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

Percentage Accurate: 99.9% → 99.9%

Time: 2.5s

Alternatives: 11

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: 83.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-32}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-141}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|-y\right|, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.6e-32)
   (* (- x y) 0.5)
   (if (<= x 2e-141) (fma 0.5 (fabs (- y)) x) (fma 1.5 x (* -0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.6e-32) {
		tmp = (x - y) * 0.5;
	} else if (x <= 2e-141) {
		tmp = fma(0.5, fabs(-y), x);
	} else {
		tmp = fma(1.5, x, (-0.5 * y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.6e-32)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (x <= 2e-141)
		tmp = fma(0.5, abs(Float64(-y)), x);
	else
		tmp = fma(1.5, x, Float64(-0.5 * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.6e-32], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 2e-141], N[(0.5 * N[Abs[(-y)], $MachinePrecision] + x), $MachinePrecision], N[(1.5 * x + N[(-0.5 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6000000000000001e-32

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. flip3-+ N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 26.6%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. sqrt-unprod N/A

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

      5. rem-square-sqrt N/A

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

      6. lift--.f64 83.0

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

   6. Applied rewrites 83.0%

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

   if -1.6000000000000001e-32 < x < 2.0000000000000001e-141

   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. 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 84.2

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

   7. Applied rewrites 84.2%

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

   if 2.0000000000000001e-141 < 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. 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 66.1%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 66.0

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

   9. Applied rewrites 66.0%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 83.1%

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

Alternative 3: 83.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-32}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-141}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|-y\right|, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.6e-32)
   (* (- x y) 0.5)
   (if (<= x 2e-141) (fma 0.5 (fabs (- y)) x) (fma (- x y) 0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.6e-32) {
		tmp = (x - y) * 0.5;
	} else if (x <= 2e-141) {
		tmp = fma(0.5, fabs(-y), x);
	} else {
		tmp = fma((x - y), 0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.6e-32)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (x <= 2e-141)
		tmp = fma(0.5, abs(Float64(-y)), x);
	else
		tmp = fma(Float64(x - y), 0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.6e-32], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 2e-141], N[(0.5 * N[Abs[(-y)], $MachinePrecision] + x), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * 0.5 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6000000000000001e-32

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. flip3-+ N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 26.6%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. sqrt-unprod N/A

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

      5. rem-square-sqrt N/A

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

      6. lift--.f64 83.0

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

   6. Applied rewrites 83.0%

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

   if -1.6000000000000001e-32 < x < 2.0000000000000001e-141

   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. 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 84.2

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

   7. Applied rewrites 84.2%

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

   if 2.0000000000000001e-141 < x

   1. Initial program 99.8%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. flip3-+ N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 33.7%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

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

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

      5. sqrt-unprod N/A

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

      6. rem-square-sqrt N/A

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

      7. lift--.f64 82.9

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

   6. Applied rewrites 82.9%

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

Alternative 4: 82.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-32}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-141}:\\ \;\;\;\;0.5 \cdot \left|x - y\right|\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.6e-32)
   (* (- x y) 0.5)
   (if (<= x 2e-141) (* 0.5 (fabs (- x y))) (fma (- x y) 0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.6e-32) {
		tmp = (x - y) * 0.5;
	} else if (x <= 2e-141) {
		tmp = 0.5 * fabs((x - y));
	} else {
		tmp = fma((x - y), 0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.6e-32)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (x <= 2e-141)
		tmp = Float64(0.5 * abs(Float64(x - y)));
	else
		tmp = fma(Float64(x - y), 0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.6e-32], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 2e-141], N[(0.5 * N[Abs[N[(x - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * 0.5 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6000000000000001e-32

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. flip3-+ N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 26.6%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. sqrt-unprod N/A

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

      5. rem-square-sqrt N/A

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

      6. lift--.f64 83.0

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

   6. Applied rewrites 83.0%

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

   if -1.6000000000000001e-32 < x < 2.0000000000000001e-141

   1. Initial program 100.0%

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

   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 82.0

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

   4. Applied rewrites 82.0%

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

   if 2.0000000000000001e-141 < x

   1. Initial program 99.8%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. flip3-+ N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 33.7%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

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

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

      5. sqrt-unprod N/A

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

      6. rem-square-sqrt N/A

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

      7. lift--.f64 82.9

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

   6. Applied rewrites 82.9%

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

Alternative 5: 82.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-32}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-150}:\\ \;\;\;\;0.5 \cdot \left|-y\right|\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.6e-32)
   (* (- x y) 0.5)
   (if (<= x 6.2e-150) (* 0.5 (fabs (- y))) (fma (- x y) 0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.6e-32) {
		tmp = (x - y) * 0.5;
	} else if (x <= 6.2e-150) {
		tmp = 0.5 * fabs(-y);
	} else {
		tmp = fma((x - y), 0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.6e-32)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (x <= 6.2e-150)
		tmp = Float64(0.5 * abs(Float64(-y)));
	else
		tmp = fma(Float64(x - y), 0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.6e-32], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 6.2e-150], N[(0.5 * N[Abs[(-y)], $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * 0.5 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6000000000000001e-32

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. flip3-+ N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 26.6%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. sqrt-unprod N/A

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

      5. rem-square-sqrt N/A

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

      6. lift--.f64 83.0

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

   6. Applied rewrites 83.0%

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

   if -1.6000000000000001e-32 < x < 6.19999999999999996e-150

   1. Initial program 100.0%

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

   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 82.0

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

   4. Applied rewrites 82.0%

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

   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 81.6

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

   7. Applied rewrites 81.6%

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

   if 6.19999999999999996e-150 < x

   1. Initial program 99.8%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. flip3-+ N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 33.8%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

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

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

      5. sqrt-unprod N/A

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

      6. rem-square-sqrt N/A

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

      7. lift--.f64 82.7

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

   6. Applied rewrites 82.7%

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

Alternative 6: 78.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-146}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|x\right|, x\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left|-y\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.4e-146)
   (* (- x y) 0.5)
   (if (<= y 3.2e-53) (fma 0.5 (fabs x) x) (* 0.5 (fabs (- y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.4e-146) {
		tmp = (x - y) * 0.5;
	} else if (y <= 3.2e-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 <= -2.4e-146)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (y <= 3.2e-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, -2.4e-146], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[y, 3.2e-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 < -2.4000000000000002e-146

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. flip3-+ N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 34.7%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. sqrt-unprod N/A

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

      5. rem-square-sqrt N/A

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

      6. lift--.f64 82.0

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

   6. Applied rewrites 82.0%

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

   if -2.4000000000000002e-146 < y < 3.2000000000000001e-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 80.8%

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

   if 3.2000000000000001e-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 71.8

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

   4. Applied rewrites 71.8%

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

   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 70.1

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

   7. Applied rewrites 70.1%

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

Alternative 7: 66.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-64}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|x\right|, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 8e-64) (* (- x y) 0.5) (fma 0.5 (fabs x) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 8e-64) {
		tmp = (x - y) * 0.5;
	} else {
		tmp = fma(0.5, fabs(x), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 8e-64)
		tmp = Float64(Float64(x - y) * 0.5);
	else
		tmp = fma(0.5, abs(x), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 8e-64], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[Abs[x], $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.99999999999999972e-64

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. flip3-+ N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 35.3%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. sqrt-unprod N/A

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

      5. rem-square-sqrt N/A

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

      6. lift--.f64 64.6

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

   6. Applied rewrites 64.6%

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

   if 7.99999999999999972e-64 < 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. 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 72.2%

      \[\leadsto \mathsf{fma}\left(0.5, \left|x\right|, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 62.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-45}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|x\right|, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.2e-45) (* -0.5 y) (fma 0.5 (fabs x) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.2e-45) {
		tmp = -0.5 * y;
	} else {
		tmp = fma(0.5, fabs(x), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.2e-45)
		tmp = Float64(-0.5 * y);
	else
		tmp = fma(0.5, abs(x), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.2e-45], N[(-0.5 * y), $MachinePrecision], N[(0.5 * N[Abs[x], $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.20000000000000007e-45

   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 29.2%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 15.8

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

   9. Applied rewrites 15.8%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 71.5%

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

   if -3.20000000000000007e-45 < 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 inf

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

   7. Applied rewrites 58.8%

      \[\leadsto \mathsf{fma}\left(0.5, \left|x\right|, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 45.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{-144}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.75e-144) (* -0.5 y) (* 1.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.75e-144) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.75d-144)) then
        tmp = (-0.5d0) * y
    else
        tmp = 1.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.75e-144) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.75e-144:
		tmp = -0.5 * y
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.75e-144)
		tmp = Float64(-0.5 * y);
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.75e-144)
		tmp = -0.5 * y;
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.75e-144], N[(-0.5 * y), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.7499999999999999e-144

   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 37.4%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 20.4

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

   9. Applied rewrites 20.4%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 63.4%

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

   if -1.7499999999999999e-144 < 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 inf

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

   7. Applied rewrites 58.0%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 29.2

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

   9. Applied rewrites 29.2%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 34.8%

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

Alternative 10: 31.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{-208}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -9.5e-208) (* -0.5 y) x))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9.5e-208) {
		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 <= (-9.5d-208)) 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 <= -9.5e-208) {
		tmp = -0.5 * y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9.5e-208:
		tmp = -0.5 * y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9.5e-208)
		tmp = Float64(-0.5 * y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -9.5e-208)
		tmp = -0.5 * y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9.5e-208], N[(-0.5 * y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -9.5000000000000001e-208

   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 42.1%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 22.8

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

   9. Applied rewrites 22.8%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 58.8%

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

   if -9.5000000000000001e-208 < 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 12.3%

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

Alternative 11: 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 2025106 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))