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

Percentage Accurate: 100.0% → 100.0%

Time: 7.3s

Alternatives: 7

Speedup: 1.0×

• 20.2% 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. Final simplification 100.0%

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

Alternative 2: 57.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+132}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{-141}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{-210}:\\ \;\;\;\;\frac{-x}{z}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+54} \lor \neg \left(z \leq 1.9 \cdot 10^{+142}\right) \land z \leq 6.6 \cdot 10^{+168}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -8.2e+132)
   x
   (if (<= z -1.6e-141)
     (/ y z)
     (if (<= z -2.1e-210)
       (/ (- x) z)
       (if (or (<= z 3.2e+54) (and (not (<= z 1.9e+142)) (<= z 6.6e+168)))
         (/ y z)
         x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -8.2e+132) {
		tmp = x;
	} else if (z <= -1.6e-141) {
		tmp = y / z;
	} else if (z <= -2.1e-210) {
		tmp = -x / z;
	} else if ((z <= 3.2e+54) || (!(z <= 1.9e+142) && (z <= 6.6e+168))) {
		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 <= (-8.2d+132)) then
        tmp = x
    else if (z <= (-1.6d-141)) then
        tmp = y / z
    else if (z <= (-2.1d-210)) then
        tmp = -x / z
    else if ((z <= 3.2d+54) .or. (.not. (z <= 1.9d+142)) .and. (z <= 6.6d+168)) 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 <= -8.2e+132) {
		tmp = x;
	} else if (z <= -1.6e-141) {
		tmp = y / z;
	} else if (z <= -2.1e-210) {
		tmp = -x / z;
	} else if ((z <= 3.2e+54) || (!(z <= 1.9e+142) && (z <= 6.6e+168))) {
		tmp = y / z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -8.2e+132:
		tmp = x
	elif z <= -1.6e-141:
		tmp = y / z
	elif z <= -2.1e-210:
		tmp = -x / z
	elif (z <= 3.2e+54) or (not (z <= 1.9e+142) and (z <= 6.6e+168)):
		tmp = y / z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -8.2e+132)
		tmp = x;
	elseif (z <= -1.6e-141)
		tmp = Float64(y / z);
	elseif (z <= -2.1e-210)
		tmp = Float64(Float64(-x) / z);
	elseif ((z <= 3.2e+54) || (!(z <= 1.9e+142) && (z <= 6.6e+168)))
		tmp = Float64(y / z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -8.2e+132)
		tmp = x;
	elseif (z <= -1.6e-141)
		tmp = y / z;
	elseif (z <= -2.1e-210)
		tmp = -x / z;
	elseif ((z <= 3.2e+54) || (~((z <= 1.9e+142)) && (z <= 6.6e+168)))
		tmp = y / z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -8.2e+132], x, If[LessEqual[z, -1.6e-141], N[(y / z), $MachinePrecision], If[LessEqual[z, -2.1e-210], N[((-x) / z), $MachinePrecision], If[Or[LessEqual[z, 3.2e+54], And[N[Not[LessEqual[z, 1.9e+142]], $MachinePrecision], LessEqual[z, 6.6e+168]]], N[(y / z), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.19999999999999983e132 or 3.2e54 < z < 1.89999999999999995e142 or 6.5999999999999997e168 < 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. associate-+r- 100.0%

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

      3. remove-double-neg 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. associate--r+ 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-frac-neg 100.0%

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

      9. sub-neg 100.0%

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

      10. 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 84.3%

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

   if -8.19999999999999983e132 < z < -1.6000000000000001e-141 or -2.10000000000000016e-210 < z < 3.2e54 or 1.89999999999999995e142 < z < 6.5999999999999997e168

   1. Initial program 99.9%

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

      1. div-sub 96.1%

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

      2. associate-+r- 96.1%

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

      3. remove-double-neg 96.1%

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

      4. distribute-frac-neg 96.1%

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

      5. unsub-neg 96.1%

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

      6. associate--r+ 96.1%

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

      7. +-commutative 96.1%

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

      8. distribute-frac-neg 96.1%

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

      9. sub-neg 96.1%

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

      10. 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 x around 0 59.2%

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

   if -1.6000000000000001e-141 < z < -2.10000000000000016e-210

   1. Initial program 99.9%

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

      1. div-sub 100.0%

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

      2. associate-+r- 100.0%

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

      3. remove-double-neg 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. associate--r+ 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-frac-neg 100.0%

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

      9. sub-neg 100.0%

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

      10. 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 x around inf 83.9%

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

      1. distribute-lft-out-- 83.9%

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

      2. *-rgt-identity 83.9%

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

      3. associate-*r/ 83.9%

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

      4. *-rgt-identity 83.9%

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

   7. Simplified 83.9%

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

   8. Taylor expanded in z around 0 83.9%

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

      1. mul-1-neg 83.9%

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

   10. Simplified 83.9%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+132}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{-141}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{-210}:\\ \;\;\;\;\frac{-x}{z}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+54} \lor \neg \left(z \leq 1.9 \cdot 10^{+142}\right) \land z \leq 6.6 \cdot 10^{+168}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.38 \cdot 10^{+138}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+54} \lor \neg \left(z \leq 1.9 \cdot 10^{+142}\right) \land z \leq 4.5 \cdot 10^{+168}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.38e+138)
   x
   (if (or (<= z 3.5e+54) (and (not (<= z 1.9e+142)) (<= z 4.5e+168)))
     (/ y z)
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.38e+138) {
		tmp = x;
	} else if ((z <= 3.5e+54) || (!(z <= 1.9e+142) && (z <= 4.5e+168))) {
		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 <= (-1.38d+138)) then
        tmp = x
    else if ((z <= 3.5d+54) .or. (.not. (z <= 1.9d+142)) .and. (z <= 4.5d+168)) 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 <= -1.38e+138) {
		tmp = x;
	} else if ((z <= 3.5e+54) || (!(z <= 1.9e+142) && (z <= 4.5e+168))) {
		tmp = y / z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.38e+138:
		tmp = x
	elif (z <= 3.5e+54) or (not (z <= 1.9e+142) and (z <= 4.5e+168)):
		tmp = y / z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.38e+138)
		tmp = x;
	elseif ((z <= 3.5e+54) || (!(z <= 1.9e+142) && (z <= 4.5e+168)))
		tmp = Float64(y / z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.38e+138)
		tmp = x;
	elseif ((z <= 3.5e+54) || (~((z <= 1.9e+142)) && (z <= 4.5e+168)))
		tmp = y / z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.38e+138], x, If[Or[LessEqual[z, 3.5e+54], And[N[Not[LessEqual[z, 1.9e+142]], $MachinePrecision], LessEqual[z, 4.5e+168]]], N[(y / z), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -1.38000000000000008e138 or 3.5000000000000001e54 < z < 1.89999999999999995e142 or 4.50000000000000012e168 < 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. associate-+r- 100.0%

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

      3. remove-double-neg 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. associate--r+ 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-frac-neg 100.0%

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

      9. sub-neg 100.0%

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

      10. 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 84.3%

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

   if -1.38000000000000008e138 < z < 3.5000000000000001e54 or 1.89999999999999995e142 < z < 4.50000000000000012e168

   1. Initial program 99.9%

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

      1. div-sub 96.4%

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

      2. associate-+r- 96.4%

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

      3. remove-double-neg 96.4%

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

      4. distribute-frac-neg 96.4%

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

      5. unsub-neg 96.4%

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

      6. associate--r+ 96.4%

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

      7. +-commutative 96.4%

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

      8. distribute-frac-neg 96.4%

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

      9. sub-neg 96.4%

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

      10. 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 x around 0 55.8%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.38 \cdot 10^{+138}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+54} \lor \neg \left(z \leq 1.9 \cdot 10^{+142}\right) \land z \leq 4.5 \cdot 10^{+168}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-45} \lor \neg \left(y \leq 5.8 \cdot 10^{-68}\right):\\ \;\;\;\;x + \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.6e-45) (not (<= y 5.8e-68))) (+ x (/ y z)) (- x (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.6e-45) || !(y <= 5.8e-68)) {
		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 <= (-1.6d-45)) .or. (.not. (y <= 5.8d-68))) 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 <= -1.6e-45) || !(y <= 5.8e-68)) {
		tmp = x + (y / z);
	} else {
		tmp = x - (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.6e-45) or not (y <= 5.8e-68):
		tmp = x + (y / z)
	else:
		tmp = x - (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.6e-45) || !(y <= 5.8e-68))
		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 <= -1.6e-45) || ~((y <= 5.8e-68)))
		tmp = x + (y / z);
	else
		tmp = x - (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.6e-45], N[Not[LessEqual[y, 5.8e-68]], $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 < -1.60000000000000004e-45 or 5.8000000000000001e-68 < y

   1. Initial program 99.9%

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

      1. div-sub 96.2%

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

      2. associate-+r- 96.2%

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

      3. remove-double-neg 96.2%

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

      4. distribute-frac-neg 96.2%

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

      5. unsub-neg 96.2%

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

      6. associate--r+ 96.2%

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

      7. +-commutative 96.2%

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

      8. distribute-frac-neg 96.2%

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

      9. sub-neg 96.2%

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

      10. 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 x around 0 88.1%

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

      1. neg-mul-1 88.1%

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

      2. distribute-neg-frac 88.1%

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

   7. Simplified 88.1%

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

      1. div-inv 87.9%

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

      2. cancel-sign-sub 87.9%

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

      3. div-inv 88.1%

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

      4. +-commutative 88.1%

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

   9. Applied egg-rr 88.1%

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

   if -1.60000000000000004e-45 < y < 5.8000000000000001e-68

   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. associate-+r- 100.0%

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

      3. remove-double-neg 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. associate--r+ 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-frac-neg 100.0%

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

      9. sub-neg 100.0%

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

      10. 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 90.3%

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

      1. distribute-lft-out-- 90.3%

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

      2. *-rgt-identity 90.3%

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

      3. associate-*r/ 90.4%

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

      4. *-rgt-identity 90.4%

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

   7. Simplified 90.4%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-45} \lor \neg \left(y \leq 5.8 \cdot 10^{-68}\right):\\ \;\;\;\;x + \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% 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. associate-+r- 100.0%

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

      3. remove-double-neg 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. associate--r+ 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-frac-neg 100.0%

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

      9. sub-neg 100.0%

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

      10. 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 98.0%

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

      1. neg-mul-1 98.0%

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

      2. distribute-neg-frac 98.0%

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

   7. Simplified 98.0%

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

      1. div-inv 97.9%

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

      2. cancel-sign-sub 97.9%

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

      3. div-inv 98.0%

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

      4. +-commutative 98.0%

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

   9. Applied egg-rr 98.0%

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

   if -1 < z < 1

   1. Initial program 99.9%

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

      1. div-sub 95.1%

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

      2. associate-+r- 95.1%

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

      3. remove-double-neg 95.1%

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

      4. distribute-frac-neg 95.1%

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

      5. unsub-neg 95.1%

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

      6. associate--r+ 95.1%

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

      7. +-commutative 95.1%

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

      8. distribute-frac-neg 95.1%

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

      9. sub-neg 95.1%

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

      10. 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 98.9%

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

4. Final simplification 98.4%

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

Alternative 6: 75.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ x + \frac{y}{z} \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return x + (y / 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 / z)
end function

Java

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

Python

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

Julia

function code(x, y, z)
	return Float64(x + Float64(y / z))
end

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x + N[(y / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. div-sub 97.6%

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

   2. associate-+r- 97.6%

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

   3. remove-double-neg 97.6%

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

   4. distribute-frac-neg 97.6%

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

   5. unsub-neg 97.6%

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

   6. associate--r+ 97.6%

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

   7. +-commutative 97.6%

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

   8. distribute-frac-neg 97.6%

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

   9. sub-neg 97.6%

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

   10. 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 77.3%

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

   1. neg-mul-1 77.3%

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

   2. distribute-neg-frac 77.3%

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

7. Simplified 77.3%

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

   1. div-inv 77.1%

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

   2. cancel-sign-sub 77.1%

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

   3. div-inv 77.3%

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

   4. +-commutative 77.3%

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

9. Applied egg-rr 77.3%

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

10. Final simplification 77.3%

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

Alternative 7: 36.4% 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 97.6%

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

   2. associate-+r- 97.6%

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

   3. remove-double-neg 97.6%

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

   4. distribute-frac-neg 97.6%

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

   5. unsub-neg 97.6%

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

   6. associate--r+ 97.6%

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

   7. +-commutative 97.6%

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

   8. distribute-frac-neg 97.6%

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

   9. sub-neg 97.6%

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

   10. 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 36.8%

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

6. Final simplification 36.8%

   \[\leadsto x \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024027 
(FPCore (x y z)
  :name "Statistics.Sample:$swelfordMean from math-functions-0.1.5.2"
  :precision binary64
  (+ x (/ (- y x) z)))