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

Percentage Accurate: 99.9% → 99.9%

Time: 2.6s

Alternatives: 12

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. fabs-sub N/A

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

   4. *-lft-identity N/A

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

   5. metadata-eval N/A

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

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

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

   7. lower-fabs.f64 N/A

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

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

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

   9. metadata-eval N/A

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

   10. *-lft-identity N/A

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

   11. lower--.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 84.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+24}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-181}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|-y\right|, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.45e+24)
   (* 0.5 (+ x y))
   (if (<= x 5.2e-181) (fma 0.5 (fabs (- y)) x) (fma 1.5 x (* -0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.45e+24) {
		tmp = 0.5 * (x + y);
	} else if (x <= 5.2e-181) {
		tmp = fma(0.5, fabs(-y), x);
	} else {
		tmp = fma(1.5, x, (-0.5 * y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.45e+24)
		tmp = Float64(0.5 * Float64(x + y));
	elseif (x <= 5.2e-181)
		tmp = fma(0.5, abs(Float64(-y)), x);
	else
		tmp = fma(1.5, x, Float64(-0.5 * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.45e+24], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.2e-181], N[(0.5 * N[Abs[(-y)], $MachinePrecision] + x), $MachinePrecision], N[(1.5 * x + N[(-0.5 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4499999999999999e24

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower-fabs.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. neg-fabs N/A

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

      5. flip-- N/A

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

      6. distribute-neg-frac N/A

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

      7. sqr-neg-rev N/A

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

      8. mul-1-neg N/A

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. distribute-neg-frac N/A

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

      14. flip-+ N/A

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

      15. neg-fabs N/A

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

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

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

      17. sqrt-prod N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 87.5%

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

   7. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 88.2

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

   9. Applied rewrites 88.2%

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

   if -1.4499999999999999e24 < x < 5.19999999999999998e-181

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower-fabs.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 80.8

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

   7. Applied rewrites 80.8%

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

   if 5.19999999999999998e-181 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower-fabs.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. neg-fabs N/A

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

      5. flip-- N/A

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

      6. distribute-neg-frac N/A

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

      7. sqr-neg-rev N/A

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

      8. mul-1-neg N/A

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. distribute-neg-frac N/A

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

      14. flip-+ N/A

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

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

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

      16. sqrt-prod N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 80.6%

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

   7. Taylor expanded in x around 0

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. sqrt-unprod N/A

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

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

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 81.1

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

   9. Applied rewrites 81.1%

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

Alternative 3: 82.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+24}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-181}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|-y\right|, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.45e+24)
   (* 0.5 (+ x y))
   (if (<= x 5.2e-181) (fma 0.5 (fabs (- y)) x) (fma (- x y) 0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.45e+24) {
		tmp = 0.5 * (x + y);
	} else if (x <= 5.2e-181) {
		tmp = fma(0.5, fabs(-y), x);
	} else {
		tmp = fma((x - y), 0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.45e+24)
		tmp = Float64(0.5 * Float64(x + y));
	elseif (x <= 5.2e-181)
		tmp = fma(0.5, abs(Float64(-y)), x);
	else
		tmp = fma(Float64(x - y), 0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.45e+24], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.2e-181], N[(0.5 * N[Abs[(-y)], $MachinePrecision] + x), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * 0.5 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4499999999999999e24

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower-fabs.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. neg-fabs N/A

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

      5. flip-- N/A

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

      6. distribute-neg-frac N/A

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

      7. sqr-neg-rev N/A

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

      8. mul-1-neg N/A

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. distribute-neg-frac N/A

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

      14. flip-+ N/A

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

      15. neg-fabs N/A

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

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

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

      17. sqrt-prod N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 87.5%

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

   7. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 88.2

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

   9. Applied rewrites 88.2%

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

   if -1.4499999999999999e24 < x < 5.19999999999999998e-181

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower-fabs.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 80.8

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

   7. Applied rewrites 80.8%

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

   if 5.19999999999999998e-181 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower-fabs.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. neg-fabs N/A

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

      5. flip-- N/A

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

      6. distribute-neg-frac N/A

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

      7. sqr-neg-rev N/A

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

      8. mul-1-neg N/A

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. distribute-neg-frac N/A

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

      14. flip-+ N/A

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

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

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

      16. sqrt-prod N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 80.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift--.f64 80.9

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

   8. Applied rewrites 80.9%

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

Alternative 4: 82.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-96}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-106}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|x\right|, x\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.5e-96)
   (* (- x y) 0.5)
   (if (<= y 6.5e-106) (fma 0.5 (fabs x) x) (* 0.5 (+ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.5e-96) {
		tmp = (x - y) * 0.5;
	} else if (y <= 6.5e-106) {
		tmp = fma(0.5, fabs(x), x);
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.5e-96)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (y <= 6.5e-106)
		tmp = fma(0.5, abs(x), x);
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.5e-96], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[y, 6.5e-106], N[(0.5 * N[Abs[x], $MachinePrecision] + x), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.49999999999999997e-96

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower-fabs.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 N/A

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

      7. lift-/.f64 81.1

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

   7. Applied rewrites 81.1%

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

   8. Taylor expanded in x around 0

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

      1. fabs-sub N/A

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

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

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

      3. sqrt-unprod N/A

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

      4. rem-square-sqrt N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 84.4

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

   10. Applied rewrites 84.4%

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

   if -2.49999999999999997e-96 < y < 6.4999999999999997e-106

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower-fabs.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 83.3%

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

   if 6.4999999999999997e-106 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower-fabs.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. neg-fabs N/A

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

      5. flip-- N/A

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

      6. distribute-neg-frac N/A

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

      7. sqr-neg-rev N/A

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

      8. mul-1-neg N/A

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. distribute-neg-frac N/A

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

      14. flip-+ N/A

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

      15. neg-fabs N/A

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

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

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

      17. sqrt-prod N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 83.1%

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

   7. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 86.6

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

   9. Applied rewrites 86.6%

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

Alternative 5: 78.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-293}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2e-293) (* (- x y) 0.5) (* 0.5 (+ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2e-293) {
		tmp = (x - y) * 0.5;
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2d-293)) then
        tmp = (x - y) * 0.5d0
    else
        tmp = 0.5d0 * (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2e-293) {
		tmp = (x - y) * 0.5;
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2e-293:
		tmp = (x - y) * 0.5
	else:
		tmp = 0.5 * (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2e-293)
		tmp = Float64(Float64(x - y) * 0.5);
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2e-293)
		tmp = (x - y) * 0.5;
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2e-293], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.0000000000000001e-293

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower-fabs.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 N/A

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

      7. lift-/.f64 87.2

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

   7. Applied rewrites 87.2%

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

   8. Taylor expanded in x around 0

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

      1. fabs-sub N/A

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

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

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

      3. sqrt-unprod N/A

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

      4. rem-square-sqrt N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 78.4

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

   10. Applied rewrites 78.4%

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

   if -2.0000000000000001e-293 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower-fabs.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. neg-fabs N/A

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

      5. flip-- N/A

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

      6. distribute-neg-frac N/A

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

      7. sqr-neg-rev N/A

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

      8. mul-1-neg N/A

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. distribute-neg-frac N/A

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

      14. flip-+ N/A

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

      15. neg-fabs N/A

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

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

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

      17. sqrt-prod N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 73.5%

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

   7. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 78.7

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

   9. Applied rewrites 78.7%

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

Alternative 6: 71.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{-288}:\\ \;\;\;\;\left(x - y\right) \cdot 0.5\\ \mathbf{elif}\;y \leq 7.9 \cdot 10^{-107}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.05e-288)
   (* (- x y) 0.5)
   (if (<= y 7.9e-107) (* 1.5 x) (fma 0.5 y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.05e-288) {
		tmp = (x - y) * 0.5;
	} else if (y <= 7.9e-107) {
		tmp = 1.5 * x;
	} else {
		tmp = fma(0.5, y, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.05e-288)
		tmp = Float64(Float64(x - y) * 0.5);
	elseif (y <= 7.9e-107)
		tmp = Float64(1.5 * x);
	else
		tmp = fma(0.5, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.05e-288], N[(N[(x - y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[y, 7.9e-107], N[(1.5 * x), $MachinePrecision], N[(0.5 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.04999999999999998e-288

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower-fabs.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 N/A

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

      7. lift-/.f64 87.9

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

   7. Applied rewrites 87.9%

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

   8. Taylor expanded in x around 0

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

      1. fabs-sub N/A

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

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

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

      3. sqrt-unprod N/A

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

      4. rem-square-sqrt N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 77.2

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

   10. Applied rewrites 77.2%

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

   if 1.04999999999999998e-288 < y < 7.9000000000000001e-107

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower-fabs.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. neg-fabs N/A

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

      5. flip-- N/A

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

      6. distribute-neg-frac N/A

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

      7. sqr-neg-rev N/A

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

      8. mul-1-neg N/A

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. distribute-neg-frac N/A

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

      14. flip-+ N/A

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

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

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

      16. sqrt-prod N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 42.3%

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

   7. Taylor expanded in x around inf

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      2. lift--.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      4. lift--.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      7. lift--.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{3}{2} \cdot x \]

      11. sqrt-unprod N/A

          \[\leadsto \frac{3}{2} \cdot x \]

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

          \[\leadsto \frac{3}{2} \cdot x \]

      13. lower-*.f64 50.2

          \[\leadsto 1.5 \cdot x \]

   9. Applied rewrites 50.2%

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

   if 7.9000000000000001e-107 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower-fabs.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. neg-fabs N/A

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

      5. flip-- N/A

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

      6. distribute-neg-frac N/A

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

      7. sqr-neg-rev N/A

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

      8. mul-1-neg N/A

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. distribute-neg-frac N/A

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

      14. flip-+ N/A

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

      15. neg-fabs N/A

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

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

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

      17. sqrt-prod N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 83.0%

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

   7. Taylor expanded in x around 0

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

      1. sqrt-unprod 71.7

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

      2. rem-sqrt-square-rev 71.7

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

      3. fabs-sub 71.7

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

   9. Applied rewrites 71.7%

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

Alternative 7: 62.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.18 \cdot 10^{-11}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-148}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 7.9 \cdot 10^{-107}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.18e-11)
   (* -0.5 y)
   (if (<= y -6e-148) (* x 0.5) (if (<= y 7.9e-107) (* 1.5 x) (fma 0.5 y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.18e-11) {
		tmp = -0.5 * y;
	} else if (y <= -6e-148) {
		tmp = x * 0.5;
	} else if (y <= 7.9e-107) {
		tmp = 1.5 * x;
	} else {
		tmp = fma(0.5, y, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.18e-11)
		tmp = Float64(-0.5 * y);
	elseif (y <= -6e-148)
		tmp = Float64(x * 0.5);
	elseif (y <= 7.9e-107)
		tmp = Float64(1.5 * x);
	else
		tmp = fma(0.5, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.18e-11], N[(-0.5 * y), $MachinePrecision], If[LessEqual[y, -6e-148], N[(x * 0.5), $MachinePrecision], If[LessEqual[y, 7.9e-107], N[(1.5 * x), $MachinePrecision], N[(0.5 * y + x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.18e-11

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower-fabs.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. neg-fabs N/A

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

      5. flip-- N/A

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

      6. distribute-neg-frac N/A

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

      7. sqr-neg-rev N/A

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

      8. mul-1-neg N/A

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. distribute-neg-frac N/A

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

      14. flip-+ N/A

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

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

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

      16. sqrt-prod N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 86.4%

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

   7. Taylor expanded in x around 0

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. sqrt-unprod N/A

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

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

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

      13. lower-*.f64 73.6

          \[\leadsto -0.5 \cdot y \]

   9. Applied rewrites 73.6%

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

   if -1.18e-11 < y < -5.99999999999999996e-148

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower-fabs.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 N/A

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

      7. lift-/.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around 0

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

      1. fabs-sub N/A

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

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

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

      3. sqrt-unprod N/A

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

      4. rem-square-sqrt N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 72.6

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

   10. Applied rewrites 72.6%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 37.2%

       \[\leadsto x \cdot 0.5 \]

   if -5.99999999999999996e-148 < y < 7.9000000000000001e-107

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower-fabs.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. neg-fabs N/A

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

      5. flip-- N/A

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

      6. distribute-neg-frac N/A

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

      7. sqr-neg-rev N/A

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

      8. mul-1-neg N/A

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. distribute-neg-frac N/A

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

      14. flip-+ N/A

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

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

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

      16. sqrt-prod N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 49.2%

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

   7. Taylor expanded in x around inf

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      2. lift--.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      4. lift--.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      7. lift--.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{3}{2} \cdot x \]

      11. sqrt-unprod N/A

          \[\leadsto \frac{3}{2} \cdot x \]

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

          \[\leadsto \frac{3}{2} \cdot x \]

      13. lower-*.f64 51.2

          \[\leadsto 1.5 \cdot x \]

   9. Applied rewrites 51.2%

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

   if 7.9000000000000001e-107 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower-fabs.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. neg-fabs N/A

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

      5. flip-- N/A

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

      6. distribute-neg-frac N/A

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

      7. sqr-neg-rev N/A

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

      8. mul-1-neg N/A

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. distribute-neg-frac N/A

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

      14. flip-+ N/A

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

      15. neg-fabs N/A

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

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

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

      17. sqrt-prod N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 83.0%

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

   7. Taylor expanded in x around 0

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

      1. sqrt-unprod 71.7

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

      2. rem-sqrt-square-rev 71.7

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

      3. fabs-sub 71.7

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

   9. Applied rewrites 71.7%

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

Alternative 8: 61.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.18 \cdot 10^{-11}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-148}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-71}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.18e-11)
   (* -0.5 y)
   (if (<= y -6e-148) (* x 0.5) (if (<= y 4.3e-71) (* 1.5 x) (* 0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.18e-11) {
		tmp = -0.5 * y;
	} else if (y <= -6e-148) {
		tmp = x * 0.5;
	} else if (y <= 4.3e-71) {
		tmp = 1.5 * x;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.18d-11)) then
        tmp = (-0.5d0) * y
    else if (y <= (-6d-148)) then
        tmp = x * 0.5d0
    else if (y <= 4.3d-71) then
        tmp = 1.5d0 * x
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.18e-11) {
		tmp = -0.5 * y;
	} else if (y <= -6e-148) {
		tmp = x * 0.5;
	} else if (y <= 4.3e-71) {
		tmp = 1.5 * x;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.18e-11:
		tmp = -0.5 * y
	elif y <= -6e-148:
		tmp = x * 0.5
	elif y <= 4.3e-71:
		tmp = 1.5 * x
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.18e-11)
		tmp = Float64(-0.5 * y);
	elseif (y <= -6e-148)
		tmp = Float64(x * 0.5);
	elseif (y <= 4.3e-71)
		tmp = Float64(1.5 * x);
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.18e-11)
		tmp = -0.5 * y;
	elseif (y <= -6e-148)
		tmp = x * 0.5;
	elseif (y <= 4.3e-71)
		tmp = 1.5 * x;
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.18e-11], N[(-0.5 * y), $MachinePrecision], If[LessEqual[y, -6e-148], N[(x * 0.5), $MachinePrecision], If[LessEqual[y, 4.3e-71], N[(1.5 * x), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.18e-11

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower-fabs.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. neg-fabs N/A

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

      5. flip-- N/A

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

      6. distribute-neg-frac N/A

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

      7. sqr-neg-rev N/A

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

      8. mul-1-neg N/A

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. distribute-neg-frac N/A

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

      14. flip-+ N/A

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

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

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

      16. sqrt-prod N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 86.4%

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

   7. Taylor expanded in x around 0

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. sqrt-unprod N/A

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

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

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

      13. lower-*.f64 73.6

          \[\leadsto -0.5 \cdot y \]

   9. Applied rewrites 73.6%

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

   if -1.18e-11 < y < -5.99999999999999996e-148

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. fabs-sub N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower-fabs.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\left(1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{y}{x}\right) \cdot x\right|, x\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\left(1 - 1 \cdot \frac{y}{x}\right) \cdot x\right|, x\right) \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\left(1 - \frac{y}{x}\right) \cdot x\right|, x\right) \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\left(1 - \frac{y}{x}\right) \cdot x\right|, x\right) \]

      7. lift-/.f64 99.8

         \[\leadsto \mathsf{fma}\left(0.5, \left|\left(1 - \frac{y}{x}\right) \cdot x\right|, x\right) \]

   7. Applied rewrites 99.8%

      \[\leadsto \mathsf{fma}\left(0.5, \left|\left(1 - \frac{y}{x}\right) \cdot x\right|, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left|y - x\right|} \]
   9. Step-by-step derivation

      1. fabs-sub N/A

         \[\leadsto \frac{1}{2} \cdot \left|x - y\right| \]

      2. rem-sqrt-square-rev N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(x - y\right) \cdot \left(x - y\right)} \]

      3. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \left(x - \color{blue}{y}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(x - y\right) \cdot \color{blue}{\frac{1}{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(x - y\right) \cdot \color{blue}{\frac{1}{2}} \]

      7. lift--.f64 72.6

         \[\leadsto \left(x - y\right) \cdot 0.5 \]

   10. Applied rewrites 72.6%

       \[\leadsto \color{blue}{\left(x - y\right) \cdot 0.5} \]

   11. Taylor expanded in x around inf

       \[\leadsto x \cdot \frac{1}{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 37.2%

       \[\leadsto x \cdot 0.5 \]

   if -5.99999999999999996e-148 < y < 4.2999999999999997e-71

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left|y - x\right| + \color{blue}{x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left|y - x\right|}, x\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      7. lower-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      8. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      11. lower--.f64 99.9

          \[\leadsto \mathsf{fma}\left(0.5, \left|x - y\right|, x\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|x - y\right|, x\right)} \]
   5. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      2. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|y - x\right|, x\right) \]

      4. neg-fabs N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\left(y - x\right)\right)\right|, x\right) \]

      5. flip-- N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\frac{y \cdot y - x \cdot x}{y + x}\right)\right|, x\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - x \cdot x\right)\right)}{y + x}\right|, x\right) \]

      7. sqr-neg-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)\right)}{y + x}\right|, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)\right)}{y + x}\right|, x\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)\right)\right)}{y + x}\right|, x\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)\right)\right)}{y + 1 \cdot x}\right|, x\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)\right)\right)}{y + \left(\mathsf{neg}\left(-1\right)\right) \cdot x}\right|, x\right) \]

      12. fp-cancel-sub-sign-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)\right)\right)}{y - -1 \cdot x}\right|, x\right) \]

      13. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\frac{y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)}{y - -1 \cdot x}\right)\right|, x\right) \]

      14. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)\right|, x\right) \]

      15. rem-sqrt-square-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\left(\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)\right)}, x\right) \]

      16. sqrt-prod N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)} \cdot \color{blue}{\sqrt{\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)}}, x\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)} \cdot \color{blue}{\sqrt{\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)}}, x\right) \]

   6. Applied rewrites 47.7%

      \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

   7. Taylor expanded in x around inf

      \[\leadsto \frac{3}{2} \cdot \color{blue}{x} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      2. lift--.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      4. lift--.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      7. lift--.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{3}{2} \cdot x \]

      11. sqrt-unprod N/A

          \[\leadsto \frac{3}{2} \cdot x \]

      12. rem-sqrt-square-rev N/A

          \[\leadsto \frac{3}{2} \cdot x \]

      13. lower-*.f64 50.2

          \[\leadsto 1.5 \cdot x \]

   9. Applied rewrites 50.2%

      \[\leadsto 1.5 \cdot \color{blue}{x} \]

   if 4.2999999999999997e-71 < y

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left|y - x\right| + \color{blue}{x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left|y - x\right|}, x\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      7. lower-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      8. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      11. lower--.f64 99.9

          \[\leadsto \mathsf{fma}\left(0.5, \left|x - y\right|, x\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|x - y\right|, x\right)} \]
   5. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      2. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|y - x\right|, x\right) \]

      4. neg-fabs N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\left(y - x\right)\right)\right|, x\right) \]

      5. flip-- N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\frac{y \cdot y - x \cdot x}{y + x}\right)\right|, x\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - x \cdot x\right)\right)}{y + x}\right|, x\right) \]

      7. sqr-neg-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)\right)}{y + x}\right|, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)\right)}{y + x}\right|, x\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)\right)\right)}{y + x}\right|, x\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)\right)\right)}{y + 1 \cdot x}\right|, x\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)\right)\right)}{y + \left(\mathsf{neg}\left(-1\right)\right) \cdot x}\right|, x\right) \]

      12. fp-cancel-sub-sign-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)\right)\right)}{y - -1 \cdot x}\right|, x\right) \]

      13. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\frac{y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)}{y - -1 \cdot x}\right)\right|, x\right) \]

      14. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)\right|, x\right) \]

      15. neg-fabs N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|y + -1 \cdot x\right|, x\right) \]

      16. rem-sqrt-square-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\left(y + -1 \cdot x\right) \cdot \left(y + -1 \cdot x\right)}, x\right) \]

      17. sqrt-prod N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{y + -1 \cdot x} \cdot \color{blue}{\sqrt{y + -1 \cdot x}}, x\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{y + -1 \cdot x} \cdot \color{blue}{\sqrt{y + -1 \cdot x}}, x\right) \]

   6. Applied rewrites 84.7%

      \[\leadsto \mathsf{fma}\left(0.5, \sqrt{y - x} \cdot \color{blue}{\sqrt{y - x}}, x\right) \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{y} \]
   8. Step-by-step derivation

      1. lower-*.f64 69.5

         \[\leadsto 0.5 \cdot y \]

   9. Applied rewrites 69.5%

      \[\leadsto 0.5 \cdot \color{blue}{y} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 9: 60.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-11}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-71}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.4e-11) (* -0.5 y) (if (<= y 4.3e-71) (* 1.5 x) (* 0.5 y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.4e-11) {
		tmp = -0.5 * y;
	} else if (y <= 4.3e-71) {
		tmp = 1.5 * x;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.4d-11)) then
        tmp = (-0.5d0) * y
    else if (y <= 4.3d-71) then
        tmp = 1.5d0 * x
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.4e-11) {
		tmp = -0.5 * y;
	} else if (y <= 4.3e-71) {
		tmp = 1.5 * x;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.4e-11:
		tmp = -0.5 * y
	elif y <= 4.3e-71:
		tmp = 1.5 * x
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.4e-11)
		tmp = Float64(-0.5 * y);
	elseif (y <= 4.3e-71)
		tmp = Float64(1.5 * x);
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.4e-11)
		tmp = -0.5 * y;
	elseif (y <= 4.3e-71)
		tmp = 1.5 * x;
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.4e-11], N[(-0.5 * y), $MachinePrecision], If[LessEqual[y, 4.3e-71], N[(1.5 * x), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.4e-11

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left|y - x\right| + \color{blue}{x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left|y - x\right|}, x\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      7. lower-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      8. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      11. lower--.f64 100.0

          \[\leadsto \mathsf{fma}\left(0.5, \left|x - y\right|, x\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|x - y\right|, x\right)} \]
   5. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      2. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|y - x\right|, x\right) \]

      4. neg-fabs N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\left(y - x\right)\right)\right|, x\right) \]

      5. flip-- N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\frac{y \cdot y - x \cdot x}{y + x}\right)\right|, x\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - x \cdot x\right)\right)}{y + x}\right|, x\right) \]

      7. sqr-neg-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)\right)}{y + x}\right|, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)\right)}{y + x}\right|, x\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)\right)\right)}{y + x}\right|, x\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)\right)\right)}{y + 1 \cdot x}\right|, x\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)\right)\right)}{y + \left(\mathsf{neg}\left(-1\right)\right) \cdot x}\right|, x\right) \]

      12. fp-cancel-sub-sign-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)\right)\right)}{y - -1 \cdot x}\right|, x\right) \]

      13. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\frac{y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)}{y - -1 \cdot x}\right)\right|, x\right) \]

      14. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)\right|, x\right) \]

      15. rem-sqrt-square-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\left(\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)\right)}, x\right) \]

      16. sqrt-prod N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)} \cdot \color{blue}{\sqrt{\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)}}, x\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)} \cdot \color{blue}{\sqrt{\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)}}, x\right) \]

   6. Applied rewrites 86.4%

      \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{2} \cdot \color{blue}{y} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot y \]

      2. lift--.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot y \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot y \]

      4. lift--.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot y \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot y \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot y \]

      7. lift--.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot y \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot y \]

      9. lift--.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot y \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{-1}{2} \cdot y \]

      11. sqrt-unprod N/A

          \[\leadsto \frac{-1}{2} \cdot y \]

      12. rem-sqrt-square-rev N/A

          \[\leadsto \frac{-1}{2} \cdot y \]

      13. lower-*.f64 73.7

          \[\leadsto -0.5 \cdot y \]

   9. Applied rewrites 73.7%

      \[\leadsto -0.5 \cdot \color{blue}{y} \]

   if -1.4e-11 < y < 4.2999999999999997e-71

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left|y - x\right| + \color{blue}{x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left|y - x\right|}, x\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      7. lower-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      8. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      11. lower--.f64 99.9

          \[\leadsto \mathsf{fma}\left(0.5, \left|x - y\right|, x\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|x - y\right|, x\right)} \]
   5. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      2. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|y - x\right|, x\right) \]

      4. neg-fabs N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\left(y - x\right)\right)\right|, x\right) \]

      5. flip-- N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\frac{y \cdot y - x \cdot x}{y + x}\right)\right|, x\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - x \cdot x\right)\right)}{y + x}\right|, x\right) \]

      7. sqr-neg-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)\right)}{y + x}\right|, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)\right)}{y + x}\right|, x\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)\right)\right)}{y + x}\right|, x\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)\right)\right)}{y + 1 \cdot x}\right|, x\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)\right)\right)}{y + \left(\mathsf{neg}\left(-1\right)\right) \cdot x}\right|, x\right) \]

      12. fp-cancel-sub-sign-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)\right)\right)}{y - -1 \cdot x}\right|, x\right) \]

      13. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\frac{y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)}{y - -1 \cdot x}\right)\right|, x\right) \]

      14. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)\right|, x\right) \]

      15. rem-sqrt-square-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\left(\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)\right)}, x\right) \]

      16. sqrt-prod N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)} \cdot \color{blue}{\sqrt{\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)}}, x\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)} \cdot \color{blue}{\sqrt{\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)}}, x\right) \]

   6. Applied rewrites 53.2%

      \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

   7. Taylor expanded in x around inf

      \[\leadsto \frac{3}{2} \cdot \color{blue}{x} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      2. lift--.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      4. lift--.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      7. lift--.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{3}{2} \cdot x \]

      11. sqrt-unprod N/A

          \[\leadsto \frac{3}{2} \cdot x \]

      12. rem-sqrt-square-rev N/A

          \[\leadsto \frac{3}{2} \cdot x \]

      13. lower-*.f64 47.1

          \[\leadsto 1.5 \cdot x \]

   9. Applied rewrites 47.1%

      \[\leadsto 1.5 \cdot \color{blue}{x} \]

   if 4.2999999999999997e-71 < y

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left|y - x\right| + \color{blue}{x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left|y - x\right|}, x\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      7. lower-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      8. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      11. lower--.f64 99.9

          \[\leadsto \mathsf{fma}\left(0.5, \left|x - y\right|, x\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|x - y\right|, x\right)} \]
   5. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      2. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|y - x\right|, x\right) \]

      4. neg-fabs N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\left(y - x\right)\right)\right|, x\right) \]

      5. flip-- N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\frac{y \cdot y - x \cdot x}{y + x}\right)\right|, x\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - x \cdot x\right)\right)}{y + x}\right|, x\right) \]

      7. sqr-neg-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)\right)}{y + x}\right|, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)\right)}{y + x}\right|, x\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)\right)\right)}{y + x}\right|, x\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)\right)\right)}{y + 1 \cdot x}\right|, x\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)\right)\right)}{y + \left(\mathsf{neg}\left(-1\right)\right) \cdot x}\right|, x\right) \]

      12. fp-cancel-sub-sign-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)\right)\right)}{y - -1 \cdot x}\right|, x\right) \]

      13. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\frac{y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)}{y - -1 \cdot x}\right)\right|, x\right) \]

      14. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)\right|, x\right) \]

      15. neg-fabs N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|y + -1 \cdot x\right|, x\right) \]

      16. rem-sqrt-square-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\left(y + -1 \cdot x\right) \cdot \left(y + -1 \cdot x\right)}, x\right) \]

      17. sqrt-prod N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{y + -1 \cdot x} \cdot \color{blue}{\sqrt{y + -1 \cdot x}}, x\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{y + -1 \cdot x} \cdot \color{blue}{\sqrt{y + -1 \cdot x}}, x\right) \]

   6. Applied rewrites 84.7%

      \[\leadsto \mathsf{fma}\left(0.5, \sqrt{y - x} \cdot \color{blue}{\sqrt{y - x}}, x\right) \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{y} \]
   8. Step-by-step derivation

      1. lower-*.f64 69.5

         \[\leadsto 0.5 \cdot y \]

   9. Applied rewrites 69.5%

      \[\leadsto 0.5 \cdot \color{blue}{y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 45.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-11}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.4e-11) (* -0.5 y) (* 1.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.4e-11) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.4d-11)) then
        tmp = (-0.5d0) * y
    else
        tmp = 1.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.4e-11) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.4e-11:
		tmp = -0.5 * y
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.4e-11)
		tmp = Float64(-0.5 * y);
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.4e-11)
		tmp = -0.5 * y;
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.4e-11], N[(-0.5 * y), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4e-11

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left|y - x\right| + \color{blue}{x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left|y - x\right|}, x\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      7. lower-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      8. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      11. lower--.f64 100.0

          \[\leadsto \mathsf{fma}\left(0.5, \left|x - y\right|, x\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|x - y\right|, x\right)} \]
   5. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      2. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|y - x\right|, x\right) \]

      4. neg-fabs N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\left(y - x\right)\right)\right|, x\right) \]

      5. flip-- N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\frac{y \cdot y - x \cdot x}{y + x}\right)\right|, x\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - x \cdot x\right)\right)}{y + x}\right|, x\right) \]

      7. sqr-neg-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)\right)}{y + x}\right|, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)\right)}{y + x}\right|, x\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)\right)\right)}{y + x}\right|, x\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)\right)\right)}{y + 1 \cdot x}\right|, x\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)\right)\right)}{y + \left(\mathsf{neg}\left(-1\right)\right) \cdot x}\right|, x\right) \]

      12. fp-cancel-sub-sign-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)\right)\right)}{y - -1 \cdot x}\right|, x\right) \]

      13. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\frac{y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)}{y - -1 \cdot x}\right)\right|, x\right) \]

      14. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)\right|, x\right) \]

      15. rem-sqrt-square-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\left(\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)\right)}, x\right) \]

      16. sqrt-prod N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)} \cdot \color{blue}{\sqrt{\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)}}, x\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)} \cdot \color{blue}{\sqrt{\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)}}, x\right) \]

   6. Applied rewrites 86.4%

      \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{2} \cdot \color{blue}{y} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot y \]

      2. lift--.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot y \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot y \]

      4. lift--.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot y \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot y \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot y \]

      7. lift--.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot y \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot y \]

      9. lift--.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot y \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{-1}{2} \cdot y \]

      11. sqrt-unprod N/A

          \[\leadsto \frac{-1}{2} \cdot y \]

      12. rem-sqrt-square-rev N/A

          \[\leadsto \frac{-1}{2} \cdot y \]

      13. lower-*.f64 73.7

          \[\leadsto -0.5 \cdot y \]

   9. Applied rewrites 73.7%

      \[\leadsto -0.5 \cdot \color{blue}{y} \]

   if -1.4e-11 < y

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left|y - x\right| + \color{blue}{x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left|y - x\right|}, x\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      7. lower-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      8. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      11. lower--.f64 99.9

          \[\leadsto \mathsf{fma}\left(0.5, \left|x - y\right|, x\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|x - y\right|, x\right)} \]
   5. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      2. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|y - x\right|, x\right) \]

      4. neg-fabs N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\left(y - x\right)\right)\right|, x\right) \]

      5. flip-- N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\frac{y \cdot y - x \cdot x}{y + x}\right)\right|, x\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - x \cdot x\right)\right)}{y + x}\right|, x\right) \]

      7. sqr-neg-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)\right)}{y + x}\right|, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)\right)}{y + x}\right|, x\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)\right)\right)}{y + x}\right|, x\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)\right)\right)}{y + 1 \cdot x}\right|, x\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)\right)\right)}{y + \left(\mathsf{neg}\left(-1\right)\right) \cdot x}\right|, x\right) \]

      12. fp-cancel-sub-sign-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)\right)\right)}{y - -1 \cdot x}\right|, x\right) \]

      13. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\frac{y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)}{y - -1 \cdot x}\right)\right|, x\right) \]

      14. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)\right|, x\right) \]

      15. rem-sqrt-square-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\left(\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)\right)}, x\right) \]

      16. sqrt-prod N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)} \cdot \color{blue}{\sqrt{\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)}}, x\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)} \cdot \color{blue}{\sqrt{\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)}}, x\right) \]

   6. Applied rewrites 37.2%

      \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

   7. Taylor expanded in x around inf

      \[\leadsto \frac{3}{2} \cdot \color{blue}{x} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      2. lift--.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      4. lift--.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      7. lift--.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{3}{2} \cdot x \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{3}{2} \cdot x \]

      11. sqrt-unprod N/A

          \[\leadsto \frac{3}{2} \cdot x \]

      12. rem-sqrt-square-rev N/A

          \[\leadsto \frac{3}{2} \cdot x \]

      13. lower-*.f64 35.4

          \[\leadsto 1.5 \cdot x \]

   9. Applied rewrites 35.4%

      \[\leadsto 1.5 \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 31.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{-268}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -3.9e-268) (* -0.5 y) x))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.9e-268) {
		tmp = -0.5 * y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.9d-268)) then
        tmp = (-0.5d0) * y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.9e-268) {
		tmp = -0.5 * y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.9e-268:
		tmp = -0.5 * y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.9e-268)
		tmp = Float64(-0.5 * y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.9e-268)
		tmp = -0.5 * y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.9e-268], N[(-0.5 * y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -3.8999999999999998e-268

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \left|y - x\right|} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left|y - x\right| + \color{blue}{x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left|y - x\right|}, x\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      7. lower-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x + -1 \cdot y\right|, x\right) \]

      8. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right|, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - 1 \cdot y\right|, x\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      11. lower--.f64 99.9

          \[\leadsto \mathsf{fma}\left(0.5, \left|x - y\right|, x\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \left|x - y\right|, x\right)} \]
   5. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      2. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|x - y\right|, x\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|y - x\right|, x\right) \]

      4. neg-fabs N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\left(y - x\right)\right)\right|, x\right) \]

      5. flip-- N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\frac{y \cdot y - x \cdot x}{y + x}\right)\right|, x\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - x \cdot x\right)\right)}{y + x}\right|, x\right) \]

      7. sqr-neg-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)\right)}{y + x}\right|, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)\right)}{y + x}\right|, x\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)\right)\right)}{y + x}\right|, x\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)\right)\right)}{y + 1 \cdot x}\right|, x\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)\right)\right)}{y + \left(\mathsf{neg}\left(-1\right)\right) \cdot x}\right|, x\right) \]

      12. fp-cancel-sub-sign-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\frac{\mathsf{neg}\left(\left(y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)\right)\right)}{y - -1 \cdot x}\right|, x\right) \]

      13. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\frac{y \cdot y - \left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)}{y - -1 \cdot x}\right)\right|, x\right) \]

      14. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left|\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)\right|, x\right) \]

      15. rem-sqrt-square-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\left(\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)\right)}, x\right) \]

      16. sqrt-prod N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)} \cdot \color{blue}{\sqrt{\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)}}, x\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)} \cdot \color{blue}{\sqrt{\mathsf{neg}\left(\left(y + -1 \cdot x\right)\right)}}, x\right) \]

   6. Applied rewrites 76.1%

      \[\leadsto \mathsf{fma}\left(0.5, \sqrt{x - y} \cdot \color{blue}{\sqrt{x - y}}, x\right) \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{2} \cdot \color{blue}{y} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot y \]

      2. lift--.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot y \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot y \]

      4. lift--.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot y \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot y \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot y \]

      7. lift--.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot y \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot y \]

      9. lift--.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot y \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{-1}{2} \cdot y \]

      11. sqrt-unprod N/A

          \[\leadsto \frac{-1}{2} \cdot y \]

      12. rem-sqrt-square-rev N/A

          \[\leadsto \frac{-1}{2} \cdot y \]

      13. lower-*.f64 54.1

          \[\leadsto -0.5 \cdot y \]

   9. Applied rewrites 54.1%

      \[\leadsto -0.5 \cdot \color{blue}{y} \]

   if -3.8999999999999998e-268 < y

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 11.9%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 11.4% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 x)

C

double code(double x, double y) {
	return x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x
end function

Java

public static double code(double x, double y) {
	return x;
}

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

function tmp = code(x, y)
	tmp = x;
end

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 99.9%

   \[x + \frac{\left|y - x\right|}{2} \]

2. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} \]
3. Step-by-step derivation

4. Applied rewrites 11.4%

   \[\leadsto \color{blue}{x} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025115 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))