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

Percentage Accurate: 100.0% → 100.0%

Time: 5.6s

Alternatives: 9

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * (y - 1.0d0)) - (y * 0.5d0)) + 0.918938533204673d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * (y - 1.0d0)) - (y * 0.5d0)) + 0.918938533204673d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 100.0%

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

Alternative 2: 74.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+257}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq -1300:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.18 \cdot 10^{-8}:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+53}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.8e+257)
   (* -0.5 y)
   (if (<= y -1300.0)
     (* y x)
     (if (<= y 1.18e-8)
       (- 0.918938533204673 x)
       (if (<= y 1.65e+53) (fma -0.5 y 0.918938533204673) (* y x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.8e+257) {
		tmp = -0.5 * y;
	} else if (y <= -1300.0) {
		tmp = y * x;
	} else if (y <= 1.18e-8) {
		tmp = 0.918938533204673 - x;
	} else if (y <= 1.65e+53) {
		tmp = fma(-0.5, y, 0.918938533204673);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.8e+257)
		tmp = Float64(-0.5 * y);
	elseif (y <= -1300.0)
		tmp = Float64(y * x);
	elseif (y <= 1.18e-8)
		tmp = Float64(0.918938533204673 - x);
	elseif (y <= 1.65e+53)
		tmp = fma(-0.5, y, 0.918938533204673);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.8e+257], N[(-0.5 * y), $MachinePrecision], If[LessEqual[y, -1300.0], N[(y * x), $MachinePrecision], If[LessEqual[y, 1.18e-8], N[(0.918938533204673 - x), $MachinePrecision], If[LessEqual[y, 1.65e+53], N[(-0.5 * y + 0.918938533204673), $MachinePrecision], N[(y * x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -4.8000000000000001e257

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

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

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

      2. *-commutative N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. remove-double-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. associate-*r* N/A

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

      15. mul-1-neg N/A

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

      16. *-commutative N/A

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

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

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

      18. metadata-eval N/A

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

      19. mul-1-neg N/A

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

      20. distribute-neg-out N/A

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

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

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

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

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

      23. remove-double-neg N/A

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

      24. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 90.2%

      \[\leadsto -0.5 \cdot y \]

   if -4.8000000000000001e257 < y < -1300 or 1.6500000000000001e53 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

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

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

      2. *-commutative N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. remove-double-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. associate-*r* N/A

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

      15. mul-1-neg N/A

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

      16. *-commutative N/A

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

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

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

      18. metadata-eval N/A

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

      19. mul-1-neg N/A

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

      20. distribute-neg-out N/A

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

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

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

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

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

      23. remove-double-neg N/A

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

      24. lower-*.f64 N/A

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 32.4%

      \[\leadsto -0.5 \cdot y \]

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 66.7%

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

   if -1300 < y < 1.18e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 97.9

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

   5. Applied rewrites 97.9%

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

   if 1.18e-8 < y < 1.6500000000000001e53

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 70.1

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

   5. Applied rewrites 70.1%

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

Alternative 3: 74.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+257}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq -1300 \lor \neg \left(y \leq 1.1\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.8e+257)
   (* -0.5 y)
   (if (or (<= y -1300.0) (not (<= y 1.1))) (* y x) (- 0.918938533204673 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.8e+257) {
		tmp = -0.5 * y;
	} else if ((y <= -1300.0) || !(y <= 1.1)) {
		tmp = y * x;
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.8d+257)) then
        tmp = (-0.5d0) * y
    else if ((y <= (-1300.0d0)) .or. (.not. (y <= 1.1d0))) then
        tmp = y * x
    else
        tmp = 0.918938533204673d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.8e+257) {
		tmp = -0.5 * y;
	} else if ((y <= -1300.0) || !(y <= 1.1)) {
		tmp = y * x;
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.8e+257:
		tmp = -0.5 * y
	elif (y <= -1300.0) or not (y <= 1.1):
		tmp = y * x
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.8e+257)
		tmp = Float64(-0.5 * y);
	elseif ((y <= -1300.0) || !(y <= 1.1))
		tmp = Float64(y * x);
	else
		tmp = Float64(0.918938533204673 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.8e+257)
		tmp = -0.5 * y;
	elseif ((y <= -1300.0) || ~((y <= 1.1)))
		tmp = y * x;
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.8e+257], N[(-0.5 * y), $MachinePrecision], If[Or[LessEqual[y, -1300.0], N[Not[LessEqual[y, 1.1]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.8000000000000001e257

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

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

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

      2. *-commutative N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. remove-double-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. associate-*r* N/A

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

      15. mul-1-neg N/A

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

      16. *-commutative N/A

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

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

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

      18. metadata-eval N/A

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

      19. mul-1-neg N/A

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

      20. distribute-neg-out N/A

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

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

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

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

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

      23. remove-double-neg N/A

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

      24. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 90.2%

      \[\leadsto -0.5 \cdot y \]

   if -4.8000000000000001e257 < y < -1300 or 1.1000000000000001 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

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

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

      2. *-commutative N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. remove-double-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. associate-*r* N/A

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

      15. mul-1-neg N/A

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

      16. *-commutative N/A

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

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

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

      18. metadata-eval N/A

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

      19. mul-1-neg N/A

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

      20. distribute-neg-out N/A

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

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

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

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

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

      23. remove-double-neg N/A

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

      24. lower-*.f64 N/A

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

   5. Applied rewrites 96.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 34.3%

      \[\leadsto -0.5 \cdot y \]

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 62.4%

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

   if -1300 < y < 1.1000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 97.6

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

   5. Applied rewrites 97.6%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+257}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq -1300 \lor \neg \left(y \leq 1.1\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-15} \lor \neg \left(x \leq 9.5 \cdot 10^{-10}\right):\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, y, -x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.2e-15) (not (<= x 9.5e-10)))
   (fma (- x 0.5) y (- x))
   (fma -0.5 y 0.918938533204673)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.2e-15) || !(x <= 9.5e-10)) {
		tmp = fma((x - 0.5), y, -x);
	} else {
		tmp = fma(-0.5, y, 0.918938533204673);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3.2e-15) || !(x <= 9.5e-10))
		tmp = fma(Float64(x - 0.5), y, Float64(-x));
	else
		tmp = fma(-0.5, y, 0.918938533204673);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3.2e-15], N[Not[LessEqual[x, 9.5e-10]], $MachinePrecision]], N[(N[(x - 0.5), $MachinePrecision] * y + (-x)), $MachinePrecision], N[(-0.5 * y + 0.918938533204673), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.1999999999999999e-15 or 9.50000000000000028e-10 < 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 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.0%

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

   if -3.1999999999999999e-15 < x < 9.50000000000000028e-10

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-15} \lor \neg \left(x \leq 9.5 \cdot 10^{-10}\right):\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, y, -x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.7 \lor \neg \left(x \leq 0.92\right):\\ \;\;\;\;\left(y - 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.7) (not (<= x 0.92)))
   (* (- y 1.0) x)
   (fma -0.5 y 0.918938533204673)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.7) || !(x <= 0.92)) {
		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 <= 0.92))
		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, 0.92]], $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 0.92000000000000004 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 97.3%

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

   if -0.69999999999999996 < x < 0.92000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 99.1

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

   5. Applied rewrites 99.1%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.7 \lor \neg \left(x \leq 0.92\right):\\ \;\;\;\;\left(y - 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1300 \lor \neg \left(y \leq 1.1\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1300.0) (not (<= y 1.1))) (* y x) (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1300.0) || !(y <= 1.1)) {
		tmp = y * x;
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1300.0d0)) .or. (.not. (y <= 1.1d0))) then
        tmp = y * x
    else
        tmp = 0.918938533204673d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1300.0) || !(y <= 1.1)) {
		tmp = y * x;
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1300.0) or not (y <= 1.1):
		tmp = y * x
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1300.0) || !(y <= 1.1))
		tmp = Float64(y * x);
	else
		tmp = Float64(0.918938533204673 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1300.0) || ~((y <= 1.1)))
		tmp = y * x;
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1300.0], N[Not[LessEqual[y, 1.1]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1300 or 1.1000000000000001 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

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

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

      2. *-commutative N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. remove-double-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. associate-*r* N/A

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

      15. mul-1-neg N/A

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

      16. *-commutative N/A

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

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

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

      18. metadata-eval N/A

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

      19. mul-1-neg N/A

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

      20. distribute-neg-out N/A

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

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

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

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

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

      23. remove-double-neg N/A

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

      24. lower-*.f64 N/A

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

   5. Applied rewrites 97.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 39.8%

      \[\leadsto -0.5 \cdot y \]

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 57.6%

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

   if -1300 < y < 1.1000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 97.6

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

   5. Applied rewrites 97.6%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1300 \lor \neg \left(y \leq 1.1\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.92 \lor \neg \left(x \leq 17\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.92) (not (<= x 17.0))) (- x) 0.918938533204673))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.92) || !(x <= 17.0)) {
		tmp = -x;
	} else {
		tmp = 0.918938533204673;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-0.92d0)) .or. (.not. (x <= 17.0d0))) then
        tmp = -x
    else
        tmp = 0.918938533204673d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -0.92) || !(x <= 17.0)) {
		tmp = -x;
	} else {
		tmp = 0.918938533204673;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -0.92) or not (x <= 17.0):
		tmp = -x
	else:
		tmp = 0.918938533204673
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.92) || !(x <= 17.0))
		tmp = Float64(-x);
	else
		tmp = 0.918938533204673;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -0.92) || ~((x <= 17.0)))
		tmp = -x;
	else
		tmp = 0.918938533204673;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.92], N[Not[LessEqual[x, 17.0]], $MachinePrecision]], (-x), 0.918938533204673]

Derivation

1. Split input into 2 regimes

2. if x < -0.92000000000000004 or 17 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 53.0

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

   5. Applied rewrites 53.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 52.0%

      \[\leadsto -x \]

   if -0.92000000000000004 < x < 17

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 59.3

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

   5. Applied rewrites 59.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 59.3%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.92 \lor \neg \left(x \leq 17\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower--.f64 55.9

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

5. Applied rewrites 55.9%

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

Alternative 9: 26.1% accurate, 20.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 0.918938533204673)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, y):
	return 0.918938533204673

Julia

function code(x, y)
	return 0.918938533204673
end

MATLAB

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

Wolfram

code[x_, y_] := 0.918938533204673

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

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

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower--.f64 55.9

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

5. Applied rewrites 55.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 28.6%

   \[\leadsto 0.918938533204673 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025007 
(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))