Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 4.0s

Alternatives: 6

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: 58.0% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot x}{y \cdot y}\\ \mathbf{if}\;x \leq -9.5 \cdot 10^{+76}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+79}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;x \leq 1.18 \cdot 10^{+243}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x x) (* y y))))
   (if (<= x -9.5e+76)
     t_0
     (if (<= x 7.5e+79)
       (/ y (+ x y))
       (if (<= x 1.18e+243) (+ (/ x y) -1.0) t_0)))))

C

double code(double x, double y) {
	double t_0 = (x * x) / (y * y);
	double tmp;
	if (x <= -9.5e+76) {
		tmp = t_0;
	} else if (x <= 7.5e+79) {
		tmp = y / (x + y);
	} else if (x <= 1.18e+243) {
		tmp = (x / y) + -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * x) / (y * y)
    if (x <= (-9.5d+76)) then
        tmp = t_0
    else if (x <= 7.5d+79) then
        tmp = y / (x + y)
    else if (x <= 1.18d+243) then
        tmp = (x / y) + (-1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * x) / (y * y);
	double tmp;
	if (x <= -9.5e+76) {
		tmp = t_0;
	} else if (x <= 7.5e+79) {
		tmp = y / (x + y);
	} else if (x <= 1.18e+243) {
		tmp = (x / y) + -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * x) / (y * y)
	tmp = 0
	if x <= -9.5e+76:
		tmp = t_0
	elif x <= 7.5e+79:
		tmp = y / (x + y)
	elif x <= 1.18e+243:
		tmp = (x / y) + -1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * x) / Float64(y * y))
	tmp = 0.0
	if (x <= -9.5e+76)
		tmp = t_0;
	elseif (x <= 7.5e+79)
		tmp = Float64(y / Float64(x + y));
	elseif (x <= 1.18e+243)
		tmp = Float64(Float64(x / y) + -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * x) / (y * y);
	tmp = 0.0;
	if (x <= -9.5e+76)
		tmp = t_0;
	elseif (x <= 7.5e+79)
		tmp = y / (x + y);
	elseif (x <= 1.18e+243)
		tmp = (x / y) + -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9.5e+76], t$95$0, If[LessEqual[x, 7.5e+79], N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.18e+243], N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.5000000000000003e76 or 1.1799999999999999e243 < x

   1. Initial program 100.0%

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

      1. div-inv 99.8%

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

      2. add-sqr-sqrt 38.8%

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

      3. fabs-sqr 38.8%

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

      4. add-sqr-sqrt 39.1%

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

      5. *-commutative 39.1%

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

      6. add-sqr-sqrt 15.0%

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

      7. fabs-sqr 15.0%

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

      8. add-sqr-sqrt 39.2%

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

   3. Applied egg-rr 39.2%

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

      1. flip-- 33.5%

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

      2. associate-*r/ 33.5%

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

      3. +-commutative 33.5%

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

   5. Applied egg-rr 33.5%

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

   6. Taylor expanded in y around 0 33.2%

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

      1. unpow2 33.2%

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

      2. associate-/l* 35.3%

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

      3. associate-/r/ 35.3%

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

   8. Simplified 35.3%

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

   9. Taylor expanded in x around 0 57.6%

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

       1. unpow2 57.6%

          \[\leadsto \frac{\color{blue}{x \cdot x}}{{y}^{2}} \]

       2. unpow2 57.6%

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

   11. Simplified 57.6%

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

   if -9.5000000000000003e76 < x < 7.49999999999999967e79

   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 57.4%

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

      3. fabs-sqr 57.4%

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

      4. add-sqr-sqrt 58.3%

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

      5. *-commutative 58.3%

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

      6. add-sqr-sqrt 8.3%

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

      7. fabs-sqr 8.3%

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

      8. add-sqr-sqrt 16.4%

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

   3. Applied egg-rr 16.4%

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

      1. flip-- 13.3%

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

      2. associate-*r/ 12.7%

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

      3. +-commutative 12.7%

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

   5. Applied egg-rr 12.7%

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

   6. Taylor expanded in x around 0 1.8%

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

      1. unpow2 1.8%

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

      2. mul-1-neg 1.8%

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

      3. distribute-rgt-neg-out 1.8%

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

   8. Simplified 1.8%

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

      1. associate-*r* 2.0%

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

      2. lft-mult-inverse 2.0%

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

      3. *-un-lft-identity 2.0%

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

      4. neg-sub0 2.0%

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

      5. metadata-eval 2.0%

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

      6. sub-neg 2.0%

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

      7. metadata-eval 2.0%

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

      8. add-sqr-sqrt 1.2%

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

      9. sqrt-unprod 14.4%

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

      10. sqr-neg 14.4%

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

      11. sqrt-unprod 28.3%

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

      12. add-sqr-sqrt 67.9%

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

   10. Applied egg-rr 67.9%

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

       1. +-lft-identity 67.9%

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

   12. Simplified 67.9%

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

   if 7.49999999999999967e79 < x < 1.1799999999999999e243

   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-sub 100.0%

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

      5. fabs-div 100.0%

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

      6. rem-square-sqrt 52.3%

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

      7. fabs-sqr 52.3%

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

      8. rem-square-sqrt 53.0%

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

      9. div-sub 53.0%

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

      10. sub-neg 53.0%

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

      11. *-inverses 53.0%

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

      12. metadata-eval 53.0%

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

      13. +-commutative 53.0%

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

   4. Simplified 53.0%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+76}:\\ \;\;\;\;\frac{x \cdot x}{y \cdot y}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+79}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;x \leq 1.18 \cdot 10^{+243}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{y \cdot y}\\ \end{array} \]

Alternative 3: 57.7% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-159} \lor \neg \left(y \leq 1.7 \cdot 10^{-102}\right):\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.1e-159) (not (<= y 1.7e-102)))
   (/ y (+ x y))
   (+ (/ x y) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.1e-159) || !(y <= 1.7e-102)) {
		tmp = y / (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 ((y <= (-1.1d-159)) .or. (.not. (y <= 1.7d-102))) then
        tmp = y / (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 ((y <= -1.1e-159) || !(y <= 1.7e-102)) {
		tmp = y / (x + y);
	} else {
		tmp = (x / y) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.1e-159) or not (y <= 1.7e-102):
		tmp = y / (x + y)
	else:
		tmp = (x / y) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.1e-159) || !(y <= 1.7e-102))
		tmp = Float64(y / 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 ((y <= -1.1e-159) || ~((y <= 1.7e-102)))
		tmp = y / (x + y);
	else
		tmp = (x / y) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.1e-159], N[Not[LessEqual[y, 1.7e-102]], $MachinePrecision]], N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.1e-159 or 1.70000000000000006e-102 < y

   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 63.8%

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

      3. fabs-sqr 63.8%

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

      4. add-sqr-sqrt 64.6%

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

      5. *-commutative 64.6%

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

      6. add-sqr-sqrt 10.1%

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

      7. fabs-sqr 10.1%

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

      8. add-sqr-sqrt 16.6%

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

   3. Applied egg-rr 16.6%

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

      1. flip-- 10.1%

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

      2. associate-*r/ 10.1%

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

      3. +-commutative 10.1%

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

   5. Applied egg-rr 10.1%

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

   6. Taylor expanded in x around 0 1.6%

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

      1. unpow2 1.6%

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

      2. mul-1-neg 1.6%

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

      3. distribute-rgt-neg-out 1.6%

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

   8. Simplified 1.6%

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

      1. associate-*r* 2.0%

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

      2. lft-mult-inverse 2.0%

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

      3. *-un-lft-identity 2.0%

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

      4. neg-sub0 2.0%

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

      5. metadata-eval 2.0%

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

      6. sub-neg 2.0%

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

      7. metadata-eval 2.0%

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

      8. add-sqr-sqrt 1.2%

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

      9. sqrt-unprod 13.4%

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

      10. sqr-neg 13.4%

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

      11. sqrt-unprod 24.4%

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

      12. add-sqr-sqrt 65.2%

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

   10. Applied egg-rr 65.2%

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

       1. +-lft-identity 65.2%

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

   12. Simplified 65.2%

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

   if -1.1e-159 < y < 1.70000000000000006e-102

   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-sub 100.0%

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

      5. fabs-div 100.0%

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

      6. rem-square-sqrt 47.3%

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

      7. fabs-sqr 47.3%

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

      8. rem-square-sqrt 47.9%

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

      9. div-sub 47.9%

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

      10. sub-neg 47.9%

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

      11. *-inverses 47.9%

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

      12. metadata-eval 47.9%

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

      13. +-commutative 47.9%

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

   4. Simplified 47.9%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-159} \lor \neg \left(y \leq 1.7 \cdot 10^{-102}\right):\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \]

Alternative 4: 25.9% 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. 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-sub 100.0%

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

   5. fabs-div 100.0%

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

   6. rem-square-sqrt 25.6%

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

   7. fabs-sqr 25.6%

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

   8. rem-square-sqrt 26.7%

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

   9. div-sub 26.7%

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

   10. sub-neg 26.7%

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

   11. *-inverses 26.7%

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

   12. metadata-eval 26.7%

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

   13. +-commutative 26.7%

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

4. Simplified 26.7%

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

5. Final simplification 26.7%

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

Alternative 5: 26.5% 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 58.5%

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

   3. fabs-sqr 58.5%

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

   4. add-sqr-sqrt 59.2%

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

   5. *-commutative 59.2%

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

   6. add-sqr-sqrt 16.3%

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

   7. fabs-sqr 16.3%

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

   8. add-sqr-sqrt 26.6%

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

3. Applied egg-rr 26.6%

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

4. Taylor expanded in y around 0 26.4%

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

5. Final simplification 26.4%

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

Alternative 6: 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 58.5%

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

   3. fabs-sqr 58.5%

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

   4. add-sqr-sqrt 59.2%

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

   5. *-commutative 59.2%

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

   6. add-sqr-sqrt 16.3%

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

   7. fabs-sqr 16.3%

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

   8. add-sqr-sqrt 26.6%

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

3. Applied egg-rr 26.6%

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

4. Taylor expanded in y around inf 1.3%

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

5. Final simplification 1.3%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023171 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))