Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B

Percentage Accurate: 100.0% → 100.0%

Time: 1.8s

Alternatives: 8

Speedup: 1.2×

Specification

Math

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

FPCore

(FPCore (x y) :precision binary64 (+ (+ (* x y) x) y))

C

double code(double x, double y) {
	return ((x * y) + x) + y;
}

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) + x) + y
end function

Java

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

Python

def code(x, y):
	return ((x * y) + x) + y

Julia

function code(x, y)
	return Float64(Float64(Float64(x * y) + x) + y)
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * y) + x) + y;
end

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (+ (+ (* x y) x) y))

C

double code(double x, double y) {
	return ((x * y) + x) + y;
}

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) + x) + y
end function

Java

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

Python

def code(x, y):
	return ((x * y) + x) + y

Julia

function code(x, y)
	return Float64(Float64(Float64(x * y) + x) + y)
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * y) + x) + y;
end

Wolfram

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

Alternative 1: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \mathsf{fma}\left(y - -1, x, y\right) \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (fma (- y -1.0) x y))

C

assert(x < y);
double code(double x, double y) {
	return fma((y - -1.0), x, y);
}

Julia

x, y = sort([x, y])
function code(x, y)
	return fma(Float64(y - -1.0), x, y)
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(N[(y - -1.0), $MachinePrecision] * x + y), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. distribute-lft1-in N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. metadata-eval N/A

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

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

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower--.f64 100.0

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

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y - -1, x, y\right)} \]
4. Add Preprocessing

Alternative 2: 86.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(x \cdot y + x\right) + y\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+307}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;t\_0 \leq 10^{+291}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ (* x y) x) y)))
   (if (<= t_0 -1e+307) (* y x) (if (<= t_0 1e+291) (+ x y) (* y x)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = ((x * y) + x) + y;
	double tmp;
	if (t_0 <= -1e+307) {
		tmp = y * x;
	} else if (t_0 <= 1e+291) {
		tmp = x + y;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x * y) + x) + y
    if (t_0 <= (-1d+307)) then
        tmp = y * x
    else if (t_0 <= 1d+291) then
        tmp = x + y
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = ((x * y) + x) + y;
	double tmp;
	if (t_0 <= -1e+307) {
		tmp = y * x;
	} else if (t_0 <= 1e+291) {
		tmp = x + y;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = ((x * y) + x) + y
	tmp = 0
	if t_0 <= -1e+307:
		tmp = y * x
	elif t_0 <= 1e+291:
		tmp = x + y
	else:
		tmp = y * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(Float64(x * y) + x) + y)
	tmp = 0.0
	if (t_0 <= -1e+307)
		tmp = Float64(y * x);
	elseif (t_0 <= 1e+291)
		tmp = Float64(x + y);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = ((x * y) + x) + y;
	tmp = 0.0;
	if (t_0 <= -1e+307)
		tmp = y * x;
	elseif (t_0 <= 1e+291)
		tmp = x + y;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x * y), $MachinePrecision] + x), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+307], N[(y * x), $MachinePrecision], If[LessEqual[t$95$0, 1e+291], N[(x + y), $MachinePrecision], N[(y * x), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x y) x) y) < -9.99999999999999986e306 or 9.9999999999999996e290 < (+.f64 (+.f64 (*.f64 x y) x) y)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 91.5

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

   4. Applied rewrites 91.5%

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

      1. lift-fma.f64 N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 91.5

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

   6. Applied rewrites 91.5%

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

   7. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto y \cdot x \]

      2. lower-*.f64 90.4

         \[\leadsto y \cdot x \]

   9. Applied rewrites 90.4%

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

   if -9.99999999999999986e306 < (+.f64 (+.f64 (*.f64 x y) x) y) < 9.9999999999999996e290

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 85.6%

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

Alternative 3: 97.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot y + x\right) + y \leq -2 \cdot 10^{-202}:\\ \;\;\;\;\left(y - -1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= (+ (+ (* x y) x) y) -2e-202) (* (- y -1.0) x) (fma y x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if ((((x * y) + x) + y) <= -2e-202) {
		tmp = (y - -1.0) * x;
	} else {
		tmp = fma(y, x, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (Float64(Float64(Float64(x * y) + x) + y) <= -2e-202)
		tmp = Float64(Float64(y - -1.0) * x);
	else
		tmp = fma(y, x, y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[N[(N[(N[(x * y), $MachinePrecision] + x), $MachinePrecision] + y), $MachinePrecision], -2e-202], N[(N[(y - -1.0), $MachinePrecision] * x), $MachinePrecision], N[(y * x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x y) x) y) < -2.0000000000000001e-202

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 98.7

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

   4. Applied rewrites 98.7%

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

      1. lift-fma.f64 N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 98.7

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

   6. Applied rewrites 98.7%

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

   if -2.0000000000000001e-202 < (+.f64 (+.f64 (*.f64 x y) x) y)

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

         \[\leadsto y \cdot x + y \]

      4. lower-fma.f64 95.8

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

   4. Applied rewrites 95.8%

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

Alternative 4: 97.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot y + x\right) + y \leq -2 \cdot 10^{-202}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= (+ (+ (* x y) x) y) -2e-202) (fma y x x) (fma y x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if ((((x * y) + x) + y) <= -2e-202) {
		tmp = fma(y, x, x);
	} else {
		tmp = fma(y, x, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (Float64(Float64(Float64(x * y) + x) + y) <= -2e-202)
		tmp = fma(y, x, x);
	else
		tmp = fma(y, x, y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[N[(N[(N[(x * y), $MachinePrecision] + x), $MachinePrecision] + y), $MachinePrecision], -2e-202], N[(y * x + x), $MachinePrecision], N[(y * x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x y) x) y) < -2.0000000000000001e-202

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 98.7

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

   4. Applied rewrites 98.7%

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

   if -2.0000000000000001e-202 < (+.f64 (+.f64 (*.f64 x y) x) y)

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

         \[\leadsto y \cdot x + y \]

      4. lower-fma.f64 95.8

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

   4. Applied rewrites 95.8%

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

Alternative 5: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{elif}\;x \leq 4.2:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) (fma y x x) (if (<= x 4.2) (+ x y) (* y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = fma(y, x, x);
	} else if (x <= 4.2) {
		tmp = x + y;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = fma(y, x, x);
	elseif (x <= 4.2)
		tmp = Float64(x + y);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.0], N[(y * x + x), $MachinePrecision], If[LessEqual[x, 4.2], N[(x + y), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 99.4

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

   4. Applied rewrites 99.4%

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

   if -1 < x < 4.20000000000000018

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 99.0%

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

   if 4.20000000000000018 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 95.5

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

   4. Applied rewrites 95.5%

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

      1. lift-fma.f64 N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 95.5

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

   6. Applied rewrites 95.5%

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

   7. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto y \cdot x \]

      2. lower-*.f64 95.5

         \[\leadsto y \cdot x \]

   9. Applied rewrites 95.5%

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

Alternative 6: 73.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot y + x\right) + y \leq -2 \cdot 10^{-202}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= (+ (+ (* x y) x) y) -2e-202) x y))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if ((((x * y) + x) + y) <= -2e-202) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 * y) + x) + y) <= (-2d-202)) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if ((((x * y) + x) + y) <= -2e-202) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if (((x * y) + x) + y) <= -2e-202:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (Float64(Float64(Float64(x * y) + x) + y) <= -2e-202)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((((x * y) + x) + y) <= -2e-202)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[N[(N[(N[(x * y), $MachinePrecision] + x), $MachinePrecision] + y), $MachinePrecision], -2e-202], x, y]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x y) x) y) < -2.0000000000000001e-202

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 75.0%

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

   if -2.0000000000000001e-202 < (+.f64 (+.f64 (*.f64 x y) x) y)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 71.8%

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

Alternative 7: 75.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ x + y \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (+ x y))

C

assert(x < y);
double code(double x, double y) {
	return x + y;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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
end function

Java

assert x < y;
public static double code(double x, double y) {
	return x + y;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return x + y

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x + y)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x + y;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x + y), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in y around 0

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

4. Applied rewrites 75.2%

   \[\leadsto \color{blue}{x} + y \]
5. Add Preprocessing

Alternative 8: 39.5% accurate, 12.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ x \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 x)

C

assert(x < y);
double code(double x, double y) {
	return x;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

[x, y] = sort([x, y])
def code(x, y):
	return x

Julia

x, y = sort([x, y])
function code(x, y)
	return x
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := x

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in y around 0

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

4. Applied rewrites 39.5%

   \[\leadsto \color{blue}{x} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025092 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))