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

Percentage Accurate: 100.0% → 100.0%

Time: 5.8s

Alternatives: 7

Speedup: 1.0×

• 20.9% 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: 61.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.12 \cdot 10^{-77}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-297}:\\ \;\;\;\;\frac{x}{-z}\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+76}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.2)
   x
   (if (<= z -1.12e-77)
     (/ y z)
     (if (<= z 9.5e-297) (/ x (- z)) (if (<= z 1.95e+76) (/ y z) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.2) {
		tmp = x;
	} else if (z <= -1.12e-77) {
		tmp = y / z;
	} else if (z <= 9.5e-297) {
		tmp = x / -z;
	} else if (z <= 1.95e+76) {
		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 <= (-2.2d0)) then
        tmp = x
    else if (z <= (-1.12d-77)) then
        tmp = y / z
    else if (z <= 9.5d-297) then
        tmp = x / -z
    else if (z <= 1.95d+76) 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 <= -2.2) {
		tmp = x;
	} else if (z <= -1.12e-77) {
		tmp = y / z;
	} else if (z <= 9.5e-297) {
		tmp = x / -z;
	} else if (z <= 1.95e+76) {
		tmp = y / z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.2:
		tmp = x
	elif z <= -1.12e-77:
		tmp = y / z
	elif z <= 9.5e-297:
		tmp = x / -z
	elif z <= 1.95e+76:
		tmp = y / z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.2)
		tmp = x;
	elseif (z <= -1.12e-77)
		tmp = Float64(y / z);
	elseif (z <= 9.5e-297)
		tmp = Float64(x / Float64(-z));
	elseif (z <= 1.95e+76)
		tmp = Float64(y / z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.2)
		tmp = x;
	elseif (z <= -1.12e-77)
		tmp = y / z;
	elseif (z <= 9.5e-297)
		tmp = x / -z;
	elseif (z <= 1.95e+76)
		tmp = y / z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.2], x, If[LessEqual[z, -1.12e-77], N[(y / z), $MachinePrecision], If[LessEqual[z, 9.5e-297], N[(x / (-z)), $MachinePrecision], If[LessEqual[z, 1.95e+76], N[(y / z), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.2000000000000002 or 1.94999999999999995e76 < 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 76.6%

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

   if -2.2000000000000002 < z < -1.12000000000000009e-77 or 9.5000000000000005e-297 < z < 1.94999999999999995e76

   1. Initial program 100.0%

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

      1. div-sub 97.0%

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

      2. sub-neg 97.0%

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

      3. distribute-frac-neg 97.0%

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

      4. +-commutative 97.0%

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

      5. associate-+r+ 97.0%

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

      6. distribute-frac-neg 97.0%

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

      7. sub-neg 97.0%

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

      8. associate--r- 97.0%

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

      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 0 58.1%

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

   if -1.12000000000000009e-77 < z < 9.5000000000000005e-297

   1. Initial program 100.0%

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

      1. div-sub 88.9%

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

      2. sub-neg 88.9%

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

      3. distribute-frac-neg 88.9%

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

      4. +-commutative 88.9%

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

      5. associate-+r+ 88.9%

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

      6. distribute-frac-neg 88.9%

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

      7. sub-neg 88.9%

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

      8. associate--r- 88.9%

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

      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 65.1%

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

   6. Taylor expanded in z around 0 65.1%

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

      1. mul-1-neg 65.1%

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

      2. distribute-frac-neg 65.1%

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

   8. Simplified 65.1%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.12 \cdot 10^{-77}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-297}:\\ \;\;\;\;\frac{x}{-z}\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+76}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x + \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 1.0))) (+ x (/ y z)) (/ (- y x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = x + (y / 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 ((z <= (-1.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = x + (y / 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 ((z <= -1.0) || !(z <= 1.0)) {
		tmp = x + (y / z);
	} else {
		tmp = (y - x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 1.0):
		tmp = x + (y / z)
	else:
		tmp = (y - x) / z
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(x + N[(y / z), $MachinePrecision]), $MachinePrecision], N[(N[(y - x), $MachinePrecision] / z), $MachinePrecision]]

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. 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 0 97.3%

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

      1. neg-mul-1 97.3%

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

      2. distribute-neg-frac 97.3%

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

   7. Simplified 97.3%

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

      1. sub-neg 97.3%

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

      2. +-commutative 97.3%

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

      3. distribute-frac-neg 97.3%

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

      4. remove-double-neg 97.3%

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

   9. Applied egg-rr 97.3%

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

   if -1 < z < 1

   1. Initial program 100.0%

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

      1. div-sub 93.3%

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

      2. sub-neg 93.3%

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

      3. distribute-frac-neg 93.3%

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

      4. +-commutative 93.3%

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

      5. associate-+r+ 93.3%

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

      6. distribute-frac-neg 93.3%

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

      7. sub-neg 93.3%

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

      8. associate--r- 93.3%

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

      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 99.7%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x + \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-79} \lor \neg \left(x \leq 8.8 \cdot 10^{+89}\right):\\ \;\;\;\;x - \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.9e-79) (not (<= x 8.8e+89))) (- x (/ x z)) (+ x (/ y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.9e-79) || !(x <= 8.8e+89)) {
		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 <= (-2.9d-79)) .or. (.not. (x <= 8.8d+89))) 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 <= -2.9e-79) || !(x <= 8.8e+89)) {
		tmp = x - (x / z);
	} else {
		tmp = x + (y / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.9e-79) or not (x <= 8.8e+89):
		tmp = x - (x / z)
	else:
		tmp = x + (y / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.9e-79) || !(x <= 8.8e+89))
		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 <= -2.9e-79) || ~((x <= 8.8e+89)))
		tmp = x - (x / z);
	else
		tmp = x + (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.9e-79], N[Not[LessEqual[x, 8.8e+89]], $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 < -2.9000000000000001e-79 or 8.8000000000000001e89 < x

   1. Initial program 100.0%

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

      1. div-sub 94.2%

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

      2. sub-neg 94.2%

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

      3. distribute-frac-neg 94.2%

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

      4. +-commutative 94.2%

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

      5. associate-+r+ 94.2%

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

      6. distribute-frac-neg 94.2%

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

      7. sub-neg 94.2%

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

      8. associate--r- 94.2%

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

      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 85.5%

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

   if -2.9000000000000001e-79 < x < 8.8000000000000001e89

   1. Initial program 100.0%

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

      1. div-sub 99.1%

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

      2. sub-neg 99.1%

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

      3. distribute-frac-neg 99.1%

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

      4. +-commutative 99.1%

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

      5. associate-+r+ 99.1%

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

      6. distribute-frac-neg 99.1%

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

      7. sub-neg 99.1%

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

      8. associate--r- 99.1%

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

      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 88.7%

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

      1. neg-mul-1 88.7%

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

      2. distribute-neg-frac 88.7%

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

   7. Simplified 88.7%

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

      1. sub-neg 88.7%

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

      2. +-commutative 88.7%

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

      3. distribute-frac-neg 88.7%

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

      4. remove-double-neg 88.7%

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

   9. Applied egg-rr 88.7%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-79} \lor \neg \left(x \leq 8.8 \cdot 10^{+89}\right):\\ \;\;\;\;x - \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.26 \cdot 10^{-77} \lor \neg \left(z \leq 8 \cdot 10^{-268}\right):\\ \;\;\;\;x + \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.26e-77) (not (<= z 8e-268))) (+ x (/ y z)) (/ x (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.26e-77) || !(z <= 8e-268)) {
		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 <= (-1.26d-77)) .or. (.not. (z <= 8d-268))) 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 <= -1.26e-77) || !(z <= 8e-268)) {
		tmp = x + (y / z);
	} else {
		tmp = x / -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.26e-77) or not (z <= 8e-268):
		tmp = x + (y / z)
	else:
		tmp = x / -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.26e-77) || !(z <= 8e-268))
		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 ((z <= -1.26e-77) || ~((z <= 8e-268)))
		tmp = x + (y / z);
	else
		tmp = x / -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.26e-77], N[Not[LessEqual[z, 8e-268]], $MachinePrecision]], N[(x + N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x / (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2599999999999999e-77 or 7.99999999999999966e-268 < z

   1. Initial program 100.0%

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

      1. div-sub 99.0%

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

      2. sub-neg 99.0%

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

      3. distribute-frac-neg 99.0%

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

      4. +-commutative 99.0%

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

      5. associate-+r+ 99.0%

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

      6. distribute-frac-neg 99.0%

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

      7. sub-neg 99.0%

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

      8. associate--r- 99.0%

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

      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 0 82.4%

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

      1. neg-mul-1 82.4%

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

      2. distribute-neg-frac 82.4%

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

   7. Simplified 82.4%

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

      1. sub-neg 82.4%

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

      2. +-commutative 82.4%

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

      3. distribute-frac-neg 82.4%

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

      4. remove-double-neg 82.4%

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

   9. Applied egg-rr 82.4%

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

   if -1.2599999999999999e-77 < z < 7.99999999999999966e-268

   1. Initial program 100.0%

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

      1. div-sub 88.5%

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

      2. sub-neg 88.5%

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

      3. distribute-frac-neg 88.5%

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

      4. +-commutative 88.5%

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

      5. associate-+r+ 88.5%

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

      6. distribute-frac-neg 88.5%

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

      7. sub-neg 88.5%

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

      8. associate--r- 88.5%

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

      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 64.3%

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

   6. Taylor expanded in z around 0 64.3%

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

      1. mul-1-neg 64.3%

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

      2. distribute-frac-neg 64.3%

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

   8. Simplified 64.3%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.26 \cdot 10^{-77} \lor \neg \left(z \leq 8 \cdot 10^{-268}\right):\\ \;\;\;\;x + \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+76}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.2) x (if (<= z 5.5e+76) (/ y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.2) {
		tmp = x;
	} else if (z <= 5.5e+76) {
		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 <= (-2.2d0)) then
        tmp = x
    else if (z <= 5.5d+76) 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 <= -2.2) {
		tmp = x;
	} else if (z <= 5.5e+76) {
		tmp = y / z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.2:
		tmp = x
	elif z <= 5.5e+76:
		tmp = y / z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.2)
		tmp = x;
	elseif (z <= 5.5e+76)
		tmp = Float64(y / z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.2)
		tmp = x;
	elseif (z <= 5.5e+76)
		tmp = y / z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.2], x, If[LessEqual[z, 5.5e+76], N[(y / z), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -2.2000000000000002 or 5.5000000000000001e76 < 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 76.6%

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

   if -2.2000000000000002 < z < 5.5000000000000001e76

   1. Initial program 100.0%

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

      1. div-sub 94.2%

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

      2. sub-neg 94.2%

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

      3. distribute-frac-neg 94.2%

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

      4. +-commutative 94.2%

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

      5. associate-+r+ 94.2%

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

      6. distribute-frac-neg 94.2%

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

      7. sub-neg 94.2%

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

      8. associate--r- 94.2%

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

      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 51.3%

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

Alternative 7: 36.7% 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 96.5%

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

   2. sub-neg 96.5%

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

   3. distribute-frac-neg 96.5%

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

   4. +-commutative 96.5%

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

   5. associate-+r+ 96.5%

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

   6. distribute-frac-neg 96.5%

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

   7. sub-neg 96.5%

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

   8. associate--r- 96.5%

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

   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 34.9%

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

Reproduce

herbie shell --seed 2024118 
(FPCore (x y z)
  :name "Statistics.Sample:$swelfordMean from math-functions-0.1.5.2"
  :precision binary64
  (+ x (/ (- y x) z)))