Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 1.8s

Alternatives: 3

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

real(8) function code(x, y)
    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

real(8) function code(x, y)
    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

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

FPCore

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

C

double code(double x, double y) {
	return fabs((y - x)) / fabs(y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = abs((y - x)) / abs(y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 72.6% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+159} \lor \neg \left(x \leq -1.7 \cdot 10^{+123}\right) \land \left(x \leq 1.5 \cdot 10^{+154} \lor \neg \left(x \leq 2.35 \cdot 10^{+260}\right) \land x \leq 4.1 \cdot 10^{+295}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -4e+159)
         (and (not (<= x -1.7e+123))
              (or (<= x 1.5e+154)
                  (and (not (<= x 2.35e+260)) (<= x 4.1e+295)))))
   (- 1.0 (/ x y))
   (+ (/ x y) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -4e+159) || (!(x <= -1.7e+123) && ((x <= 1.5e+154) || (!(x <= 2.35e+260) && (x <= 4.1e+295))))) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = (x / y) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-4d+159)) .or. (.not. (x <= (-1.7d+123))) .and. (x <= 1.5d+154) .or. (.not. (x <= 2.35d+260)) .and. (x <= 4.1d+295)) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = (x / y) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -4e+159) || (!(x <= -1.7e+123) && ((x <= 1.5e+154) || (!(x <= 2.35e+260) && (x <= 4.1e+295))))) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = (x / y) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -4e+159) or (not (x <= -1.7e+123) and ((x <= 1.5e+154) or (not (x <= 2.35e+260) and (x <= 4.1e+295)))):
		tmp = 1.0 - (x / y)
	else:
		tmp = (x / y) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -4e+159) || (!(x <= -1.7e+123) && ((x <= 1.5e+154) || (!(x <= 2.35e+260) && (x <= 4.1e+295)))))
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = Float64(Float64(x / y) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -4e+159) || (~((x <= -1.7e+123)) && ((x <= 1.5e+154) || (~((x <= 2.35e+260)) && (x <= 4.1e+295)))))
		tmp = 1.0 - (x / y);
	else
		tmp = (x / y) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -4e+159], And[N[Not[LessEqual[x, -1.7e+123]], $MachinePrecision], Or[LessEqual[x, 1.5e+154], And[N[Not[LessEqual[x, 2.35e+260]], $MachinePrecision], LessEqual[x, 4.1e+295]]]]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.9999999999999997e159 or -1.70000000000000001e123 < x < 1.50000000000000013e154 or 2.35000000000000011e260 < x < 4.09999999999999983e295

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. fabs-div 100.0%

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

      6. rem-square-sqrt 81.7%

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

      7. fabs-sqr 81.7%

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

      8. rem-square-sqrt 82.0%

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

      9. div-sub 82.0%

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

      10. *-inverses 82.0%

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

   6. Simplified 82.0%

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

   if -3.9999999999999997e159 < x < -1.70000000000000001e123 or 1.50000000000000013e154 < x < 2.35000000000000011e260 or 4.09999999999999983e295 < x

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. fabs-div 100.0%

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

      6. rem-square-sqrt 28.5%

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

      7. fabs-sqr 28.5%

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

      8. rem-square-sqrt 28.8%

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

      9. rem-square-sqrt 5.7%

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

      10. fabs-sqr 5.7%

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

      11. rem-square-sqrt 57.4%

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

      12. fabs-sub 57.4%

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

      13. unpow1 57.4%

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

      14. sqr-pow 51.3%

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

      15. fabs-sqr 51.3%

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

      16. sqr-pow 71.7%

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

      17. unpow1 71.7%

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

      18. div-sub 71.7%

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

      19. *-inverses 71.7%

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

      20. sub-neg 71.7%

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

      21. metadata-eval 71.7%

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

      22. +-commutative 71.7%

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

   6. Simplified 71.7%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+159} \lor \neg \left(x \leq -1.7 \cdot 10^{+123}\right) \land \left(x \leq 1.5 \cdot 10^{+154} \lor \neg \left(x \leq 2.35 \cdot 10^{+260}\right) \land x \leq 4.1 \cdot 10^{+295}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \]

Alternative 3: 25.2% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x / y) + (-1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. fabs-sub 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in y around -inf 100.0%

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

   1. fabs-neg 100.0%

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

   2. neg-mul-1 100.0%

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

   3. sub-neg 100.0%

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

   4. fabs-sub 100.0%

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

   5. fabs-div 100.0%

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

   6. rem-square-sqrt 74.5%

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

   7. fabs-sqr 74.5%

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

   8. rem-square-sqrt 74.7%

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

   9. rem-square-sqrt 36.8%

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

   10. fabs-sqr 36.8%

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

   11. rem-square-sqrt 49.7%

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

   12. fabs-sub 49.7%

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

   13. unpow1 49.7%

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

   14. sqr-pow 12.5%

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

   15. fabs-sqr 12.5%

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

   16. sqr-pow 26.3%

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

   17. unpow1 26.3%

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

   18. div-sub 26.3%

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

   19. *-inverses 26.3%

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

   20. sub-neg 26.3%

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

   21. metadata-eval 26.3%

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

   22. +-commutative 26.3%

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

6. Simplified 26.3%

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

7. Final simplification 26.3%

   \[\leadsto \frac{x}{y} + -1 \]

Reproduce

herbie shell --seed 2023336 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))