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

Percentage Accurate: 100.0% → 100.0%

Time: 1.4s

Alternatives: 6

Speedup: 1.0×

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

Specification

Math

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

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, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, 2, x \cdot x\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. associate-+l+ 100.0%

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

   2. +-commutative 100.0%

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

   3. count-2 100.0%

      \[\leadsto \color{blue}{2 \cdot y} + x \cdot x \]

   4. *-commutative 100.0%

      \[\leadsto \color{blue}{y \cdot 2} + x \cdot x \]

   5. fma-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(y, 2, x \cdot x\right) \]

Alternative 2: 88.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5.4 \cdot 10^{+52}:\\ \;\;\;\;y + y\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 5.4e+52) (+ y y) (+ y (* x x))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 5.4e+52) {
		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) <= 5.4d+52) 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) <= 5.4e+52) {
		tmp = y + y;
	} else {
		tmp = y + (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 5.4e+52:
		tmp = y + y
	else:
		tmp = y + (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 5.4e+52)
		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) <= 5.4e+52)
		tmp = y + y;
	else
		tmp = y + (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 5.4e+52], N[(y + y), $MachinePrecision], N[(y + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.4e52

   1. Initial program 100.0%

      \[\left(x \cdot x + y\right) + y \]

   2. Taylor expanded in x around 0 90.5%

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

   if 5.4e52 < (*.f64 x x)

   1. Initial program 100.0%

      \[\left(x \cdot x + y\right) + y \]

   2. Taylor expanded in x around inf 90.8%

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

      1. unpow2 90.8%

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

   4. Simplified 90.8%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5.4 \cdot 10^{+52}:\\ \;\;\;\;y + y\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot x\\ \end{array} \]

Alternative 3: 55.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2.02 \cdot 10^{-120}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* x x) 2.02e-120) y (* x x)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 2.02e-120) {
		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) <= 2.02d-120) 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) <= 2.02e-120) {
		tmp = y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 2.02e-120:
		tmp = y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 2.02e-120)
		tmp = y;
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * x) <= 2.02e-120)
		tmp = y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 2.02e-120], y, N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.0200000000000001e-120

   1. Initial program 100.0%

      \[\left(x \cdot x + y\right) + y \]

   2. Taylor expanded in x around inf 20.8%

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

      1. unpow2 20.8%

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

   4. Simplified 20.8%

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

   5. Taylor expanded in x around 0 18.4%

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

   if 2.0200000000000001e-120 < (*.f64 x x)

   1. Initial program 100.0%

      \[\left(x \cdot x + y\right) + y \]

   2. Taylor expanded in x around inf 80.0%

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

      1. unpow2 80.0%

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

   4. Simplified 80.0%

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

   5. Taylor expanded in x around inf 76.0%

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

      1. unpow2 80.0%

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

   7. Simplified 76.0%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2.02 \cdot 10^{-120}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 4: 87.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.85 \cdot 10^{+52}:\\ \;\;\;\;y + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* x x) 1.85e+52) (+ y y) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 1.85e+52) {
		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) <= 1.85d+52) 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) <= 1.85e+52) {
		tmp = y + y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 1.85e+52:
		tmp = y + y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 1.85e+52)
		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) <= 1.85e+52)
		tmp = y + y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 1.85e+52], N[(y + y), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.85e52

   1. Initial program 100.0%

      \[\left(x \cdot x + y\right) + y \]

   2. Taylor expanded in x around 0 90.5%

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

   if 1.85e52 < (*.f64 x x)

   1. Initial program 100.0%

      \[\left(x \cdot x + y\right) + y \]

   2. Taylor expanded in x around inf 90.8%

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

      1. unpow2 90.8%

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

   4. Simplified 90.8%

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

   5. Taylor expanded in x around inf 89.1%

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

      1. unpow2 90.8%

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

   7. Simplified 89.1%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.85 \cdot 10^{+52}:\\ \;\;\;\;y + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 5: 100.0% accurate, 1.0× speedup

Math

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

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(y + Float64(y + Float64(x * x)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(x \cdot x + y\right) + y \]

2. Final simplification 100.0%

   \[\leadsto y + \left(y + x \cdot x\right) \]

Alternative 6: 10.8% 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. Taylor expanded in x around inf 55.5%

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

   1. unpow2 55.5%

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

4. Simplified 55.5%

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

5. Taylor expanded in x around 0 11.5%

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

6. Final simplification 11.5%

   \[\leadsto y \]

Developer target: 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 2023290 
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalTail from random-fu-0.2.6.2"
  :precision binary64

  :herbie-target
  (+ (+ y y) (* x x))

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