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

Percentage Accurate: 100.0% → 100.0%

Time: 3.8s

Alternatives: 12

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(0.918938533204673 - N[(N[(0.5 - x), $MachinePrecision] * y + 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 y around 0

   \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + \left(-1 \cdot x + y \cdot \left(x - \frac{1}{2}\right)\right)} \]
4. Step-by-step derivation

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower--.f64 N/A

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

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

      \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, y, \frac{918938533204673}{1000000000000000} - \left(\mathsf{neg}\left(-1\right)\right) \cdot x\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, y, \frac{918938533204673}{1000000000000000} - 1 \cdot x\right) \]

   8. *-lft-identity N/A

      \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, y, \frac{918938533204673}{1000000000000000} - x\right) \]

   9. lower--.f64 100.0

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

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, y, 0.918938533204673 - x\right)} \]
6. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, y, \frac{918938533204673}{1000000000000000} - x\right) \]

   2. lift--.f64 N/A

      \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, y, \frac{918938533204673}{1000000000000000} - x\right) \]

   3. lift-fma.f64 N/A

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

   4. associate-+r- N/A

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

   5. *-commutative N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

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

      \[\leadsto \left(\frac{918938533204673}{1000000000000000} - \left(\mathsf{neg}\left(\left(x - \frac{1}{2}\right)\right)\right) \cdot y\right) - x \]

   9. mul-1-neg N/A

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

   10. associate--l- N/A

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

   11. lower--.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. mul-1-neg N/A

       \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(\mathsf{neg}\left(\left(x - \frac{1}{2}\right)\right), y, x\right) \]

   14. metadata-eval N/A

       \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(\mathsf{neg}\left(\left(x - \frac{1}{2} \cdot 1\right)\right), y, x\right) \]

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

       \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(\mathsf{neg}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot 1\right)\right), y, x\right) \]

   16. metadata-eval N/A

       \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(\mathsf{neg}\left(\left(x + \frac{-1}{2} \cdot 1\right)\right), y, x\right) \]

   17. metadata-eval N/A

       \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(\mathsf{neg}\left(\left(x + \frac{-1}{2}\right)\right), y, x\right) \]

   18. distribute-neg-out N/A

       \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(\left(\mathsf{neg}\left(x\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right), y, x\right) \]

   19. mul-1-neg N/A

       \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(-1 \cdot x + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right), y, x\right) \]

   20. metadata-eval N/A

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

   21. +-commutative N/A

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

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

       \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(\frac{1}{2} - \left(\mathsf{neg}\left(-1\right)\right) \cdot x, y, x\right) \]

   23. metadata-eval N/A

       \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(\frac{1}{2} - 1 \cdot x, y, x\right) \]

   24. *-lft-identity N/A

       \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(\frac{1}{2} - x, y, x\right) \]

   25. lower--.f64 100.0

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

7. Applied rewrites 100.0%

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

Alternative 2: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+15} \lor \neg \left(t\_0 \leq 10000000000000\right):\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, y, -x\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - \mathsf{fma}\left(0.5, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (- (* x (- y 1.0)) (* y 0.5)) 0.918938533204673)))
   (if (or (<= t_0 -2e+15) (not (<= t_0 10000000000000.0)))
     (fma (- x 0.5) y (- x))
     (- 0.918938533204673 (fma 0.5 y x)))))

C

double code(double x, double y) {
	double t_0 = ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
	double tmp;
	if ((t_0 <= -2e+15) || !(t_0 <= 10000000000000.0)) {
		tmp = fma((x - 0.5), y, -x);
	} else {
		tmp = 0.918938533204673 - fma(0.5, y, x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x * Float64(y - 1.0)) - Float64(y * 0.5)) + 0.918938533204673)
	tmp = 0.0
	if ((t_0 <= -2e+15) || !(t_0 <= 10000000000000.0))
		tmp = fma(Float64(x - 0.5), y, Float64(-x));
	else
		tmp = Float64(0.918938533204673 - fma(0.5, y, x));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x * N[(y - 1.0), $MachinePrecision]), $MachinePrecision] - N[(y * 0.5), $MachinePrecision]), $MachinePrecision] + 0.918938533204673), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e+15], N[Not[LessEqual[t$95$0, 10000000000000.0]], $MachinePrecision]], N[(N[(x - 0.5), $MachinePrecision] * y + (-x)), $MachinePrecision], N[(0.918938533204673 - N[(0.5 * y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (*.f64 x (-.f64 y #s(literal 1 binary64))) (*.f64 y #s(literal 1/2 binary64))) #s(literal 918938533204673/1000000000000000 binary64)) < -2e15 or 1e13 < (+.f64 (-.f64 (*.f64 x (-.f64 y #s(literal 1 binary64))) (*.f64 y #s(literal 1/2 binary64))) #s(literal 918938533204673/1000000000000000 binary64))

   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

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + \left(-1 \cdot x + y \cdot \left(x - \frac{1}{2}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

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

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, y, \frac{918938533204673}{1000000000000000} - \left(\mathsf{neg}\left(-1\right)\right) \cdot x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, y, \frac{918938533204673}{1000000000000000} - 1 \cdot x\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, y, \frac{918938533204673}{1000000000000000} - x\right) \]

      9. lower--.f64 100.0

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

   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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -2e15 < (+.f64 (-.f64 (*.f64 x (-.f64 y #s(literal 1 binary64))) (*.f64 y #s(literal 1/2 binary64))) #s(literal 918938533204673/1000000000000000 binary64)) < 1e13

   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

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + \left(-1 \cdot x + y \cdot \left(x - \frac{1}{2}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

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

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, y, \frac{918938533204673}{1000000000000000} - \left(\mathsf{neg}\left(-1\right)\right) \cdot x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, y, \frac{918938533204673}{1000000000000000} - 1 \cdot x\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, y, \frac{918938533204673}{1000000000000000} - x\right) \]

      9. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, y, 0.918938533204673 - x\right)} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, y, \frac{918938533204673}{1000000000000000} - x\right) \]

      2. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, y, \frac{918938533204673}{1000000000000000} - x\right) \]

      3. lift-fma.f64 N/A

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

      4. associate-+r- N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

         \[\leadsto \left(\frac{918938533204673}{1000000000000000} - \left(\mathsf{neg}\left(\left(x - \frac{1}{2}\right)\right)\right) \cdot y\right) - x \]

      9. mul-1-neg N/A

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

      10. associate--l- N/A

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

      11. lower--.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. mul-1-neg N/A

          \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(\mathsf{neg}\left(\left(x - \frac{1}{2}\right)\right), y, x\right) \]

      14. metadata-eval N/A

          \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(\mathsf{neg}\left(\left(x - \frac{1}{2} \cdot 1\right)\right), y, x\right) \]

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

          \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(\mathsf{neg}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot 1\right)\right), y, x\right) \]

      16. metadata-eval N/A

          \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(\mathsf{neg}\left(\left(x + \frac{-1}{2} \cdot 1\right)\right), y, x\right) \]

      17. metadata-eval N/A

          \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(\mathsf{neg}\left(\left(x + \frac{-1}{2}\right)\right), y, x\right) \]

      18. distribute-neg-out N/A

          \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(\left(\mathsf{neg}\left(x\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right), y, x\right) \]

      19. mul-1-neg N/A

          \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(-1 \cdot x + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right), y, x\right) \]

      20. metadata-eval N/A

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

      21. +-commutative N/A

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

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

          \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(\frac{1}{2} - \left(\mathsf{neg}\left(-1\right)\right) \cdot x, y, x\right) \]

      23. metadata-eval N/A

          \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(\frac{1}{2} - 1 \cdot x, y, x\right) \]

      24. *-lft-identity N/A

          \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(\frac{1}{2} - x, y, x\right) \]

      25. lower--.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 100.0%

       \[\leadsto 0.918938533204673 - \mathsf{fma}\left(0.5, y, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \leq -2 \cdot 10^{+15} \lor \neg \left(\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \leq 10000000000000\right):\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, y, -x\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - \mathsf{fma}\left(0.5, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 48.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673\\ \mathbf{if}\;t\_0 \leq -50 \lor \neg \left(t\_0 \leq 2 \cdot 10^{+48}\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (- (* x (- y 1.0)) (* y 0.5)) 0.918938533204673)))
   (if (or (<= t_0 -50.0) (not (<= t_0 2e+48))) (- x) 0.918938533204673)))

C

double code(double x, double y) {
	double t_0 = ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
	double tmp;
	if ((t_0 <= -50.0) || !(t_0 <= 2e+48)) {
		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) :: t_0
    real(8) :: tmp
    t_0 = ((x * (y - 1.0d0)) - (y * 0.5d0)) + 0.918938533204673d0
    if ((t_0 <= (-50.0d0)) .or. (.not. (t_0 <= 2d+48))) then
        tmp = -x
    else
        tmp = 0.918938533204673d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
	double tmp;
	if ((t_0 <= -50.0) || !(t_0 <= 2e+48)) {
		tmp = -x;
	} else {
		tmp = 0.918938533204673;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673
	tmp = 0
	if (t_0 <= -50.0) or not (t_0 <= 2e+48):
		tmp = -x
	else:
		tmp = 0.918938533204673
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x * Float64(y - 1.0)) - Float64(y * 0.5)) + 0.918938533204673)
	tmp = 0.0
	if ((t_0 <= -50.0) || !(t_0 <= 2e+48))
		tmp = Float64(-x);
	else
		tmp = 0.918938533204673;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
	tmp = 0.0;
	if ((t_0 <= -50.0) || ~((t_0 <= 2e+48)))
		tmp = -x;
	else
		tmp = 0.918938533204673;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x * N[(y - 1.0), $MachinePrecision]), $MachinePrecision] - N[(y * 0.5), $MachinePrecision]), $MachinePrecision] + 0.918938533204673), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -50.0], N[Not[LessEqual[t$95$0, 2e+48]], $MachinePrecision]], (-x), 0.918938533204673]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (*.f64 x (-.f64 y #s(literal 1 binary64))) (*.f64 y #s(literal 1/2 binary64))) #s(literal 918938533204673/1000000000000000 binary64)) < -50 or 2.00000000000000009e48 < (+.f64 (-.f64 (*.f64 x (-.f64 y #s(literal 1 binary64))) (*.f64 y #s(literal 1/2 binary64))) #s(literal 918938533204673/1000000000000000 binary64))

   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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 35.4

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

   5. Applied rewrites 35.4%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(x\right) \]

      2. lower-neg.f64 34.8

         \[\leadsto -x \]

   8. Applied rewrites 34.8%

      \[\leadsto -x \]

   if -50 < (+.f64 (-.f64 (*.f64 x (-.f64 y #s(literal 1 binary64))) (*.f64 y #s(literal 1/2 binary64))) #s(literal 918938533204673/1000000000000000 binary64)) < 2.00000000000000009e48

   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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 94.2

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

   5. Applied rewrites 94.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 93.5%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \leq -50 \lor \neg \left(\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \leq 2 \cdot 10^{+48}\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -120000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.4:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+43} \lor \neg \left(y \leq 3.9 \cdot 10^{+148}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -120000.0)
   (* y x)
   (if (<= y 1.4)
     (- 0.918938533204673 x)
     (if (or (<= y 8.5e+43) (not (<= y 3.9e+148))) (* y x) (* -0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -120000.0) {
		tmp = y * x;
	} else if (y <= 1.4) {
		tmp = 0.918938533204673 - x;
	} else if ((y <= 8.5e+43) || !(y <= 3.9e+148)) {
		tmp = y * 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 <= (-120000.0d0)) then
        tmp = y * x
    else if (y <= 1.4d0) then
        tmp = 0.918938533204673d0 - x
    else if ((y <= 8.5d+43) .or. (.not. (y <= 3.9d+148))) then
        tmp = y * 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 <= -120000.0) {
		tmp = y * x;
	} else if (y <= 1.4) {
		tmp = 0.918938533204673 - x;
	} else if ((y <= 8.5e+43) || !(y <= 3.9e+148)) {
		tmp = y * x;
	} else {
		tmp = -0.5 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -120000.0:
		tmp = y * x
	elif y <= 1.4:
		tmp = 0.918938533204673 - x
	elif (y <= 8.5e+43) or not (y <= 3.9e+148):
		tmp = y * x
	else:
		tmp = -0.5 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -120000.0)
		tmp = Float64(y * x);
	elseif (y <= 1.4)
		tmp = Float64(0.918938533204673 - x);
	elseif ((y <= 8.5e+43) || !(y <= 3.9e+148))
		tmp = Float64(y * x);
	else
		tmp = Float64(-0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -120000.0)
		tmp = y * x;
	elseif (y <= 1.4)
		tmp = 0.918938533204673 - x;
	elseif ((y <= 8.5e+43) || ~((y <= 3.9e+148)))
		tmp = y * x;
	else
		tmp = -0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -120000.0], N[(y * x), $MachinePrecision], If[LessEqual[y, 1.4], N[(0.918938533204673 - x), $MachinePrecision], If[Or[LessEqual[y, 8.5e+43], N[Not[LessEqual[y, 3.9e+148]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(-0.5 * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2e5 or 1.3999999999999999 < y < 8.5e43 or 3.90000000000000002e148 < 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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift--.f64 61.9

         \[\leadsto \left(y - 1\right) \cdot x \]

   5. Applied rewrites 61.9%

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

   6. Taylor expanded in y around inf

      \[\leadsto y \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 61.7%

      \[\leadsto y \cdot x \]

   if -1.2e5 < y < 1.3999999999999999

   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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 97.2

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

   5. Applied rewrites 97.2%

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

   if 8.5e43 < y < 3.90000000000000002e148

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 100.0

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

   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 73.5%

      \[\leadsto -0.5 \cdot y \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -120000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.4:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+43} \lor \neg \left(y \leq 3.9 \cdot 10^{+148}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -120000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.037:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -120000.0)
   (* y x)
   (if (<= y 0.037)
     (- 0.918938533204673 x)
     (if (<= y 3.9e+148) (fma -0.5 y 0.918938533204673) (* y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -120000.0) {
		tmp = y * x;
	} else if (y <= 0.037) {
		tmp = 0.918938533204673 - x;
	} else if (y <= 3.9e+148) {
		tmp = fma(-0.5, y, 0.918938533204673);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -120000.0)
		tmp = Float64(y * x);
	elseif (y <= 0.037)
		tmp = Float64(0.918938533204673 - x);
	elseif (y <= 3.9e+148)
		tmp = fma(-0.5, y, 0.918938533204673);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -120000.0], N[(y * x), $MachinePrecision], If[LessEqual[y, 0.037], N[(0.918938533204673 - x), $MachinePrecision], If[LessEqual[y, 3.9e+148], N[(-0.5 * y + 0.918938533204673), $MachinePrecision], N[(y * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2e5 or 3.90000000000000002e148 < 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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift--.f64 61.0

         \[\leadsto \left(y - 1\right) \cdot x \]

   5. Applied rewrites 61.0%

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

   6. Taylor expanded in y around inf

      \[\leadsto y \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 60.7%

      \[\leadsto y \cdot x \]

   if -1.2e5 < y < 0.0369999999999999982

   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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 97.2

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

   5. Applied rewrites 97.2%

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

   if 0.0369999999999999982 < y < 3.90000000000000002e148

   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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 65.7

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

   5. Applied rewrites 65.7%

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

Alternative 6: 98.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -25000000000 \lor \neg \left(y \leq 4100000000\right):\\ \;\;\;\;\left(x - 0.5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - \mathsf{fma}\left(0.5, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -25000000000.0) (not (<= y 4100000000.0)))
   (* (- x 0.5) y)
   (- 0.918938533204673 (fma 0.5 y x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -25000000000.0) || !(y <= 4100000000.0)) {
		tmp = (x - 0.5) * y;
	} else {
		tmp = 0.918938533204673 - fma(0.5, y, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -25000000000.0) || !(y <= 4100000000.0))
		tmp = Float64(Float64(x - 0.5) * y);
	else
		tmp = Float64(0.918938533204673 - fma(0.5, y, x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -25000000000.0], N[Not[LessEqual[y, 4100000000.0]], $MachinePrecision]], N[(N[(x - 0.5), $MachinePrecision] * y), $MachinePrecision], N[(0.918938533204673 - N[(0.5 * y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.5e10 or 4.1e9 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if -2.5e10 < y < 4.1e9

   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

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + \left(-1 \cdot x + y \cdot \left(x - \frac{1}{2}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

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

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, y, \frac{918938533204673}{1000000000000000} - \left(\mathsf{neg}\left(-1\right)\right) \cdot x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, y, \frac{918938533204673}{1000000000000000} - 1 \cdot x\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, y, \frac{918938533204673}{1000000000000000} - x\right) \]

      9. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, y, 0.918938533204673 - x\right)} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, y, \frac{918938533204673}{1000000000000000} - x\right) \]

      2. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, y, \frac{918938533204673}{1000000000000000} - x\right) \]

      3. lift-fma.f64 N/A

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

      4. associate-+r- N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

         \[\leadsto \left(\frac{918938533204673}{1000000000000000} - \left(\mathsf{neg}\left(\left(x - \frac{1}{2}\right)\right)\right) \cdot y\right) - x \]

      9. mul-1-neg N/A

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

      10. associate--l- N/A

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

      11. lower--.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. mul-1-neg N/A

          \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(\mathsf{neg}\left(\left(x - \frac{1}{2}\right)\right), y, x\right) \]

      14. metadata-eval N/A

          \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(\mathsf{neg}\left(\left(x - \frac{1}{2} \cdot 1\right)\right), y, x\right) \]

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

          \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(\mathsf{neg}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot 1\right)\right), y, x\right) \]

      16. metadata-eval N/A

          \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(\mathsf{neg}\left(\left(x + \frac{-1}{2} \cdot 1\right)\right), y, x\right) \]

      17. metadata-eval N/A

          \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(\mathsf{neg}\left(\left(x + \frac{-1}{2}\right)\right), y, x\right) \]

      18. distribute-neg-out N/A

          \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(\left(\mathsf{neg}\left(x\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right), y, x\right) \]

      19. mul-1-neg N/A

          \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(-1 \cdot x + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right), y, x\right) \]

      20. metadata-eval N/A

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

      21. +-commutative N/A

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

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

          \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(\frac{1}{2} - \left(\mathsf{neg}\left(-1\right)\right) \cdot x, y, x\right) \]

      23. metadata-eval N/A

          \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(\frac{1}{2} - 1 \cdot x, y, x\right) \]

      24. *-lft-identity N/A

          \[\leadsto \frac{918938533204673}{1000000000000000} - \mathsf{fma}\left(\frac{1}{2} - x, y, x\right) \]

      25. lower--.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.2%

       \[\leadsto 0.918938533204673 - \mathsf{fma}\left(0.5, y, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -25000000000 \lor \neg \left(y \leq 4100000000\right):\\ \;\;\;\;\left(x - 0.5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - \mathsf{fma}\left(0.5, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.7 \lor \neg \left(x \leq 0.72\right):\\ \;\;\;\;\left(y - 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.7) (not (<= x 0.72)))
   (* (- y 1.0) x)
   (fma -0.5 y 0.918938533204673)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.7) || !(x <= 0.72)) {
		tmp = (y - 1.0) * x;
	} else {
		tmp = fma(-0.5, y, 0.918938533204673);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.7) || !(x <= 0.72))
		tmp = Float64(Float64(y - 1.0) * x);
	else
		tmp = fma(-0.5, y, 0.918938533204673);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.7], N[Not[LessEqual[x, 0.72]], $MachinePrecision]], N[(N[(y - 1.0), $MachinePrecision] * x), $MachinePrecision], N[(-0.5 * y + 0.918938533204673), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.69999999999999996 or 0.71999999999999997 < 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 x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift--.f64 98.0

         \[\leadsto \left(y - 1\right) \cdot x \]

   5. Applied rewrites 98.0%

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

   if -0.69999999999999996 < x < 0.71999999999999997

   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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 98.7

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

   5. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.7 \lor \neg \left(x \leq 0.72\right):\\ \;\;\;\;\left(y - 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \lor \neg \left(y \leq 1.4\right):\\ \;\;\;\;\left(x - 0.5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.4) (not (<= y 1.4)))
   (* (- x 0.5) y)
   (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.4) || !(y <= 1.4)) {
		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 <= (-1.4d0)) .or. (.not. (y <= 1.4d0))) 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 <= -1.4) || !(y <= 1.4)) {
		tmp = (x - 0.5) * y;
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.4) or not (y <= 1.4):
		tmp = (x - 0.5) * y
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.4) || !(y <= 1.4))
		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 <= -1.4) || ~((y <= 1.4)))
		tmp = (x - 0.5) * y;
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.4], N[Not[LessEqual[y, 1.4]], $MachinePrecision]], N[(N[(x - 0.5), $MachinePrecision] * y), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3999999999999999 or 1.3999999999999999 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 97.5

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

   5. Applied rewrites 97.5%

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

   if -1.3999999999999999 < y < 1.3999999999999999

   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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 98.5

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

   5. Applied rewrites 98.5%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \lor \neg \left(y \leq 1.4\right):\\ \;\;\;\;\left(x - 0.5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.7:\\ \;\;\;\;\mathsf{fma}\left(y, x, -x\right)\\ \mathbf{elif}\;x \leq 0.72:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.7)
   (fma y x (- x))
   (if (<= x 0.72) (fma -0.5 y 0.918938533204673) (* (- y 1.0) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.7) {
		tmp = fma(y, x, -x);
	} else if (x <= 0.72) {
		tmp = fma(-0.5, y, 0.918938533204673);
	} else {
		tmp = (y - 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.7)
		tmp = fma(y, x, Float64(-x));
	elseif (x <= 0.72)
		tmp = fma(-0.5, y, 0.918938533204673);
	else
		tmp = Float64(Float64(y - 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.7], N[(y * x + (-x)), $MachinePrecision], If[LessEqual[x, 0.72], N[(-0.5 * y + 0.918938533204673), $MachinePrecision], N[(N[(y - 1.0), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.69999999999999996

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift--.f64 100.0

         \[\leadsto \left(y - 1\right) \cdot x \]

   5. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

         \[\leadsto \left(y - 1\right) \cdot x \]

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

         \[\leadsto x \cdot \left(y + -1 \cdot 1\right) \]

      7. metadata-eval N/A

         \[\leadsto x \cdot \left(y + -1\right) \]

      8. distribute-rgt-out N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -0.69999999999999996 < x < 0.71999999999999997

   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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if 0.71999999999999997 < 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 x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift--.f64 96.1

         \[\leadsto \left(y - 1\right) \cdot x \]

   5. Applied rewrites 96.1%

      \[\leadsto \color{blue}{\left(y - 1\right) \cdot x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 74.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -520 \lor \neg \left(y \leq 1.85\right):\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -520.0) (not (<= y 1.85))) (* -0.5 y) (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -520.0) || !(y <= 1.85)) {
		tmp = -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 <= (-520.0d0)) .or. (.not. (y <= 1.85d0))) then
        tmp = (-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 <= -520.0) || !(y <= 1.85)) {
		tmp = -0.5 * y;
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -520.0) or not (y <= 1.85):
		tmp = -0.5 * y
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -520.0) || !(y <= 1.85))
		tmp = Float64(-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 <= -520.0) || ~((y <= 1.85)))
		tmp = -0.5 * y;
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -520.0], N[Not[LessEqual[y, 1.85]], $MachinePrecision]], N[(-0.5 * y), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -520 or 1.8500000000000001 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 97.5

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 48.0%

      \[\leadsto -0.5 \cdot y \]

   if -520 < y < 1.8500000000000001

   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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 98.5

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

   5. Applied rewrites 98.5%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -520 \lor \neg \left(y \leq 1.85\right):\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.7% 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

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower--.f64 53.8

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

5. Applied rewrites 53.8%

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

Alternative 12: 26.0% 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

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower--.f64 53.8

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

5. Applied rewrites 53.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 30.9%

   \[\leadsto 0.918938533204673 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025037 
(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))