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

Percentage Accurate: 100.0% → 100.0%

Time: 6.6s

Alternatives: 7

Speedup: 1.0×

• 21.6% 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: 61.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{z}\\ \mathbf{if}\;z \leq -1.2 \cdot 10^{+61}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.02 \cdot 10^{-132}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq -3.45 \cdot 10^{-217}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-268}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-262}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-229}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-215}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{+47}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x) z)))
   (if (<= z -1.2e+61)
     x
     (if (<= z -1.02e-132)
       (/ y z)
       (if (<= z -3.45e-217)
         t_0
         (if (<= z -7e-268)
           (/ y z)
           (if (<= z 4.4e-262)
             t_0
             (if (<= z 1.05e-229)
               (/ y z)
               (if (<= z 1.2e-215) t_0 (if (<= z 4.7e+47) (/ y z) x))))))))))

C

double code(double x, double y, double z) {
	double t_0 = -x / z;
	double tmp;
	if (z <= -1.2e+61) {
		tmp = x;
	} else if (z <= -1.02e-132) {
		tmp = y / z;
	} else if (z <= -3.45e-217) {
		tmp = t_0;
	} else if (z <= -7e-268) {
		tmp = y / z;
	} else if (z <= 4.4e-262) {
		tmp = t_0;
	} else if (z <= 1.05e-229) {
		tmp = y / z;
	} else if (z <= 1.2e-215) {
		tmp = t_0;
	} else if (z <= 4.7e+47) {
		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) :: t_0
    real(8) :: tmp
    t_0 = -x / z
    if (z <= (-1.2d+61)) then
        tmp = x
    else if (z <= (-1.02d-132)) then
        tmp = y / z
    else if (z <= (-3.45d-217)) then
        tmp = t_0
    else if (z <= (-7d-268)) then
        tmp = y / z
    else if (z <= 4.4d-262) then
        tmp = t_0
    else if (z <= 1.05d-229) then
        tmp = y / z
    else if (z <= 1.2d-215) then
        tmp = t_0
    else if (z <= 4.7d+47) 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 t_0 = -x / z;
	double tmp;
	if (z <= -1.2e+61) {
		tmp = x;
	} else if (z <= -1.02e-132) {
		tmp = y / z;
	} else if (z <= -3.45e-217) {
		tmp = t_0;
	} else if (z <= -7e-268) {
		tmp = y / z;
	} else if (z <= 4.4e-262) {
		tmp = t_0;
	} else if (z <= 1.05e-229) {
		tmp = y / z;
	} else if (z <= 1.2e-215) {
		tmp = t_0;
	} else if (z <= 4.7e+47) {
		tmp = y / z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -x / z
	tmp = 0
	if z <= -1.2e+61:
		tmp = x
	elif z <= -1.02e-132:
		tmp = y / z
	elif z <= -3.45e-217:
		tmp = t_0
	elif z <= -7e-268:
		tmp = y / z
	elif z <= 4.4e-262:
		tmp = t_0
	elif z <= 1.05e-229:
		tmp = y / z
	elif z <= 1.2e-215:
		tmp = t_0
	elif z <= 4.7e+47:
		tmp = y / z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-x) / z)
	tmp = 0.0
	if (z <= -1.2e+61)
		tmp = x;
	elseif (z <= -1.02e-132)
		tmp = Float64(y / z);
	elseif (z <= -3.45e-217)
		tmp = t_0;
	elseif (z <= -7e-268)
		tmp = Float64(y / z);
	elseif (z <= 4.4e-262)
		tmp = t_0;
	elseif (z <= 1.05e-229)
		tmp = Float64(y / z);
	elseif (z <= 1.2e-215)
		tmp = t_0;
	elseif (z <= 4.7e+47)
		tmp = Float64(y / z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -x / z;
	tmp = 0.0;
	if (z <= -1.2e+61)
		tmp = x;
	elseif (z <= -1.02e-132)
		tmp = y / z;
	elseif (z <= -3.45e-217)
		tmp = t_0;
	elseif (z <= -7e-268)
		tmp = y / z;
	elseif (z <= 4.4e-262)
		tmp = t_0;
	elseif (z <= 1.05e-229)
		tmp = y / z;
	elseif (z <= 1.2e-215)
		tmp = t_0;
	elseif (z <= 4.7e+47)
		tmp = y / z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-x) / z), $MachinePrecision]}, If[LessEqual[z, -1.2e+61], x, If[LessEqual[z, -1.02e-132], N[(y / z), $MachinePrecision], If[LessEqual[z, -3.45e-217], t$95$0, If[LessEqual[z, -7e-268], N[(y / z), $MachinePrecision], If[LessEqual[z, 4.4e-262], t$95$0, If[LessEqual[z, 1.05e-229], N[(y / z), $MachinePrecision], If[LessEqual[z, 1.2e-215], t$95$0, If[LessEqual[z, 4.7e+47], N[(y / z), $MachinePrecision], x]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.1999999999999999e61 or 4.69999999999999964e47 < 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 81.2%

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

   if -1.1999999999999999e61 < z < -1.01999999999999998e-132 or -3.44999999999999987e-217 < z < -7.00000000000000011e-268 or 4.39999999999999977e-262 < z < 1.04999999999999992e-229 or 1.20000000000000005e-215 < z < 4.69999999999999964e47

   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. associate-+r- 99.1%

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

      3. remove-double-neg 99.1%

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

      4. distribute-frac-neg 99.1%

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

      5. unsub-neg 99.1%

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

      6. associate--r+ 99.1%

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

      7. +-commutative 99.1%

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

      8. distribute-frac-neg 99.1%

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

      9. sub-neg 99.1%

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

      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 64.4%

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

   if -1.01999999999999998e-132 < z < -3.44999999999999987e-217 or -7.00000000000000011e-268 < z < 4.39999999999999977e-262 or 1.04999999999999992e-229 < z < 1.20000000000000005e-215

   1. Initial program 100.0%

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

      1. div-sub 94.6%

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

      2. associate-+r- 94.6%

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

      3. remove-double-neg 94.6%

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

      4. distribute-frac-neg 94.6%

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

      5. unsub-neg 94.6%

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

      6. associate--r+ 94.6%

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

      7. +-commutative 94.6%

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

      8. distribute-frac-neg 94.6%

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

      9. sub-neg 94.6%

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

      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 inf 77.9%

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

   6. Taylor expanded in z around 0 77.9%

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

      1. associate-*r/ 77.9%

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

      2. mul-1-neg 77.9%

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

   8. Simplified 77.9%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+61}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.02 \cdot 10^{-132}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq -3.45 \cdot 10^{-217}:\\ \;\;\;\;\frac{-x}{z}\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-268}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-262}:\\ \;\;\;\;\frac{-x}{z}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-229}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-215}:\\ \;\;\;\;\frac{-x}{z}\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{+47}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{-67} \lor \neg \left(y \leq 2.2 \cdot 10^{+19}\right):\\ \;\;\;\;x + \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4.1e-67) (not (<= y 2.2e+19))) (+ x (/ y z)) (- x (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.1e-67) || !(y <= 2.2e+19)) {
		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 <= (-4.1d-67)) .or. (.not. (y <= 2.2d+19))) 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 <= -4.1e-67) || !(y <= 2.2e+19)) {
		tmp = x + (y / z);
	} else {
		tmp = x - (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4.1e-67) or not (y <= 2.2e+19):
		tmp = x + (y / z)
	else:
		tmp = x - (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4.1e-67) || !(y <= 2.2e+19))
		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 <= -4.1e-67) || ~((y <= 2.2e+19)))
		tmp = x + (y / z);
	else
		tmp = x - (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4.1e-67], N[Not[LessEqual[y, 2.2e+19]], $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 < -4.0999999999999997e-67 or 2.2e19 < y

   1. Initial program 100.0%

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

      1. div-sub 97.8%

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

      2. associate-+r- 97.8%

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

      3. remove-double-neg 97.8%

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

      4. distribute-frac-neg 97.8%

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

      5. unsub-neg 97.8%

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

      6. associate--r+ 97.8%

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

      7. +-commutative 97.8%

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

      8. distribute-frac-neg 97.8%

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

      9. sub-neg 97.8%

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

      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 90.3%

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

      1. neg-mul-1 90.3%

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

      2. distribute-neg-frac 90.3%

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

   7. Simplified 90.3%

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

   8. Taylor expanded in x around 0 90.3%

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

      1. +-commutative 90.3%

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

   10. Simplified 90.3%

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

   if -4.0999999999999997e-67 < y < 2.2e19

   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 90.4%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{-67} \lor \neg \left(y \leq 2.2 \cdot 10^{+19}\right):\\ \;\;\;\;x + \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -22000000000 \lor \neg \left(z \leq 4.5 \cdot 10^{-19}\right):\\ \;\;\;\;x + \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -22000000000.0) (not (<= z 4.5e-19)))
   (+ x (/ y z))
   (/ (- y x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -22000000000.0) || !(z <= 4.5e-19)) {
		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 <= (-22000000000.0d0)) .or. (.not. (z <= 4.5d-19))) 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 <= -22000000000.0) || !(z <= 4.5e-19)) {
		tmp = x + (y / z);
	} else {
		tmp = (y - x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -22000000000.0) or not (z <= 4.5e-19):
		tmp = x + (y / z)
	else:
		tmp = (y - x) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -22000000000.0) || !(z <= 4.5e-19))
		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 <= -22000000000.0) || ~((z <= 4.5e-19)))
		tmp = x + (y / z);
	else
		tmp = (y - x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -22000000000.0], N[Not[LessEqual[z, 4.5e-19]], $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 < -2.2e10 or 4.50000000000000013e-19 < 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 99.8%

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

      1. neg-mul-1 99.8%

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

      2. distribute-neg-frac 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around 0 99.8%

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

      1. +-commutative 99.8%

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

   10. Simplified 99.8%

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

   if -2.2e10 < z < 4.50000000000000013e-19

   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 0 99.5%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -22000000000 \lor \neg \left(z \leq 4.5 \cdot 10^{-19}\right):\\ \;\;\;\;x + \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+61}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+42}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.2e+61) x (if (<= z 5.5e+42) (/ y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.2e+61) {
		tmp = x;
	} else if (z <= 5.5e+42) {
		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 <= (-1.2d+61)) then
        tmp = x
    else if (z <= 5.5d+42) 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 <= -1.2e+61) {
		tmp = x;
	} else if (z <= 5.5e+42) {
		tmp = y / z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.2e+61:
		tmp = x
	elif z <= 5.5e+42:
		tmp = y / z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.2e+61)
		tmp = x;
	elseif (z <= 5.5e+42)
		tmp = Float64(y / z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.2e+61)
		tmp = x;
	elseif (z <= 5.5e+42)
		tmp = y / z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.2e+61], x, If[LessEqual[z, 5.5e+42], N[(y / z), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -1.1999999999999999e61 or 5.50000000000000001e42 < 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 81.2%

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

   if -1.1999999999999999e61 < z < 5.50000000000000001e42

   1. Initial program 100.0%

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

      1. div-sub 98.0%

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

      2. associate-+r- 98.0%

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

      3. remove-double-neg 98.0%

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

      4. distribute-frac-neg 98.0%

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

      5. unsub-neg 98.0%

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

      6. associate--r+ 98.0%

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

      7. +-commutative 98.0%

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

      8. distribute-frac-neg 98.0%

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

      9. sub-neg 98.0%

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

      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 56.3%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+61}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+42}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.6% 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 98.8%

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

   2. associate-+r- 98.8%

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

   3. remove-double-neg 98.8%

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

   4. distribute-frac-neg 98.8%

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

   5. unsub-neg 98.8%

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

   6. associate--r+ 98.8%

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

   7. +-commutative 98.8%

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

   8. distribute-frac-neg 98.8%

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

   9. sub-neg 98.8%

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

   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 77.6%

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

   1. neg-mul-1 77.6%

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

   2. distribute-neg-frac 77.6%

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

7. Simplified 77.6%

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

8. Taylor expanded in x around 0 77.6%

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

   1. +-commutative 77.6%

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

10. Simplified 77.6%

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

11. Final simplification 77.6%

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

Alternative 7: 37.1% 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 98.8%

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

   2. associate-+r- 98.8%

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

   3. remove-double-neg 98.8%

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

   4. distribute-frac-neg 98.8%

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

   5. unsub-neg 98.8%

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

   6. associate--r+ 98.8%

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

   7. +-commutative 98.8%

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

   8. distribute-frac-neg 98.8%

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

   9. sub-neg 98.8%

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

   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 38.5%

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

6. Final simplification 38.5%

   \[\leadsto x \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(FPCore (x y z)
  :name "Statistics.Sample:$swelfordMean from math-functions-0.1.5.2"
  :precision binary64
  (+ x (/ (- y x) z)))