Data.Random.Distribution.Normal:normalTail from random-fu-0.2.6.2

Percentage Accurate: 100.0% → 100.0%

Time: 3.0s

Alternatives: 6

Speedup: 1.2×

• 24.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x y) :precision binary64 (+ (+ (* x x) y) y))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * x) + y) + y
end function

Java

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

Python

def code(x, y):
	return ((x * x) + y) + y

Julia

function code(x, y)
	return Float64(Float64(Float64(x * x) + y) + y)
end

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (+ (+ (* x x) y) y))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * x) + y) + y
end function

Java

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

Python

def code(x, y):
	return ((x * x) + y) + y

Julia

function code(x, y)
	return Float64(Float64(Float64(x * x) + y) + y)
end

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (fma x x (+ y y)))

C

double code(double x, double y) {
	return fma(x, x, (y + y));
}

Julia

function code(x, y)
	return fma(x, x, Float64(y + y))
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. associate-+l+ N/A

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

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

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

   3. +-lowering-+.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 89.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-30}:\\ \;\;\;\;y + y\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 5e-30) (+ y y) (+ y (* x x))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 5e-30) {
		tmp = y + y;
	} else {
		tmp = y + (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) <= 5d-30) then
        tmp = y + y
    else
        tmp = y + (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x * x) <= 5e-30) {
		tmp = y + y;
	} else {
		tmp = y + (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 5e-30:
		tmp = y + y
	else:
		tmp = y + (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 5e-30)
		tmp = Float64(y + y);
	else
		tmp = Float64(y + Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * x) <= 5e-30)
		tmp = y + y;
	else
		tmp = y + (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-30], N[(y + y), $MachinePrecision], N[(y + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.99999999999999972e-30

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y} + y \]
   4. Step-by-step derivation

   5. Simplified 90.8%

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

   if 4.99999999999999972e-30 < (*.f64 x x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 87.9

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

   5. Simplified 87.9%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-30}:\\ \;\;\;\;y + y\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-30}:\\ \;\;\;\;y + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* x x) 5e-30) (+ y y) (fma x x y)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 5e-30) {
		tmp = y + y;
	} else {
		tmp = fma(x, x, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 5e-30)
		tmp = Float64(y + y);
	else
		tmp = fma(x, x, y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-30], N[(y + y), $MachinePrecision], N[(x * x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.99999999999999972e-30

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y} + y \]
   4. Step-by-step derivation

   5. Simplified 90.8%

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

   if 4.99999999999999972e-30 < (*.f64 x x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 87.9

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

   5. Simplified 87.9%

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

      1. accelerator-lowering-fma.f64 87.9

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

   7. Applied egg-rr 87.9%

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

Alternative 4: 88.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 9.6 \cdot 10^{-29}:\\ \;\;\;\;y + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* x x) 9.6e-29) (+ y y) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 9.6e-29) {
		tmp = y + y;
	} else {
		tmp = 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) <= 9.6d-29) then
        tmp = y + y
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x * x) <= 9.6e-29) {
		tmp = y + y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 9.6e-29:
		tmp = y + y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 9.6e-29)
		tmp = Float64(y + y);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * x) <= 9.6e-29)
		tmp = y + y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 9.6e-29], N[(y + y), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 9.59999999999999968e-29

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y} + y \]
   4. Step-by-step derivation

   5. Simplified 90.8%

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

   if 9.59999999999999968e-29 < (*.f64 x x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 85.6

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

   5. Simplified 85.6%

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

Alternative 5: 49.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ y + y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ y y))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return y + y

Julia

function code(x, y)
	return Float64(y + y)
end

MATLAB

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

Wolfram

code[x_, y_] := N[(y + y), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{y} + y \]
4. Step-by-step derivation

5. Simplified 50.6%

   \[\leadsto \color{blue}{y} + y \]
6. Add Preprocessing

Alternative 6: 10.6% accurate, 12.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 y)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return y

Julia

function code(x, y)
	return y
end

MATLAB

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

Wolfram

code[x_, y_] := y

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 58.6

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

5. Simplified 58.6%

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

6. Taylor expanded in x around 0

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

8. Simplified 10.9%

   \[\leadsto \color{blue}{y} \]
9. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (+ (+ y y) (* x x)))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (y + y) + (x * x)
end function

Java

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

Python

def code(x, y):
	return (y + y) + (x * x)

Julia

function code(x, y)
	return Float64(Float64(y + y) + Float64(x * x))
end

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024199 
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalTail from random-fu-0.2.6.2"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (+ y y) (* x x)))

  (+ (+ (* x x) y) y))