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

Percentage Accurate: 99.9% → 99.9%

Time: 2.2s

Alternatives: 10

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. fabs-sub N/A

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

   4. *-lft-identity N/A

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

   5. metadata-eval N/A

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

   6. 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

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

4. Applied rewrites 99.9%

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

Alternative 2: 79.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-231}:\\ \;\;\;\;\mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-54}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|x\right|, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|-y\right|, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.3e-231)
   (fma 1.5 x (* -0.5 y))
   (if (<= y 6.5e-54) (fma 0.5 (fabs x) x) (fma 0.5 (fabs (- y)) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.3e-231) {
		tmp = fma(1.5, x, (-0.5 * y));
	} else if (y <= 6.5e-54) {
		tmp = fma(0.5, fabs(x), x);
	} else {
		tmp = fma(0.5, fabs(-y), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.3e-231)
		tmp = fma(1.5, x, Float64(-0.5 * y));
	elseif (y <= 6.5e-54)
		tmp = fma(0.5, abs(x), x);
	else
		tmp = fma(0.5, abs(Float64(-y)), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.3e-231], N[(1.5 * x + N[(-0.5 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.5e-54], N[(0.5 * N[Abs[x], $MachinePrecision] + x), $MachinePrecision], N[(0.5 * N[Abs[(-y)], $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.30000000000000028e-231

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. 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

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

   4. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. flip-- N/A

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

      4. fabs-div N/A

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

      5. sqr-neg-rev N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

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

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

      11. fabs-div N/A

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

      12. flip-+ N/A

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

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

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

      14. sqrt-prod N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

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

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lift--.f64 N/A

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

   6. Applied rewrites 77.9%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 82.2

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

   9. Applied rewrites 82.2%

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

   if -3.30000000000000028e-231 < y < 6.49999999999999991e-54

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. 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

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 81.2%

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

   if 6.49999999999999991e-54 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. 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

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 74.2

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

   7. Applied rewrites 74.2%

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

Alternative 3: 79.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-231}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-54}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|x\right|, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|-y\right|, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.3e-231)
   (fma (- x y) 0.5 x)
   (if (<= y 6.5e-54) (fma 0.5 (fabs x) x) (fma 0.5 (fabs (- y)) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.3e-231) {
		tmp = fma((x - y), 0.5, x);
	} else if (y <= 6.5e-54) {
		tmp = fma(0.5, fabs(x), x);
	} else {
		tmp = fma(0.5, fabs(-y), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.3e-231)
		tmp = fma(Float64(x - y), 0.5, x);
	elseif (y <= 6.5e-54)
		tmp = fma(0.5, abs(x), x);
	else
		tmp = fma(0.5, abs(Float64(-y)), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.3e-231], N[(N[(x - y), $MachinePrecision] * 0.5 + x), $MachinePrecision], If[LessEqual[y, 6.5e-54], N[(0.5 * N[Abs[x], $MachinePrecision] + x), $MachinePrecision], N[(0.5 * N[Abs[(-y)], $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.30000000000000028e-231

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. 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

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

   4. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. flip-- N/A

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

      4. fabs-div N/A

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

      5. sqr-neg-rev N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

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

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

      11. fabs-div N/A

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

      12. flip-+ N/A

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

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

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

      14. sqrt-prod N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

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

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lift--.f64 N/A

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

   6. Applied rewrites 77.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift--.f64 82.2

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

   8. Applied rewrites 82.2%

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

   if -3.30000000000000028e-231 < y < 6.49999999999999991e-54

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. 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

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 81.2%

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

   if 6.49999999999999991e-54 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. 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

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 74.2

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

   7. Applied rewrites 74.2%

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

Alternative 4: 78.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-231}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-54}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|x\right|, x\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left|x - y\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.3e-231)
   (fma (- x y) 0.5 x)
   (if (<= y 6.5e-54) (fma 0.5 (fabs x) x) (* 0.5 (fabs (- x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.3e-231) {
		tmp = fma((x - y), 0.5, x);
	} else if (y <= 6.5e-54) {
		tmp = fma(0.5, fabs(x), x);
	} else {
		tmp = 0.5 * fabs((x - y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.3e-231)
		tmp = fma(Float64(x - y), 0.5, x);
	elseif (y <= 6.5e-54)
		tmp = fma(0.5, abs(x), x);
	else
		tmp = Float64(0.5 * abs(Float64(x - y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.3e-231], N[(N[(x - y), $MachinePrecision] * 0.5 + x), $MachinePrecision], If[LessEqual[y, 6.5e-54], N[(0.5 * N[Abs[x], $MachinePrecision] + x), $MachinePrecision], N[(0.5 * N[Abs[N[(x - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.30000000000000028e-231

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. 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

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

   4. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. flip-- N/A

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

      4. fabs-div N/A

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

      5. sqr-neg-rev N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

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

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

      11. fabs-div N/A

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

      12. flip-+ N/A

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

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

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

      14. sqrt-prod N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

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

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lift--.f64 N/A

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

   6. Applied rewrites 77.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift--.f64 82.2

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

   8. Applied rewrites 82.2%

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

   if -3.30000000000000028e-231 < y < 6.49999999999999991e-54

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. 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

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 81.2%

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

   if 6.49999999999999991e-54 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. fabs-sub N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. 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 70.9

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

   4. Applied rewrites 70.9%

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

Alternative 5: 78.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-231}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|x\right|, x\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left|-y\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.3e-231)
   (fma (- x y) 0.5 x)
   (if (<= y 5.2e-53) (fma 0.5 (fabs x) x) (* 0.5 (fabs (- y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.3e-231) {
		tmp = fma((x - y), 0.5, x);
	} else if (y <= 5.2e-53) {
		tmp = fma(0.5, fabs(x), x);
	} else {
		tmp = 0.5 * fabs(-y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.3e-231)
		tmp = fma(Float64(x - y), 0.5, x);
	elseif (y <= 5.2e-53)
		tmp = fma(0.5, abs(x), x);
	else
		tmp = Float64(0.5 * abs(Float64(-y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.3e-231], N[(N[(x - y), $MachinePrecision] * 0.5 + x), $MachinePrecision], If[LessEqual[y, 5.2e-53], N[(0.5 * N[Abs[x], $MachinePrecision] + x), $MachinePrecision], N[(0.5 * N[Abs[(-y)], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.30000000000000028e-231

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. 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

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

   4. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. flip-- N/A

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

      4. fabs-div N/A

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

      5. sqr-neg-rev N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

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

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

      11. fabs-div N/A

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

      12. flip-+ N/A

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

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

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

      14. sqrt-prod N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

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

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lift--.f64 N/A

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

   6. Applied rewrites 77.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift--.f64 82.2

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

   8. Applied rewrites 82.2%

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

   if -3.30000000000000028e-231 < y < 5.19999999999999993e-53

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. 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

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 81.2%

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

   if 5.19999999999999993e-53 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. fabs-sub N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. 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 71.0

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

   4. Applied rewrites 71.0%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 69.4

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

   7. Applied rewrites 69.4%

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

Alternative 6: 77.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-231}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|x\right|, x\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left|-y\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.3e-231)
   (* (- x y) 0.5)
   (if (<= y 5.2e-53) (fma 0.5 (fabs x) x) (* 0.5 (fabs (- y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.3e-231) {
		tmp = (x - y) * 0.5;
	} else if (y <= 5.2e-53) {
		tmp = fma(0.5, fabs(x), x);
	} else {
		tmp = 0.5 * fabs(-y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.3e-231)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (y <= 5.2e-53)
		tmp = fma(0.5, abs(x), x);
	else
		tmp = Float64(0.5 * abs(Float64(-y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.3e-231], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[y, 5.2e-53], N[(0.5 * N[Abs[x], $MachinePrecision] + x), $MachinePrecision], N[(0.5 * N[Abs[(-y)], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.30000000000000028e-231

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 86.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, \frac{0.5}{x}, 1\right) \cdot x} \]

   6. Applied rewrites 65.0%

      \[\leadsto \frac{\left({\left(\frac{0.5}{x} \cdot \left|y - x\right|\right)}^{2} - 1\right) \cdot x}{\color{blue}{\mathsf{fma}\left(\frac{0.5}{x}, \left|y - x\right|, -1\right)}} \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub N/A

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

      10. metadata-eval N/A

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

      11. unpow2 N/A

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

      12. associate-*l/ N/A

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

      13. associate-*r/ N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. lower-fma.f64 N/A

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

   8. Applied rewrites 65.0%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left|y - x\right|}{x}, 0.5 \cdot \left(\frac{0.5}{x} \cdot \left|y - x\right|\right), -1\right) \cdot x}{\mathsf{fma}\left(\frac{\color{blue}{0.5}}{x}, \left|y - x\right|, -1\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left|y - x\right|} \]
   10. 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. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lift--.f64 81.0

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

   11. Applied rewrites 81.0%

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

   if -3.30000000000000028e-231 < y < 5.19999999999999993e-53

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. 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

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 81.2%

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

   if 5.19999999999999993e-53 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. fabs-sub N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. 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 71.0

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

   4. Applied rewrites 71.0%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 69.4

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

   7. Applied rewrites 69.4%

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

Alternative 7: 67.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.85 \cdot 10^{-95}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.85e-95) (* (- x y) 0.5) (* 1.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.85e-95) {
		tmp = (x - y) * 0.5;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 2.85d-95) then
        tmp = (x - y) * 0.5d0
    else
        tmp = 1.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.85e-95) {
		tmp = (x - y) * 0.5;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.85e-95:
		tmp = (x - y) * 0.5
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.85e-95)
		tmp = Float64(Float64(x - y) * 0.5);
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.85e-95)
		tmp = (x - y) * 0.5;
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.85e-95], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.85e-95

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 83.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, \frac{0.5}{x}, 1\right) \cdot x} \]

   6. Applied rewrites 62.9%

      \[\leadsto \frac{\left({\left(\frac{0.5}{x} \cdot \left|y - x\right|\right)}^{2} - 1\right) \cdot x}{\color{blue}{\mathsf{fma}\left(\frac{0.5}{x}, \left|y - x\right|, -1\right)}} \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub N/A

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

      10. metadata-eval N/A

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

      11. unpow2 N/A

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

      12. associate-*l/ N/A

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

      13. associate-*r/ N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. lower-fma.f64 N/A

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

   8. Applied rewrites 62.9%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left|y - x\right|}{x}, 0.5 \cdot \left(\frac{0.5}{x} \cdot \left|y - x\right|\right), -1\right) \cdot x}{\mathsf{fma}\left(\frac{\color{blue}{0.5}}{x}, \left|y - x\right|, -1\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left|y - x\right|} \]
   10. 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. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lift--.f64 66.7

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

   11. Applied rewrites 66.7%

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

   if 2.85e-95 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. 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

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

   4. Applied rewrites 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. flip-- N/A

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

      4. fabs-div N/A

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

      5. sqr-neg-rev N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

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

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

      11. fabs-div N/A

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

      12. flip-+ N/A

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

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

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

      14. sqrt-prod N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

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

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lift--.f64 N/A

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

   6. Applied rewrites 83.1%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 67.8

         \[\leadsto 1.5 \cdot x \]

   9. Applied rewrites 67.8%

      \[\leadsto 1.5 \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 42.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.75 \cdot 10^{-95}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 2.75e-95) (* -0.5 y) (* 1.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.75e-95) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 2.75d-95) then
        tmp = (-0.5d0) * y
    else
        tmp = 1.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.75e-95) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.75e-95:
		tmp = -0.5 * y
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.75e-95)
		tmp = Float64(-0.5 * y);
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.75e-95)
		tmp = -0.5 * y;
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.75e-95], N[(-0.5 * y), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.75000000000000001e-95

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. 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

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

   4. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. flip-- N/A

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

      4. fabs-div N/A

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

      5. sqr-neg-rev N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

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

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

      11. fabs-div N/A

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

      12. flip-+ N/A

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

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

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

      14. sqrt-prod N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

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

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lift--.f64 N/A

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

   6. Applied rewrites 34.0%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 31.1

         \[\leadsto -0.5 \cdot y \]

   9. Applied rewrites 31.1%

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

   if 2.75000000000000001e-95 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. 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

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

   4. Applied rewrites 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. flip-- N/A

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

      4. fabs-div N/A

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

      5. sqr-neg-rev N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

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

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

      11. fabs-div N/A

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

      12. flip-+ N/A

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

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

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

      14. sqrt-prod N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

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

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lift--.f64 N/A

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

   6. Applied rewrites 83.1%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 67.8

         \[\leadsto 1.5 \cdot x \]

   9. Applied rewrites 67.8%

      \[\leadsto 1.5 \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 32.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{-231}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.25e-231) (* -0.5 y) x))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.25e-231) {
		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 <= (-1.25d-231)) 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 <= -1.25e-231) {
		tmp = -0.5 * y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.25e-231:
		tmp = -0.5 * y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.25e-231)
		tmp = Float64(-0.5 * y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.25e-231)
		tmp = -0.5 * y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.25e-231], N[(-0.5 * y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -1.25000000000000006e-231

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. 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

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

   4. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. flip-- N/A

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

      4. fabs-div N/A

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

      5. sqr-neg-rev N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

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

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

      11. fabs-div N/A

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

      12. flip-+ N/A

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

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

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

      14. sqrt-prod N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

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

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lift--.f64 N/A

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

   6. Applied rewrites 77.8%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 57.5

         \[\leadsto -0.5 \cdot y \]

   9. Applied rewrites 57.5%

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

   if -1.25000000000000006e-231 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 12.2%

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

Alternative 10: 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. Taylor expanded in x around inf

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

4. Applied rewrites 11.3%

   \[\leadsto \color{blue}{x} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025105 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))