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

Percentage Accurate: 100.0% → 100.0%

Time: 1.5s

Alternatives: 6

Speedup: 1.0×

• 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, 0.1× 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. Step-by-step derivation

   1. associate-+l+ 100.0%

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

   2. fma-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 84.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 3.2 \cdot 10^{-127} \lor \neg \left(x \cdot x \leq 1.5 \cdot 10^{-91}\right) \land \left(x \cdot x \leq 10^{-62} \lor \neg \left(x \cdot x \leq 63000000\right) \land x \cdot x \leq 2.6 \cdot 10^{+83}\right):\\ \;\;\;\;y + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* x x) 3.2e-127)
         (and (not (<= (* x x) 1.5e-91))
              (or (<= (* x x) 1e-62)
                  (and (not (<= (* x x) 63000000.0)) (<= (* x x) 2.6e+83)))))
   (+ y y)
   (* x x)))

C

double code(double x, double y) {
	double tmp;
	if (((x * x) <= 3.2e-127) || (!((x * x) <= 1.5e-91) && (((x * x) <= 1e-62) || (!((x * x) <= 63000000.0) && ((x * x) <= 2.6e+83))))) {
		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) <= 3.2d-127) .or. (.not. ((x * x) <= 1.5d-91)) .and. ((x * x) <= 1d-62) .or. (.not. ((x * x) <= 63000000.0d0)) .and. ((x * x) <= 2.6d+83)) 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) <= 3.2e-127) || (!((x * x) <= 1.5e-91) && (((x * x) <= 1e-62) || (!((x * x) <= 63000000.0) && ((x * x) <= 2.6e+83))))) {
		tmp = y + y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * x) <= 3.2e-127) or (not ((x * x) <= 1.5e-91) and (((x * x) <= 1e-62) or (not ((x * x) <= 63000000.0) and ((x * x) <= 2.6e+83)))):
		tmp = y + y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(x * x) <= 3.2e-127) || (!(Float64(x * x) <= 1.5e-91) && ((Float64(x * x) <= 1e-62) || (!(Float64(x * x) <= 63000000.0) && (Float64(x * x) <= 2.6e+83)))))
		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) <= 3.2e-127) || (~(((x * x) <= 1.5e-91)) && (((x * x) <= 1e-62) || (~(((x * x) <= 63000000.0)) && ((x * x) <= 2.6e+83)))))
		tmp = y + y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(x * x), $MachinePrecision], 3.2e-127], And[N[Not[LessEqual[N[(x * x), $MachinePrecision], 1.5e-91]], $MachinePrecision], Or[LessEqual[N[(x * x), $MachinePrecision], 1e-62], And[N[Not[LessEqual[N[(x * x), $MachinePrecision], 63000000.0]], $MachinePrecision], LessEqual[N[(x * x), $MachinePrecision], 2.6e+83]]]]], N[(y + y), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 3.20000000000000017e-127 or 1.5000000000000001e-91 < (*.f64 x x) < 1e-62 or 6.3e7 < (*.f64 x x) < 2.6000000000000001e83

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 92.1%

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

   if 3.20000000000000017e-127 < (*.f64 x x) < 1.5000000000000001e-91 or 1e-62 < (*.f64 x x) < 6.3e7 or 2.6000000000000001e83 < (*.f64 x x)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 92.6%

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

      1. unpow2 92.6%

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

   4. Simplified 92.6%

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

   5. Taylor expanded in x around inf 91.3%

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

      1. unpow2 92.6%

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

   7. Simplified 91.3%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 3.2 \cdot 10^{-127} \lor \neg \left(x \cdot x \leq 1.5 \cdot 10^{-91}\right) \land \left(x \cdot x \leq 10^{-62} \lor \neg \left(x \cdot x \leq 63000000\right) \land x \cdot x \leq 2.6 \cdot 10^{+83}\right):\\ \;\;\;\;y + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 3: 87.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2.65 \cdot 10^{-135}:\\ \;\;\;\;y + y\\ \mathbf{elif}\;x \cdot x \leq 1.06 \cdot 10^{-91}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \cdot x \leq 4.8 \cdot 10^{-62}:\\ \;\;\;\;y + y\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 2.65e-135)
   (+ y y)
   (if (<= (* x x) 1.06e-91)
     (* x x)
     (if (<= (* x x) 4.8e-62) (+ y y) (+ y (* x x))))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 2.65e-135) {
		tmp = y + y;
	} else if ((x * x) <= 1.06e-91) {
		tmp = x * x;
	} else if ((x * x) <= 4.8e-62) {
		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) <= 2.65d-135) then
        tmp = y + y
    else if ((x * x) <= 1.06d-91) then
        tmp = x * x
    else if ((x * x) <= 4.8d-62) 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) <= 2.65e-135) {
		tmp = y + y;
	} else if ((x * x) <= 1.06e-91) {
		tmp = x * x;
	} else if ((x * x) <= 4.8e-62) {
		tmp = y + y;
	} else {
		tmp = y + (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 2.65e-135:
		tmp = y + y
	elif (x * x) <= 1.06e-91:
		tmp = x * x
	elif (x * x) <= 4.8e-62:
		tmp = y + y
	else:
		tmp = y + (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 2.65e-135)
		tmp = Float64(y + y);
	elseif (Float64(x * x) <= 1.06e-91)
		tmp = Float64(x * x);
	elseif (Float64(x * x) <= 4.8e-62)
		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) <= 2.65e-135)
		tmp = y + y;
	elseif ((x * x) <= 1.06e-91)
		tmp = x * x;
	elseif ((x * x) <= 4.8e-62)
		tmp = y + y;
	else
		tmp = y + (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 2.65e-135], N[(y + y), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 1.06e-91], N[(x * x), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 4.8e-62], N[(y + y), $MachinePrecision], N[(y + N[(x * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 2.65e-135 or 1.06000000000000006e-91 < (*.f64 x x) < 4.79999999999999967e-62

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 94.2%

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

   if 2.65e-135 < (*.f64 x x) < 1.06000000000000006e-91

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. unpow2 100.0%

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

   7. Simplified 100.0%

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

   if 4.79999999999999967e-62 < (*.f64 x x)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 87.7%

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

      1. unpow2 87.7%

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

   4. Simplified 87.7%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2.65 \cdot 10^{-135}:\\ \;\;\;\;y + y\\ \mathbf{elif}\;x \cdot x \leq 1.06 \cdot 10^{-91}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \cdot x \leq 4.8 \cdot 10^{-62}:\\ \;\;\;\;y + y\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot x\\ \end{array} \]

Alternative 4: 55.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 3 \cdot 10^{-244}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* x x) 3e-244) y (* x x)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 3e-244) {
		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) <= 3d-244) 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) <= 3e-244) {
		tmp = y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 3e-244:
		tmp = y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 3e-244)
		tmp = y;
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * x) <= 3e-244)
		tmp = y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 3e-244], y, N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 3.0000000000000001e-244

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 18.8%

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

      1. unpow2 18.8%

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

   4. Simplified 18.8%

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

   5. Taylor expanded in x around 0 18.8%

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

   if 3.0000000000000001e-244 < (*.f64 x x)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 78.8%

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

      1. unpow2 78.8%

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

   4. Simplified 78.8%

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

   5. Taylor expanded in x around inf 75.1%

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

      1. unpow2 78.8%

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

   7. Simplified 75.1%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 3 \cdot 10^{-244}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

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 60.5%

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

   1. unpow2 60.5%

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

4. Simplified 60.5%

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

5. Taylor expanded in x around 0 10.6%

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

6. Final simplification 10.6%

   \[\leadsto y \]

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 2023271 
(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))