Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, B

Percentage Accurate: 99.9% → 100.0%

Time: 5.1s

Alternatives: 6

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ x - \frac{y}{1 + \frac{x \cdot y}{2}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- x (/ y (+ 1.0 (/ (* x y) 2.0)))))

C

double code(double x, double y) {
	return x - (y / (1.0 + ((x * y) / 2.0)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x - (y / (1.0d0 + ((x * y) / 2.0d0)))
end function

Java

public static double code(double x, double y) {
	return x - (y / (1.0 + ((x * y) / 2.0)));
}

Python

def code(x, y):
	return x - (y / (1.0 + ((x * y) / 2.0)))

Julia

function code(x, y)
	return Float64(x - Float64(y / Float64(1.0 + Float64(Float64(x * y) / 2.0))))
end

MATLAB

function tmp = code(x, y)
	tmp = x - (y / (1.0 + ((x * y) / 2.0)));
end

Wolfram

code[x_, y_] := N[(x - N[(y / N[(1.0 + N[(N[(x * y), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \frac{y}{1 + \frac{x \cdot y}{2}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- x (/ y (+ 1.0 (/ (* x y) 2.0)))))

C

double code(double x, double y) {
	return x - (y / (1.0 + ((x * y) / 2.0)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x - (y / (1.0d0 + ((x * y) / 2.0d0)))
end function

Java

public static double code(double x, double y) {
	return x - (y / (1.0 + ((x * y) / 2.0)));
}

Python

def code(x, y):
	return x - (y / (1.0 + ((x * y) / 2.0)))

Julia

function code(x, y)
	return Float64(x - Float64(y / Float64(1.0 + Float64(Float64(x * y) / 2.0))))
end

MATLAB

function tmp = code(x, y)
	tmp = x - (y / (1.0 + ((x * y) / 2.0)));
end

Wolfram

code[x_, y_] := N[(x - N[(y / N[(1.0 + N[(N[(x * y), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \frac{y}{1 + \frac{y}{\frac{2}{x}}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- x (/ y (+ 1.0 (/ y (/ 2.0 x))))))

C

double code(double x, double y) {
	return x - (y / (1.0 + (y / (2.0 / x))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x - (y / (1.0d0 + (y / (2.0d0 / x))))
end function

Java

public static double code(double x, double y) {
	return x - (y / (1.0 + (y / (2.0 / x))));
}

Python

def code(x, y):
	return x - (y / (1.0 + (y / (2.0 / x))))

Julia

function code(x, y)
	return Float64(x - Float64(y / Float64(1.0 + Float64(y / Float64(2.0 / x)))))
end

MATLAB

function tmp = code(x, y)
	tmp = x - (y / (1.0 + (y / (2.0 / x))));
end

Wolfram

code[x_, y_] := N[(x - N[(y / N[(1.0 + N[(y / N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
2. Step-by-step derivation

   1. *-commutative 99.8%

      \[\leadsto x - \frac{y}{1 + \frac{\color{blue}{y \cdot x}}{2}} \]

   2. associate-/l* 99.8%

      \[\leadsto x - \frac{y}{1 + \color{blue}{\frac{y}{\frac{2}{x}}}} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{x - \frac{y}{1 + \frac{y}{\frac{2}{x}}}} \]

4. Final simplification 99.8%

   \[\leadsto x - \frac{y}{1 + \frac{y}{\frac{2}{x}}} \]

Alternative 2: 87.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-18}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;x \leq 1.58 \cdot 10^{-12}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6e-8)
   x
   (if (<= x -3.1e-18) (/ -2.0 x) (if (<= x 1.58e-12) (- x y) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6e-8) {
		tmp = x;
	} else if (x <= -3.1e-18) {
		tmp = -2.0 / x;
	} else if (x <= 1.58e-12) {
		tmp = x - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-6d-8)) then
        tmp = x
    else if (x <= (-3.1d-18)) then
        tmp = (-2.0d0) / x
    else if (x <= 1.58d-12) then
        tmp = x - y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -6e-8) {
		tmp = x;
	} else if (x <= -3.1e-18) {
		tmp = -2.0 / x;
	} else if (x <= 1.58e-12) {
		tmp = x - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6e-8:
		tmp = x
	elif x <= -3.1e-18:
		tmp = -2.0 / x
	elif x <= 1.58e-12:
		tmp = x - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6e-8)
		tmp = x;
	elseif (x <= -3.1e-18)
		tmp = Float64(-2.0 / x);
	elseif (x <= 1.58e-12)
		tmp = Float64(x - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6e-8)
		tmp = x;
	elseif (x <= -3.1e-18)
		tmp = -2.0 / x;
	elseif (x <= 1.58e-12)
		tmp = x - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6e-8], x, If[LessEqual[x, -3.1e-18], N[(-2.0 / x), $MachinePrecision], If[LessEqual[x, 1.58e-12], N[(x - y), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.99999999999999946e-8 or 1.57999999999999993e-12 < x

   1. Initial program 99.8%

      \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
   2. Step-by-step derivation

      1. *-commutative 99.8%

         \[\leadsto x - \frac{y}{1 + \frac{\color{blue}{y \cdot x}}{2}} \]

      2. associate-/l* 99.8%

         \[\leadsto x - \frac{y}{1 + \color{blue}{\frac{y}{\frac{2}{x}}}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x - \frac{y}{1 + \frac{y}{\frac{2}{x}}}} \]

   4. Taylor expanded in x around inf 98.8%

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

   if -5.99999999999999946e-8 < x < -3.10000000000000007e-18

   1. Initial program 99.6%

      \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
   2. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto x - \frac{y}{1 + \frac{\color{blue}{y \cdot x}}{2}} \]

      2. associate-/l* 99.8%

         \[\leadsto x - \frac{y}{1 + \color{blue}{\frac{y}{\frac{2}{x}}}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x - \frac{y}{1 + \frac{y}{\frac{2}{x}}}} \]

   4. Taylor expanded in y around inf 79.1%

      \[\leadsto x - \color{blue}{\frac{2}{x}} \]

   5. Taylor expanded in x around 0 78.7%

      \[\leadsto \color{blue}{\frac{-2}{x}} \]

   if -3.10000000000000007e-18 < x < 1.57999999999999993e-12

   1. Initial program 99.9%

      \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
   2. Step-by-step derivation

      1. *-commutative 99.9%

         \[\leadsto x - \frac{y}{1 + \frac{\color{blue}{y \cdot x}}{2}} \]

      2. associate-/l* 100.0%

         \[\leadsto x - \frac{y}{1 + \color{blue}{\frac{y}{\frac{2}{x}}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x - \frac{y}{1 + \frac{y}{\frac{2}{x}}}} \]

   4. Taylor expanded in y around 0 81.1%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-18}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;x \leq 1.58 \cdot 10^{-12}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 90.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+70} \lor \neg \left(y \leq 5.1 \cdot 10^{+104}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.9e+70) (not (<= y 5.1e+104))) (- x (/ 2.0 x)) (- x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.9e+70) || !(y <= 5.1e+104)) {
		tmp = x - (2.0 / x);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.9d+70)) .or. (.not. (y <= 5.1d+104))) then
        tmp = x - (2.0d0 / x)
    else
        tmp = x - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.9e+70) || !(y <= 5.1e+104)) {
		tmp = x - (2.0 / x);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.9e+70) or not (y <= 5.1e+104):
		tmp = x - (2.0 / x)
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.9e+70) || !(y <= 5.1e+104))
		tmp = Float64(x - Float64(2.0 / x));
	else
		tmp = Float64(x - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.9e+70) || ~((y <= 5.1e+104)))
		tmp = x - (2.0 / x);
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.9e+70], N[Not[LessEqual[y, 5.1e+104]], $MachinePrecision]], N[(x - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(x - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8999999999999999e70 or 5.1000000000000002e104 < y

   1. Initial program 99.5%

      \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
   2. Step-by-step derivation

      1. *-commutative 99.5%

         \[\leadsto x - \frac{y}{1 + \frac{\color{blue}{y \cdot x}}{2}} \]

      2. associate-/l* 99.6%

         \[\leadsto x - \frac{y}{1 + \color{blue}{\frac{y}{\frac{2}{x}}}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{x - \frac{y}{1 + \frac{y}{\frac{2}{x}}}} \]

   4. Taylor expanded in y around inf 83.7%

      \[\leadsto x - \color{blue}{\frac{2}{x}} \]

   if -1.8999999999999999e70 < y < 5.1000000000000002e104

   1. Initial program 100.0%

      \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto x - \frac{y}{1 + \frac{\color{blue}{y \cdot x}}{2}} \]

      2. associate-/l* 100.0%

         \[\leadsto x - \frac{y}{1 + \color{blue}{\frac{y}{\frac{2}{x}}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x - \frac{y}{1 + \frac{y}{\frac{2}{x}}}} \]

   4. Taylor expanded in y around 0 96.9%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+70} \lor \neg \left(y \leq 5.1 \cdot 10^{+104}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]

Alternative 4: 87.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.58 \cdot 10^{-12}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.4e-8) x (if (<= x 1.58e-12) (- x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.4e-8) {
		tmp = x;
	} else if (x <= 1.58e-12) {
		tmp = x - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.4d-8)) then
        tmp = x
    else if (x <= 1.58d-12) then
        tmp = x - y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.4e-8) {
		tmp = x;
	} else if (x <= 1.58e-12) {
		tmp = x - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.4e-8:
		tmp = x
	elif x <= 1.58e-12:
		tmp = x - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.4e-8)
		tmp = x;
	elseif (x <= 1.58e-12)
		tmp = Float64(x - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.4e-8)
		tmp = x;
	elseif (x <= 1.58e-12)
		tmp = x - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.4e-8], x, If[LessEqual[x, 1.58e-12], N[(x - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4e-8 or 1.57999999999999993e-12 < x

   1. Initial program 99.8%

      \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
   2. Step-by-step derivation

      1. *-commutative 99.8%

         \[\leadsto x - \frac{y}{1 + \frac{\color{blue}{y \cdot x}}{2}} \]

      2. associate-/l* 99.7%

         \[\leadsto x - \frac{y}{1 + \color{blue}{\frac{y}{\frac{2}{x}}}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x - \frac{y}{1 + \frac{y}{\frac{2}{x}}}} \]

   4. Taylor expanded in x around inf 97.4%

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

   if -1.4e-8 < x < 1.57999999999999993e-12

   1. Initial program 99.9%

      \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
   2. Step-by-step derivation

      1. *-commutative 99.9%

         \[\leadsto x - \frac{y}{1 + \frac{\color{blue}{y \cdot x}}{2}} \]

      2. associate-/l* 100.0%

         \[\leadsto x - \frac{y}{1 + \color{blue}{\frac{y}{\frac{2}{x}}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x - \frac{y}{1 + \frac{y}{\frac{2}{x}}}} \]

   4. Taylor expanded in y around 0 78.5%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.58 \cdot 10^{-12}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 78.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-134}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-113}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7.8e-134) x (if (<= x 2e-113) (- y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.8e-134) {
		tmp = x;
	} else if (x <= 2e-113) {
		tmp = -y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7.8d-134)) then
        tmp = x
    else if (x <= 2d-113) then
        tmp = -y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -7.8e-134) {
		tmp = x;
	} else if (x <= 2e-113) {
		tmp = -y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7.8e-134:
		tmp = x
	elif x <= 2e-113:
		tmp = -y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7.8e-134)
		tmp = x;
	elseif (x <= 2e-113)
		tmp = Float64(-y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.8e-134)
		tmp = x;
	elseif (x <= 2e-113)
		tmp = -y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7.8e-134], x, If[LessEqual[x, 2e-113], (-y), x]]

Derivation

1. Split input into 2 regimes

2. if x < -7.8000000000000002e-134 or 1.99999999999999996e-113 < x

   1. Initial program 99.8%

      \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
   2. Step-by-step derivation

      1. *-commutative 99.8%

         \[\leadsto x - \frac{y}{1 + \frac{\color{blue}{y \cdot x}}{2}} \]

      2. associate-/l* 99.8%

         \[\leadsto x - \frac{y}{1 + \color{blue}{\frac{y}{\frac{2}{x}}}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x - \frac{y}{1 + \frac{y}{\frac{2}{x}}}} \]

   4. Taylor expanded in x around inf 84.7%

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

   if -7.8000000000000002e-134 < x < 1.99999999999999996e-113

   1. Initial program 100.0%

      \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto x - \frac{y}{1 + \frac{\color{blue}{y \cdot x}}{2}} \]

      2. associate-/l* 100.0%

         \[\leadsto x - \frac{y}{1 + \color{blue}{\frac{y}{\frac{2}{x}}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x - \frac{y}{1 + \frac{y}{\frac{2}{x}}}} \]

   4. Taylor expanded in x around 0 80.5%

      \[\leadsto \color{blue}{-1 \cdot y} \]
   5. Step-by-step derivation

      1. neg-mul-1 80.5%

         \[\leadsto \color{blue}{-y} \]

   6. Simplified 80.5%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-134}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-113}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 62.5% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 x)

C

double code(double x, double y) {
	return x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x
end function

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

function tmp = code(x, y)
	tmp = x;
end

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 99.8%

   \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
2. Step-by-step derivation

   1. *-commutative 99.8%

      \[\leadsto x - \frac{y}{1 + \frac{\color{blue}{y \cdot x}}{2}} \]

   2. associate-/l* 99.8%

      \[\leadsto x - \frac{y}{1 + \color{blue}{\frac{y}{\frac{2}{x}}}} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{x - \frac{y}{1 + \frac{y}{\frac{2}{x}}}} \]

4. Taylor expanded in x around inf 64.9%

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

5. Final simplification 64.9%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023279 
(FPCore (x y)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, B"
  :precision binary64
  (- x (/ y (+ 1.0 (/ (* x y) 2.0)))))