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

Percentage Accurate: 100.0% → 100.0%

Time: 2.6s

Alternatives: 5

Speedup: 1.0×

• 25.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot x + y \cdot 2 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (* x x) (* y 2.0)))

C

double code(double x, double y) {
	return (x * x) + (y * 2.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * x) + (y * 2.0d0)
end function

Java

public static double code(double x, double y) {
	return (x * x) + (y * 2.0);
}

Python

def code(x, y):
	return (x * x) + (y * 2.0)

Julia

function code(x, y)
	return Float64(Float64(x * x) + Float64(y * 2.0))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * x) + (y * 2.0);
end

Wolfram

code[x_, y_] := N[(N[(x * x), $MachinePrecision] + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. associate-+l+ N/A

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

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

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{\left(y + y\right)}\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{y} + y\right)\right) \]

   4. count-2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(2 \cdot \color{blue}{y}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot \color{blue}{2}\right)\right) \]

   6. *-lowering-*.f64 100.0%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \color{blue}{2}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot x + y \cdot 2} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 88.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 6.1 \cdot 10^{-69}:\\ \;\;\;\;y + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 6.1e-69) (+ y y) (+ (* x x) y)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 6.1e-69) {
		tmp = y + y;
	} else {
		tmp = (x * 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 * x) <= 6.1d-69) then
        tmp = y + y
    else
        tmp = (x * x) + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x * x) <= 6.1e-69) {
		tmp = y + y;
	} else {
		tmp = (x * x) + y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 6.1e-69:
		tmp = y + y
	else:
		tmp = (x * x) + y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 6.1e-69)
		tmp = Float64(y + y);
	else
		tmp = Float64(Float64(x * x) + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * x) <= 6.1e-69)
		tmp = y + y;
	else
		tmp = (x * x) + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 6.1e-69], N[(y + y), $MachinePrecision], N[(N[(x * x), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 6.1000000000000003e-69

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{y}, y\right) \]
   4. Step-by-step derivation

   5. Simplified 89.2%

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

   if 6.1000000000000003e-69 < (*.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

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({x}^{2}\right)}, y\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), y\right) \]

      2. *-lowering-*.f64 88.5%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), y\right) \]

   5. Simplified 88.5%

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

Alternative 3: 87.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 6.5 \cdot 10^{-69}:\\ \;\;\;\;y + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* x x) 6.5e-69) (+ y y) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 6.5e-69) {
		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) <= 6.5d-69) 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) <= 6.5e-69) {
		tmp = y + y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 6.5e-69:
		tmp = y + y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 6.5e-69)
		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) <= 6.5e-69)
		tmp = y + y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 6.5e-69], N[(y + y), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 6.49999999999999951e-69

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{y}, y\right) \]
   4. Step-by-step derivation

   5. Simplified 89.2%

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

   if 6.49999999999999951e-69 < (*.f64 x x)

   1. Initial program 100.0%

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

      1. associate-+l+ N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{\left(y + y\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{y} + y\right)\right) \]

      4. count-2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(2 \cdot \color{blue}{y}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot \color{blue}{2}\right)\right) \]

      6. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \color{blue}{2}\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot x + y \cdot 2} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 86.7%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

   7. Simplified 86.7%

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

Alternative 4: 56.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.25 \cdot 10^{-212}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* x x) 1.25e-212) y (* x x)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 1.25e-212) {
		tmp = 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) <= 1.25d-212) then
        tmp = 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) <= 1.25e-212) {
		tmp = y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 1.25e-212:
		tmp = y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 1.25e-212)
		tmp = y;
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * x) <= 1.25e-212)
		tmp = y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 1.25e-212], y, N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.25000000000000011e-212

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({x}^{2}\right)}, y\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), y\right) \]

      2. *-lowering-*.f64 19.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), y\right) \]

   5. Simplified 19.6%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 18.6%

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

   if 1.25000000000000011e-212 < (*.f64 x x)

   1. Initial program 100.0%

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

      1. associate-+l+ N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{\left(y + y\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{y} + y\right)\right) \]

      4. count-2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(2 \cdot \color{blue}{y}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot \color{blue}{2}\right)\right) \]

      6. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \color{blue}{2}\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot x + y \cdot 2} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 75.1%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

   7. Simplified 75.1%

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

Alternative 5: 10.7% accurate, 7.0× 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

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({x}^{2}\right)}, y\right) \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), y\right) \]

   2. *-lowering-*.f64 57.1%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), y\right) \]

5. Simplified 57.1%

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

6. Taylor expanded in x around 0

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

8. Simplified 11.1%

   \[\leadsto \color{blue}{y} \]
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 2024160 
(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))