Statistics.Sample:$swelfordMean from math-functions-0.1.5.2

Percentage Accurate: 100.0% → 100.0%

Time: 4.9s

Alternatives: 5

Speedup: 1.0×

• 20.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x + \frac{y - x}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x (/ (- y x) z)))

C

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

Fortran

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

Java

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

Python

def code(x, y, z):
	return x + ((y - x) / z)

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(y - x) / z))
end

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(y - x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{y - x}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x (/ (- y x) z)))

C

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

Fortran

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

Java

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

Python

def code(x, y, z):
	return x + ((y - x) / z)

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(y - x) / z))
end

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(y - x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{y - x}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x (/ (- y x) z)))

C

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

Fortran

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

Java

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

Python

def code(x, y, z):
	return x + ((y - x) / z)

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(y - x) / z))
end

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(y - x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x + \frac{y - x}{z} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 80.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-51} \lor \neg \left(x \leq 3.5 \cdot 10^{+95}\right):\\ \;\;\;\;x - \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.4e-51) (not (<= x 3.5e+95))) (- x (/ x z)) (/ (- y x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.4e-51) || !(x <= 3.5e+95)) {
		tmp = x - (x / z);
	} else {
		tmp = (y - x) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-2.4d-51)) .or. (.not. (x <= 3.5d+95))) then
        tmp = x - (x / z)
    else
        tmp = (y - x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.4e-51) || !(x <= 3.5e+95)) {
		tmp = x - (x / z);
	} else {
		tmp = (y - x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.4e-51) or not (x <= 3.5e+95):
		tmp = x - (x / z)
	else:
		tmp = (y - x) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.4e-51) || !(x <= 3.5e+95))
		tmp = Float64(x - Float64(x / z));
	else
		tmp = Float64(Float64(y - x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.4e-51) || ~((x <= 3.5e+95)))
		tmp = x - (x / z);
	else
		tmp = (y - x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.4e-51], N[Not[LessEqual[x, 3.5e+95]], $MachinePrecision]], N[(x - N[(x / z), $MachinePrecision]), $MachinePrecision], N[(N[(y - x), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4e-51 or 3.5e95 < x

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{1}{z}\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-out-- N/A

         \[\leadsto \color{blue}{1 \cdot x - \frac{1}{z} \cdot x} \]

      2. *-lft-identity N/A

         \[\leadsto \color{blue}{x} - \frac{1}{z} \cdot x \]

      3. associate-*l/ N/A

         \[\leadsto x - \color{blue}{\frac{1 \cdot x}{z}} \]

      4. *-lft-identity N/A

         \[\leadsto x - \frac{\color{blue}{x}}{z} \]

      5. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{x}{z}} \]

      6. lower-/.f64 87.3

         \[\leadsto x - \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 87.3%

      \[\leadsto \color{blue}{x - \frac{x}{z}} \]

   if -2.4e-51 < x < 3.5e95

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y - x}{z}} \]

      2. lower--.f64 79.4

         \[\leadsto \frac{\color{blue}{y - x}}{z} \]

   5. Applied rewrites 79.4%

      \[\leadsto \color{blue}{\frac{y - x}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-51} \lor \neg \left(x \leq 3.5 \cdot 10^{+95}\right):\\ \;\;\;\;x - \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+37} \lor \neg \left(y \leq 4 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.3e+37) (not (<= y 4e-15))) (/ y z) (- x (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.3e+37) || !(y <= 4e-15)) {
		tmp = y / z;
	} else {
		tmp = x - (x / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-3.3d+37)) .or. (.not. (y <= 4d-15))) then
        tmp = y / z
    else
        tmp = x - (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.3e+37) || !(y <= 4e-15)) {
		tmp = y / z;
	} else {
		tmp = x - (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.3e+37) or not (y <= 4e-15):
		tmp = y / z
	else:
		tmp = x - (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.3e+37) || !(y <= 4e-15))
		tmp = Float64(y / z);
	else
		tmp = Float64(x - Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -3.3e+37) || ~((y <= 4e-15)))
		tmp = y / z;
	else
		tmp = x - (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.3e+37], N[Not[LessEqual[y, 4e-15]], $MachinePrecision]], N[(y / z), $MachinePrecision], N[(x - N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.3000000000000001e37 or 4.0000000000000003e-15 < y

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 76.4

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

   5. Applied rewrites 76.4%

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

   if -3.3000000000000001e37 < y < 4.0000000000000003e-15

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{1}{z}\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-out-- N/A

         \[\leadsto \color{blue}{1 \cdot x - \frac{1}{z} \cdot x} \]

      2. *-lft-identity N/A

         \[\leadsto \color{blue}{x} - \frac{1}{z} \cdot x \]

      3. associate-*l/ N/A

         \[\leadsto x - \color{blue}{\frac{1 \cdot x}{z}} \]

      4. *-lft-identity N/A

         \[\leadsto x - \frac{\color{blue}{x}}{z} \]

      5. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{x}{z}} \]

      6. lower-/.f64 79.5

         \[\leadsto x - \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 79.5%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+37} \lor \neg \left(y \leq 4 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-156} \lor \neg \left(y \leq 9.5 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.9e-156) (not (<= y 9.5e-16))) (/ y z) (/ (- x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.9e-156) || !(y <= 9.5e-16)) {
		tmp = y / z;
	} else {
		tmp = -x / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-1.9d-156)) .or. (.not. (y <= 9.5d-16))) then
        tmp = y / z
    else
        tmp = -x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.9e-156) || !(y <= 9.5e-16)) {
		tmp = y / z;
	} else {
		tmp = -x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.9e-156) or not (y <= 9.5e-16):
		tmp = y / z
	else:
		tmp = -x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.9e-156) || !(y <= 9.5e-16))
		tmp = Float64(y / z);
	else
		tmp = Float64(Float64(-x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.9e-156) || ~((y <= 9.5e-16)))
		tmp = y / z;
	else
		tmp = -x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.9e-156], N[Not[LessEqual[y, 9.5e-16]], $MachinePrecision]], N[(y / z), $MachinePrecision], N[((-x) / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.90000000000000004e-156 or 9.5000000000000005e-16 < y

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 66.5

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

   5. Applied rewrites 66.5%

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

   if -1.90000000000000004e-156 < y < 9.5000000000000005e-16

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y - x}{z}} \]

      2. lower--.f64 56.6

         \[\leadsto \frac{\color{blue}{y - x}}{z} \]

   5. Applied rewrites 56.6%

      \[\leadsto \color{blue}{\frac{y - x}{z}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-1 \cdot x}{z} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.7%

      \[\leadsto \frac{-x}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-156} \lor \neg \left(y \leq 9.5 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 41.8% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (/ y z))

C

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

Fortran

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

Java

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

Python

def code(x, y, z):
	return y / z

Julia

function code(x, y, z)
	return Float64(y / z)
end

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(y / z), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x + \frac{y - x}{z} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. lower-/.f64 48.1

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

5. Applied rewrites 48.1%

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

Reproduce

herbie shell --seed 2024322 
(FPCore (x y z)
  :name "Statistics.Sample:$swelfordMean from math-functions-0.1.5.2"
  :precision binary64
  (+ x (/ (- y x) z)))