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

Percentage Accurate: 100.0% → 100.0%

Time: 6.6s

Alternatives: 7

Speedup: 1.0×

• 21.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + ((y - x) / z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + ((y - x) / z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + ((y - x) / z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x + \frac{y - x}{z} \]

2. Final simplification 100.0%

   \[\leadsto x + \frac{y - x}{z} \]

Alternative 2: 62.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{z}\\ \mathbf{if}\;z \leq -7.4 \cdot 10^{+22}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-137}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq -2.5 \cdot 10^{-185}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -4.4 \cdot 10^{-266}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-306}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-294}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-228}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x) z)))
   (if (<= z -7.4e+22)
     x
     (if (<= z -1.7e-137)
       (/ y z)
       (if (<= z -2.5e-185)
         t_0
         (if (<= z -4.4e-266)
           (/ y z)
           (if (<= z -5e-306)
             t_0
             (if (<= z 5e-294)
               (/ y z)
               (if (<= z 2.5e-228)
                 t_0
                 (if (<= z 4.5e-83) (/ y z) (if (<= z 1.0) t_0 x)))))))))))

C

double code(double x, double y, double z) {
	double t_0 = -x / z;
	double tmp;
	if (z <= -7.4e+22) {
		tmp = x;
	} else if (z <= -1.7e-137) {
		tmp = y / z;
	} else if (z <= -2.5e-185) {
		tmp = t_0;
	} else if (z <= -4.4e-266) {
		tmp = y / z;
	} else if (z <= -5e-306) {
		tmp = t_0;
	} else if (z <= 5e-294) {
		tmp = y / z;
	} else if (z <= 2.5e-228) {
		tmp = t_0;
	} else if (z <= 4.5e-83) {
		tmp = y / z;
	} else if (z <= 1.0) {
		tmp = t_0;
	} 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 <= (-7.4d+22)) then
        tmp = x
    else if (z <= (-1.7d-137)) then
        tmp = y / z
    else if (z <= (-2.5d-185)) then
        tmp = t_0
    else if (z <= (-4.4d-266)) then
        tmp = y / z
    else if (z <= (-5d-306)) then
        tmp = t_0
    else if (z <= 5d-294) then
        tmp = y / z
    else if (z <= 2.5d-228) then
        tmp = t_0
    else if (z <= 4.5d-83) then
        tmp = y / z
    else if (z <= 1.0d0) then
        tmp = t_0
    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 <= -7.4e+22) {
		tmp = x;
	} else if (z <= -1.7e-137) {
		tmp = y / z;
	} else if (z <= -2.5e-185) {
		tmp = t_0;
	} else if (z <= -4.4e-266) {
		tmp = y / z;
	} else if (z <= -5e-306) {
		tmp = t_0;
	} else if (z <= 5e-294) {
		tmp = y / z;
	} else if (z <= 2.5e-228) {
		tmp = t_0;
	} else if (z <= 4.5e-83) {
		tmp = y / z;
	} else if (z <= 1.0) {
		tmp = t_0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -x / z
	tmp = 0
	if z <= -7.4e+22:
		tmp = x
	elif z <= -1.7e-137:
		tmp = y / z
	elif z <= -2.5e-185:
		tmp = t_0
	elif z <= -4.4e-266:
		tmp = y / z
	elif z <= -5e-306:
		tmp = t_0
	elif z <= 5e-294:
		tmp = y / z
	elif z <= 2.5e-228:
		tmp = t_0
	elif z <= 4.5e-83:
		tmp = y / z
	elif z <= 1.0:
		tmp = t_0
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-x) / z)
	tmp = 0.0
	if (z <= -7.4e+22)
		tmp = x;
	elseif (z <= -1.7e-137)
		tmp = Float64(y / z);
	elseif (z <= -2.5e-185)
		tmp = t_0;
	elseif (z <= -4.4e-266)
		tmp = Float64(y / z);
	elseif (z <= -5e-306)
		tmp = t_0;
	elseif (z <= 5e-294)
		tmp = Float64(y / z);
	elseif (z <= 2.5e-228)
		tmp = t_0;
	elseif (z <= 4.5e-83)
		tmp = Float64(y / z);
	elseif (z <= 1.0)
		tmp = t_0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -x / z;
	tmp = 0.0;
	if (z <= -7.4e+22)
		tmp = x;
	elseif (z <= -1.7e-137)
		tmp = y / z;
	elseif (z <= -2.5e-185)
		tmp = t_0;
	elseif (z <= -4.4e-266)
		tmp = y / z;
	elseif (z <= -5e-306)
		tmp = t_0;
	elseif (z <= 5e-294)
		tmp = y / z;
	elseif (z <= 2.5e-228)
		tmp = t_0;
	elseif (z <= 4.5e-83)
		tmp = y / z;
	elseif (z <= 1.0)
		tmp = t_0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-x) / z), $MachinePrecision]}, If[LessEqual[z, -7.4e+22], x, If[LessEqual[z, -1.7e-137], N[(y / z), $MachinePrecision], If[LessEqual[z, -2.5e-185], t$95$0, If[LessEqual[z, -4.4e-266], N[(y / z), $MachinePrecision], If[LessEqual[z, -5e-306], t$95$0, If[LessEqual[z, 5e-294], N[(y / z), $MachinePrecision], If[LessEqual[z, 2.5e-228], t$95$0, If[LessEqual[z, 4.5e-83], N[(y / z), $MachinePrecision], If[LessEqual[z, 1.0], t$95$0, x]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.3999999999999996e22 or 1 < z

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]

   2. Taylor expanded in z around inf 76.1%

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

   if -7.3999999999999996e22 < z < -1.70000000000000007e-137 or -2.5000000000000001e-185 < z < -4.3999999999999999e-266 or -4.99999999999999998e-306 < z < 5.0000000000000003e-294 or 2.49999999999999986e-228 < z < 4.49999999999999997e-83

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. clear-num 99.8%

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

      3. associate-/r/ 98.7%

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

      4. fma-def 98.7%

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

   3. Applied egg-rr 98.7%

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

   4. Taylor expanded in y around inf 68.2%

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

   if -1.70000000000000007e-137 < z < -2.5000000000000001e-185 or -4.3999999999999999e-266 < z < -4.99999999999999998e-306 or 5.0000000000000003e-294 < z < 2.49999999999999986e-228 or 4.49999999999999997e-83 < z < 1

   1. Initial program 99.9%

      \[x + \frac{y - x}{z} \]

   2. Taylor expanded in y around 0 75.4%

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

   3. Taylor expanded in z around 0 73.1%

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

      1. mul-1-neg 73.1%

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

      2. distribute-frac-neg 73.1%

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

   5. Simplified 73.1%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.4 \cdot 10^{+22}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-137}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq -2.5 \cdot 10^{-185}:\\ \;\;\;\;\frac{-x}{z}\\ \mathbf{elif}\;z \leq -4.4 \cdot 10^{-266}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-306}:\\ \;\;\;\;\frac{-x}{z}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-294}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-228}:\\ \;\;\;\;\frac{-x}{z}\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{-x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 86.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+36} \lor \neg \left(y \leq 2.7 \cdot 10^{-131}\right):\\ \;\;\;\;x + \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.8e+36) (not (<= y 2.7e-131))) (+ x (/ y z)) (- x (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.8e+36) || !(y <= 2.7e-131)) {
		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 <= (-1.8d+36)) .or. (.not. (y <= 2.7d-131))) 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 <= -1.8e+36) || !(y <= 2.7e-131)) {
		tmp = x + (y / z);
	} else {
		tmp = x - (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.8e+36) or not (y <= 2.7e-131):
		tmp = x + (y / z)
	else:
		tmp = x - (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.8e+36) || !(y <= 2.7e-131))
		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 <= -1.8e+36) || ~((y <= 2.7e-131)))
		tmp = x + (y / z);
	else
		tmp = x - (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.8e+36], N[Not[LessEqual[y, 2.7e-131]], $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 < -1.7999999999999999e36 or 2.70000000000000021e-131 < y

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]

   2. Taylor expanded in y around inf 89.6%

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

   if -1.7999999999999999e36 < y < 2.70000000000000021e-131

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]

   2. Taylor expanded in y around 0 87.6%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+36} \lor \neg \left(y \leq 2.7 \cdot 10^{-131}\right):\\ \;\;\;\;x + \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{z}\\ \end{array} \]

Alternative 4: 98.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -32000000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x + \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -32000000000.0) (not (<= z 1.0))) (+ x (/ y z)) (/ (- y x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -32000000000.0) || !(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 <= (-32000000000.0d0)) .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 <= -32000000000.0) || !(z <= 1.0)) {
		tmp = x + (y / z);
	} else {
		tmp = (y - x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -32000000000.0) 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 <= -32000000000.0) || !(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 <= -32000000000.0) || ~((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, -32000000000.0], 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 < -3.2e10 or 1 < z

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]

   2. Taylor expanded in y around inf 99.9%

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

   if -3.2e10 < z < 1

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. clear-num 99.7%

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

      3. associate-/r/ 99.0%

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

      4. fma-def 99.0%

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

   3. Applied egg-rr 99.0%

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

   4. Taylor expanded in z around 0 98.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -32000000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x + \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - x}{z}\\ \end{array} \]

Alternative 5: 61.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+16}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{+35}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -9e+16) x (if (<= z 2.45e+35) (/ y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -9e+16) {
		tmp = x;
	} else if (z <= 2.45e+35) {
		tmp = y / z;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-9d+16)) then
        tmp = x
    else if (z <= 2.45d+35) then
        tmp = y / z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -9e+16) {
		tmp = x;
	} else if (z <= 2.45e+35) {
		tmp = y / z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -9e+16:
		tmp = x
	elif z <= 2.45e+35:
		tmp = y / z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -9e+16)
		tmp = x;
	elseif (z <= 2.45e+35)
		tmp = Float64(y / z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -9e+16)
		tmp = x;
	elseif (z <= 2.45e+35)
		tmp = y / z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -9e+16], x, If[LessEqual[z, 2.45e+35], N[(y / z), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -9e16 or 2.45000000000000013e35 < z

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]

   2. Taylor expanded in z around inf 77.6%

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

   if -9e16 < z < 2.45000000000000013e35

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. clear-num 99.7%

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

      3. associate-/r/ 99.1%

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

      4. fma-def 99.1%

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

   3. Applied egg-rr 99.1%

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

   4. Taylor expanded in y around inf 54.6%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+16}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{+35}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.02 \cdot 10^{+269}:\\ \;\;\;\;x + \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.02e+269) (+ x (/ y z)) (/ (- x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.02e+269) {
		tmp = x + (y / z);
	} else {
		tmp = -x / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 1.02d+269) then
        tmp = x + (y / z)
    else
        tmp = -x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.02e+269) {
		tmp = x + (y / z);
	} else {
		tmp = -x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 1.02e+269:
		tmp = x + (y / z)
	else:
		tmp = -x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.02e+269)
		tmp = Float64(x + Float64(y / z));
	else
		tmp = Float64(Float64(-x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 1.02e+269)
		tmp = x + (y / z);
	else
		tmp = -x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.02e+269], N[(x + N[(y / z), $MachinePrecision]), $MachinePrecision], N[((-x) / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.01999999999999994e269

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]

   2. Taylor expanded in y around inf 77.0%

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

   if 1.01999999999999994e269 < x

   1. Initial program 100.0%

      \[x + \frac{y - x}{z} \]

   2. Taylor expanded in y around 0 100.0%

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

   3. Taylor expanded in z around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.02 \cdot 10^{+269}:\\ \;\;\;\;x + \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z}\\ \end{array} \]

Alternative 7: 37.3% 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. Taylor expanded in z around inf 36.2%

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

3. Final simplification 36.2%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023279 
(FPCore (x y z)
  :name "Statistics.Sample:$swelfordMean from math-functions-0.1.5.2"
  :precision binary64
  (+ x (/ (- y x) z)))