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

Percentage Accurate: 100.0% → 100.0%

Time: 1.7s

Alternatives: 6

Speedup: 1.1×

Specification

Math

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

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

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

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, 1.1× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (fma (- x -1.0) y x))

C

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

Julia

function code(x, y)
	return fma(Float64(x - -1.0), y, x)
end

Wolfram

code[x_, y_] := N[(N[(x - -1.0), $MachinePrecision] * y + x), $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. +-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. associate-+l+ N/A

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

   5. lift-*.f64 N/A

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

   6. distribute-rgt1-in N/A

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

   7. lower-fma.f64 N/A

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

   8. add-flip N/A

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

   9. lower--.f64 N/A

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

   10. metadata-eval 100.0%

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

3. Applied rewrites 100.0%

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

Alternative 2: 98.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(\mathsf{min}\left(x, y\right) \cdot \mathsf{max}\left(x, y\right) + \mathsf{min}\left(x, y\right)\right) + \mathsf{max}\left(x, y\right) \leq -1 \cdot 10^{-258}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{max}\left(x, y\right), \mathsf{min}\left(x, y\right), \mathsf{min}\left(x, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{max}\left(x, y\right), \mathsf{min}\left(x, y\right), \mathsf{max}\left(x, y\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (+ (* (fmin x y) (fmax x y)) (fmin x y)) (fmax x y)) -1e-258)
   (fma (fmax x y) (fmin x y) (fmin x y))
   (fma (fmax x y) (fmin x y) (fmax x y))))

C

double code(double x, double y) {
	double tmp;
	if ((((fmin(x, y) * fmax(x, y)) + fmin(x, y)) + fmax(x, y)) <= -1e-258) {
		tmp = fma(fmax(x, y), fmin(x, y), fmin(x, y));
	} else {
		tmp = fma(fmax(x, y), fmin(x, y), fmax(x, y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(Float64(fmin(x, y) * fmax(x, y)) + fmin(x, y)) + fmax(x, y)) <= -1e-258)
		tmp = fma(fmax(x, y), fmin(x, y), fmin(x, y));
	else
		tmp = fma(fmax(x, y), fmin(x, y), fmax(x, y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[(N[Min[x, y], $MachinePrecision] * N[Max[x, y], $MachinePrecision]), $MachinePrecision] + N[Min[x, y], $MachinePrecision]), $MachinePrecision] + N[Max[x, y], $MachinePrecision]), $MachinePrecision], -1e-258], N[(N[Max[x, y], $MachinePrecision] * N[Min[x, y], $MachinePrecision] + N[Min[x, y], $MachinePrecision]), $MachinePrecision], N[(N[Max[x, y], $MachinePrecision] * N[Min[x, y], $MachinePrecision] + N[Max[x, y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x y) x) y) < -9.9999999999999995e-259

   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 37.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 64.1%

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

   7. Applied rewrites 64.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-fma.f64 64.1%

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

   9. Applied rewrites 64.1%

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

   if -9.9999999999999995e-259 < (+.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. lower-*.f64 N/A

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

      2. lower-+.f64 62.4%

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

   4. Applied rewrites 62.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. add-flip-rev N/A

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

      7. sub-flip N/A

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

          \[\leadsto x \cdot y + \left(\mathsf{neg}\left(y\right)\right) \cdot \left(-1 + \color{blue}{0}\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto x \cdot y + \left(-1 \cdot \left(\mathsf{neg}\left(y\right)\right) + \color{blue}{0 \cdot \left(\mathsf{neg}\left(y\right)\right)}\right) \]

      12. mul-1-neg N/A

          \[\leadsto x \cdot y + \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + \color{blue}{0} \cdot \left(\mathsf{neg}\left(y\right)\right)\right) \]

      13. remove-double-neg N/A

          \[\leadsto x \cdot y + \left(y + \color{blue}{0} \cdot \left(\mathsf{neg}\left(y\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto x \cdot y + \left(y + \left(0 \cdot -1\right) \cdot \left(\mathsf{neg}\left(\color{blue}{y}\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto x \cdot y + \left(y + 0 \cdot \color{blue}{\left(-1 \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}\right) \]

      16. mul-1-neg N/A

          \[\leadsto x \cdot y + \left(y + 0 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right) \]

      17. remove-double-neg N/A

          \[\leadsto x \cdot y + \left(y + 0 \cdot y\right) \]

      18. mul0-lft N/A

          \[\leadsto x \cdot y + \left(y + 0\right) \]

      19. remove-double-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot y\right)\right)\right)\right) + \left(\color{blue}{y} + 0\right) \]

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

          \[\leadsto \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right) \cdot y\right)\right) + \left(y + 0\right) \]

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

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right) + \left(\color{blue}{y} + 0\right) \]

      22. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(-1 \cdot y\right) + \left(y + 0\right) \]

      23. remove-double-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(-1 \cdot y\right) + \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + 0\right) \]

      24. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(-1 \cdot y\right) + \left(-1 \cdot \left(\mathsf{neg}\left(y\right)\right) + 0\right) \]

      25. mul0-lft N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(-1 \cdot y\right) + \left(-1 \cdot \left(\mathsf{neg}\left(y\right)\right) + 0 \cdot \color{blue}{y}\right) \]

      26. remove-double-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(-1 \cdot y\right) + \left(-1 \cdot \left(\mathsf{neg}\left(y\right)\right) + 0 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right) \]

      27. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(-1 \cdot y\right) + \left(-1 \cdot \left(\mathsf{neg}\left(y\right)\right) + 0 \cdot \left(-1 \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      28. associate-*r* N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(-1 \cdot y\right) + \left(-1 \cdot \left(\mathsf{neg}\left(y\right)\right) + \left(0 \cdot -1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) \]

      29. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(-1 \cdot y\right) + \left(-1 \cdot \left(\mathsf{neg}\left(y\right)\right) + 0 \cdot \left(\mathsf{neg}\left(\color{blue}{y}\right)\right)\right) \]

   6. Applied rewrites 62.4%

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

Alternative 3: 89.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(x, y\right) \cdot \mathsf{max}\left(x, y\right)\\ t_1 := \left(t\_0 + \mathsf{min}\left(x, y\right)\right) + \mathsf{max}\left(x, y\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-258}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{max}\left(x, y\right), \mathsf{min}\left(x, y\right), \mathsf{min}\left(x, y\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+270}:\\ \;\;\;\;\mathsf{max}\left(x, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fmin x y) (fmax x y)))
        (t_1 (+ (+ t_0 (fmin x y)) (fmax x y))))
   (if (<= t_1 -1e-258)
     (fma (fmax x y) (fmin x y) (fmin x y))
     (if (<= t_1 2e+270) (fmax x y) t_0))))

C

double code(double x, double y) {
	double t_0 = fmin(x, y) * fmax(x, y);
	double t_1 = (t_0 + fmin(x, y)) + fmax(x, y);
	double tmp;
	if (t_1 <= -1e-258) {
		tmp = fma(fmax(x, y), fmin(x, y), fmin(x, y));
	} else if (t_1 <= 2e+270) {
		tmp = fmax(x, y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fmin(x, y) * fmax(x, y))
	t_1 = Float64(Float64(t_0 + fmin(x, y)) + fmax(x, y))
	tmp = 0.0
	if (t_1 <= -1e-258)
		tmp = fma(fmax(x, y), fmin(x, y), fmin(x, y));
	elseif (t_1 <= 2e+270)
		tmp = fmax(x, y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Min[x, y], $MachinePrecision] * N[Max[x, y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + N[Min[x, y], $MachinePrecision]), $MachinePrecision] + N[Max[x, y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-258], N[(N[Max[x, y], $MachinePrecision] * N[Min[x, y], $MachinePrecision] + N[Min[x, y], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+270], N[Max[x, y], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (*.f64 x y) x) y) < -9.9999999999999995e-259

   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 37.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 64.1%

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

   7. Applied rewrites 64.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-fma.f64 64.1%

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

   9. Applied rewrites 64.1%

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

   if -9.9999999999999995e-259 < (+.f64 (+.f64 (*.f64 x y) x) y) < 2.0000000000000001e270

   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 37.3%

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

   if 2.0000000000000001e270 < (+.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. lower-*.f64 N/A

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

      2. lower-+.f64 62.4%

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

   4. Applied rewrites 62.4%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 27.5%

         \[\leadsto x \cdot y \]

   7. Applied rewrites 27.5%

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

Alternative 4: 83.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(x, y\right) \cdot \mathsf{max}\left(x, y\right)\\ t_1 := \left(t\_0 + \mathsf{min}\left(x, y\right)\right) + \mathsf{max}\left(x, y\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+300}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-258}:\\ \;\;\;\;\mathsf{min}\left(x, y\right) \cdot 1\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+270}:\\ \;\;\;\;\mathsf{max}\left(x, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fmin x y) (fmax x y)))
        (t_1 (+ (+ t_0 (fmin x y)) (fmax x y))))
   (if (<= t_1 -2e+300)
     t_0
     (if (<= t_1 -1e-258)
       (* (fmin x y) 1.0)
       (if (<= t_1 2e+270) (fmax x y) t_0)))))

C

double code(double x, double y) {
	double t_0 = fmin(x, y) * fmax(x, y);
	double t_1 = (t_0 + fmin(x, y)) + fmax(x, y);
	double tmp;
	if (t_1 <= -2e+300) {
		tmp = t_0;
	} else if (t_1 <= -1e-258) {
		tmp = fmin(x, y) * 1.0;
	} else if (t_1 <= 2e+270) {
		tmp = fmax(x, y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = fmin(x, y) * fmax(x, y)
    t_1 = (t_0 + fmin(x, y)) + fmax(x, y)
    if (t_1 <= (-2d+300)) then
        tmp = t_0
    else if (t_1 <= (-1d-258)) then
        tmp = fmin(x, y) * 1.0d0
    else if (t_1 <= 2d+270) then
        tmp = fmax(x, y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = fmin(x, y) * fmax(x, y);
	double t_1 = (t_0 + fmin(x, y)) + fmax(x, y);
	double tmp;
	if (t_1 <= -2e+300) {
		tmp = t_0;
	} else if (t_1 <= -1e-258) {
		tmp = fmin(x, y) * 1.0;
	} else if (t_1 <= 2e+270) {
		tmp = fmax(x, y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = fmin(x, y) * fmax(x, y)
	t_1 = (t_0 + fmin(x, y)) + fmax(x, y)
	tmp = 0
	if t_1 <= -2e+300:
		tmp = t_0
	elif t_1 <= -1e-258:
		tmp = fmin(x, y) * 1.0
	elif t_1 <= 2e+270:
		tmp = fmax(x, y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(fmin(x, y) * fmax(x, y))
	t_1 = Float64(Float64(t_0 + fmin(x, y)) + fmax(x, y))
	tmp = 0.0
	if (t_1 <= -2e+300)
		tmp = t_0;
	elseif (t_1 <= -1e-258)
		tmp = Float64(fmin(x, y) * 1.0);
	elseif (t_1 <= 2e+270)
		tmp = fmax(x, y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = min(x, y) * max(x, y);
	t_1 = (t_0 + min(x, y)) + max(x, y);
	tmp = 0.0;
	if (t_1 <= -2e+300)
		tmp = t_0;
	elseif (t_1 <= -1e-258)
		tmp = min(x, y) * 1.0;
	elseif (t_1 <= 2e+270)
		tmp = max(x, y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Min[x, y], $MachinePrecision] * N[Max[x, y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + N[Min[x, y], $MachinePrecision]), $MachinePrecision] + N[Max[x, y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+300], t$95$0, If[LessEqual[t$95$1, -1e-258], N[(N[Min[x, y], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$1, 2e+270], N[Max[x, y], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (*.f64 x y) x) y) < -2.0000000000000001e300 or 2.0000000000000001e270 < (+.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. lower-*.f64 N/A

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

      2. lower-+.f64 62.4%

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

   4. Applied rewrites 62.4%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 27.5%

         \[\leadsto x \cdot y \]

   7. Applied rewrites 27.5%

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

   if -2.0000000000000001e300 < (+.f64 (+.f64 (*.f64 x y) x) y) < -9.9999999999999995e-259

   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 37.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 64.1%

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

   7. Applied rewrites 64.1%

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

   8. Taylor expanded in y around 0

      \[\leadsto x \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 38.9%

       \[\leadsto x \cdot 1 \]

   if -9.9999999999999995e-259 < (+.f64 (+.f64 (*.f64 x y) x) y) < 2.0000000000000001e270

   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 37.3%

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

Alternative 5: 62.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(x, y\right) \cdot \mathsf{max}\left(x, y\right)\\ \mathbf{if}\;\mathsf{min}\left(x, y\right) \leq -2.6:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\mathsf{min}\left(x, y\right) \leq 26000000000:\\ \;\;\;\;\mathsf{max}\left(x, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fmin x y) (fmax x y))))
   (if (<= (fmin x y) -2.6)
     t_0
     (if (<= (fmin x y) 26000000000.0) (fmax x y) t_0))))

C

double code(double x, double y) {
	double t_0 = fmin(x, y) * fmax(x, y);
	double tmp;
	if (fmin(x, y) <= -2.6) {
		tmp = t_0;
	} else if (fmin(x, y) <= 26000000000.0) {
		tmp = fmax(x, y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = fmin(x, y) * fmax(x, y)
    if (fmin(x, y) <= (-2.6d0)) then
        tmp = t_0
    else if (fmin(x, y) <= 26000000000.0d0) then
        tmp = fmax(x, y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = fmin(x, y) * fmax(x, y);
	double tmp;
	if (fmin(x, y) <= -2.6) {
		tmp = t_0;
	} else if (fmin(x, y) <= 26000000000.0) {
		tmp = fmax(x, y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = fmin(x, y) * fmax(x, y)
	tmp = 0
	if fmin(x, y) <= -2.6:
		tmp = t_0
	elif fmin(x, y) <= 26000000000.0:
		tmp = fmax(x, y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(fmin(x, y) * fmax(x, y))
	tmp = 0.0
	if (fmin(x, y) <= -2.6)
		tmp = t_0;
	elseif (fmin(x, y) <= 26000000000.0)
		tmp = fmax(x, y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = min(x, y) * max(x, y);
	tmp = 0.0;
	if (min(x, y) <= -2.6)
		tmp = t_0;
	elseif (min(x, y) <= 26000000000.0)
		tmp = max(x, y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Min[x, y], $MachinePrecision] * N[Max[x, y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Min[x, y], $MachinePrecision], -2.6], t$95$0, If[LessEqual[N[Min[x, y], $MachinePrecision], 26000000000.0], N[Max[x, y], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.6000000000000001 or 2.6e10 < x

   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. lower-*.f64 N/A

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

      2. lower-+.f64 62.4%

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

   4. Applied rewrites 62.4%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 27.5%

         \[\leadsto x \cdot y \]

   7. Applied rewrites 27.5%

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

   if -2.6000000000000001 < x < 2.6e10

   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 37.3%

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

Alternative 6: 37.8% accurate, 2.4× speedup

Math

\[\mathsf{max}\left(x, y\right) \]

FPCore

(FPCore (x y) :precision binary64 (fmax x y))

C

double code(double x, double y) {
	return fmax(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 = fmax(x, y)
end function

Java

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

Python

def code(x, y):
	return fmax(x, y)

Julia

function code(x, y)
	return fmax(x, y)
end

MATLAB

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

Wolfram

code[x_, y_] := N[Max[x, y], $MachinePrecision]

Derivation

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 37.3%

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

Reproduce

herbie shell --seed 2025192 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))