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

Percentage Accurate: 99.9% → 99.9%

Time: 2.6s

Alternatives: 7

Speedup: 0.9×

Specification

Math

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

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

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

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, 0.9× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (* 0.5 (+ (+ (fabs (- y x)) x) x)))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.5d0 * ((abs((y - x)) + x) + x)
end function

Java

public static double code(double x, double y) {
	return 0.5 * ((Math.abs((y - x)) + x) + x);
}

Python

def code(x, y):
	return 0.5 * ((math.fabs((y - x)) + x) + x)

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = 0.5 * ((abs((y - x)) + x) + x);
end

Wolfram

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

Derivation

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. add-to-fraction N/A

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

   4. mult-flip N/A

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

   5. *-commutative N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. count-2-rev N/A

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

   9. associate-+r+ N/A

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

   10. distribute-rgt-in N/A

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

   11. mult-flip N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-+.f64 N/A

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

   14. lift-fabs.f64 N/A

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

   15. lift--.f64 N/A

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

   16. fabs-sub N/A

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

   17. lower-fabs.f64 N/A

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

   18. lower--.f64 N/A

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

   19. metadata-eval N/A

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

   20. *-lft-identity N/A

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

   21. metadata-eval N/A

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

   22. times-frac N/A

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

   23. /-rgt-identity N/A

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

   24. lower-*.f64 N/A

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

   25. metadata-eval 99.9%

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

3. Applied rewrites 99.9%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. distribute-rgt-out N/A

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

   5. lower-*.f64 N/A

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

   6. lower-+.f64 99.9%

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

   7. lift-fabs.f64 N/A

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

   8. lift--.f64 N/A

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

   9. sub-negate-rev N/A

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

   10. sub-to-mult-rev N/A

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

   11. lift-/.f64 N/A

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

   12. lift--.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. neg-fabs N/A

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

   15. lift-fabs.f64 88.0%

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

   16. lift-*.f64 N/A

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

   17. lift--.f64 N/A

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

   18. lift-/.f64 N/A

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

   19. sub-to-mult-rev N/A

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

   20. lower--.f64 99.9%

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

5. Applied rewrites 99.9%

   \[\leadsto \color{blue}{0.5 \cdot \left(\left(\left|y - x\right| + x\right) + x\right)} \]
6. Add Preprocessing

Alternative 2: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. add-to-fraction N/A

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

   4. +-commutative N/A

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

   5. div-add N/A

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

   6. mult-flip N/A

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

   7. associate-/l* N/A

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

   8. metadata-eval N/A

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

   9. *-rgt-identity N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-fabs.f64 N/A

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

   12. lift--.f64 N/A

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

   13. fabs-sub N/A

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

   14. lower-fabs.f64 N/A

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

   15. lower--.f64 N/A

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

   16. metadata-eval 99.9%

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

3. Applied rewrites 99.9%

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

Alternative 3: 71.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(\left|y\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, -0.5, x\right)\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.1e-24) (fma (fabs y) 0.5 x) (fma (- y x) -0.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.1e-24) {
		tmp = fma(fabs(y), 0.5, x);
	} else {
		tmp = fma((y - x), -0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.1e-24)
		tmp = fma(abs(y), 0.5, x);
	else
		tmp = fma(Float64(y - x), -0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.1e-24], N[(N[Abs[y], $MachinePrecision] * 0.5 + x), $MachinePrecision], N[(N[(y - x), $MachinePrecision] * -0.5 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0999999999999999e-24

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 58.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 58.7%

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

   6. Applied rewrites 58.7%

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

   if 2.0999999999999999e-24 < x

   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. add-to-fraction N/A

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

      4. +-commutative N/A

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

      5. div-add N/A

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

      6. mult-flip N/A

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

      7. associate-/l* N/A

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

      8. metadata-eval N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. lift--.f64 N/A

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

      13. fabs-sub N/A

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

      14. lower-fabs.f64 N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 99.9%

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

   3. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-to-mult-rev N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lower-*.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lift-/.f64 N/A

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

      12. frac-2neg N/A

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

      13. sub-to-fraction N/A

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

      14. distribute-neg-frac2 N/A

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

      15. remove-double-neg N/A

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

      16. add-flip-rev N/A

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

      17. *-lft-identity N/A

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

      18. +-commutative N/A

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

      19. sub-flip N/A

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

      20. lift--.f64 N/A

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

      21. lower-/.f64 88.0%

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

   5. Applied rewrites 88.0%

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

      1. lift-fabs.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. fabs-mul N/A

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

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

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

      9. sqrt-unprod N/A

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

      10. rem-square-sqrt N/A

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

      11. metadata-eval N/A

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

      12. *-inverses N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity 55.2%

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

      15. rem-square-sqrt N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. lift-*.f64 50.3%

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

      19. lift-fma.f64 N/A

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

   7. Applied rewrites 55.2%

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

Alternative 4: 68.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(\left|y\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.1e-24) (fma (fabs y) 0.5 x) (* 1.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.1e-24) {
		tmp = fma(fabs(y), 0.5, x);
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.1e-24)
		tmp = fma(abs(y), 0.5, x);
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.1e-24], N[(N[Abs[y], $MachinePrecision] * 0.5 + x), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0999999999999999e-24

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 58.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 58.7%

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

   6. Applied rewrites 58.7%

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

   if 2.0999999999999999e-24 < x

   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. add-to-fraction N/A

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

      4. +-commutative N/A

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

      5. div-add N/A

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

      6. mult-flip N/A

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

      7. associate-/l* N/A

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

      8. metadata-eval N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. lift--.f64 N/A

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

      13. fabs-sub N/A

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

      14. lower-fabs.f64 N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 99.9%

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

   3. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-to-mult-rev N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lower-*.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lift-/.f64 N/A

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

      12. frac-2neg N/A

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

      13. sub-to-fraction N/A

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

      14. distribute-neg-frac2 N/A

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

      15. remove-double-neg N/A

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

      16. add-flip-rev N/A

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

      17. *-lft-identity N/A

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

      18. +-commutative N/A

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

      19. sub-flip N/A

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

      20. lift--.f64 N/A

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

      21. lower-/.f64 88.0%

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

   5. Applied rewrites 88.0%

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

      1. lift-fabs.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. fabs-mul N/A

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

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

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

      9. sqrt-unprod N/A

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

      10. rem-square-sqrt N/A

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

      11. metadata-eval N/A

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

      12. *-inverses N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity 55.2%

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

      15. rem-square-sqrt N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. lift-*.f64 50.3%

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

      19. lift-fma.f64 N/A

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

   7. Applied rewrites 55.2%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 30.7%

         \[\leadsto 1.5 \cdot \color{blue}{x} \]

   10. Applied rewrites 30.7%

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

Alternative 5: 47.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 2.25 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(y, -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.25e-51) (fma y -0.5 x) (* 1.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.25e-51) {
		tmp = fma(y, -0.5, x);
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.25e-51)
		tmp = fma(y, -0.5, x);
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.25e-51], N[(y * -0.5 + x), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2499999999999999e-51

   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. add-to-fraction N/A

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

      4. +-commutative N/A

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

      5. div-add N/A

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

      6. mult-flip N/A

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

      7. associate-/l* N/A

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

      8. metadata-eval N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. lift--.f64 N/A

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

      13. fabs-sub N/A

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

      14. lower-fabs.f64 N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 99.9%

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

   3. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-to-mult-rev N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lower-*.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lift-/.f64 N/A

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

      12. frac-2neg N/A

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

      13. sub-to-fraction N/A

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

      14. distribute-neg-frac2 N/A

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

      15. remove-double-neg N/A

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

      16. add-flip-rev N/A

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

      17. *-lft-identity N/A

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

      18. +-commutative N/A

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

      19. sub-flip N/A

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

      20. lift--.f64 N/A

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

      21. lower-/.f64 88.0%

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

   5. Applied rewrites 88.0%

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

      1. lift-fabs.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. fabs-mul N/A

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

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

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

      9. sqrt-unprod N/A

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

      10. rem-square-sqrt N/A

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

      11. metadata-eval N/A

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

      12. *-inverses N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity 55.2%

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

      15. rem-square-sqrt N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. lift-*.f64 50.3%

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

      19. lift-fma.f64 N/A

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

   7. Applied rewrites 55.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 35.0%

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

   if 2.2499999999999999e-51 < x

   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. add-to-fraction N/A

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

      4. +-commutative N/A

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

      5. div-add N/A

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

      6. mult-flip N/A

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

      7. associate-/l* N/A

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

      8. metadata-eval N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. lift--.f64 N/A

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

      13. fabs-sub N/A

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

      14. lower-fabs.f64 N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 99.9%

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

   3. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-to-mult-rev N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lower-*.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lift-/.f64 N/A

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

      12. frac-2neg N/A

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

      13. sub-to-fraction N/A

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

      14. distribute-neg-frac2 N/A

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

      15. remove-double-neg N/A

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

      16. add-flip-rev N/A

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

      17. *-lft-identity N/A

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

      18. +-commutative N/A

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

      19. sub-flip N/A

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

      20. lift--.f64 N/A

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

      21. lower-/.f64 88.0%

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

   5. Applied rewrites 88.0%

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

      1. lift-fabs.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. fabs-mul N/A

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

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

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

      9. sqrt-unprod N/A

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

      10. rem-square-sqrt N/A

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

      11. metadata-eval N/A

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

      12. *-inverses N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity 55.2%

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

      15. rem-square-sqrt N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. lift-*.f64 50.3%

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

      19. lift-fma.f64 N/A

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

   7. Applied rewrites 55.2%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 30.7%

         \[\leadsto 1.5 \cdot \color{blue}{x} \]

   10. Applied rewrites 30.7%

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

Alternative 6: 44.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -1.06 \cdot 10^{+125}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-113}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.06e+125) (* 1.5 x) (if (<= x 2.7e-113) (* -0.5 y) (* 1.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.06e+125) {
		tmp = 1.5 * x;
	} else if (x <= 2.7e-113) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.06d+125)) then
        tmp = 1.5d0 * x
    else if (x <= 2.7d-113) then
        tmp = (-0.5d0) * y
    else
        tmp = 1.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.06e+125) {
		tmp = 1.5 * x;
	} else if (x <= 2.7e-113) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.06e+125:
		tmp = 1.5 * x
	elif x <= 2.7e-113:
		tmp = -0.5 * y
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.06e+125)
		tmp = Float64(1.5 * x);
	elseif (x <= 2.7e-113)
		tmp = Float64(-0.5 * y);
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.06e+125)
		tmp = 1.5 * x;
	elseif (x <= 2.7e-113)
		tmp = -0.5 * y;
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.06e+125], N[(1.5 * x), $MachinePrecision], If[LessEqual[x, 2.7e-113], N[(-0.5 * y), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0600000000000001e125 or 2.7e-113 < x

   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. add-to-fraction N/A

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

      4. +-commutative N/A

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

      5. div-add N/A

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

      6. mult-flip N/A

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

      7. associate-/l* N/A

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

      8. metadata-eval N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. lift--.f64 N/A

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

      13. fabs-sub N/A

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

      14. lower-fabs.f64 N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 99.9%

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

   3. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-to-mult-rev N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lower-*.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lift-/.f64 N/A

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

      12. frac-2neg N/A

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

      13. sub-to-fraction N/A

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

      14. distribute-neg-frac2 N/A

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

      15. remove-double-neg N/A

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

      16. add-flip-rev N/A

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

      17. *-lft-identity N/A

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

      18. +-commutative N/A

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

      19. sub-flip N/A

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

      20. lift--.f64 N/A

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

      21. lower-/.f64 88.0%

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

   5. Applied rewrites 88.0%

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

      1. lift-fabs.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. fabs-mul N/A

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

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

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

      9. sqrt-unprod N/A

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

      10. rem-square-sqrt N/A

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

      11. metadata-eval N/A

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

      12. *-inverses N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity 55.2%

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

      15. rem-square-sqrt N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. lift-*.f64 50.3%

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

      19. lift-fma.f64 N/A

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

   7. Applied rewrites 55.2%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 30.7%

         \[\leadsto 1.5 \cdot \color{blue}{x} \]

   10. Applied rewrites 30.7%

       \[\leadsto \color{blue}{1.5 \cdot x} \]

   if -1.0600000000000001e125 < x < 2.7e-113

   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. add-to-fraction N/A

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

      4. +-commutative N/A

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

      5. div-add N/A

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

      6. mult-flip N/A

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

      7. associate-/l* N/A

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

      8. metadata-eval N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. lift--.f64 N/A

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

      13. fabs-sub N/A

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

      14. lower-fabs.f64 N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 99.9%

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

   3. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-to-mult-rev N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lower-*.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lift-/.f64 N/A

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

      12. frac-2neg N/A

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

      13. sub-to-fraction N/A

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

      14. distribute-neg-frac2 N/A

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

      15. remove-double-neg N/A

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

      16. add-flip-rev N/A

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

      17. *-lft-identity N/A

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

      18. +-commutative N/A

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

      19. sub-flip N/A

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

      20. lift--.f64 N/A

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

      21. lower-/.f64 88.0%

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

   5. Applied rewrites 88.0%

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

      1. lift-fabs.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. fabs-mul N/A

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

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

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

      9. sqrt-unprod N/A

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

      10. rem-square-sqrt N/A

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

      11. metadata-eval N/A

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

      12. *-inverses N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity 55.2%

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

      15. rem-square-sqrt N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. lift-*.f64 50.3%

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

      19. lift-fma.f64 N/A

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

   7. Applied rewrites 55.2%

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

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 26.9%

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

   10. Applied rewrites 26.9%

       \[\leadsto \color{blue}{-0.5 \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 26.9% accurate, 2.7× speedup

Math

\[-0.5 \cdot y \]

FPCore

(FPCore (x y) :precision binary64 (* -0.5 y))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (-0.5d0) * y
end function

Java

public static double code(double x, double y) {
	return -0.5 * y;
}

Python

def code(x, y):
	return -0.5 * y

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = -0.5 * y;
end

Wolfram

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

Derivation

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. add-to-fraction N/A

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

   4. +-commutative N/A

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

   5. div-add N/A

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

   6. mult-flip N/A

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

   7. associate-/l* N/A

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

   8. metadata-eval N/A

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

   9. *-rgt-identity N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-fabs.f64 N/A

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

   12. lift--.f64 N/A

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

   13. fabs-sub N/A

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

   14. lower-fabs.f64 N/A

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

   15. lower--.f64 N/A

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

   16. metadata-eval 99.9%

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

3. Applied rewrites 99.9%

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

   1. lift--.f64 N/A

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

   2. sub-negate-rev N/A

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

   3. sub-to-mult-rev N/A

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

   4. lift-/.f64 N/A

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

   5. lift--.f64 N/A

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

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

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

   7. lift--.f64 N/A

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

   8. sub-negate-rev N/A

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

   9. lower-*.f64 N/A

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

   10. sub-negate-rev N/A

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

   11. lift-/.f64 N/A

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

   12. frac-2neg N/A

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

   13. sub-to-fraction N/A

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

   14. distribute-neg-frac2 N/A

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

   15. remove-double-neg N/A

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

   16. add-flip-rev N/A

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

   17. *-lft-identity N/A

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

   18. +-commutative N/A

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

   19. sub-flip N/A

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

   20. lift--.f64 N/A

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

   21. lower-/.f64 88.0%

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

5. Applied rewrites 88.0%

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

   1. lift-fabs.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. associate-/l* N/A

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

   6. *-inverses N/A

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

   7. fabs-mul N/A

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

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

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

   9. sqrt-unprod N/A

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

   10. rem-square-sqrt N/A

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

   11. metadata-eval N/A

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

   12. *-inverses N/A

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

   13. *-inverses N/A

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

   14. *-rgt-identity 55.2%

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

   15. rem-square-sqrt N/A

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

   16. lift-sqrt.f64 N/A

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

   17. lift-sqrt.f64 N/A

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

   18. lift-*.f64 50.3%

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

   19. lift-fma.f64 N/A

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

7. Applied rewrites 55.2%

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

8. Taylor expanded in x around 0

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

   1. lower-*.f64 26.9%

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

10. Applied rewrites 26.9%

    \[\leadsto \color{blue}{-0.5 \cdot y} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025207 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))