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

Percentage Accurate: 100.0% → 100.0%

Time: 4.0s

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. Add Preprocessing

3. Final simplification 100.0%

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

Alternative 2: 87.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-95} \lor \neg \left(y \leq 1.1 \cdot 10^{-60}\right):\\ \;\;\;\;x + \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -7e-95) (not (<= y 1.1e-60))) (+ x (/ y z)) (- x (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -7e-95) || !(y <= 1.1e-60)) {
		tmp = x + (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 <= (-7d-95)) .or. (.not. (y <= 1.1d-60))) then
        tmp = x + (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 <= -7e-95) || !(y <= 1.1e-60)) {
		tmp = x + (y / z);
	} else {
		tmp = x - (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -7e-95) or not (y <= 1.1e-60):
		tmp = x + (y / z)
	else:
		tmp = x - (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -7e-95) || !(y <= 1.1e-60))
		tmp = Float64(x + 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 <= -7e-95) || ~((y <= 1.1e-60)))
		tmp = x + (y / z);
	else
		tmp = x - (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -7e-95], N[Not[LessEqual[y, 1.1e-60]], $MachinePrecision]], N[(x + N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x - N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.9999999999999994e-95 or 1.0999999999999999e-60 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 87.8%

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

   if -6.9999999999999994e-95 < y < 1.0999999999999999e-60

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 85.7%

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

      1. neg-mul-1 85.7%

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

      2. distribute-neg-frac 85.7%

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

   5. Simplified 85.7%

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

   6. Taylor expanded in x around 0 85.5%

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

      1. sub-neg 85.5%

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

      2. distribute-rgt-in 85.5%

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

      3. *-lft-identity 85.5%

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

      4. cancel-sign-sub-inv 85.5%

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

      5. associate-*l/ 85.7%

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

      6. *-lft-identity 85.7%

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

   8. Simplified 85.7%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-95} \lor \neg \left(y \leq 1.1 \cdot 10^{-60}\right):\\ \;\;\;\;x + \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{-z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0) x (if (<= z 1.0) (/ x (- z)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = x;
	} else if (z <= 1.0) {
		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 <= (-1.0d0)) then
        tmp = x
    else if (z <= 1.0d0) 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 <= -1.0) {
		tmp = x;
	} else if (z <= 1.0) {
		tmp = x / -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = x
	elif z <= 1.0:
		tmp = x / -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = x;
	elseif (z <= 1.0)
		tmp = Float64(x / Float64(-z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = x;
	elseif (z <= 1.0)
		tmp = x / -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.0], x, If[LessEqual[z, 1.0], N[(x / (-z)), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 65.1%

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

      1. neg-mul-1 65.1%

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

      2. distribute-neg-frac 65.1%

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

   5. Simplified 65.1%

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

   6. Taylor expanded in z around inf 62.4%

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

   if -1 < z < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 47.5%

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

      1. neg-mul-1 47.5%

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

      2. distribute-neg-frac 47.5%

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

   5. Simplified 47.5%

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

   6. Taylor expanded in x around 0 47.4%

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

      1. sub-neg 47.4%

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

      2. distribute-rgt-in 47.4%

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

      3. *-lft-identity 47.4%

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

      4. cancel-sign-sub-inv 47.4%

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

      5. associate-*l/ 47.5%

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

      6. *-lft-identity 47.5%

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

   8. Simplified 47.5%

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

   9. Taylor expanded in z around 0 47.1%

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

       1. metadata-eval 47.1%

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

       2. times-frac 47.1%

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

       3. *-lft-identity 47.1%

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

       4. neg-mul-1 47.1%

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

   11. Simplified 47.1%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{-z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{+196}:\\ \;\;\;\;x + \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.45e+196) (+ x (/ y z)) (/ x (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.45e+196) {
		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 (x <= 1.45d+196) 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 (x <= 1.45e+196) {
		tmp = x + (y / z);
	} else {
		tmp = x / -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 1.45e+196:
		tmp = x + (y / z)
	else:
		tmp = x / -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.45e+196)
		tmp = Float64(x + Float64(y / z));
	else
		tmp = Float64(x / Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 1.45e+196)
		tmp = x + (y / z);
	else
		tmp = x / -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.45e+196], N[(x + N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x / (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.45e196

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 77.6%

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

   if 1.45e196 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 98.4%

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

      1. neg-mul-1 98.4%

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

      2. distribute-neg-frac 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in x around 0 98.4%

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

      1. sub-neg 98.4%

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

      2. distribute-rgt-in 98.4%

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

      3. *-lft-identity 98.4%

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

      4. cancel-sign-sub-inv 98.4%

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

      5. associate-*l/ 98.4%

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

      6. *-lft-identity 98.4%

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

   8. Simplified 98.4%

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

   9. Taylor expanded in z around 0 65.7%

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

       1. metadata-eval 65.7%

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

       2. times-frac 65.7%

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

       3. *-lft-identity 65.7%

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

       4. neg-mul-1 65.7%

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

   11. Simplified 65.7%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{+196}:\\ \;\;\;\;x + \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 37.2% 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. Add Preprocessing

3. Taylor expanded in y around 0 54.7%

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

   1. neg-mul-1 54.7%

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

   2. distribute-neg-frac 54.7%

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

5. Simplified 54.7%

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

6. Taylor expanded in z around inf 27.4%

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

7. Final simplification 27.4%

   \[\leadsto x \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(FPCore (x y z)
  :name "Statistics.Sample:$swelfordMean from math-functions-0.1.5.2"
  :precision binary64
  (+ x (/ (- y x) z)))