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

Percentage Accurate: 100.0% → 100.0%

Time: 4.7s

Alternatives: 4

Speedup: 1.0×

• 25.4% 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, 0.9× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \mathsf{fma}\left(x, x, 0 - y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* 0.5 (fma x x (- 0.0 y))))

C

double code(double x, double y) {
	return 0.5 * fma(x, x, (0.0 - y));
}

Julia

function code(x, y)
	return Float64(0.5 * fma(x, x, Float64(0.0 - y)))
end

Wolfram

code[x_, y_] := N[(0.5 * N[(x * x + N[(0.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[0.5 \cdot \left(x \cdot x - y\right) \]
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-sub0 N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, x, \color{blue}{0 - y}\right) \]

   4. --lowering--.f64 100.0

      \[\leadsto 0.5 \cdot \mathsf{fma}\left(x, x, \color{blue}{0 - y}\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(x, x, 0 - y\right)} \]
5. Step-by-step derivation

   1. sub0-neg N/A

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

   2. +-lft-identity N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left(\color{blue}{\left(0 + y\right)}\right)\right) \]

   3. neg-lowering-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, x, \color{blue}{\mathsf{neg}\left(\left(0 + y\right)\right)}\right) \]

   4. +-lft-identity 100.0

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

   \[\leadsto 0.5 \cdot \mathsf{fma}\left(x, x, 0 - y\right) \]
8. Add Preprocessing

Alternative 2: 87.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-23}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 2e-23) (* y -0.5) (* x (* 0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 2e-23) {
		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-23) 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-23) {
		tmp = y * -0.5;
	} else {
		tmp = x * (0.5 * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 2e-23:
		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-23)
		tmp = Float64(y * -0.5);
	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-23)
		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-23], N[(y * -0.5), $MachinePrecision], N[(x * N[(0.5 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.99999999999999992e-23

   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. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot y + 0} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{-1}{2}} + 0 \]

      3. accelerator-lowering-fma.f64 83.4

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, -0.5, 0\right)} \]

   5. Simplified 83.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, -0.5, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{y \cdot \frac{-1}{2}} \]

      2. +-lft-identity N/A

         \[\leadsto \color{blue}{\left(0 + y\right)} \cdot \frac{-1}{2} \]

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

         \[\leadsto \color{blue}{\left(0 + y\right) \cdot \frac{-1}{2}} \]

      4. +-lft-identity 83.4

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

   7. Applied egg-rr 83.4%

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

   if 1.99999999999999992e-23 < (*.f64 x x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left({x}^{2} + 0\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{x \cdot x} + 0\right) \]

      3. accelerator-lowering-fma.f64 84.5

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]

   5. Simplified 84.5%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-lowering-*.f64 84.5

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

   7. Applied egg-rr 84.5%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-23}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 100.0%

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

Alternative 4: 51.1% 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 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. +-rgt-identity N/A

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot y + 0} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{y \cdot \frac{-1}{2}} + 0 \]

   3. accelerator-lowering-fma.f64 46.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, -0.5, 0\right)} \]

5. Simplified 46.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, -0.5, 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{y \cdot \frac{-1}{2}} \]

   2. +-lft-identity N/A

      \[\leadsto \color{blue}{\left(0 + y\right)} \cdot \frac{-1}{2} \]

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

      \[\leadsto \color{blue}{\left(0 + y\right) \cdot \frac{-1}{2}} \]

   4. +-lft-identity 46.8

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

7. Applied egg-rr 46.8%

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

Reproduce

herbie shell --seed 2024195 
(FPCore (x y)
  :name "System.Random.MWC.Distributions:standard from mwc-random-0.13.3.2"
  :precision binary64
  (* 0.5 (- (* x x) y)))