Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 3.5s

Alternatives: 7

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. div-fabs 100.0%

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

   2. div-sub 100.0%

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

   3. pow1 100.0%

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

   4. pow1 100.0%

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

   5. pow-div 100.0%

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

   6. metadata-eval 100.0%

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

   7. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 74.2% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-212} \lor \neg \left(y \leq -6.6 \cdot 10^{-257}\right) \land y \leq 2.5 \cdot 10^{-172}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.6e-212) (and (not (<= y -6.6e-257)) (<= y 2.5e-172)))
   (- 1.0 (/ x y))
   (/ (+ x y) y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.6e-212) || (!(y <= -6.6e-257) && (y <= 2.5e-172))) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = (x + y) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.6d-212)) .or. (.not. (y <= (-6.6d-257))) .and. (y <= 2.5d-172)) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = (x + y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.6e-212) || (!(y <= -6.6e-257) && (y <= 2.5e-172))) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = (x + y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.6e-212) or (not (y <= -6.6e-257) and (y <= 2.5e-172)):
		tmp = 1.0 - (x / y)
	else:
		tmp = (x + y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.6e-212) || (!(y <= -6.6e-257) && (y <= 2.5e-172)))
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = Float64(Float64(x + y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.6e-212) || (~((y <= -6.6e-257)) && (y <= 2.5e-172)))
		tmp = 1.0 - (x / y);
	else
		tmp = (x + y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.6e-212], And[N[Not[LessEqual[y, -6.6e-257]], $MachinePrecision], LessEqual[y, 2.5e-172]]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.5999999999999999e-212 or -6.6e-257 < y < 2.5e-172

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 73.8%

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

      2. fabs-sqr 73.8%

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

      3. add-sqr-sqrt 7.3%

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

      4. fabs-sqr 7.3%

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

      5. add-sqr-sqrt 7.4%

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

      6. add-sqr-sqrt 20.4%

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

      7. clear-num 20.3%

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

      8. associate-/r/ 20.3%

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

   3. Applied egg-rr 20.3%

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

      1. associate-*l/ 20.4%

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

      2. *-un-lft-identity 20.4%

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

      3. frac-2neg 20.4%

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

      4. sub-neg 20.4%

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

      5. distribute-neg-in 20.4%

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

      6. remove-double-neg 20.4%

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

      7. add-sqr-sqrt 12.9%

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

      8. sqrt-unprod 14.0%

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

      9. sqr-neg 14.0%

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

      10. sqrt-unprod 13.4%

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

      11. add-sqr-sqrt 80.6%

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

   5. Applied egg-rr 80.6%

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

   6. Taylor expanded in x around 0 80.7%

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

      1. mul-1-neg 80.7%

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

      2. unsub-neg 80.7%

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

   8. Simplified 80.7%

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

   if -1.5999999999999999e-212 < y < -6.6e-257 or 2.5e-172 < y

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 22.3%

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

      2. fabs-sqr 22.3%

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

      3. add-sqr-sqrt 21.3%

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

      4. fabs-sqr 21.3%

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

      5. add-sqr-sqrt 22.4%

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

      6. add-sqr-sqrt 29.9%

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

      7. clear-num 29.8%

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

      8. associate-/r/ 29.8%

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

   3. Applied egg-rr 29.8%

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

      1. associate-*l/ 29.9%

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

      2. *-un-lft-identity 29.9%

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

      3. sub-neg 29.9%

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

      4. add-sqr-sqrt 7.5%

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

      5. sqrt-unprod 59.9%

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

      6. sqr-neg 59.9%

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

      7. sqrt-unprod 77.5%

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

      8. add-sqr-sqrt 86.3%

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

   5. Applied egg-rr 86.3%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-212} \lor \neg \left(y \leq -6.6 \cdot 10^{-257}\right) \land y \leq 2.5 \cdot 10^{-172}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{y}\\ \end{array} \]

Alternative 3: 57.3% accurate, 28.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+231} \lor \neg \left(x \leq 2.9 \cdot 10^{+143}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -7e+231) (not (<= x 2.9e+143))) (/ x y) 1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -7e+231) || !(x <= 2.9e+143)) {
		tmp = x / y;
	} else {
		tmp = 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 <= (-7d+231)) .or. (.not. (x <= 2.9d+143))) then
        tmp = x / y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -7e+231) || !(x <= 2.9e+143)) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -7e+231) or not (x <= 2.9e+143):
		tmp = x / y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -7e+231) || !(x <= 2.9e+143))
		tmp = Float64(x / y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -7e+231) || ~((x <= 2.9e+143)))
		tmp = x / y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -7e+231], N[Not[LessEqual[x, 2.9e+143]], $MachinePrecision]], N[(x / y), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -6.9999999999999997e231 or 2.8999999999999998e143 < x

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 68.1%

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

      2. fabs-sqr 68.1%

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

      3. add-sqr-sqrt 38.6%

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

      4. fabs-sqr 38.6%

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

      5. add-sqr-sqrt 38.8%

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

      6. add-sqr-sqrt 52.1%

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

      7. clear-num 52.0%

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

      8. associate-/r/ 52.0%

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

   3. Applied egg-rr 52.0%

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

   4. Taylor expanded in y around 0 51.8%

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

   if -6.9999999999999997e231 < x < 2.8999999999999998e143

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 48.1%

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

      2. fabs-sqr 48.1%

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

      3. add-sqr-sqrt 6.3%

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

      4. fabs-sqr 6.3%

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

      5. add-sqr-sqrt 6.9%

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

      6. add-sqr-sqrt 17.0%

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

      7. clear-num 16.9%

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

      8. associate-/r/ 16.9%

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

   3. Applied egg-rr 16.9%

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

      1. associate-*l/ 17.0%

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

      2. *-un-lft-identity 17.0%

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

      3. sub-neg 17.0%

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

      4. add-sqr-sqrt 10.0%

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

      5. sqrt-unprod 32.8%

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

      6. sqr-neg 32.8%

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

      7. sqrt-unprod 37.0%

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

      8. add-sqr-sqrt 81.2%

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

   5. Applied egg-rr 81.2%

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

   6. Taylor expanded in x around 0 67.8%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+231} \lor \neg \left(x \leq 2.9 \cdot 10^{+143}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 58.2% accurate, 28.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+179}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+143}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6.5e+179) (/ (- x) y) (if (<= x 2.8e+143) 1.0 (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.5e+179) {
		tmp = -x / y;
	} else if (x <= 2.8e+143) {
		tmp = 1.0;
	} 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 <= (-6.5d+179)) then
        tmp = -x / y
    else if (x <= 2.8d+143) then
        tmp = 1.0d0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -6.5e+179) {
		tmp = -x / y;
	} else if (x <= 2.8e+143) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6.5e+179:
		tmp = -x / y
	elif x <= 2.8e+143:
		tmp = 1.0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6.5e+179)
		tmp = Float64(Float64(-x) / y);
	elseif (x <= 2.8e+143)
		tmp = 1.0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6.5e+179)
		tmp = -x / y;
	elseif (x <= 2.8e+143)
		tmp = 1.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.5e+179], N[((-x) / y), $MachinePrecision], If[LessEqual[x, 2.8e+143], 1.0, N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.50000000000000052e179

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 3.9%

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

      2. fabs-sqr 3.9%

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

      3. add-sqr-sqrt 0.0%

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

      4. fabs-sqr 0.0%

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

      5. add-sqr-sqrt 0.2%

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

      6. add-sqr-sqrt 36.2%

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

      7. clear-num 36.1%

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

      8. associate-/r/ 36.1%

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

   3. Applied egg-rr 36.1%

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

      1. associate-*l/ 36.2%

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

      2. *-un-lft-identity 36.2%

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

      3. frac-2neg 36.2%

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

      4. sub-neg 36.2%

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

      5. distribute-neg-in 36.2%

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

      6. remove-double-neg 36.2%

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

      7. add-sqr-sqrt 35.8%

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

      8. sqrt-unprod 76.7%

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

      9. sqr-neg 76.7%

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

      10. sqrt-unprod 59.8%

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

      11. add-sqr-sqrt 64.2%

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

   5. Applied egg-rr 64.2%

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

   6. Taylor expanded in x around inf 53.0%

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

      1. associate-*r/ 53.0%

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

      2. mul-1-neg 53.0%

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

   8. Simplified 53.0%

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

   if -6.50000000000000052e179 < x < 2.79999999999999998e143

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 50.3%

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

      2. fabs-sqr 50.3%

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

      3. add-sqr-sqrt 6.6%

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

      4. fabs-sqr 6.6%

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

      5. add-sqr-sqrt 7.2%

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

      6. add-sqr-sqrt 16.7%

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

      7. clear-num 16.7%

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

      8. associate-/r/ 16.7%

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

   3. Applied egg-rr 16.7%

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

      1. associate-*l/ 16.7%

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

      2. *-un-lft-identity 16.7%

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

      3. sub-neg 16.7%

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

      4. add-sqr-sqrt 9.4%

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

      5. sqrt-unprod 33.3%

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

      6. sqr-neg 33.3%

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

      7. sqrt-unprod 37.7%

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

      8. add-sqr-sqrt 82.9%

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

   5. Applied egg-rr 82.9%

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

   6. Taylor expanded in x around 0 69.8%

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

   if 2.79999999999999998e143 < x

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 94.2%

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

      2. fabs-sqr 94.2%

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

      3. add-sqr-sqrt 54.9%

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

      4. fabs-sqr 54.9%

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

      5. add-sqr-sqrt 55.1%

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

      6. add-sqr-sqrt 55.5%

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

      7. clear-num 55.4%

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

      8. associate-/r/ 55.4%

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

   3. Applied egg-rr 55.4%

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

   4. Taylor expanded in y around 0 56.0%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+179}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+143}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5: 74.7% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{+260}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 6.5e+260) (- 1.0 (/ x y)) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6.5e+260) {
		tmp = 1.0 - (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 <= 6.5d+260) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 6.5e+260) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 6.5e+260:
		tmp = 1.0 - (x / y)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 6.5e+260)
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 6.5e+260)
		tmp = 1.0 - (x / y);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 6.5e+260], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.49999999999999954e260

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 50.0%

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

      2. fabs-sqr 50.0%

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

      3. add-sqr-sqrt 10.1%

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

      4. fabs-sqr 10.1%

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

      5. add-sqr-sqrt 10.6%

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

      6. add-sqr-sqrt 21.9%

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

      7. clear-num 21.8%

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

      8. associate-/r/ 21.8%

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

   3. Applied egg-rr 21.8%

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

      1. associate-*l/ 21.9%

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

      2. *-un-lft-identity 21.9%

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

      3. frac-2neg 21.9%

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

      4. sub-neg 21.9%

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

      5. distribute-neg-in 21.9%

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

      6. remove-double-neg 21.9%

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

      7. add-sqr-sqrt 11.1%

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

      8. sqrt-unprod 28.5%

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

      9. sqr-neg 28.5%

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

      10. sqrt-unprod 38.7%

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

      11. add-sqr-sqrt 79.2%

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

   5. Applied egg-rr 79.2%

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

   6. Taylor expanded in x around 0 79.2%

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

      1. mul-1-neg 79.2%

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

      2. unsub-neg 79.2%

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

   8. Simplified 79.2%

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

   if 6.49999999999999954e260 < x

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 99.7%

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

      2. fabs-sqr 99.7%

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

      3. add-sqr-sqrt 74.7%

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

      4. fabs-sqr 74.7%

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

      5. add-sqr-sqrt 74.9%

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

      6. add-sqr-sqrt 75.0%

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

      7. clear-num 75.0%

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

      8. associate-/r/ 75.0%

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

   3. Applied egg-rr 75.0%

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

   4. Taylor expanded in y around 0 75.0%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{+260}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

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. add-sqr-sqrt 52.3%

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

   2. fabs-sqr 52.3%

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

   3. add-sqr-sqrt 13.2%

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

   4. fabs-sqr 13.2%

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

   5. add-sqr-sqrt 13.7%

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

   6. add-sqr-sqrt 24.4%

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

   7. clear-num 24.3%

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

   8. associate-/r/ 24.3%

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

3. Applied egg-rr 24.3%

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

4. Taylor expanded in y around inf 1.3%

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

5. Final simplification 1.3%

   \[\leadsto -1 \]

Alternative 7: 51.3% 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. add-sqr-sqrt 52.3%

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

   2. fabs-sqr 52.3%

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

   3. add-sqr-sqrt 13.2%

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

   4. fabs-sqr 13.2%

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

   5. add-sqr-sqrt 13.7%

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

   6. add-sqr-sqrt 24.4%

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

   7. clear-num 24.3%

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

   8. associate-/r/ 24.3%

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

3. Applied egg-rr 24.3%

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

   1. associate-*l/ 24.4%

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

   2. *-un-lft-identity 24.4%

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

   3. sub-neg 24.4%

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

   4. add-sqr-sqrt 10.7%

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

   5. sqrt-unprod 36.1%

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

   6. sqr-neg 36.1%

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

   7. sqrt-unprod 38.0%

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

   8. add-sqr-sqrt 75.9%

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

5. Applied egg-rr 75.9%

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

6. Taylor expanded in x around 0 55.4%

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

7. Final simplification 55.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023334 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))