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

Percentage Accurate: 100.0% → 100.0%

Time: 2.0s

Alternatives: 9

Speedup: 1.3×

Specification

Math

\[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]

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

\[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]

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, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. sub-negate-rev N/A

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

   5. sub-flip-reverse N/A

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

   6. lower--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. sub-flip N/A

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

   10. distribute-rgt-in N/A

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

   11. associate--r+ N/A

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

   12. *-commutative N/A

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

   13. metadata-eval N/A

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

   14. mul-1-neg N/A

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

   15. add-flip-rev N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

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

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

   19. *-rgt-identity N/A

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

   20. remove-double-neg N/A

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

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

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

   22. *-commutative N/A

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

   23. lower-fma.f64 N/A

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

3. Applied rewrites 100.0%

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

Alternative 2: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+18}:\\ \;\;\;\;x \cdot y - x\\ \mathbf{elif}\;x \leq 95000000000:\\ \;\;\;\;0.918938533204673 - y \cdot \left(0.5 - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y - 1\right)\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9.2e+18)
   (- (* x y) x)
   (if (<= x 95000000000.0)
     (- 0.918938533204673 (* y (- 0.5 x)))
     (* x (- y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9.2e+18) {
		tmp = (x * y) - x;
	} else if (x <= 95000000000.0) {
		tmp = 0.918938533204673 - (y * (0.5 - x));
	} else {
		tmp = x * (y - 1.0);
	}
	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 <= (-9.2d+18)) then
        tmp = (x * y) - x
    else if (x <= 95000000000.0d0) then
        tmp = 0.918938533204673d0 - (y * (0.5d0 - x))
    else
        tmp = x * (y - 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -9.2e+18) {
		tmp = (x * y) - x;
	} else if (x <= 95000000000.0) {
		tmp = 0.918938533204673 - (y * (0.5 - x));
	} else {
		tmp = x * (y - 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9.2e+18:
		tmp = (x * y) - x
	elif x <= 95000000000.0:
		tmp = 0.918938533204673 - (y * (0.5 - x))
	else:
		tmp = x * (y - 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9.2e+18)
		tmp = Float64(Float64(x * y) - x);
	elseif (x <= 95000000000.0)
		tmp = Float64(0.918938533204673 - Float64(y * Float64(0.5 - x)));
	else
		tmp = Float64(x * Float64(y - 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9.2e+18)
		tmp = (x * y) - x;
	elseif (x <= 95000000000.0)
		tmp = 0.918938533204673 - (y * (0.5 - x));
	else
		tmp = x * (y - 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9.2e+18], N[(N[(x * y), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[x, 95000000000.0], N[(0.918938533204673 - N[(y * N[(0.5 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y - 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.2e18

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 50.8%

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 51.5%

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

   7. Applied rewrites 51.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. distribute-lft-in N/A

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

      5. distribute-rgt-neg-out N/A

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

      6. sub-flip-reverse N/A

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

      7. lower--.f64 N/A

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

      8. *-rgt-identity N/A

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

      9. lower-*.f64 N/A

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

      10. *-rgt-identity N/A

          \[\leadsto x \cdot y - x \cdot 1 \]

      11. *-rgt-identity 51.5%

          \[\leadsto x \cdot y - x \]

   9. Applied rewrites 51.5%

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

   if -9.2e18 < x < 9.5e10

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. sub-flip-reverse N/A

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

      6. lower--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-flip N/A

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

      10. distribute-rgt-in N/A

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

      11. associate--r+ N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. mul-1-neg N/A

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

      15. add-flip-rev N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

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

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

      19. *-rgt-identity N/A

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

      20. remove-double-neg N/A

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

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

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

      22. *-commutative N/A

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

      23. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in y around inf

      \[\leadsto 0.918938533204673 - \color{blue}{y \cdot \left(\frac{1}{2} - x\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower--.f64 74.5%

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

   6. Applied rewrites 74.5%

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

   if 9.5e10 < x

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 50.8%

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 51.5%

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

   7. Applied rewrites 51.5%

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

Alternative 3: 97.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -0.59:\\ \;\;\;\;x \cdot y - x\\ \mathbf{elif}\;x \leq 7400000:\\ \;\;\;\;\mathsf{fma}\left(y, -0.5, 0.918938533204673\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y - 1\right)\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.59)
   (- (* x y) x)
   (if (<= x 7400000.0) (fma y -0.5 0.918938533204673) (* x (- y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.59) {
		tmp = (x * y) - x;
	} else if (x <= 7400000.0) {
		tmp = fma(y, -0.5, 0.918938533204673);
	} else {
		tmp = x * (y - 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.59)
		tmp = Float64(Float64(x * y) - x);
	elseif (x <= 7400000.0)
		tmp = fma(y, -0.5, 0.918938533204673);
	else
		tmp = Float64(x * Float64(y - 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.59], N[(N[(x * y), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[x, 7400000.0], N[(y * -0.5 + 0.918938533204673), $MachinePrecision], N[(x * N[(y - 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.589999999999999969

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 50.8%

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 51.5%

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

   7. Applied rewrites 51.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. distribute-lft-in N/A

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

      5. distribute-rgt-neg-out N/A

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

      6. sub-flip-reverse N/A

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

      7. lower--.f64 N/A

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

      8. *-rgt-identity N/A

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

      9. lower-*.f64 N/A

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

      10. *-rgt-identity N/A

          \[\leadsto x \cdot y - x \cdot 1 \]

      11. *-rgt-identity 51.5%

          \[\leadsto x \cdot y - x \]

   9. Applied rewrites 51.5%

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

   if -0.589999999999999969 < x < 7.4e6

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 49.8%

         \[\leadsto 0.918938533204673 - 0.5 \cdot \color{blue}{y} \]

   4. Applied rewrites 49.8%

      \[\leadsto \color{blue}{0.918938533204673 - 0.5 \cdot y} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval 49.8%

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

   6. Applied rewrites 49.8%

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

   if 7.4e6 < x

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 50.8%

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 51.5%

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

   7. Applied rewrites 51.5%

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

Alternative 4: 97.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := x \cdot \left(y - 1\right)\\ \mathbf{if}\;x \leq -0.59:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7400000:\\ \;\;\;\;\mathsf{fma}\left(y, -0.5, 0.918938533204673\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (- y 1.0))))
   (if (<= x -0.59)
     t_0
     (if (<= x 7400000.0) (fma y -0.5 0.918938533204673) t_0))))

C

double code(double x, double y) {
	double t_0 = x * (y - 1.0);
	double tmp;
	if (x <= -0.59) {
		tmp = t_0;
	} else if (x <= 7400000.0) {
		tmp = fma(y, -0.5, 0.918938533204673);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x * Float64(y - 1.0))
	tmp = 0.0
	if (x <= -0.59)
		tmp = t_0;
	elseif (x <= 7400000.0)
		tmp = fma(y, -0.5, 0.918938533204673);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(y - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.59], t$95$0, If[LessEqual[x, 7400000.0], N[(y * -0.5 + 0.918938533204673), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.589999999999999969 or 7.4e6 < x

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 50.8%

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 51.5%

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

   7. Applied rewrites 51.5%

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

   if -0.589999999999999969 < x < 7.4e6

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 49.8%

         \[\leadsto 0.918938533204673 - 0.5 \cdot \color{blue}{y} \]

   4. Applied rewrites 49.8%

      \[\leadsto \color{blue}{0.918938533204673 - 0.5 \cdot y} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval 49.8%

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

   6. Applied rewrites 49.8%

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

Alternative 5: 73.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+249}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{+34}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -21000:\\ \;\;\;\;\mathsf{fma}\left(y, -0.5, 0.918938533204673\right)\\ \mathbf{elif}\;y \leq 980000000:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+180}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.5e+249)
   (* y -0.5)
   (if (<= y -5.6e+34)
     (* x y)
     (if (<= y -21000.0)
       (fma y -0.5 0.918938533204673)
       (if (<= y 980000000.0)
         (- 0.918938533204673 x)
         (if (<= y 4.1e+180) (* x y) (* y -0.5)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6.5e+249) {
		tmp = y * -0.5;
	} else if (y <= -5.6e+34) {
		tmp = x * y;
	} else if (y <= -21000.0) {
		tmp = fma(y, -0.5, 0.918938533204673);
	} else if (y <= 980000000.0) {
		tmp = 0.918938533204673 - x;
	} else if (y <= 4.1e+180) {
		tmp = x * y;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6.5e+249)
		tmp = Float64(y * -0.5);
	elseif (y <= -5.6e+34)
		tmp = Float64(x * y);
	elseif (y <= -21000.0)
		tmp = fma(y, -0.5, 0.918938533204673);
	elseif (y <= 980000000.0)
		tmp = Float64(0.918938533204673 - x);
	elseif (y <= 4.1e+180)
		tmp = Float64(x * y);
	else
		tmp = Float64(y * -0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6.5e+249], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, -5.6e+34], N[(x * y), $MachinePrecision], If[LessEqual[y, -21000.0], N[(y * -0.5 + 0.918938533204673), $MachinePrecision], If[LessEqual[y, 980000000.0], N[(0.918938533204673 - x), $MachinePrecision], If[LessEqual[y, 4.1e+180], N[(x * y), $MachinePrecision], N[(y * -0.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -6.50000000000000028e249 or 4.1e180 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 50.8%

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 26.1%

      \[\leadsto y \cdot -0.5 \]

   if -6.50000000000000028e249 < y < -5.60000000000000016e34 or 9.8e8 < y < 4.1e180

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 50.8%

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 26.8%

         \[\leadsto x \cdot y \]

   7. Applied rewrites 26.8%

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

   if -5.60000000000000016e34 < y < -21000

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 49.8%

         \[\leadsto 0.918938533204673 - 0.5 \cdot \color{blue}{y} \]

   4. Applied rewrites 49.8%

      \[\leadsto \color{blue}{0.918938533204673 - 0.5 \cdot y} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval 49.8%

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

   6. Applied rewrites 49.8%

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

   if -21000 < y < 9.8e8

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. sub-flip-reverse N/A

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

      6. lower--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-flip N/A

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

      10. distribute-rgt-in N/A

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

      11. associate--r+ N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. mul-1-neg N/A

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

      15. add-flip-rev N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

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

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

      19. *-rgt-identity N/A

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

      20. remove-double-neg N/A

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

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

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

      22. *-commutative N/A

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

      23. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]
   5. Step-by-step derivation

      1. lower--.f64 50.3%

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

   6. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 73.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+249}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{+34}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -21000:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 980000000:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+180}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.5e+249)
   (* y -0.5)
   (if (<= y -5.6e+34)
     (* x y)
     (if (<= y -21000.0)
       (* y -0.5)
       (if (<= y 980000000.0)
         (- 0.918938533204673 x)
         (if (<= y 4.1e+180) (* x y) (* y -0.5)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6.5e+249) {
		tmp = y * -0.5;
	} else if (y <= -5.6e+34) {
		tmp = x * y;
	} else if (y <= -21000.0) {
		tmp = y * -0.5;
	} else if (y <= 980000000.0) {
		tmp = 0.918938533204673 - x;
	} else if (y <= 4.1e+180) {
		tmp = x * y;
	} else {
		tmp = y * -0.5;
	}
	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 <= (-6.5d+249)) then
        tmp = y * (-0.5d0)
    else if (y <= (-5.6d+34)) then
        tmp = x * y
    else if (y <= (-21000.0d0)) then
        tmp = y * (-0.5d0)
    else if (y <= 980000000.0d0) then
        tmp = 0.918938533204673d0 - x
    else if (y <= 4.1d+180) then
        tmp = x * y
    else
        tmp = y * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -6.5e+249) {
		tmp = y * -0.5;
	} else if (y <= -5.6e+34) {
		tmp = x * y;
	} else if (y <= -21000.0) {
		tmp = y * -0.5;
	} else if (y <= 980000000.0) {
		tmp = 0.918938533204673 - x;
	} else if (y <= 4.1e+180) {
		tmp = x * y;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6.5e+249:
		tmp = y * -0.5
	elif y <= -5.6e+34:
		tmp = x * y
	elif y <= -21000.0:
		tmp = y * -0.5
	elif y <= 980000000.0:
		tmp = 0.918938533204673 - x
	elif y <= 4.1e+180:
		tmp = x * y
	else:
		tmp = y * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6.5e+249)
		tmp = Float64(y * -0.5);
	elseif (y <= -5.6e+34)
		tmp = Float64(x * y);
	elseif (y <= -21000.0)
		tmp = Float64(y * -0.5);
	elseif (y <= 980000000.0)
		tmp = Float64(0.918938533204673 - x);
	elseif (y <= 4.1e+180)
		tmp = Float64(x * y);
	else
		tmp = Float64(y * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -6.5e+249)
		tmp = y * -0.5;
	elseif (y <= -5.6e+34)
		tmp = x * y;
	elseif (y <= -21000.0)
		tmp = y * -0.5;
	elseif (y <= 980000000.0)
		tmp = 0.918938533204673 - x;
	elseif (y <= 4.1e+180)
		tmp = x * y;
	else
		tmp = y * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6.5e+249], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, -5.6e+34], N[(x * y), $MachinePrecision], If[LessEqual[y, -21000.0], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 980000000.0], N[(0.918938533204673 - x), $MachinePrecision], If[LessEqual[y, 4.1e+180], N[(x * y), $MachinePrecision], N[(y * -0.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.50000000000000028e249 or -5.60000000000000016e34 < y < -21000 or 4.1e180 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 50.8%

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 26.1%

      \[\leadsto y \cdot -0.5 \]

   if -6.50000000000000028e249 < y < -5.60000000000000016e34 or 9.8e8 < y < 4.1e180

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 50.8%

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 26.8%

         \[\leadsto x \cdot y \]

   7. Applied rewrites 26.8%

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

   if -21000 < y < 9.8e8

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. sub-flip-reverse N/A

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

      6. lower--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-flip N/A

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

      10. distribute-rgt-in N/A

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

      11. associate--r+ N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. mul-1-neg N/A

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

      15. add-flip-rev N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

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

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

      19. *-rgt-identity N/A

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

      20. remove-double-neg N/A

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

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

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

      22. *-commutative N/A

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

      23. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]
   5. Step-by-step derivation

      1. lower--.f64 50.3%

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

   6. Applied rewrites 50.3%

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

Alternative 7: 73.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y \leq -520000000:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 980000000:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -520000000.0)
   (* x y)
   (if (<= y 980000000.0) (- 0.918938533204673 x) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -520000000.0) {
		tmp = x * y;
	} else if (y <= 980000000.0) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-520000000.0d0)) then
        tmp = x * y
    else if (y <= 980000000.0d0) then
        tmp = 0.918938533204673d0 - x
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -520000000.0) {
		tmp = x * y;
	} else if (y <= 980000000.0) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -520000000.0:
		tmp = x * y
	elif y <= 980000000.0:
		tmp = 0.918938533204673 - x
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -520000000.0)
		tmp = Float64(x * y);
	elseif (y <= 980000000.0)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -520000000.0)
		tmp = x * y;
	elseif (y <= 980000000.0)
		tmp = 0.918938533204673 - x;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -520000000.0], N[(x * y), $MachinePrecision], If[LessEqual[y, 980000000.0], N[(0.918938533204673 - x), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.2e8 or 9.8e8 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 50.8%

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 26.8%

         \[\leadsto x \cdot y \]

   7. Applied rewrites 26.8%

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

   if -5.2e8 < y < 9.8e8

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. sub-flip-reverse N/A

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

      6. lower--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-flip N/A

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

      10. distribute-rgt-in N/A

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

      11. associate--r+ N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. mul-1-neg N/A

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

      15. add-flip-rev N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

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

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

      19. *-rgt-identity N/A

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

      20. remove-double-neg N/A

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

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

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

      22. *-commutative N/A

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

      23. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]
   5. Step-by-step derivation

      1. lower--.f64 50.3%

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

   6. Applied rewrites 50.3%

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

Alternative 8: 50.3% accurate, 4.1× speedup

Math

\[0.918938533204673 - x \]

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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. sub-negate-rev N/A

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

   5. sub-flip-reverse N/A

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

   6. lower--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. sub-flip N/A

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

   10. distribute-rgt-in N/A

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

   11. associate--r+ N/A

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

   12. *-commutative N/A

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

   13. metadata-eval N/A

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

   14. mul-1-neg N/A

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

   15. add-flip-rev N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

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

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

   19. *-rgt-identity N/A

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

   20. remove-double-neg N/A

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

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

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

   22. *-commutative N/A

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

   23. lower-fma.f64 N/A

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]
5. Step-by-step derivation

   1. lower--.f64 50.3%

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

6. Applied rewrites 50.3%

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

Alternative 9: 25.5% accurate, 14.9× speedup

Math

\[0.918938533204673 \]

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. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-*.f64 49.8%

      \[\leadsto 0.918938533204673 - 0.5 \cdot \color{blue}{y} \]

4. Applied rewrites 49.8%

   \[\leadsto \color{blue}{0.918938533204673 - 0.5 \cdot y} \]

5. Taylor expanded in y around 0

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

7. Applied rewrites 25.5%

   \[\leadsto 0.918938533204673 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025183 
(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))