System.Random.MWC.Distributions:standard from mwc-random-0.13.3.2

Percentage Accurate: 100.0% → 100.0%

Time: 3.8s

Alternatives: 5

Speedup: 1.0×

• 25.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(x \cdot x - y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* 0.5 (- (* x x) y)))

C

double code(double x, double y) {
	return 0.5 * ((x * x) - y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.5d0 * ((x * x) - y)
end function

Java

public static double code(double x, double y) {
	return 0.5 * ((x * x) - y);
}

Python

def code(x, y):
	return 0.5 * ((x * x) - y)

Julia

function code(x, y)
	return Float64(0.5 * Float64(Float64(x * x) - y))
end

MATLAB

function tmp = code(x, y)
	tmp = 0.5 * ((x * x) - y);
end

Wolfram

code[x_, y_] := N[(0.5 * N[(N[(x * x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(x \cdot x - y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* 0.5 (- (* x x) y)))

C

double code(double x, double y) {
	return 0.5 * ((x * x) - y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.5d0 * ((x * x) - y)
end function

Java

public static double code(double x, double y) {
	return 0.5 * ((x * x) - y);
}

Python

def code(x, y):
	return 0.5 * ((x * x) - y)

Julia

function code(x, y)
	return Float64(0.5 * Float64(Float64(x * x) - y))
end

MATLAB

function tmp = code(x, y)
	tmp = 0.5 * ((x * x) - y);
end

Wolfram

code[x_, y_] := N[(0.5 * N[(N[(x * x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot 0.5, x, y \cdot -0.5\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma (* x 0.5) x (* y -0.5)))

C

double code(double x, double y) {
	return fma((x * 0.5), x, (y * -0.5));
}

Julia

function code(x, y)
	return fma(Float64(x * 0.5), x, Float64(y * -0.5))
end

Wolfram

code[x_, y_] := N[(N[(x * 0.5), $MachinePrecision] * x + N[(y * -0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[0.5 \cdot \left(x \cdot x - y\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. distribute-lft-in N/A

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

   5. lift-*.f64 N/A

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

   6. sqr-neg-rev N/A

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

   7. associate-*r* N/A

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

   8. neg-mul-1 N/A

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

   9. metadata-eval N/A

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

   10. unpow1 N/A

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

   11. unpow-prod-down N/A

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

   12. neg-mul-1 N/A

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

   13. metadata-eval N/A

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

   14. sqrt-pow1 N/A

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

   15. pow2 N/A

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

   16. sqr-neg-rev N/A

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

   17. sqrt-prod N/A

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

   18. rem-square-sqrt N/A

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

   19. *-commutative N/A

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

   20. lower-fma.f64 N/A

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot x, x, -0.5 \cdot y\right)} \]

5. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(x \cdot 0.5, x, y \cdot -0.5\right) \]
6. Add Preprocessing

Alternative 2: 87.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+39}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 2e+39) (* y -0.5) (* (* x 0.5) x)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 2e+39) {
		tmp = y * -0.5;
	} else {
		tmp = (x * 0.5) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x * x) <= 2d+39) then
        tmp = y * (-0.5d0)
    else
        tmp = (x * 0.5d0) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x * x) <= 2e+39) {
		tmp = y * -0.5;
	} else {
		tmp = (x * 0.5) * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 2e+39:
		tmp = y * -0.5
	else:
		tmp = (x * 0.5) * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 2e+39)
		tmp = Float64(y * -0.5);
	else
		tmp = Float64(Float64(x * 0.5) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * x) <= 2e+39)
		tmp = y * -0.5;
	else
		tmp = (x * 0.5) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e+39], N[(y * -0.5), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.99999999999999988e39

   1. Initial program 100.0%

      \[0.5 \cdot \left(x \cdot x - y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 85.8

         \[\leadsto \color{blue}{-0.5 \cdot y} \]

   5. Applied rewrites 85.8%

      \[\leadsto \color{blue}{-0.5 \cdot y} \]

   if 1.99999999999999988e39 < (*.f64 x x)

   1. Initial program 99.3%

      \[0.5 \cdot \left(x \cdot x - y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \frac{1}{2}} \]

      3. unpow2 N/A

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

      4. lower-*.f64 92.2

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot 0.5 \]

   5. Applied rewrites 92.2%

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 92.9%

      \[\leadsto \color{blue}{\left(x \cdot 0.5\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+39}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, x, -y\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (fma x x (- y)) 0.5))

C

double code(double x, double y) {
	return fma(x, x, -y) * 0.5;
}

Julia

function code(x, y)
	return Float64(fma(x, x, Float64(-y)) * 0.5)
end

Wolfram

code[x_, y_] := N[(N[(x * x + (-y)), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[0.5 \cdot \left(x \cdot x - y\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-neg.f64 99.7

      \[\leadsto 0.5 \cdot \mathsf{fma}\left(x, x, \color{blue}{-y}\right) \]

4. Applied rewrites 99.7%

   \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(x, x, -y\right)} \]

5. Final simplification 99.7%

   \[\leadsto \mathsf{fma}\left(x, x, -y\right) \cdot 0.5 \]
6. Add Preprocessing

Alternative 4: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot x - y\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (- (* x x) y) 0.5))

C

double code(double x, double y) {
	return ((x * x) - y) * 0.5;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * x) - y) * 0.5d0
end function

Java

public static double code(double x, double y) {
	return ((x * x) - y) * 0.5;
}

Python

def code(x, y):
	return ((x * x) - y) * 0.5

Julia

function code(x, y)
	return Float64(Float64(Float64(x * x) - y) * 0.5)
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * x) - y) * 0.5;
end

Wolfram

code[x_, y_] := N[(N[(N[(x * x), $MachinePrecision] - y), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[0.5 \cdot \left(x \cdot x - y\right) \]
2. Add Preprocessing

3. Final simplification 99.7%

   \[\leadsto \left(x \cdot x - y\right) \cdot 0.5 \]
4. Add Preprocessing

Alternative 5: 50.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ y \cdot -0.5 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* y -0.5))

C

double code(double x, double y) {
	return y * -0.5;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y * (-0.5d0)
end function

Java

public static double code(double x, double y) {
	return y * -0.5;
}

Python

def code(x, y):
	return y * -0.5

Julia

function code(x, y)
	return Float64(y * -0.5)
end

MATLAB

function tmp = code(x, y)
	tmp = y * -0.5;
end

Wolfram

code[x_, y_] := N[(y * -0.5), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[0.5 \cdot \left(x \cdot x - y\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot y} \]
4. Step-by-step derivation

   1. lower-*.f64 51.2

      \[\leadsto \color{blue}{-0.5 \cdot y} \]

5. Applied rewrites 51.2%

   \[\leadsto \color{blue}{-0.5 \cdot y} \]

6. Final simplification 51.2%

   \[\leadsto y \cdot -0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024298 
(FPCore (x y)
  :name "System.Random.MWC.Distributions:standard from mwc-random-0.13.3.2"
  :precision binary64
  (* 0.5 (- (* x x) y)))