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

Percentage Accurate: 100.0% → 100.0%

Time: 5.0s

Alternatives: 7

Speedup: 1.0×

• 24.6% 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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(0.5 * x), $MachinePrecision] * x + N[(-0.5 * y), $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. sub-neg N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. *-commutative N/A

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

   8. neg-mul-1 N/A

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

   9. associate-*r* N/A

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

   10. *-lowering-*.f64 N/A

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

   11. metadata-eval 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 86.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-80}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 2e-80) (* -0.5 y) (* x (* 0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 2e-80) {
		tmp = -0.5 * y;
	} 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-80) then
        tmp = (-0.5d0) * y
    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-80) {
		tmp = -0.5 * y;
	} else {
		tmp = x * (0.5 * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 2e-80:
		tmp = -0.5 * y
	else:
		tmp = x * (0.5 * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 2e-80)
		tmp = Float64(-0.5 * y);
	else
		tmp = Float64(x * Float64(0.5 * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * x) <= 2e-80)
		tmp = -0.5 * y;
	else
		tmp = x * (0.5 * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e-80], N[(-0.5 * y), $MachinePrecision], N[(x * N[(0.5 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.99999999999999992e-80

   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. *-lowering-*.f64 89.2

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

   5. Simplified 89.2%

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

   if 1.99999999999999992e-80 < (*.f64 x x)

   1. Initial program 99.4%

      \[0.5 \cdot \left(x \cdot x - y\right) \]
   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. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 85.1

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

   5. Simplified 85.1%

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 85.7

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

   7. Applied egg-rr 85.7%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-80}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-80}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 2e-80) (* -0.5 y) (* 0.5 (* x x))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 2e-80) {
		tmp = -0.5 * y;
	} else {
		tmp = 0.5 * (x * 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-80) then
        tmp = (-0.5d0) * y
    else
        tmp = 0.5d0 * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x * x) <= 2e-80) {
		tmp = -0.5 * y;
	} else {
		tmp = 0.5 * (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 2e-80:
		tmp = -0.5 * y
	else:
		tmp = 0.5 * (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 2e-80)
		tmp = Float64(-0.5 * y);
	else
		tmp = Float64(0.5 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * x) <= 2e-80)
		tmp = -0.5 * y;
	else
		tmp = 0.5 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e-80], N[(-0.5 * y), $MachinePrecision], N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.99999999999999992e-80

   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. *-lowering-*.f64 89.2

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

   5. Simplified 89.2%

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

   if 1.99999999999999992e-80 < (*.f64 x x)

   1. Initial program 99.4%

      \[0.5 \cdot \left(x \cdot x - y\right) \]
   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. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 85.1

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

   5. Simplified 85.1%

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

Alternative 4: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(0.5 * N[(x * x + (-y)), $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. sub-neg N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. neg-lowering-neg.f64 99.7

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

4. Applied egg-rr 99.7%

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

Alternative 5: 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]

Derivation

1. Initial program 99.7%

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

Alternative 6: 52.4% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(-0.5 * y), $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. *-lowering-*.f64 50.3

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

5. Simplified 50.3%

   \[\leadsto \color{blue}{-0.5 \cdot y} \]
6. Add Preprocessing

Alternative 7: 3.8% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 0.5

Julia

function code(x, y)
	return 0.5
end

MATLAB

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

Wolfram

code[x_, y_] := 0.5

Derivation

1. Initial program 99.7%

   \[0.5 \cdot \left(x \cdot x - y\right) \]
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. *-lowering-*.f64 N/A

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 51.0

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

5. Simplified 51.0%

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

   1. associate-*r* N/A

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

   2. *-lowering-*.f64 N/A

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

   3. *-lowering-*.f64 51.3

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

7. Applied egg-rr 51.3%

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

9. Applied egg-rr 3.8%

   \[\leadsto \color{blue}{0.5} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024204 
(FPCore (x y)
  :name "System.Random.MWC.Distributions:standard from mwc-random-0.13.3.2"
  :precision binary64
  (* 0.5 (- (* x x) y)))