \cdot \left(x - y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.1e-166) (* 0.5 (- x y)) (fma y 0.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.1e-166) {
		tmp = 0.5 * (x - y);
	} else {
		tmp = fma(y, 0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.1e-166)
		tmp = Float64(0.5 * Float64(x - y));
	else
		tmp = fma(y, 0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.1e-166], N[(0.5 * N[(x - y), $MachinePrecision]), $MachinePrecision], N[(y * 0.5 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.1000000000000002e-166

   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. fabs-sub N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. cancel-sign-sub N/A

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

      10. lower-fabs.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. lower--.f64 42.7

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

   5. Applied rewrites 42.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. fabs-sub N/A

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

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

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

      8. unpow2 N/A

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

      9. sqrt-pow1 N/A

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

      10. metadata-eval N/A

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

      11. unpow1 N/A

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

      12. lift--.f64 2.9

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

   7. Applied rewrites 2.9%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 2.3

         \[\leadsto -0.5 \cdot x \]

   10. Applied rewrites 2.3%

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

   11. Taylor expanded in x around 0

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

       1. fabs-sub N/A

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

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

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

       3. sqrt-unprod N/A

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

       4. rem-square-sqrt N/A

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

       5. lower-*.f64 N/A

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

       6. lift--.f64 72.6

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

   13. Applied rewrites 72.6%

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

   if 5.1000000000000002e-166 < 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

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub N/A

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

      11. lower-fabs.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. lower--.f64 99.9

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

   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. lift--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. fabs-sub N/A

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

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

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

      8. unpow2 N/A

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

      9. sqrt-pow1 N/A

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

      10. metadata-eval N/A

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

      11. unpow1 N/A

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

      12. lift--.f64 89.5

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

   7. Applied rewrites 89.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 73.0%

       \[\leadsto \mathsf{fma}\left(y, 0.5, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 45.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-101}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -7.5e-101) (* -0.5 y) (* 1.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.5e-101) {
		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 <= (-7.5d-101)) 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 <= -7.5e-101) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.5e-101:
		tmp = -0.5 * y
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.5e-101)
		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 <= -7.5e-101)
		tmp = -0.5 * y;
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.5e-101], N[(-0.5 * y), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.5000000000000001e-101

   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

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub N/A

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

      11. lower-fabs.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. lower--.f64 100.0

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

   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-fma.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. fabs-sub N/A

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

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

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

      8. unpow2 N/A

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

      9. sqrt-pow1 N/A

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

      10. metadata-eval N/A

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

      11. unpow1 N/A

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

      12. lift--.f64 21.1

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

   7. Applied rewrites 21.1%

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

      1. lift--.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot \left(y - x\right)}, \frac{1}{2}, x\right) \]

      6. rem-sqrt-square-rev N/A

         \[\leadsto \mathsf{fma}\left(\left|y - x\right|, \frac{1}{2}, x\right) \]

      7. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\left|x - y\right|, \frac{1}{2}, x\right) \]

      8. rem-sqrt-square-rev N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\left(x - y\right) \cdot \left(x - y\right)}, \frac{1}{2}, x\right) \]

      9. sqrt-prod N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, \frac{1}{2}, x\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, \frac{1}{2}, x\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, \frac{1}{2}, x\right) \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, \frac{1}{2}, x\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, \frac{1}{2}, x\right) \]

      14. lower--.f64 81.6

          \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, 0.5, x\right) \]

   9. Applied rewrites 81.6%

      \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, 0.5, x\right) \]

   10. Taylor expanded in x around 0

       \[\leadsto \frac{-1}{2} \cdot \color{blue}{y} \]
   11. Step-by-step derivation

   12. Applied rewrites 65.9%

       \[\leadsto -0.5 \cdot \color{blue}{y} \]

   if -7.5000000000000001e-101 < 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

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left|y - x\right|}, x\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|1 \cdot x - y\right|, x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - y\right|, x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|1 \cdot x - y\right|, x\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      10. cancel-sign-sub N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      11. lower-fabs.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      12. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      14. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      15. lower--.f64 99.9

          \[\leadsto \mathsf{fma}\left(0.5, \left|x - y\right|, x\right) \]

   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. lift--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left|x - y\right| + x \]

      3. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left|x - y\right| + x \]

      4. *-commutative N/A

         \[\leadsto \left|x - y\right| \cdot \frac{1}{2} + x \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left|x - y\right|, \color{blue}{\frac{1}{2}}, x\right) \]

      6. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\left|y - x\right|, \frac{1}{2}, x\right) \]

      7. rem-sqrt-square-rev N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot \left(y - x\right)}, \frac{1}{2}, x\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{{\left(y - x\right)}^{2}}, \frac{1}{2}, x\right) \]

      9. sqrt-pow1 N/A

         \[\leadsto \mathsf{fma}\left({\left(y - x\right)}^{\left(\frac{2}{2}\right)}, \frac{1}{2}, x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({\left(y - x\right)}^{1}, \frac{1}{2}, x\right) \]

      11. unpow1 N/A

          \[\leadsto \mathsf{fma}\left(y - x, \frac{1}{2}, x\right) \]

      12. lift--.f64 75.1

          \[\leadsto \mathsf{fma}\left(y - x, 0.5, x\right) \]

   7. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 0.5, x\right)} \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - x, \frac{1}{2}, x\right) \]

      2. unpow1 N/A

         \[\leadsto \mathsf{fma}\left({\left(y - x\right)}^{1}, \frac{1}{2}, x\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({\left(y - x\right)}^{\left(\frac{2}{2}\right)}, \frac{1}{2}, x\right) \]

      4. sqrt-pow1 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{{\left(y - x\right)}^{2}}, \frac{1}{2}, x\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot \left(y - x\right)}, \frac{1}{2}, x\right) \]

      6. rem-sqrt-square-rev N/A

         \[\leadsto \mathsf{fma}\left(\left|y - x\right|, \frac{1}{2}, x\right) \]

      7. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\left|x - y\right|, \frac{1}{2}, x\right) \]

      8. rem-sqrt-square-rev N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\left(x - y\right) \cdot \left(x - y\right)}, \frac{1}{2}, x\right) \]

      9. sqrt-prod N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, \frac{1}{2}, x\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, \frac{1}{2}, x\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, \frac{1}{2}, x\right) \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, \frac{1}{2}, x\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, \frac{1}{2}, x\right) \]

      14. lower--.f64 29.4

          \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, 0.5, x\right) \]

   9. Applied rewrites 29.4%

      \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, 0.5, x\right) \]

   10. Taylor expanded in x around inf

       \[\leadsto \frac{3}{2} \cdot \color{blue}{x} \]
   11. Step-by-step derivation

   12. Applied rewrites 32.0%

       \[\leadsto 1.5 \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 31.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-152}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.8e-152) (* -0.5 y) x))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.8e-152) {
		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 <= (-1.8d-152)) 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 <= -1.8e-152) {
		tmp = -0.5 * y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.8e-152:
		tmp = -0.5 * y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.8e-152)
		tmp = Float64(-0.5 * y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.8e-152)
		tmp = -0.5 * y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.8e-152], N[(-0.5 * y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -1.8e-152

   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

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left|y - x\right|}, x\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|1 \cdot x - y\right|, x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\left(\mathsf{neg}\left(-1\right)\right) \cdot x - y\right|, x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|1 \cdot x - y\right|, x\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      10. cancel-sign-sub N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      11. lower-fabs.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      12. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      14. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      15. lower--.f64 100.0

          \[\leadsto \mathsf{fma}\left(0.5, \left|x - y\right|, x\right) \]

   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-fma.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left|x - y\right| + \color{blue}{x} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left|x - y\right| + x \]

      3. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left|x - y\right| + x \]

      4. *-commutative N/A

         \[\leadsto \left|x - y\right| \cdot \frac{1}{2} + x \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left|x - y\right|, \color{blue}{\frac{1}{2}}, x\right) \]

      6. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\left|y - x\right|, \frac{1}{2}, x\right) \]

      7. rem-sqrt-square-rev N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot \left(y - x\right)}, \frac{1}{2}, x\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{{\left(y - x\right)}^{2}}, \frac{1}{2}, x\right) \]

      9. sqrt-pow1 N/A

         \[\leadsto \mathsf{fma}\left({\left(y - x\right)}^{\left(\frac{2}{2}\right)}, \frac{1}{2}, x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({\left(y - x\right)}^{1}, \frac{1}{2}, x\right) \]

      11. unpow1 N/A

          \[\leadsto \mathsf{fma}\left(y - x, \frac{1}{2}, x\right) \]

      12. lift--.f64 22.5

          \[\leadsto \mathsf{fma}\left(y - x, 0.5, x\right) \]

   7. Applied rewrites 22.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 0.5, x\right)} \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - x, \frac{1}{2}, x\right) \]

      2. unpow1 N/A

         \[\leadsto \mathsf{fma}\left({\left(y - x\right)}^{1}, \frac{1}{2}, x\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({\left(y - x\right)}^{\left(\frac{2}{2}\right)}, \frac{1}{2}, x\right) \]

      4. sqrt-pow1 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{{\left(y - x\right)}^{2}}, \frac{1}{2}, x\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\left(y - x\right) \cdot \left(y - x\right)}, \frac{1}{2}, x\right) \]

      6. rem-sqrt-square-rev N/A

         \[\leadsto \mathsf{fma}\left(\left|y - x\right|, \frac{1}{2}, x\right) \]

      7. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\left|x - y\right|, \frac{1}{2}, x\right) \]

      8. rem-sqrt-square-rev N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\left(x - y\right) \cdot \left(x - y\right)}, \frac{1}{2}, x\right) \]

      9. sqrt-prod N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, \frac{1}{2}, x\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, \frac{1}{2}, x\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, \frac{1}{2}, x\right) \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, \frac{1}{2}, x\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, \frac{1}{2}, x\right) \]

      14. lower--.f64 81.3

          \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, 0.5, x\right) \]

   9. Applied rewrites 81.3%

      \[\leadsto \mathsf{fma}\left(\sqrt{x - y} \cdot \sqrt{x - y}, 0.5, x\right) \]

   10. Taylor expanded in x around 0

       \[\leadsto \frac{-1}{2} \cdot \color{blue}{y} \]
   11. Step-by-step derivation

   12. Applied rewrites 60.8%

       \[\leadsto -0.5 \cdot \color{blue}{y} \]

   if -1.8e-152 < 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 12.3%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 11.3% 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 2025037 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))