Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 6.1s

Alternatives: 5

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \frac{\left|x - y\right|}{\left|y\right|} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (fabs (- x y)) (fabs y)))

C

double code(double x, double y) {
	return fabs((x - y)) / fabs(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 = abs((x - y)) / abs(y)
end function

Java

public static double code(double x, double y) {
	return Math.abs((x - y)) / Math.abs(y);
}

Python

def code(x, y):
	return math.fabs((x - y)) / math.fabs(y)

Julia

function code(x, y)
	return Float64(abs(Float64(x - y)) / abs(y))
end

MATLAB

function tmp = code(x, y)
	tmp = abs((x - y)) / abs(y);
end

Wolfram

code[x_, y_] := N[(N[Abs[N[(x - y), $MachinePrecision]], $MachinePrecision] / N[Abs[y], $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left|x - y\right|}{\left|y\right|} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (fabs (- x y)) (fabs y)))

C

double code(double x, double y) {
	return fabs((x - y)) / fabs(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 = abs((x - y)) / abs(y)
end function

Java

public static double code(double x, double y) {
	return Math.abs((x - y)) / Math.abs(y);
}

Python

def code(x, y):
	return math.fabs((x - y)) / math.fabs(y)

Julia

function code(x, y)
	return Float64(abs(Float64(x - y)) / abs(y))
end

MATLAB

function tmp = code(x, y)
	tmp = abs((x - y)) / abs(y);
end

Wolfram

code[x_, y_] := N[(N[Abs[N[(x - y), $MachinePrecision]], $MachinePrecision] / N[Abs[y], $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left|\frac{y - x}{y}\right| \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fabs (/ (- y x) y)))

C

double code(double x, double y) {
	return fabs(((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 = abs(((y - x) / y))
end function

Java

public static double code(double x, double y) {
	return Math.abs(((y - x) / y));
}

Python

def code(x, y):
	return math.fabs(((y - x) / y))

Julia

function code(x, y)
	return abs(Float64(Float64(y - x) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = abs(((y - x) / y));
end

Wolfram

code[x_, y_] := N[Abs[N[(N[(y - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{\left|x - y\right|}{\left|y\right|} \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto \left|\frac{y - x}{y}\right| \]
4. Add Preprocessing

Alternative 2: 73.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{y - x}{y}\right|\\ \mathbf{if}\;t\_0 \leq 1000000:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_0 \leq 10^{+177}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fabs (/ (- y x) y))))
   (if (<= t_0 1000000.0) 1.0 (if (<= t_0 1e+177) (/ (- x) y) (/ x y)))))

C

double code(double x, double y) {
	double t_0 = fabs(((y - x) / y));
	double tmp;
	if (t_0 <= 1000000.0) {
		tmp = 1.0;
	} else if (t_0 <= 1e+177) {
		tmp = -x / y;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(((y - x) / y))
    if (t_0 <= 1000000.0d0) then
        tmp = 1.0d0
    else if (t_0 <= 1d+177) then
        tmp = -x / y
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.abs(((y - x) / y));
	double tmp;
	if (t_0 <= 1000000.0) {
		tmp = 1.0;
	} else if (t_0 <= 1e+177) {
		tmp = -x / y;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.fabs(((y - x) / y))
	tmp = 0
	if t_0 <= 1000000.0:
		tmp = 1.0
	elif t_0 <= 1e+177:
		tmp = -x / y
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = abs(Float64(Float64(y - x) / y))
	tmp = 0.0
	if (t_0 <= 1000000.0)
		tmp = 1.0;
	elseif (t_0 <= 1e+177)
		tmp = Float64(Float64(-x) / y);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = abs(((y - x) / y));
	tmp = 0.0;
	if (t_0 <= 1000000.0)
		tmp = 1.0;
	elseif (t_0 <= 1e+177)
		tmp = -x / y;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Abs[N[(N[(y - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 1000000.0], 1.0, If[LessEqual[t$95$0, 1e+177], N[((-x) / y), $MachinePrecision], N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (fabs.f64 (-.f64 x y)) (fabs.f64 y)) < 1e6

   1. Initial program 100.0%

      \[\frac{\left|x - y\right|}{\left|y\right|} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\left|x - y\right|}{\left|y\right|}} \cdot \sqrt{\frac{\left|x - y\right|}{\left|y\right|}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\left|x - y\right|}{\left|y\right|}} \cdot \sqrt{\frac{\left|x - y\right|}{\left|y\right|}}} \]

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   6. Step-by-step derivation

   7. Applied rewrites 93.3%

      \[\leadsto \color{blue}{1} \]

   if 1e6 < (/.f64 (fabs.f64 (-.f64 x y)) (fabs.f64 y)) < 1e177

   1. Initial program 100.0%

      \[\frac{\left|x - y\right|}{\left|y\right|} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \frac{\color{blue}{\left|x - y\right|}}{\left|y\right|} \]

      2. rem-sqrt-square-rev N/A

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

      3. pow1/2 N/A

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

      4. add-sqr-sqrt N/A

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

      5. sqrt-prod N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lift--.f64 N/A

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

      8. fabs-sub N/A

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

      9. add-sqr-sqrt N/A

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

      10. sqrt-prod N/A

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

      11. rem-sqrt-square-rev N/A

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

      12. lift--.f64 N/A

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

      13. fabs-sub N/A

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

      14. sqr-abs N/A

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

      15. unpow-prod-down N/A

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

      16. pow1/2 N/A

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

      17. pow1/2 N/A

          \[\leadsto \frac{\sqrt{y - x} \cdot \color{blue}{\sqrt{y - x}}}{\left|y\right|} \]

      18. add-sqr-sqrt N/A

          \[\leadsto \frac{\color{blue}{y - x}}{\left|y\right|} \]

      19. lower--.f64 44.4

          \[\leadsto \frac{\color{blue}{y - x}}{\left|y\right|} \]

      20. lift-fabs.f64 N/A

          \[\leadsto \frac{y - x}{\color{blue}{\left|y\right|}} \]

      21. rem-sqrt-square-rev N/A

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

      22. sqrt-prod N/A

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

      23. add-sqr-sqrt 63.4

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

   4. Applied rewrites 63.4%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 62.4%

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

   if 1e177 < (/.f64 (fabs.f64 (-.f64 x y)) (fabs.f64 y))

   1. Initial program 100.0%

      \[\frac{\left|x - y\right|}{\left|y\right|} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left|x - y\right|}{\left|y\right|}} \]

      2. lift-fabs.f64 N/A

         \[\leadsto \frac{\color{blue}{\left|x - y\right|}}{\left|y\right|} \]

      3. rem-sqrt-square-rev N/A

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

      4. sqrt-prod N/A

         \[\leadsto \frac{\color{blue}{\sqrt{x - y} \cdot \sqrt{x - y}}}{\left|y\right|} \]

      5. lift-fabs.f64 N/A

         \[\leadsto \frac{\sqrt{x - y} \cdot \sqrt{x - y}}{\color{blue}{\left|y\right|}} \]

      6. rem-sqrt-square-rev N/A

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

      7. sqrt-prod N/A

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

      8. add-sqr-sqrt N/A

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

      9. add-sqr-sqrt N/A

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

      10. lift--.f64 N/A

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

      11. div-sub N/A

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

      12. pow1 N/A

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

      13. pow1 N/A

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

      14. pow-div N/A

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

      15. metadata-eval N/A

          \[\leadsto \frac{x}{y} - {y}^{\color{blue}{0}} \]

      16. metadata-eval N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 61.5

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

   4. Applied rewrites 61.5%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 61.5%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{y - x}{y}\right| \leq 1000000:\\ \;\;\;\;1\\ \mathbf{elif}\;\left|\frac{y - x}{y}\right| \leq 10^{+177}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|\frac{y - x}{y}\right| \leq 10^{+177}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (fabs (/ (- y x) y)) 1e+177) (- 1.0 (/ x y)) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (fabs(((y - x) / y)) <= 1e+177) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (abs(((y - x) / y)) <= 1d+177) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.abs(((y - x) / y)) <= 1e+177) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.fabs(((y - x) / y)) <= 1e+177:
		tmp = 1.0 - (x / y)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (abs(Float64(Float64(y - x) / y)) <= 1e+177)
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (abs(((y - x) / y)) <= 1e+177)
		tmp = 1.0 - (x / y);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Abs[N[(N[(y - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], 1e+177], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (fabs.f64 (-.f64 x y)) (fabs.f64 y)) < 1e177

   1. Initial program 100.0%

      \[\frac{\left|x - y\right|}{\left|y\right|} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\left|x - y\right|}{\left|y\right|}} \cdot \sqrt{\frac{\left|x - y\right|}{\left|y\right|}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\left|x - y\right|}{\left|y\right|}} \cdot \sqrt{\frac{\left|x - y\right|}{\left|y\right|}}} \]

   4. Applied rewrites 85.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. add-sqr-sqrt 85.6

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. div-sub N/A

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

      8. *-inverses N/A

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

      9. lift-/.f64 N/A

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

      10. lower--.f64 85.6

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

   6. Applied rewrites 85.6%

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

   if 1e177 < (/.f64 (fabs.f64 (-.f64 x y)) (fabs.f64 y))

   1. Initial program 100.0%

      \[\frac{\left|x - y\right|}{\left|y\right|} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left|x - y\right|}{\left|y\right|}} \]

      2. lift-fabs.f64 N/A

         \[\leadsto \frac{\color{blue}{\left|x - y\right|}}{\left|y\right|} \]

      3. rem-sqrt-square-rev N/A

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

      4. sqrt-prod N/A

         \[\leadsto \frac{\color{blue}{\sqrt{x - y} \cdot \sqrt{x - y}}}{\left|y\right|} \]

      5. lift-fabs.f64 N/A

         \[\leadsto \frac{\sqrt{x - y} \cdot \sqrt{x - y}}{\color{blue}{\left|y\right|}} \]

      6. rem-sqrt-square-rev N/A

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

      7. sqrt-prod N/A

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

      8. add-sqr-sqrt N/A

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

      9. add-sqr-sqrt N/A

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

      10. lift--.f64 N/A

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

      11. div-sub N/A

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

      12. pow1 N/A

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

      13. pow1 N/A

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

      14. pow-div N/A

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

      15. metadata-eval N/A

          \[\leadsto \frac{x}{y} - {y}^{\color{blue}{0}} \]

      16. metadata-eval N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 61.5

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

   4. Applied rewrites 61.5%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 61.5%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{y - x}{y}\right| \leq 10^{+177}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|\frac{y - x}{y}\right| \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (fabs (/ (- y x) y)) 2.0) 1.0 (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (fabs(((y - x) / y)) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (abs(((y - x) / y)) <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.abs(((y - x) / y)) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.fabs(((y - x) / y)) <= 2.0:
		tmp = 1.0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (abs(Float64(Float64(y - x) / y)) <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (abs(((y - x) / y)) <= 2.0)
		tmp = 1.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Abs[N[(N[(y - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], 2.0], 1.0, N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (fabs.f64 (-.f64 x y)) (fabs.f64 y)) < 2

   1. Initial program 100.0%

      \[\frac{\left|x - y\right|}{\left|y\right|} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\left|x - y\right|}{\left|y\right|}} \cdot \sqrt{\frac{\left|x - y\right|}{\left|y\right|}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\left|x - y\right|}{\left|y\right|}} \cdot \sqrt{\frac{\left|x - y\right|}{\left|y\right|}}} \]

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   6. Step-by-step derivation

   7. Applied rewrites 94.6%

      \[\leadsto \color{blue}{1} \]

   if 2 < (/.f64 (fabs.f64 (-.f64 x y)) (fabs.f64 y))

   1. Initial program 100.0%

      \[\frac{\left|x - y\right|}{\left|y\right|} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left|x - y\right|}{\left|y\right|}} \]

      2. lift-fabs.f64 N/A

         \[\leadsto \frac{\color{blue}{\left|x - y\right|}}{\left|y\right|} \]

      3. rem-sqrt-square-rev N/A

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

      4. sqrt-prod N/A

         \[\leadsto \frac{\color{blue}{\sqrt{x - y} \cdot \sqrt{x - y}}}{\left|y\right|} \]

      5. lift-fabs.f64 N/A

         \[\leadsto \frac{\sqrt{x - y} \cdot \sqrt{x - y}}{\color{blue}{\left|y\right|}} \]

      6. rem-sqrt-square-rev N/A

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

      7. sqrt-prod N/A

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

      8. add-sqr-sqrt N/A

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

      9. add-sqr-sqrt N/A

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

      10. lift--.f64 N/A

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

      11. div-sub N/A

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

      12. pow1 N/A

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

      13. pow1 N/A

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

      14. pow-div N/A

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

      15. metadata-eval N/A

          \[\leadsto \frac{x}{y} - {y}^{\color{blue}{0}} \]

      16. metadata-eval N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 48.8

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

   4. Applied rewrites 48.8%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 47.9%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{y - x}{y}\right| \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.3% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

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 = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

   \[\frac{\left|x - y\right|}{\left|y\right|} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt N/A

      \[\leadsto \color{blue}{\sqrt{\frac{\left|x - y\right|}{\left|y\right|}} \cdot \sqrt{\frac{\left|x - y\right|}{\left|y\right|}}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{\left|x - y\right|}{\left|y\right|}} \cdot \sqrt{\frac{\left|x - y\right|}{\left|y\right|}}} \]

4. Applied rewrites 74.8%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
6. Step-by-step derivation

7. Applied rewrites 48.5%

   \[\leadsto \color{blue}{1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025021 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))