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

Percentage Accurate: 100.0% → 100.0%

Time: 2.7s

Alternatives: 4

Speedup: 1.2×

• 25.3% 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: 87.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-11}:\\ \;\;\;\;y + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 2e-11) (+ y y) (fma x x 0.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 2e-11) {
		tmp = y + y;
	} else {
		tmp = fma(x, x, 0.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 2e-11)
		tmp = Float64(y + y);
	else
		tmp = fma(x, x, 0.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e-11], N[(y + y), $MachinePrecision], N[(x * x + 0.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.99999999999999988e-11

   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 90.6%

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

   if 1.99999999999999988e-11 < (*.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. +-rgt-identity N/A

         \[\leadsto \color{blue}{{x}^{2} + 0} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{x \cdot x} + 0 \]

      3. accelerator-lowering-fma.f64 92.7

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

   5. Simplified 92.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 50.4% 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 55.5%

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

Alternative 4: 2.7% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.0)

C

double code(double x, double y) {
	return 0.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.0d0
end function

Java

public static double code(double x, double y) {
	return 0.0;
}

Python

def code(x, y):
	return 0.0

Julia

function code(x, y)
	return 0.0
end

MATLAB

function tmp = code(x, y)
	tmp = 0.0;
end

Wolfram

code[x_, y_] := 0.0

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 55.5%

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

   1. flip-+ N/A

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

   2. +-inverses N/A

      \[\leadsto \frac{\color{blue}{0}}{y - y} \]

   3. +-inverses N/A

      \[\leadsto \frac{0}{\color{blue}{0}} \]

   4. div0 3.0

      \[\leadsto \color{blue}{0} \]

7. Applied egg-rr 3.0%

   \[\leadsto \color{blue}{0} \]
8. 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 2024196 
(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))