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

Percentage Accurate: 100.0% → 100.0%

Time: 6.8s

Alternatives: 7

Speedup: 1.0×

• 20.1% 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. Add Preprocessing
3. Add Preprocessing

Alternative 2: 61.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+76}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-272}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{-z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.4e+76)
   x
   (if (<= z 1.6e-272) (/ y z) (if (<= z 1.0) (/ x (- z)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.4e+76) {
		tmp = x;
	} else if (z <= 1.6e-272) {
		tmp = y / z;
	} 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 <= (-5.4d+76)) then
        tmp = x
    else if (z <= 1.6d-272) then
        tmp = y / z
    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 <= -5.4e+76) {
		tmp = x;
	} else if (z <= 1.6e-272) {
		tmp = y / z;
	} else if (z <= 1.0) {
		tmp = x / -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -5.4e+76:
		tmp = x
	elif z <= 1.6e-272:
		tmp = y / z
	elif z <= 1.0:
		tmp = x / -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.4e+76)
		tmp = x;
	elseif (z <= 1.6e-272)
		tmp = Float64(y / z);
	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 <= -5.4e+76)
		tmp = x;
	elseif (z <= 1.6e-272)
		tmp = y / z;
	elseif (z <= 1.0)
		tmp = x / -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.4e+76], x, If[LessEqual[z, 1.6e-272], N[(y / z), $MachinePrecision], If[LessEqual[z, 1.0], N[(x / (-z)), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.3999999999999998e76 or 1 < z

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]
   2. Step-by-step derivation

      1. div-sub 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-+r+ 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. associate--r- 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 74.4%

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

   if -5.3999999999999998e76 < z < 1.6e-272

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]
   2. Step-by-step derivation

      1. div-sub 92.6%

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

      2. sub-neg 92.6%

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

      3. distribute-frac-neg 92.6%

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

      4. +-commutative 92.6%

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

      5. associate-+r+ 92.6%

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

      6. distribute-frac-neg 92.6%

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

      7. sub-neg 92.6%

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

      8. associate--r- 92.6%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 55.2%

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

   if 1.6e-272 < z < 1

   1. Initial program 99.9%

      \[x + \frac{y - x}{z} \]
   2. Step-by-step derivation

      1. div-sub 91.9%

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

      2. sub-neg 91.9%

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

      3. distribute-frac-neg 91.9%

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

      4. +-commutative 91.9%

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

      5. associate-+r+ 91.9%

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

      6. distribute-frac-neg 91.9%

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

      7. sub-neg 91.9%

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

      8. associate--r- 91.9%

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

      9. div-sub 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 99.9%

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

   6. Taylor expanded in y around 0 56.6%

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

      1. neg-mul-1 56.6%

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

   8. Simplified 56.6%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+76}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-272}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{-z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+75}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{-16}:\\ \;\;\;\;\frac{y - x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \frac{-1}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -9.5e+75)
   x
   (if (<= z 1.85e-16) (/ (- y x) z) (* x (+ 1.0 (/ -1.0 z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -9.5e+75) {
		tmp = x;
	} else if (z <= 1.85e-16) {
		tmp = (y - x) / z;
	} else {
		tmp = x * (1.0 + (-1.0 / 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 <= (-9.5d+75)) then
        tmp = x
    else if (z <= 1.85d-16) then
        tmp = (y - x) / z
    else
        tmp = x * (1.0d0 + ((-1.0d0) / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -9.5e+75) {
		tmp = x;
	} else if (z <= 1.85e-16) {
		tmp = (y - x) / z;
	} else {
		tmp = x * (1.0 + (-1.0 / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -9.5e+75:
		tmp = x
	elif z <= 1.85e-16:
		tmp = (y - x) / z
	else:
		tmp = x * (1.0 + (-1.0 / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -9.5e+75)
		tmp = x;
	elseif (z <= 1.85e-16)
		tmp = Float64(Float64(y - x) / z);
	else
		tmp = Float64(x * Float64(1.0 + Float64(-1.0 / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -9.5e+75)
		tmp = x;
	elseif (z <= 1.85e-16)
		tmp = (y - x) / z;
	else
		tmp = x * (1.0 + (-1.0 / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -9.5e+75], x, If[LessEqual[z, 1.85e-16], N[(N[(y - x), $MachinePrecision] / z), $MachinePrecision], N[(x * N[(1.0 + N[(-1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.50000000000000061e75

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]
   2. Step-by-step derivation

      1. div-sub 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-+r+ 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. associate--r- 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 79.3%

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

   if -9.50000000000000061e75 < z < 1.85e-16

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]
   2. Step-by-step derivation

      1. div-sub 92.3%

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

      2. sub-neg 92.3%

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

      3. distribute-frac-neg 92.3%

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

      4. +-commutative 92.3%

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

      5. associate-+r+ 92.3%

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

      6. distribute-frac-neg 92.3%

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

      7. sub-neg 92.3%

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

      8. associate--r- 92.3%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 95.2%

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

   if 1.85e-16 < z

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]
   2. Step-by-step derivation

      1. div-sub 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-+r+ 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. associate--r- 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 72.3%

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{1}{z}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+75}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{-16}:\\ \;\;\;\;\frac{y - x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \frac{-1}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.42 \cdot 10^{+53} \lor \neg \left(y \leq 4.6 \cdot 10^{+87}\right):\\ \;\;\;\;\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.42e+53) (not (<= y 4.6e+87))) (/ y z) (- x (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.42e+53) || !(y <= 4.6e+87)) {
		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 <= (-1.42d+53)) .or. (.not. (y <= 4.6d+87))) 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 <= -1.42e+53) || !(y <= 4.6e+87)) {
		tmp = y / z;
	} else {
		tmp = x - (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.42e+53) or not (y <= 4.6e+87):
		tmp = y / z
	else:
		tmp = x - (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.42e+53) || !(y <= 4.6e+87))
		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 <= -1.42e+53) || ~((y <= 4.6e+87)))
		tmp = y / z;
	else
		tmp = x - (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.42e+53], N[Not[LessEqual[y, 4.6e+87]], $MachinePrecision]], N[(y / z), $MachinePrecision], N[(x - N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.41999999999999999e53 or 4.6000000000000003e87 < y

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]
   2. Step-by-step derivation

      1. div-sub 90.1%

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

      2. sub-neg 90.1%

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

      3. distribute-frac-neg 90.1%

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

      4. +-commutative 90.1%

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

      5. associate-+r+ 90.1%

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

      6. distribute-frac-neg 90.1%

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

      7. sub-neg 90.1%

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

      8. associate--r- 90.1%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 74.4%

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

   if -1.41999999999999999e53 < y < 4.6000000000000003e87

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]
   2. Step-by-step derivation

      1. div-sub 99.3%

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

      2. sub-neg 99.3%

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

      3. distribute-frac-neg 99.3%

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

      4. +-commutative 99.3%

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

      5. associate-+r+ 99.3%

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

      6. distribute-frac-neg 99.3%

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

      7. sub-neg 99.3%

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

      8. associate--r- 99.3%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 81.9%

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

      1. distribute-lft-out-- 81.9%

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

      2. *-rgt-identity 81.9%

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

      3. associate-*r/ 82.0%

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

      4. *-rgt-identity 82.0%

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

   7. Simplified 82.0%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.42 \cdot 10^{+53} \lor \neg \left(y \leq 4.6 \cdot 10^{+87}\right):\\ \;\;\;\;\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.32 \cdot 10^{+76}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 300000000000:\\ \;\;\;\;\frac{y - x}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.32e+76)
   x
   (if (<= z 300000000000.0) (/ (- y x) z) (- x (/ x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.32e+76) {
		tmp = x;
	} else if (z <= 300000000000.0) {
		tmp = (y - x) / 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 (z <= (-1.32d+76)) then
        tmp = x
    else if (z <= 300000000000.0d0) then
        tmp = (y - x) / 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 (z <= -1.32e+76) {
		tmp = x;
	} else if (z <= 300000000000.0) {
		tmp = (y - x) / z;
	} else {
		tmp = x - (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.32e+76:
		tmp = x
	elif z <= 300000000000.0:
		tmp = (y - x) / z
	else:
		tmp = x - (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.32e+76)
		tmp = x;
	elseif (z <= 300000000000.0)
		tmp = Float64(Float64(y - x) / z);
	else
		tmp = Float64(x - Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.32e+76)
		tmp = x;
	elseif (z <= 300000000000.0)
		tmp = (y - x) / z;
	else
		tmp = x - (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.32e+76], x, If[LessEqual[z, 300000000000.0], N[(N[(y - x), $MachinePrecision] / z), $MachinePrecision], N[(x - N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.31999999999999999e76

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]
   2. Step-by-step derivation

      1. div-sub 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-+r+ 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. associate--r- 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 79.3%

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

   if -1.31999999999999999e76 < z < 3e11

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]
   2. Step-by-step derivation

      1. div-sub 92.6%

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

      2. sub-neg 92.6%

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

      3. distribute-frac-neg 92.6%

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

      4. +-commutative 92.6%

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

      5. associate-+r+ 92.6%

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

      6. distribute-frac-neg 92.6%

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

      7. sub-neg 92.6%

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

      8. associate--r- 92.6%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 94.1%

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

   if 3e11 < z

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]
   2. Step-by-step derivation

      1. div-sub 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-+r+ 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. associate--r- 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 73.4%

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

      1. distribute-lft-out-- 73.4%

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

      2. *-rgt-identity 73.4%

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

      3. associate-*r/ 73.4%

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

      4. *-rgt-identity 73.4%

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

   7. Simplified 73.4%

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

Alternative 6: 60.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+75}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+19}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -9e+75) x (if (<= z 5.2e+19) (/ y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -9e+75) {
		tmp = x;
	} else if (z <= 5.2e+19) {
		tmp = y / 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 <= (-9d+75)) then
        tmp = x
    else if (z <= 5.2d+19) then
        tmp = y / 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 <= -9e+75) {
		tmp = x;
	} else if (z <= 5.2e+19) {
		tmp = y / z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -9e+75:
		tmp = x
	elif z <= 5.2e+19:
		tmp = y / z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -9e+75)
		tmp = x;
	elseif (z <= 5.2e+19)
		tmp = Float64(y / z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -9e+75)
		tmp = x;
	elseif (z <= 5.2e+19)
		tmp = y / z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -9e+75], x, If[LessEqual[z, 5.2e+19], N[(y / z), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -9.0000000000000007e75 or 5.2e19 < z

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]
   2. Step-by-step derivation

      1. div-sub 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-+r+ 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. associate--r- 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 77.0%

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

   if -9.0000000000000007e75 < z < 5.2e19

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]
   2. Step-by-step derivation

      1. div-sub 92.7%

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

      2. sub-neg 92.7%

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

      3. distribute-frac-neg 92.7%

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

      4. +-commutative 92.7%

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

      5. associate-+r+ 92.7%

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

      6. distribute-frac-neg 92.7%

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

      7. sub-neg 92.7%

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

      8. associate--r- 92.7%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 51.5%

      \[\leadsto \color{blue}{\frac{y}{z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 37.8% 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. Step-by-step derivation

   1. div-sub 95.7%

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

   2. sub-neg 95.7%

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

   3. distribute-frac-neg 95.7%

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

   4. +-commutative 95.7%

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

   5. associate-+r+ 95.7%

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

   6. distribute-frac-neg 95.7%

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

   7. sub-neg 95.7%

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

   8. associate--r- 95.7%

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

   9. div-sub 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in z around inf 36.2%

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

Reproduce

herbie shell --seed 2024175 
(FPCore (x y z)
  :name "Statistics.Sample:$swelfordMean from math-functions-0.1.5.2"
  :precision binary64
  (+ x (/ (- y x) z)))