Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 3.8s

Alternatives: 5

Speedup: 2.0×

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, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around -inf 100.0%

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

   1. fabs-neg 100.0%

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

   2. mul-1-neg 100.0%

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

   3. sub-neg 100.0%

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

   4. fabs-div 100.0%

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

   5. div-sub 100.0%

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

   6. *-inverses 100.0%

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

4. Simplified 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \left|1 - \frac{x}{y}\right| \]

Alternative 2: 57.9% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+43}:\\ \;\;\;\;x \cdot \frac{\frac{x}{y}}{y - x}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+107}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.2e+43)
   (* x (/ (/ x y) (- y x)))
   (if (<= x 1.05e+107) (/ y (+ x y)) (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.2e+43) {
		tmp = x * ((x / y) / (y - x));
	} else if (x <= 1.05e+107) {
		tmp = y / (x + y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.2d+43)) then
        tmp = x * ((x / y) / (y - x))
    else if (x <= 1.05d+107) then
        tmp = y / (x + y)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.2e+43) {
		tmp = x * ((x / y) / (y - x));
	} else if (x <= 1.05e+107) {
		tmp = y / (x + y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.2e+43:
		tmp = x * ((x / y) / (y - x))
	elif x <= 1.05e+107:
		tmp = y / (x + y)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.2e+43)
		tmp = Float64(x * Float64(Float64(x / y) / Float64(y - x)));
	elseif (x <= 1.05e+107)
		tmp = Float64(y / Float64(x + y));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.2e+43)
		tmp = x * ((x / y) / (y - x));
	elseif (x <= 1.05e+107)
		tmp = y / (x + y);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.2e+43], N[(x * N[(N[(x / y), $MachinePrecision] / N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.05e+107], N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.20000000000000001e43

   1. Initial program 100.0%

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

      1. div-inv 99.7%

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

      2. add-sqr-sqrt 1.9%

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

      3. fabs-sqr 1.9%

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

      4. add-sqr-sqrt 2.4%

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

      5. *-commutative 2.4%

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

      6. add-sqr-sqrt 0.3%

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

      7. fabs-sqr 0.3%

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

      8. add-sqr-sqrt 36.1%

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

   3. Applied egg-rr 36.1%

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

      1. flip-- 20.0%

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

      2. associate-*r/ 18.2%

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

      3. +-commutative 18.2%

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

   5. Applied egg-rr 18.2%

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

   6. Taylor expanded in y around 0 18.8%

      \[\leadsto \frac{\color{blue}{\frac{{x}^{2}}{y}}}{y + x} \]
   7. Step-by-step derivation

      1. unpow2 18.8%

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

      2. associate-/l* 19.0%

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

      3. associate-/r/ 19.0%

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

   8. Simplified 19.0%

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

      1. associate-*l/ 18.8%

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

      2. *-un-lft-identity 18.8%

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

      3. associate-*l/ 18.8%

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

      4. frac-2neg 18.8%

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

      5. div-inv 18.8%

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

      6. associate-*l/ 18.7%

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

      7. *-un-lft-identity 18.7%

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

      8. distribute-neg-frac 18.7%

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

      9. add-sqr-sqrt 0.6%

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

      10. sqrt-unprod 0.5%

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

      11. sqr-neg 0.5%

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

      12. sqrt-unprod 0.1%

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

      13. add-sqr-sqrt 25.1%

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

      14. frac-2neg 25.1%

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

      15. associate-*r/ 25.2%

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

      16. distribute-neg-in 25.2%

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

      17. add-sqr-sqrt 0.2%

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

      18. sqrt-unprod 25.4%

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

      19. sqr-neg 25.4%

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

      20. sqrt-unprod 25.4%

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

      21. add-sqr-sqrt 25.7%

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

      22. sub-neg 25.7%

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

   10. Applied egg-rr 25.7%

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

       1. associate-*l* 51.7%

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

       2. associate-*r/ 51.7%

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

       3. *-rgt-identity 51.7%

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

   12. Simplified 51.7%

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

   if -2.20000000000000001e43 < x < 1.05e107

   1. Initial program 100.0%

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

      1. div-inv 99.7%

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

      2. add-sqr-sqrt 53.1%

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

      3. fabs-sqr 53.1%

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

      4. add-sqr-sqrt 54.0%

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

      5. *-commutative 54.0%

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

      6. add-sqr-sqrt 7.6%

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

      7. fabs-sqr 7.6%

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

      8. add-sqr-sqrt 13.5%

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

   3. Applied egg-rr 13.5%

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

      1. flip-- 12.1%

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

      2. associate-*r/ 11.5%

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

      3. +-commutative 11.5%

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

   5. Applied egg-rr 11.5%

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

   6. Taylor expanded in x around 0 1.7%

      \[\leadsto \frac{\frac{1}{y} \cdot \color{blue}{\left(-1 \cdot {y}^{2}\right)}}{y + x} \]
   7. Step-by-step derivation

      1. unpow2 1.7%

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

      2. mul-1-neg 1.7%

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

      3. distribute-rgt-neg-out 1.7%

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

   8. Simplified 1.7%

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

      1. associate-*r* 1.9%

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

      2. lft-mult-inverse 1.9%

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

      3. *-un-lft-identity 1.9%

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

      4. neg-sub0 1.9%

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

      5. metadata-eval 1.9%

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

      6. sub-neg 1.9%

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

      7. metadata-eval 1.9%

         \[\leadsto \frac{\color{blue}{0} + \left(-y\right)}{y + x} \]

      8. add-sqr-sqrt 1.0%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{-y} \cdot \sqrt{-y}}}{y + x} \]

      9. sqrt-unprod 22.0%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}}{y + x} \]

      10. sqr-neg 22.0%

          \[\leadsto \frac{0 + \sqrt{\color{blue}{y \cdot y}}}{y + x} \]

      11. sqrt-unprod 34.4%

          \[\leadsto \frac{0 + \color{blue}{\sqrt{y} \cdot \sqrt{y}}}{y + x} \]

      12. add-sqr-sqrt 75.1%

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

   10. Applied egg-rr 75.1%

       \[\leadsto \frac{\color{blue}{0 + y}}{y + x} \]
   11. Step-by-step derivation

       1. +-lft-identity 75.1%

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

   12. Simplified 75.1%

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

   if 1.05e107 < x

   1. Initial program 100.0%

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

      1. div-inv 99.6%

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

      2. add-sqr-sqrt 95.2%

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

      3. fabs-sqr 95.2%

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

      4. add-sqr-sqrt 95.5%

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

      5. *-commutative 95.5%

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

      6. add-sqr-sqrt 47.7%

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

      7. fabs-sqr 47.7%

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

      8. add-sqr-sqrt 48.1%

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

   3. Applied egg-rr 48.1%

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

   4. Taylor expanded in y around 0 48.4%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+43}:\\ \;\;\;\;x \cdot \frac{\frac{x}{y}}{y - x}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+107}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3: 58.8% accurate, 22.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+88}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.56 \cdot 10^{+108}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7e+88) (/ x y) (if (<= x 1.56e+108) (/ y (+ x y)) (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7e+88) {
		tmp = x / y;
	} else if (x <= 1.56e+108) {
		tmp = y / (x + y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7d+88)) then
        tmp = x / y
    else if (x <= 1.56d+108) then
        tmp = y / (x + y)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -7e+88) {
		tmp = x / y;
	} else if (x <= 1.56e+108) {
		tmp = y / (x + y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7e+88:
		tmp = x / y
	elif x <= 1.56e+108:
		tmp = y / (x + y)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7e+88)
		tmp = Float64(x / y);
	elseif (x <= 1.56e+108)
		tmp = Float64(y / Float64(x + y));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7e+88)
		tmp = x / y;
	elseif (x <= 1.56e+108)
		tmp = y / (x + y);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7e+88], N[(x / y), $MachinePrecision], If[LessEqual[x, 1.56e+108], N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.9999999999999995e88 or 1.5599999999999999e108 < x

   1. Initial program 100.0%

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

      1. div-inv 99.7%

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

      2. add-sqr-sqrt 49.7%

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

      3. fabs-sqr 49.7%

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

      4. add-sqr-sqrt 50.1%

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

      5. *-commutative 50.1%

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

      6. add-sqr-sqrt 24.5%

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

      7. fabs-sqr 24.5%

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

      8. add-sqr-sqrt 43.8%

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

   3. Applied egg-rr 43.8%

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

   4. Taylor expanded in y around 0 44.1%

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

   if -6.9999999999999995e88 < x < 1.5599999999999999e108

   1. Initial program 100.0%

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

      1. div-inv 99.7%

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

      2. add-sqr-sqrt 50.8%

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

      3. fabs-sqr 50.8%

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

      4. add-sqr-sqrt 51.7%

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

      5. *-commutative 51.7%

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

      6. add-sqr-sqrt 7.3%

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

      7. fabs-sqr 7.3%

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

      8. add-sqr-sqrt 13.5%

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

   3. Applied egg-rr 13.5%

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

      1. flip-- 12.2%

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

      2. associate-*r/ 11.7%

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

      3. +-commutative 11.7%

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

   5. Applied egg-rr 11.7%

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

   6. Taylor expanded in x around 0 1.7%

      \[\leadsto \frac{\frac{1}{y} \cdot \color{blue}{\left(-1 \cdot {y}^{2}\right)}}{y + x} \]
   7. Step-by-step derivation

      1. unpow2 1.7%

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

      2. mul-1-neg 1.7%

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

      3. distribute-rgt-neg-out 1.7%

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

   8. Simplified 1.7%

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

      1. associate-*r* 1.9%

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

      2. lft-mult-inverse 1.9%

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

      3. *-un-lft-identity 1.9%

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

      4. neg-sub0 1.9%

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

      5. metadata-eval 1.9%

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

      6. sub-neg 1.9%

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

      7. metadata-eval 1.9%

         \[\leadsto \frac{\color{blue}{0} + \left(-y\right)}{y + x} \]

      8. add-sqr-sqrt 0.9%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{-y} \cdot \sqrt{-y}}}{y + x} \]

      9. sqrt-unprod 21.7%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}}{y + x} \]

      10. sqr-neg 21.7%

          \[\leadsto \frac{0 + \sqrt{\color{blue}{y \cdot y}}}{y + x} \]

      11. sqrt-unprod 34.1%

          \[\leadsto \frac{0 + \color{blue}{\sqrt{y} \cdot \sqrt{y}}}{y + x} \]

      12. add-sqr-sqrt 73.2%

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

   10. Applied egg-rr 73.2%

       \[\leadsto \frac{\color{blue}{0 + y}}{y + x} \]
   11. Step-by-step derivation

       1. +-lft-identity 73.2%

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

   12. Simplified 73.2%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+88}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.56 \cdot 10^{+108}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4: 26.7% accurate, 68.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. div-inv 99.7%

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

   2. add-sqr-sqrt 50.4%

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

   3. fabs-sqr 50.4%

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

   4. add-sqr-sqrt 51.1%

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

   5. *-commutative 51.1%

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

   6. add-sqr-sqrt 13.6%

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

   7. fabs-sqr 13.6%

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

   8. add-sqr-sqrt 24.7%

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

3. Applied egg-rr 24.7%

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

4. Taylor expanded in y around 0 25.5%

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

5. Final simplification 25.5%

   \[\leadsto \frac{x}{y} \]

Alternative 5: 1.3% accurate, 205.0× 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

real(8) function code(x, y)
    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. div-inv 99.7%

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

   2. add-sqr-sqrt 50.4%

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

   3. fabs-sqr 50.4%

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

   4. add-sqr-sqrt 51.1%

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

   5. *-commutative 51.1%

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

   6. add-sqr-sqrt 13.6%

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

   7. fabs-sqr 13.6%

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

   8. add-sqr-sqrt 24.7%

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

3. Applied egg-rr 24.7%

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

4. Taylor expanded in y around inf 1.3%

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

5. Final simplification 1.3%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023230 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))