Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 1.3s

Alternatives: 4

Speedup: 1.0×

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, 1.0× 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]

Derivation

1. Initial program 100.0%

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

Alternative 2: 75.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} - 1\\ t_1 := \frac{\left|x - y\right|}{\left|y\right|}\\ t_2 := 1 - \frac{x}{y}\\ \mathbf{if}\;t\_1 \leq 2:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+128}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+232}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+300}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (/ x y) 1.0))
        (t_1 (/ (fabs (- x y)) (fabs y)))
        (t_2 (- 1.0 (/ x y))))
   (if (<= t_1 2.0)
     t_2
     (if (<= t_1 5e+128)
       t_0
       (if (<= t_1 5e+232) t_2 (if (<= t_1 5e+300) t_0 t_2))))))

C

double code(double x, double y) {
	double t_0 = (x / y) - 1.0;
	double t_1 = fabs((x - y)) / fabs(y);
	double t_2 = 1.0 - (x / y);
	double tmp;
	if (t_1 <= 2.0) {
		tmp = t_2;
	} else if (t_1 <= 5e+128) {
		tmp = t_0;
	} else if (t_1 <= 5e+232) {
		tmp = t_2;
	} else if (t_1 <= 5e+300) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = (x / y) - 1.0d0
    t_1 = abs((x - y)) / abs(y)
    t_2 = 1.0d0 - (x / y)
    if (t_1 <= 2.0d0) then
        tmp = t_2
    else if (t_1 <= 5d+128) then
        tmp = t_0
    else if (t_1 <= 5d+232) then
        tmp = t_2
    else if (t_1 <= 5d+300) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x / y) - 1.0;
	double t_1 = Math.abs((x - y)) / Math.abs(y);
	double t_2 = 1.0 - (x / y);
	double tmp;
	if (t_1 <= 2.0) {
		tmp = t_2;
	} else if (t_1 <= 5e+128) {
		tmp = t_0;
	} else if (t_1 <= 5e+232) {
		tmp = t_2;
	} else if (t_1 <= 5e+300) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / y) - 1.0
	t_1 = math.fabs((x - y)) / math.fabs(y)
	t_2 = 1.0 - (x / y)
	tmp = 0
	if t_1 <= 2.0:
		tmp = t_2
	elif t_1 <= 5e+128:
		tmp = t_0
	elif t_1 <= 5e+232:
		tmp = t_2
	elif t_1 <= 5e+300:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / y) - 1.0)
	t_1 = Float64(abs(Float64(x - y)) / abs(y))
	t_2 = Float64(1.0 - Float64(x / y))
	tmp = 0.0
	if (t_1 <= 2.0)
		tmp = t_2;
	elseif (t_1 <= 5e+128)
		tmp = t_0;
	elseif (t_1 <= 5e+232)
		tmp = t_2;
	elseif (t_1 <= 5e+300)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / y) - 1.0;
	t_1 = abs((x - y)) / abs(y);
	t_2 = 1.0 - (x / y);
	tmp = 0.0;
	if (t_1 <= 2.0)
		tmp = t_2;
	elseif (t_1 <= 5e+128)
		tmp = t_0;
	elseif (t_1 <= 5e+232)
		tmp = t_2;
	elseif (t_1 <= 5e+300)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[N[(x - y), $MachinePrecision]], $MachinePrecision] / N[Abs[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2.0], t$95$2, If[LessEqual[t$95$1, 5e+128], t$95$0, If[LessEqual[t$95$1, 5e+232], t$95$2, If[LessEqual[t$95$1, 5e+300], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (fabs.f64 (-.f64 x y)) (fabs.f64 y)) < 2 or 5e128 < (/.f64 (fabs.f64 (-.f64 x y)) (fabs.f64 y)) < 4.99999999999999987e232 or 5.00000000000000026e300 < (/.f64 (fabs.f64 (-.f64 x y)) (fabs.f64 y))

   1. Initial program 100.0%

      \[\frac{\left|x - y\right|}{\left|y\right|} \]
   2. 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. lift-fabs.f64 N/A

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

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

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

      6. sqrt-prod N/A

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

      7. rem-square-sqrt N/A

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

      8. lift--.f64 N/A

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

      9. flip-- N/A

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

      10. sqrt-prod N/A

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

      11. rem-square-sqrt N/A

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

      12. associate-/l/ N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-out N/A

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

      16. sum-to-mult N/A

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

      17. sqr-neg-rev N/A

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

      18. pow2 N/A

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

      19. metadata-eval N/A

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

      20. pow-plus-rev N/A

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

      21. metadata-eval N/A

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

      22. pow-prod-up N/A

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

      23. unpow-prod-down N/A

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

      24. sqr-neg-rev N/A

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

      25. pow1/2 N/A

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

      26. sqrt-prod N/A

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

      27. rem-square-sqrt N/A

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

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

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

   3. Applied rewrites 75.3%

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

   if 2 < (/.f64 (fabs.f64 (-.f64 x y)) (fabs.f64 y)) < 5e128 or 4.99999999999999987e232 < (/.f64 (fabs.f64 (-.f64 x y)) (fabs.f64 y)) < 5.00000000000000026e300

   1. Initial program 100.0%

      \[\frac{\left|x - y\right|}{\left|y\right|} \]
   2. 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. lift-fabs.f64 N/A

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

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

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

      5. sqrt-prod N/A

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

      6. rem-square-sqrt N/A

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

      7. lift--.f64 N/A

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

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

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

      9. sqrt-prod N/A

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

      10. rem-square-sqrt N/A

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

      11. div-sub N/A

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

      12. *-inverses N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 25.8

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

   3. Applied rewrites 25.8%

      \[\leadsto \color{blue}{\frac{x}{y} - 1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 75.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{x}{y} \end{array} \]

FPCore

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

C

double code(double x, double y) {
	return 1.0 - (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 = 1.0d0 - (x / y)
end function

Java

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

Python

def code(x, y):
	return 1.0 - (x / y)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{\left|x - y\right|}{\left|y\right|} \]
2. 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. lift-fabs.f64 N/A

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

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

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

   6. sqrt-prod N/A

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

   7. rem-square-sqrt N/A

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

   8. lift--.f64 N/A

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

   9. flip-- N/A

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

   10. sqrt-prod N/A

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

   11. rem-square-sqrt N/A

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

   12. associate-/l/ N/A

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

   13. *-commutative N/A

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

   14. +-commutative N/A

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

   15. distribute-rgt-out N/A

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

   16. sum-to-mult N/A

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

   17. sqr-neg-rev N/A

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

   18. pow2 N/A

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

   19. metadata-eval N/A

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

   20. pow-plus-rev N/A

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

   21. metadata-eval N/A

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

   22. pow-prod-up N/A

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

   23. unpow-prod-down N/A

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

   24. sqr-neg-rev N/A

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

   25. pow1/2 N/A

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

   26. sqrt-prod N/A

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

   27. rem-square-sqrt N/A

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

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

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

3. Applied rewrites 75.3%

   \[\leadsto \color{blue}{1 - \frac{x}{y}} \]
4. Add Preprocessing

Alternative 4: 51.8% accurate, 9.4× 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. 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. lift-fabs.f64 N/A

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

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

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

   6. sqrt-prod N/A

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

   7. rem-square-sqrt N/A

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

   8. lift--.f64 N/A

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

   9. flip-- N/A

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

   10. sqrt-prod N/A

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

   11. rem-square-sqrt N/A

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

   12. associate-/l/ N/A

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

   13. *-commutative N/A

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

   14. +-commutative N/A

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

   15. distribute-rgt-out N/A

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

   16. sum-to-mult N/A

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

   17. sqr-neg-rev N/A

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

   18. pow2 N/A

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

   19. metadata-eval N/A

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

   20. pow-plus-rev N/A

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

   21. metadata-eval N/A

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

   22. pow-prod-up N/A

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

   23. unpow-prod-down N/A

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

   24. sqr-neg-rev N/A

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

   25. pow1/2 N/A

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

   26. sqrt-prod N/A

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

   27. rem-square-sqrt N/A

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

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

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

3. Applied rewrites 75.3%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 51.8%

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

Reproduce

herbie shell --seed 2025148 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))