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

Percentage Accurate: 100.0% → 100.0%

Time: 2.6s

Alternatives: 5

Speedup: 1.0×

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. Final simplification 100.0%

   \[\leadsto x + \frac{y - x}{z} \]

Alternative 2: 75.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-134} \lor \neg \left(z \leq 1.75 \cdot 10^{-36}\right):\\ \;\;\;\;x + \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.05e-134) (not (<= z 1.75e-36))) (+ x (/ y z)) (/ (- x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.05e-134) || !(z <= 1.75e-36)) {
		tmp = x + (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 ((z <= (-1.05d-134)) .or. (.not. (z <= 1.75d-36))) then
        tmp = x + (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 ((z <= -1.05e-134) || !(z <= 1.75e-36)) {
		tmp = x + (y / z);
	} else {
		tmp = -x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.05e-134) or not (z <= 1.75e-36):
		tmp = x + (y / z)
	else:
		tmp = -x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.05e-134) || !(z <= 1.75e-36))
		tmp = Float64(x + 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 ((z <= -1.05e-134) || ~((z <= 1.75e-36)))
		tmp = x + (y / z);
	else
		tmp = -x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.05e-134], N[Not[LessEqual[z, 1.75e-36]], $MachinePrecision]], N[(x + N[(y / z), $MachinePrecision]), $MachinePrecision], N[((-x) / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.05e-134 or 1.75e-36 < z

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]

   2. Taylor expanded in y around inf 92.0%

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

   if -1.05e-134 < z < 1.75e-36

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]

   2. Taylor expanded in y around 0 70.1%

      \[\leadsto x + \color{blue}{-1 \cdot \frac{x}{z}} \]
   3. Step-by-step derivation

      1. neg-mul-1 70.1%

         \[\leadsto x + \color{blue}{\left(-\frac{x}{z}\right)} \]

      2. distribute-neg-frac 70.1%

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

   4. Simplified 70.1%

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

   5. Taylor expanded in z around 0 70.1%

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

      1. mul-1-neg 70.1%

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

      2. distribute-neg-frac 70.1%

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

   7. Simplified 70.1%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-134} \lor \neg \left(z \leq 1.75 \cdot 10^{-36}\right):\\ \;\;\;\;x + \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z}\\ \end{array} \]

Alternative 3: 84.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-93} \lor \neg \left(x \leq 7.2 \cdot 10^{-14}\right):\\ \;\;\;\;x - \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.2e-93) (not (<= x 7.2e-14))) (- x (/ x z)) (+ x (/ y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.2e-93) || !(x <= 7.2e-14)) {
		tmp = x - (x / z);
	} else {
		tmp = x + (y / 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 <= (-4.2d-93)) .or. (.not. (x <= 7.2d-14))) then
        tmp = x - (x / z)
    else
        tmp = x + (y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.2e-93) || !(x <= 7.2e-14)) {
		tmp = x - (x / z);
	} else {
		tmp = x + (y / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -4.2e-93) or not (x <= 7.2e-14):
		tmp = x - (x / z)
	else:
		tmp = x + (y / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4.2e-93) || !(x <= 7.2e-14))
		tmp = Float64(x - Float64(x / z));
	else
		tmp = Float64(x + Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -4.2e-93) || ~((x <= 7.2e-14)))
		tmp = x - (x / z);
	else
		tmp = x + (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4.2e-93], N[Not[LessEqual[x, 7.2e-14]], $MachinePrecision]], N[(x - N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.2000000000000002e-93 or 7.1999999999999996e-14 < x

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]

   2. Taylor expanded in y around 0 90.7%

      \[\leadsto x + \color{blue}{-1 \cdot \frac{x}{z}} \]
   3. Step-by-step derivation

      1. neg-mul-1 90.7%

         \[\leadsto x + \color{blue}{\left(-\frac{x}{z}\right)} \]

      2. distribute-neg-frac 90.7%

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

   4. Simplified 90.7%

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

   5. Taylor expanded in x around 0 90.6%

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

      1. *-commutative 90.6%

         \[\leadsto \color{blue}{x \cdot \left(1 - \frac{1}{z}\right)} \]

      2. distribute-lft-out-- 90.7%

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

      3. *-rgt-identity 90.7%

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

      4. associate-*r/ 90.7%

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

      5. *-rgt-identity 90.7%

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

   7. Simplified 90.7%

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

   if -4.2000000000000002e-93 < x < 7.1999999999999996e-14

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]

   2. Taylor expanded in y around inf 92.2%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-93} \lor \neg \left(x \leq 7.2 \cdot 10^{-14}\right):\\ \;\;\;\;x - \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z}\\ \end{array} \]

Alternative 4: 61.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.9:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 0.00012:\\ \;\;\;\;\frac{-x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -0.9) x (if (<= z 0.00012) (/ (- x) z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.9) {
		tmp = x;
	} else if (z <= 0.00012) {
		tmp = -x / z;
	} else {
		tmp = x;
	}
	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 (z <= (-0.9d0)) then
        tmp = x
    else if (z <= 0.00012d0) then
        tmp = -x / z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.9) {
		tmp = x;
	} else if (z <= 0.00012) {
		tmp = -x / z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -0.9:
		tmp = x
	elif z <= 0.00012:
		tmp = -x / z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -0.9)
		tmp = x;
	elseif (z <= 0.00012)
		tmp = Float64(Float64(-x) / z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -0.9)
		tmp = x;
	elseif (z <= 0.00012)
		tmp = -x / z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -0.9], x, If[LessEqual[z, 0.00012], N[((-x) / z), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -0.900000000000000022 or 1.20000000000000003e-4 < z

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]

   2. Taylor expanded in y around 0 70.6%

      \[\leadsto x + \color{blue}{-1 \cdot \frac{x}{z}} \]
   3. Step-by-step derivation

      1. neg-mul-1 70.6%

         \[\leadsto x + \color{blue}{\left(-\frac{x}{z}\right)} \]

      2. distribute-neg-frac 70.6%

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

   4. Simplified 70.6%

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

   5. Taylor expanded in z around inf 68.5%

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

   if -0.900000000000000022 < z < 1.20000000000000003e-4

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]

   2. Taylor expanded in y around 0 61.1%

      \[\leadsto x + \color{blue}{-1 \cdot \frac{x}{z}} \]
   3. Step-by-step derivation

      1. neg-mul-1 61.1%

         \[\leadsto x + \color{blue}{\left(-\frac{x}{z}\right)} \]

      2. distribute-neg-frac 61.1%

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

   4. Simplified 61.1%

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

   5. Taylor expanded in z around 0 60.3%

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

      1. mul-1-neg 60.3%

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

      2. distribute-neg-frac 60.3%

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

   7. Simplified 60.3%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.9:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 0.00012:\\ \;\;\;\;\frac{-x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 37.1% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 100.0%

   \[x + \frac{y - x}{z} \]

2. Taylor expanded in y around 0 66.0%

   \[\leadsto x + \color{blue}{-1 \cdot \frac{x}{z}} \]
3. Step-by-step derivation

   1. neg-mul-1 66.0%

      \[\leadsto x + \color{blue}{\left(-\frac{x}{z}\right)} \]

   2. distribute-neg-frac 66.0%

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

4. Simplified 66.0%

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

5. Taylor expanded in z around inf 37.1%

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

6. Final simplification 37.1%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023221 
(FPCore (x y z)
  :name "Statistics.Sample:$swelfordMean from math-functions-0.1.5.2"
  :precision binary64
  (+ x (/ (- y x) z)))