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

Percentage Accurate: 99.9% → 99.9%

Time: 11.1s

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: 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]

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 89.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+187} \lor \neg \left(y \leq 4.4 \cdot 10^{+104}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.55e+187) (not (<= y 4.4e+104))) (- x (/ 2.0 x)) (- x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.55e+187) || !(y <= 4.4e+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 <= (-1.55d+187)) .or. (.not. (y <= 4.4d+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 <= -1.55e+187) || !(y <= 4.4e+104)) {
		tmp = x - (2.0 / x);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.55e+187) or not (y <= 4.4e+104):
		tmp = x - (2.0 / x)
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.55e+187) || !(y <= 4.4e+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 <= -1.55e+187) || ~((y <= 4.4e+104)))
		tmp = x - (2.0 / x);
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.55e+187], N[Not[LessEqual[y, 4.4e+104]], $MachinePrecision]], N[(x - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(x - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.55000000000000006e187 or 4.40000000000000001e104 < y

   1. Initial program 99.7%

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

      1. associate-/l* 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 83.8%

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

   if -1.55000000000000006e187 < y < 4.40000000000000001e104

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 93.6%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+187} \lor \neg \left(y \leq 4.4 \cdot 10^{+104}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]

Alternative 3: 87.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-9}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.1e-9) x (if (<= x 1.4) (- x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.1e-9) {
		tmp = x;
	} else if (x <= 1.4) {
		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 <= (-2.1d-9)) then
        tmp = x
    else if (x <= 1.4d0) 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 <= -2.1e-9) {
		tmp = x;
	} else if (x <= 1.4) {
		tmp = x - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.1e-9:
		tmp = x
	elif x <= 1.4:
		tmp = x - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.1e-9)
		tmp = x;
	elseif (x <= 1.4)
		tmp = Float64(x - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.1e-9)
		tmp = x;
	elseif (x <= 1.4)
		tmp = x - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.1e-9], x, If[LessEqual[x, 1.4], N[(x - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.10000000000000019e-9 or 1.3999999999999999 < x

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 97.8%

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

   if -2.10000000000000019e-9 < x < 1.3999999999999999

   1. Initial program 99.8%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 77.1%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-9}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 78.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-122}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-41}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.2e-122) x (if (<= x 9.2e-41) (- y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.2e-122) {
		tmp = x;
	} else if (x <= 9.2e-41) {
		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 <= (-3.2d-122)) then
        tmp = x
    else if (x <= 9.2d-41) 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 <= -3.2e-122) {
		tmp = x;
	} else if (x <= 9.2e-41) {
		tmp = -y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.2e-122:
		tmp = x
	elif x <= 9.2e-41:
		tmp = -y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.2e-122)
		tmp = x;
	elseif (x <= 9.2e-41)
		tmp = Float64(-y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.2e-122)
		tmp = x;
	elseif (x <= 9.2e-41)
		tmp = -y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.2e-122], x, If[LessEqual[x, 9.2e-41], (-y), x]]

Derivation

1. Split input into 2 regimes

2. if x < -3.2000000000000002e-122 or 9.20000000000000041e-41 < x

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 87.8%

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

   if -3.2000000000000002e-122 < x < 9.20000000000000041e-41

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 68.5%

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

      1. mul-1-neg 68.5%

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

   6. Simplified 68.5%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-122}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-41}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{-1}{x \cdot 0.5 + \frac{1}{y}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (/ -1.0 (+ (* x 0.5) (/ 1.0 y)))))

C

double code(double x, double y) {
	return x + (-1.0 / ((x * 0.5) + (1.0 / y)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + ((-1.0d0) / ((x * 0.5d0) + (1.0d0 / y)))
end function

Java

public static double code(double x, double y) {
	return x + (-1.0 / ((x * 0.5) + (1.0 / y)));
}

Python

def code(x, y):
	return x + (-1.0 / ((x * 0.5) + (1.0 / y)))

Julia

function code(x, y)
	return Float64(x + Float64(-1.0 / Float64(Float64(x * 0.5) + Float64(1.0 / y))))
end

MATLAB

function tmp = code(x, y)
	tmp = x + (-1.0 / ((x * 0.5) + (1.0 / y)));
end

Wolfram

code[x_, y_] := N[(x + N[(-1.0 / N[(N[(x * 0.5), $MachinePrecision] + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{x - \frac{y}{1 + \frac{x}{\frac{2}{y}}}} \]
4. Step-by-step derivation

   1. clear-num 99.9%

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

   2. associate-/l* 99.9%

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

   3. inv-pow 99.9%

      \[\leadsto x - \color{blue}{{\left(\frac{1 + \frac{x \cdot y}{2}}{y}\right)}^{-1}} \]

   4. +-commutative 99.9%

      \[\leadsto x - {\left(\frac{\color{blue}{\frac{x \cdot y}{2} + 1}}{y}\right)}^{-1} \]

   5. associate-/l* 99.9%

      \[\leadsto x - {\left(\frac{\color{blue}{\frac{x}{\frac{2}{y}}} + 1}{y}\right)}^{-1} \]

   6. div-inv 99.8%

      \[\leadsto x - {\left(\frac{\color{blue}{x \cdot \frac{1}{\frac{2}{y}}} + 1}{y}\right)}^{-1} \]

   7. fma-def 99.8%

      \[\leadsto x - {\left(\frac{\color{blue}{\mathsf{fma}\left(x, \frac{1}{\frac{2}{y}}, 1\right)}}{y}\right)}^{-1} \]

   8. clear-num 99.9%

      \[\leadsto x - {\left(\frac{\mathsf{fma}\left(x, \color{blue}{\frac{y}{2}}, 1\right)}{y}\right)}^{-1} \]

   9. div-inv 99.9%

      \[\leadsto x - {\left(\frac{\mathsf{fma}\left(x, \color{blue}{y \cdot \frac{1}{2}}, 1\right)}{y}\right)}^{-1} \]

   10. metadata-eval 99.9%

       \[\leadsto x - {\left(\frac{\mathsf{fma}\left(x, y \cdot \color{blue}{0.5}, 1\right)}{y}\right)}^{-1} \]

5. Applied egg-rr 99.9%

   \[\leadsto x - \color{blue}{{\left(\frac{\mathsf{fma}\left(x, y \cdot 0.5, 1\right)}{y}\right)}^{-1}} \]
6. Step-by-step derivation

   1. unpow-1 99.9%

      \[\leadsto x - \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(x, y \cdot 0.5, 1\right)}{y}}} \]

   2. frac-2neg 99.9%

      \[\leadsto x - \frac{1}{\color{blue}{\frac{-\mathsf{fma}\left(x, y \cdot 0.5, 1\right)}{-y}}} \]

   3. associate-/r/ 99.9%

      \[\leadsto x - \color{blue}{\frac{1}{-\mathsf{fma}\left(x, y \cdot 0.5, 1\right)} \cdot \left(-y\right)} \]

   4. neg-sub0 99.9%

      \[\leadsto x - \frac{1}{\color{blue}{0 - \mathsf{fma}\left(x, y \cdot 0.5, 1\right)}} \cdot \left(-y\right) \]

   5. fma-udef 99.9%

      \[\leadsto x - \frac{1}{0 - \color{blue}{\left(x \cdot \left(y \cdot 0.5\right) + 1\right)}} \cdot \left(-y\right) \]

   6. +-commutative 99.9%

      \[\leadsto x - \frac{1}{0 - \color{blue}{\left(1 + x \cdot \left(y \cdot 0.5\right)\right)}} \cdot \left(-y\right) \]

   7. associate-*r* 99.9%

      \[\leadsto x - \frac{1}{0 - \left(1 + \color{blue}{\left(x \cdot y\right) \cdot 0.5}\right)} \cdot \left(-y\right) \]

   8. *-commutative 99.9%

      \[\leadsto x - \frac{1}{0 - \left(1 + \color{blue}{\left(y \cdot x\right)} \cdot 0.5\right)} \cdot \left(-y\right) \]

   9. associate--r+ 99.9%

      \[\leadsto x - \frac{1}{\color{blue}{\left(0 - 1\right) - \left(y \cdot x\right) \cdot 0.5}} \cdot \left(-y\right) \]

   10. metadata-eval 99.9%

       \[\leadsto x - \frac{1}{\color{blue}{-1} - \left(y \cdot x\right) \cdot 0.5} \cdot \left(-y\right) \]

   11. *-commutative 99.9%

       \[\leadsto x - \frac{1}{-1 - \color{blue}{\left(x \cdot y\right)} \cdot 0.5} \cdot \left(-y\right) \]

   12. associate-*r* 99.9%

       \[\leadsto x - \frac{1}{-1 - \color{blue}{x \cdot \left(y \cdot 0.5\right)}} \cdot \left(-y\right) \]

7. Applied egg-rr 99.9%

   \[\leadsto x - \color{blue}{\frac{1}{-1 - x \cdot \left(y \cdot 0.5\right)} \cdot \left(-y\right)} \]
8. Step-by-step derivation

   1. associate-*l/ 99.9%

      \[\leadsto x - \color{blue}{\frac{1 \cdot \left(-y\right)}{-1 - x \cdot \left(y \cdot 0.5\right)}} \]

   2. associate-/l* 99.9%

      \[\leadsto x - \color{blue}{\frac{1}{\frac{-1 - x \cdot \left(y \cdot 0.5\right)}{-y}}} \]

9. Simplified 99.9%

   \[\leadsto x - \color{blue}{\frac{1}{\frac{-1 - x \cdot \left(y \cdot 0.5\right)}{-y}}} \]

10. Taylor expanded in x around 0 99.9%

    \[\leadsto x - \frac{1}{\color{blue}{0.5 \cdot x + \frac{1}{y}}} \]

11. Final simplification 99.9%

    \[\leadsto x + \frac{-1}{x \cdot 0.5 + \frac{1}{y}} \]

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 99.9%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around inf 64.1%

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

5. Final simplification 64.1%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023336 
(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)))))