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

Percentage Accurate: 100.0% → 100.0%

Time: 3.4s

Alternatives: 5

Speedup: 1.0×

• 21.1% 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. Final simplification 100.0%

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

Alternative 2: 85.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-65} \lor \neg \left(x \leq 1.75 \cdot 10^{-10}\right):\\ \;\;\;\;x - \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.5e-65) (not (<= x 1.75e-10))) (- x (/ x z)) (+ x (/ y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.5e-65) || !(x <= 1.75e-10)) {
		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 <= (-4.5d-65)) .or. (.not. (x <= 1.75d-10))) 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 <= -4.5e-65) || !(x <= 1.75e-10)) {
		tmp = x - (x / z);
	} else {
		tmp = x + (y / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -4.5e-65) or not (x <= 1.75e-10):
		tmp = x - (x / z)
	else:
		tmp = x + (y / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4.5e-65) || !(x <= 1.75e-10))
		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 <= -4.5e-65) || ~((x <= 1.75e-10)))
		tmp = x - (x / z);
	else
		tmp = x + (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4.5e-65], N[Not[LessEqual[x, 1.75e-10]], $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 < -4.4999999999999998e-65 or 1.7499999999999999e-10 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 91.2%

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

      1. metadata-eval 91.2%

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

      2. times-frac 91.2%

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

      3. *-lft-identity 91.2%

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

      4. neg-mul-1 91.2%

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

   4. Simplified 91.2%

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

   5. Taylor expanded in x around 0 91.1%

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

      1. sub-neg 91.1%

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

      2. distribute-neg-frac 91.1%

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

      3. metadata-eval 91.1%

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

      4. distribute-lft-in 91.2%

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

      5. *-rgt-identity 91.2%

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

      6. associate-*r/ 91.2%

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

      7. *-commutative 91.2%

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

      8. associate-*r/ 91.2%

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

      9. mul-1-neg 91.2%

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

      10. sub-neg 91.2%

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

   7. Simplified 91.2%

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

   if -4.4999999999999998e-65 < x < 1.7499999999999999e-10

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 91.4%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-65} \lor \neg \left(x \leq 1.75 \cdot 10^{-10}\right):\\ \;\;\;\;x - \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z}\\ \end{array} \]

Alternative 3: 61.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -11200:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 7.2:\\ \;\;\;\;\frac{-x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -11200.0) x (if (<= z 7.2) (/ (- x) z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -11200.0) {
		tmp = x;
	} else if (z <= 7.2) {
		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 <= (-11200.0d0)) then
        tmp = x
    else if (z <= 7.2d0) 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 <= -11200.0) {
		tmp = x;
	} else if (z <= 7.2) {
		tmp = -x / z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -11200.0:
		tmp = x
	elif z <= 7.2:
		tmp = -x / z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -11200.0)
		tmp = x;
	elseif (z <= 7.2)
		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 <= -11200.0)
		tmp = x;
	elseif (z <= 7.2)
		tmp = -x / z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -11200.0], x, If[LessEqual[z, 7.2], N[((-x) / z), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -11200 or 7.20000000000000018 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 73.8%

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

      1. metadata-eval 73.8%

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

      2. times-frac 73.8%

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

      3. *-lft-identity 73.8%

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

      4. neg-mul-1 73.8%

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

   4. Simplified 73.8%

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

   5. Taylor expanded in z around inf 71.8%

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

   if -11200 < z < 7.20000000000000018

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 56.6%

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

      1. metadata-eval 56.6%

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

      2. times-frac 56.6%

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

      3. *-lft-identity 56.6%

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

      4. neg-mul-1 56.6%

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

   4. Simplified 56.6%

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

   5. Taylor expanded in x around 0 56.5%

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

      1. sub-neg 56.5%

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

      2. distribute-neg-frac 56.5%

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

      3. metadata-eval 56.5%

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

      4. distribute-lft-in 56.5%

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

      5. *-rgt-identity 56.5%

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

      6. associate-*r/ 56.6%

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

      7. *-commutative 56.6%

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

      8. associate-*r/ 56.6%

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

      9. mul-1-neg 56.6%

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

      10. sub-neg 56.6%

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

   7. Simplified 56.6%

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

   8. Taylor expanded in z around 0 56.0%

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

      1. associate-*r/ 56.0%

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

      2. neg-mul-1 56.0%

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

   10. Simplified 56.0%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -11200:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 7.2:\\ \;\;\;\;\frac{-x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+198}:\\ \;\;\;\;\frac{-x}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.35e+198) (/ (- x) z) (+ x (/ y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.35e+198) {
		tmp = -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 <= (-1.35d+198)) then
        tmp = -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 <= -1.35e+198) {
		tmp = -x / z;
	} else {
		tmp = x + (y / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.35e+198:
		tmp = -x / z
	else:
		tmp = x + (y / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.35e+198)
		tmp = Float64(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 <= -1.35e+198)
		tmp = -x / z;
	else
		tmp = x + (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.35e+198], N[((-x) / z), $MachinePrecision], N[(x + N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.35e198

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.9%

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

      1. metadata-eval 99.9%

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

      2. times-frac 99.9%

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

      3. *-lft-identity 99.9%

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

      4. neg-mul-1 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. sub-neg 99.7%

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

      2. distribute-neg-frac 99.7%

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

      3. metadata-eval 99.7%

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

      4. distribute-lft-in 99.8%

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

      5. *-rgt-identity 99.8%

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

      6. associate-*r/ 99.9%

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

      7. *-commutative 99.9%

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

      8. associate-*r/ 99.9%

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

      9. mul-1-neg 99.9%

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

      10. sub-neg 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in z around 0 63.6%

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

      1. associate-*r/ 63.6%

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

      2. neg-mul-1 63.6%

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

   10. Simplified 63.6%

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

   if -1.35e198 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 79.4%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+198}:\\ \;\;\;\;\frac{-x}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z}\\ \end{array} \]

Alternative 5: 37.0% 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 65.5%

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

   1. metadata-eval 65.5%

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

   2. times-frac 65.5%

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

   3. *-lft-identity 65.5%

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

   4. neg-mul-1 65.5%

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

4. Simplified 65.5%

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

5. Taylor expanded in z around inf 38.4%

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

6. Final simplification 38.4%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023297 
(FPCore (x y z)
  :name "Statistics.Sample:$swelfordMean from math-functions-0.1.5.2"
  :precision binary64
  (+ x (/ (- y x) z)))