Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, A

Percentage Accurate: 100.0% → 100.0%

Time: 4.8s

Alternatives: 10

Speedup: 1.5×

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (- (* x (- y 1.0)) (* y 0.5)) 0.918938533204673))

C

double code(double x, double y) {
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
}

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 * (y - 1.0d0)) - (y * 0.5d0)) + 0.918938533204673d0
end function

Java

public static double code(double x, double y) {
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
}

Python

def code(x, y):
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673

Julia

function code(x, y)
	return Float64(Float64(Float64(x * Float64(y - 1.0)) - Float64(y * 0.5)) + 0.918938533204673)
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
end

Wolfram

code[x_, y_] := N[(N[(N[(x * N[(y - 1.0), $MachinePrecision]), $MachinePrecision] - N[(y * 0.5), $MachinePrecision]), $MachinePrecision] + 0.918938533204673), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (- (* x (- y 1.0)) (* y 0.5)) 0.918938533204673))

C

double code(double x, double y) {
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
}

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 * (y - 1.0d0)) - (y * 0.5d0)) + 0.918938533204673d0
end function

Java

public static double code(double x, double y) {
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
}

Python

def code(x, y):
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673

Julia

function code(x, y)
	return Float64(Float64(Float64(x * Float64(y - 1.0)) - Float64(y * 0.5)) + 0.918938533204673)
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
end

Wolfram

code[x_, y_] := N[(N[(N[(x * N[(y - 1.0), $MachinePrecision]), $MachinePrecision] - N[(y * 0.5), $MachinePrecision]), $MachinePrecision] + 0.918938533204673), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x - 0.5, y, 0.918938533204673 - x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma (- x 0.5) y (- 0.918938533204673 x)))

C

double code(double x, double y) {
	return fma((x - 0.5), y, (0.918938533204673 - x));
}

Julia

function code(x, y)
	return fma(Float64(x - 0.5), y, Float64(0.918938533204673 - x))
end

Wolfram

code[x_, y_] := N[(N[(x - 0.5), $MachinePrecision] * y + N[(0.918938533204673 - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(\frac{918938533204673}{1000000000000000} + x \cdot \left(y - 1\right)\right) - \frac{1}{2} \cdot y} \]

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, y, 0.918938533204673 - x\right)} \]
6. Add Preprocessing

Alternative 2: 74.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+247}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{+31}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq -86:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.8:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+119} \lor \neg \left(y \leq 5.1 \cdot 10^{+278}\right):\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.4e+247)
   (* y x)
   (if (<= y -5.2e+31)
     (* -0.5 y)
     (if (<= y -86.0)
       (* y x)
       (if (<= y 1.8)
         (- 0.918938533204673 x)
         (if (or (<= y 1.7e+119) (not (<= y 5.1e+278)))
           (* -0.5 y)
           (* y x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.4e+247) {
		tmp = y * x;
	} else if (y <= -5.2e+31) {
		tmp = -0.5 * y;
	} else if (y <= -86.0) {
		tmp = y * x;
	} else if (y <= 1.8) {
		tmp = 0.918938533204673 - x;
	} else if ((y <= 1.7e+119) || !(y <= 5.1e+278)) {
		tmp = -0.5 * y;
	} else {
		tmp = y * 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+247)) then
        tmp = y * x
    else if (y <= (-5.2d+31)) then
        tmp = (-0.5d0) * y
    else if (y <= (-86.0d0)) then
        tmp = y * x
    else if (y <= 1.8d0) then
        tmp = 0.918938533204673d0 - x
    else if ((y <= 1.7d+119) .or. (.not. (y <= 5.1d+278))) then
        tmp = (-0.5d0) * y
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.4e+247) {
		tmp = y * x;
	} else if (y <= -5.2e+31) {
		tmp = -0.5 * y;
	} else if (y <= -86.0) {
		tmp = y * x;
	} else if (y <= 1.8) {
		tmp = 0.918938533204673 - x;
	} else if ((y <= 1.7e+119) || !(y <= 5.1e+278)) {
		tmp = -0.5 * y;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.4e+247:
		tmp = y * x
	elif y <= -5.2e+31:
		tmp = -0.5 * y
	elif y <= -86.0:
		tmp = y * x
	elif y <= 1.8:
		tmp = 0.918938533204673 - x
	elif (y <= 1.7e+119) or not (y <= 5.1e+278):
		tmp = -0.5 * y
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.4e+247)
		tmp = Float64(y * x);
	elseif (y <= -5.2e+31)
		tmp = Float64(-0.5 * y);
	elseif (y <= -86.0)
		tmp = Float64(y * x);
	elseif (y <= 1.8)
		tmp = Float64(0.918938533204673 - x);
	elseif ((y <= 1.7e+119) || !(y <= 5.1e+278))
		tmp = Float64(-0.5 * y);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.4e+247)
		tmp = y * x;
	elseif (y <= -5.2e+31)
		tmp = -0.5 * y;
	elseif (y <= -86.0)
		tmp = y * x;
	elseif (y <= 1.8)
		tmp = 0.918938533204673 - x;
	elseif ((y <= 1.7e+119) || ~((y <= 5.1e+278)))
		tmp = -0.5 * y;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.4e+247], N[(y * x), $MachinePrecision], If[LessEqual[y, -5.2e+31], N[(-0.5 * y), $MachinePrecision], If[LessEqual[y, -86.0], N[(y * x), $MachinePrecision], If[LessEqual[y, 1.8], N[(0.918938533204673 - x), $MachinePrecision], If[Or[LessEqual[y, 1.7e+119], N[Not[LessEqual[y, 5.1e+278]], $MachinePrecision]], N[(-0.5 * y), $MachinePrecision], N[(y * x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.3999999999999999e247 or -5.2e31 < y < -86 or 1.70000000000000007e119 < y < 5.1e278

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. distribute-rgt-out-- N/A

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

      2. *-commutative N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. remove-double-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. associate-*r* N/A

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

      15. mul-1-neg N/A

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

      16. *-commutative N/A

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

      17. distribute-rgt-neg-in N/A

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

      18. metadata-eval N/A

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

      19. mul-1-neg N/A

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

      20. distribute-neg-out N/A

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

      21. distribute-lft-neg-out N/A

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

      22. distribute-rgt-neg-in N/A

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

      23. remove-double-neg N/A

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

      24. lower-*.f64 N/A

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 74.3%

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

   if -1.3999999999999999e247 < y < -5.2e31 or 1.80000000000000004 < y < 1.70000000000000007e119 or 5.1e278 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. distribute-rgt-out-- N/A

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

      2. *-commutative N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. remove-double-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. associate-*r* N/A

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

      15. mul-1-neg N/A

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

      16. *-commutative N/A

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

      17. distribute-rgt-neg-in N/A

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

      18. metadata-eval N/A

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

      19. mul-1-neg N/A

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

      20. distribute-neg-out N/A

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

      21. distribute-lft-neg-out N/A

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

      22. distribute-rgt-neg-in N/A

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

      23. remove-double-neg N/A

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

      24. lower-*.f64 N/A

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.4%

      \[\leadsto -0.5 \cdot y \]

   if -86 < y < 1.80000000000000004

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - \left(\mathsf{neg}\left(-1\right)\right) \cdot x} \]

      2. metadata-eval N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{1} \cdot x \]

      3. *-lft-identity N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{x} \]

      4. lower--.f64 97.6

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   5. Applied rewrites 97.6%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+247}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{+31}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq -86:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.8:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+119} \lor \neg \left(y \leq 5.1 \cdot 10^{+278}\right):\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+247}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{+31}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq -86:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 6.3 \cdot 10^{-6}:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+119}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{+278}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.4e+247)
   (* y x)
   (if (<= y -5.2e+31)
     (* -0.5 y)
     (if (<= y -86.0)
       (* y x)
       (if (<= y 6.3e-6)
         (- 0.918938533204673 x)
         (if (<= y 1.7e+119)
           (fma -0.5 y 0.918938533204673)
           (if (<= y 5.1e+278) (* y x) (* -0.5 y))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.4e+247) {
		tmp = y * x;
	} else if (y <= -5.2e+31) {
		tmp = -0.5 * y;
	} else if (y <= -86.0) {
		tmp = y * x;
	} else if (y <= 6.3e-6) {
		tmp = 0.918938533204673 - x;
	} else if (y <= 1.7e+119) {
		tmp = fma(-0.5, y, 0.918938533204673);
	} else if (y <= 5.1e+278) {
		tmp = y * x;
	} else {
		tmp = -0.5 * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.4e+247)
		tmp = Float64(y * x);
	elseif (y <= -5.2e+31)
		tmp = Float64(-0.5 * y);
	elseif (y <= -86.0)
		tmp = Float64(y * x);
	elseif (y <= 6.3e-6)
		tmp = Float64(0.918938533204673 - x);
	elseif (y <= 1.7e+119)
		tmp = fma(-0.5, y, 0.918938533204673);
	elseif (y <= 5.1e+278)
		tmp = Float64(y * x);
	else
		tmp = Float64(-0.5 * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.4e+247], N[(y * x), $MachinePrecision], If[LessEqual[y, -5.2e+31], N[(-0.5 * y), $MachinePrecision], If[LessEqual[y, -86.0], N[(y * x), $MachinePrecision], If[LessEqual[y, 6.3e-6], N[(0.918938533204673 - x), $MachinePrecision], If[LessEqual[y, 1.7e+119], N[(-0.5 * y + 0.918938533204673), $MachinePrecision], If[LessEqual[y, 5.1e+278], N[(y * x), $MachinePrecision], N[(-0.5 * y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.3999999999999999e247 or -5.2e31 < y < -86 or 1.70000000000000007e119 < y < 5.1e278

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. distribute-rgt-out-- N/A

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

      2. *-commutative N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. remove-double-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. associate-*r* N/A

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

      15. mul-1-neg N/A

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

      16. *-commutative N/A

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

      17. distribute-rgt-neg-in N/A

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

      18. metadata-eval N/A

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

      19. mul-1-neg N/A

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

      20. distribute-neg-out N/A

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

      21. distribute-lft-neg-out N/A

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

      22. distribute-rgt-neg-in N/A

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

      23. remove-double-neg N/A

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

      24. lower-*.f64 N/A

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 74.3%

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

   if -1.3999999999999999e247 < y < -5.2e31 or 5.1e278 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. distribute-rgt-out-- N/A

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

      2. *-commutative N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. remove-double-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. associate-*r* N/A

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

      15. mul-1-neg N/A

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

      16. *-commutative N/A

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

      17. distribute-rgt-neg-in N/A

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

      18. metadata-eval N/A

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

      19. mul-1-neg N/A

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

      20. distribute-neg-out N/A

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

      21. distribute-lft-neg-out N/A

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

      22. distribute-rgt-neg-in N/A

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

      23. remove-double-neg N/A

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

      24. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 72.4%

      \[\leadsto -0.5 \cdot y \]

   if -86 < y < 6.29999999999999982e-6

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - \left(\mathsf{neg}\left(-1\right)\right) \cdot x} \]

      2. metadata-eval N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{1} \cdot x \]

      3. *-lft-identity N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{x} \]

      4. lower--.f64 97.6

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   5. Applied rewrites 97.6%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]

   if 6.29999999999999982e-6 < y < 1.70000000000000007e119

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - \frac{1}{2} \cdot y} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot y} \]

      2. metadata-eval N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\frac{-1}{2}} \cdot y \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot y + \frac{918938533204673}{1000000000000000}} \]

      4. lower-fma.f64 66.4

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)} \]

   5. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, y, -x\right)\\ \mathbf{elif}\;y \leq 6.3 \cdot 10^{-6}:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, y, 0.918938533204673\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6e-13)
   (fma (- x 0.5) y (- x))
   (if (<= y 6.3e-6)
     (- 0.918938533204673 x)
     (fma (- x 0.5) y 0.918938533204673))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6e-13) {
		tmp = fma((x - 0.5), y, -x);
	} else if (y <= 6.3e-6) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = fma((x - 0.5), y, 0.918938533204673);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6e-13)
		tmp = fma(Float64(x - 0.5), y, Float64(-x));
	elseif (y <= 6.3e-6)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = fma(Float64(x - 0.5), y, 0.918938533204673);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6e-13], N[(N[(x - 0.5), $MachinePrecision] * y + (-x)), $MachinePrecision], If[LessEqual[y, 6.3e-6], N[(0.918938533204673 - x), $MachinePrecision], N[(N[(x - 0.5), $MachinePrecision] * y + 0.918938533204673), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.99999999999999968e-13

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{918938533204673}{1000000000000000} + x \cdot \left(y - 1\right)\right) - \frac{1}{2} \cdot y} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, y, 0.918938533204673 - x\right)} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

   if -5.99999999999999968e-13 < y < 6.29999999999999982e-6

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - \left(\mathsf{neg}\left(-1\right)\right) \cdot x} \]

      2. metadata-eval N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{1} \cdot x \]

      3. *-lft-identity N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{x} \]

      4. lower--.f64 99.2

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]

   if 6.29999999999999982e-6 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{918938533204673}{1000000000000000} + x \cdot \left(y - 1\right)\right) - \frac{1}{2} \cdot y} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, y, 0.918938533204673 - x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, y, \frac{918938533204673}{1000000000000000}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(x - 0.5, y, 0.918938533204673\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3:\\ \;\;\;\;\left(x - 0.5\right) \cdot y\\ \mathbf{elif}\;y \leq 6.3 \cdot 10^{-6}:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, y, 0.918938533204673\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.3)
   (* (- x 0.5) y)
   (if (<= y 6.3e-6)
     (- 0.918938533204673 x)
     (fma (- x 0.5) y 0.918938533204673))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.3) {
		tmp = (x - 0.5) * y;
	} else if (y <= 6.3e-6) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = fma((x - 0.5), y, 0.918938533204673);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.3)
		tmp = Float64(Float64(x - 0.5) * y);
	elseif (y <= 6.3e-6)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = fma(Float64(x - 0.5), y, 0.918938533204673);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.3], N[(N[(x - 0.5), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 6.3e-6], N[(0.918938533204673 - x), $MachinePrecision], N[(N[(x - 0.5), $MachinePrecision] * y + 0.918938533204673), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.30000000000000004

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. distribute-rgt-out-- N/A

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

      2. *-commutative N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. remove-double-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. associate-*r* N/A

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

      15. mul-1-neg N/A

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

      16. *-commutative N/A

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

      17. distribute-rgt-neg-in N/A

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

      18. metadata-eval N/A

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

      19. mul-1-neg N/A

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

      20. distribute-neg-out N/A

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

      21. distribute-lft-neg-out N/A

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

      22. distribute-rgt-neg-in N/A

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

      23. remove-double-neg N/A

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

      24. lower-*.f64 N/A

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

   5. Applied rewrites 97.9%

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

   if -1.30000000000000004 < y < 6.29999999999999982e-6

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - \left(\mathsf{neg}\left(-1\right)\right) \cdot x} \]

      2. metadata-eval N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{1} \cdot x \]

      3. *-lft-identity N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{x} \]

      4. lower--.f64 97.6

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   5. Applied rewrites 97.6%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]

   if 6.29999999999999982e-6 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{918938533204673}{1000000000000000} + x \cdot \left(y - 1\right)\right) - \frac{1}{2} \cdot y} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, y, 0.918938533204673 - x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, y, \frac{918938533204673}{1000000000000000}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(x - 0.5, y, 0.918938533204673\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \lor \neg \left(y \leq 1.35\right):\\ \;\;\;\;\left(x - 0.5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.3) (not (<= y 1.35)))
   (* (- x 0.5) y)
   (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.3) || !(y <= 1.35)) {
		tmp = (x - 0.5) * y;
	} else {
		tmp = 0.918938533204673 - 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 <= (-1.3d0)) .or. (.not. (y <= 1.35d0))) then
        tmp = (x - 0.5d0) * y
    else
        tmp = 0.918938533204673d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.3) || !(y <= 1.35)) {
		tmp = (x - 0.5) * y;
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.3) or not (y <= 1.35):
		tmp = (x - 0.5) * y
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.3) || !(y <= 1.35))
		tmp = Float64(Float64(x - 0.5) * y);
	else
		tmp = Float64(0.918938533204673 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.3) || ~((y <= 1.35)))
		tmp = (x - 0.5) * y;
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.3], N[Not[LessEqual[y, 1.35]], $MachinePrecision]], N[(N[(x - 0.5), $MachinePrecision] * y), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.30000000000000004 or 1.3500000000000001 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. distribute-rgt-out-- N/A

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

      2. *-commutative N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. remove-double-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. associate-*r* N/A

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

      15. mul-1-neg N/A

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

      16. *-commutative N/A

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

      17. distribute-rgt-neg-in N/A

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

      18. metadata-eval N/A

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

      19. mul-1-neg N/A

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

      20. distribute-neg-out N/A

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

      21. distribute-lft-neg-out N/A

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

      22. distribute-rgt-neg-in N/A

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

      23. remove-double-neg N/A

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

      24. lower-*.f64 N/A

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

   5. Applied rewrites 96.8%

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

   if -1.30000000000000004 < y < 1.3500000000000001

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - \left(\mathsf{neg}\left(-1\right)\right) \cdot x} \]

      2. metadata-eval N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{1} \cdot x \]

      3. *-lft-identity N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{x} \]

      4. lower--.f64 97.6

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   5. Applied rewrites 97.6%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \lor \neg \left(y \leq 1.35\right):\\ \;\;\;\;\left(x - 0.5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -86 \lor \neg \left(y \leq 1.15\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -86.0) (not (<= y 1.15))) (* y x) (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -86.0) || !(y <= 1.15)) {
		tmp = y * x;
	} else {
		tmp = 0.918938533204673 - 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 <= (-86.0d0)) .or. (.not. (y <= 1.15d0))) then
        tmp = y * x
    else
        tmp = 0.918938533204673d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -86.0) || !(y <= 1.15)) {
		tmp = y * x;
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -86.0) or not (y <= 1.15):
		tmp = y * x
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -86.0) || !(y <= 1.15))
		tmp = Float64(y * x);
	else
		tmp = Float64(0.918938533204673 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -86.0) || ~((y <= 1.15)))
		tmp = y * x;
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -86.0], N[Not[LessEqual[y, 1.15]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -86 or 1.1499999999999999 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. distribute-rgt-out-- N/A

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

      2. *-commutative N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. remove-double-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. associate-*r* N/A

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

      15. mul-1-neg N/A

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

      16. *-commutative N/A

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

      17. distribute-rgt-neg-in N/A

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

      18. metadata-eval N/A

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

      19. mul-1-neg N/A

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

      20. distribute-neg-out N/A

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

      21. distribute-lft-neg-out N/A

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

      22. distribute-rgt-neg-in N/A

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

      23. remove-double-neg N/A

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

      24. lower-*.f64 N/A

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

   5. Applied rewrites 96.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 46.5%

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

   if -86 < y < 1.1499999999999999

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - \left(\mathsf{neg}\left(-1\right)\right) \cdot x} \]

      2. metadata-eval N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{1} \cdot x \]

      3. *-lft-identity N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{x} \]

      4. lower--.f64 97.6

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   5. Applied rewrites 97.6%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -86 \lor \neg \left(y \leq 1.15\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -82 \lor \neg \left(x \leq 0.92\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -82.0) (not (<= x 0.92))) (- x) 0.918938533204673))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -82.0) || !(x <= 0.92)) {
		tmp = -x;
	} else {
		tmp = 0.918938533204673;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-82.0d0)) .or. (.not. (x <= 0.92d0))) then
        tmp = -x
    else
        tmp = 0.918938533204673d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -82.0) || !(x <= 0.92)) {
		tmp = -x;
	} else {
		tmp = 0.918938533204673;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -82.0) or not (x <= 0.92):
		tmp = -x
	else:
		tmp = 0.918938533204673
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -82.0) || !(x <= 0.92))
		tmp = Float64(-x);
	else
		tmp = 0.918938533204673;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -82.0) || ~((x <= 0.92)))
		tmp = -x;
	else
		tmp = 0.918938533204673;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -82.0], N[Not[LessEqual[x, 0.92]], $MachinePrecision]], (-x), 0.918938533204673]

Derivation

1. Split input into 2 regimes

2. if x < -82 or 0.92000000000000004 < x

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - \left(\mathsf{neg}\left(-1\right)\right) \cdot x} \]

      2. metadata-eval N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{1} \cdot x \]

      3. *-lft-identity N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{x} \]

      4. lower--.f64 56.9

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   5. Applied rewrites 56.9%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 55.0%

      \[\leadsto -x \]

   if -82 < x < 0.92000000000000004

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - \left(\mathsf{neg}\left(-1\right)\right) \cdot x} \]

      2. metadata-eval N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{1} \cdot x \]

      3. *-lft-identity N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{x} \]

      4. lower--.f64 46.2

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   5. Applied rewrites 46.2%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{918938533204673}{1000000000000000} \]
   7. Step-by-step derivation

   8. Applied rewrites 44.8%

      \[\leadsto 0.918938533204673 \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -82 \lor \neg \left(x \leq 0.92\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ 0.918938533204673 - x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 0.918938533204673 x))

C

double code(double x, double y) {
	return 0.918938533204673 - x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.918938533204673d0 - x
end function

Java

public static double code(double x, double y) {
	return 0.918938533204673 - x;
}

Python

def code(x, y):
	return 0.918938533204673 - x

Julia

function code(x, y)
	return Float64(0.918938533204673 - x)
end

MATLAB

function tmp = code(x, y)
	tmp = 0.918938533204673 - x;
end

Wolfram

code[x_, y_] := N[(0.918938533204673 - x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
4. Step-by-step derivation

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

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - \left(\mathsf{neg}\left(-1\right)\right) \cdot x} \]

   2. metadata-eval N/A

      \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{1} \cdot x \]

   3. *-lft-identity N/A

      \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{x} \]

   4. lower--.f64 51.9

      \[\leadsto \color{blue}{0.918938533204673 - x} \]

5. Applied rewrites 51.9%

   \[\leadsto \color{blue}{0.918938533204673 - x} \]
6. Add Preprocessing

Alternative 10: 26.5% accurate, 20.0× speedup

Math

\[\begin{array}{l} \\ 0.918938533204673 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.918938533204673)

C

double code(double x, double y) {
	return 0.918938533204673;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.918938533204673d0
end function

Java

public static double code(double x, double y) {
	return 0.918938533204673;
}

Python

def code(x, y):
	return 0.918938533204673

Julia

function code(x, y)
	return 0.918938533204673
end

MATLAB

function tmp = code(x, y)
	tmp = 0.918938533204673;
end

Wolfram

code[x_, y_] := 0.918938533204673

Derivation

1. Initial program 100.0%

   \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
4. Step-by-step derivation

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

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - \left(\mathsf{neg}\left(-1\right)\right) \cdot x} \]

   2. metadata-eval N/A

      \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{1} \cdot x \]

   3. *-lft-identity N/A

      \[\leadsto \frac{918938533204673}{1000000000000000} - \color{blue}{x} \]

   4. lower--.f64 51.9

      \[\leadsto \color{blue}{0.918938533204673 - x} \]

5. Applied rewrites 51.9%

   \[\leadsto \color{blue}{0.918938533204673 - x} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{918938533204673}{1000000000000000} \]
7. Step-by-step derivation

8. Applied rewrites 22.3%

   \[\leadsto 0.918938533204673 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025016 
(FPCore (x y)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (* x (- y 1.0)) (* y 0.5)) 0.918938533204673))