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

Percentage Accurate: 99.9% → 99.9%

Time: 7.9s

Alternatives: 7

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. Add Preprocessing

3. Final simplification 99.9%

   \[\leadsto x + \frac{y}{-1 - \frac{x \cdot y}{2}} \]
4. Add Preprocessing

Alternative 2: 85.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{+35}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-93}:\\ \;\;\;\;x - y\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{-41}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.25e+35)
   x
   (if (<= x 3.4e-93) (- x y) (if (<= x 5.3e-41) (/ -2.0 x) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.25e+35) {
		tmp = x;
	} else if (x <= 3.4e-93) {
		tmp = x - y;
	} else if (x <= 5.3e-41) {
		tmp = -2.0 / x;
	} 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 <= (-2.25d+35)) then
        tmp = x
    else if (x <= 3.4d-93) then
        tmp = x - y
    else if (x <= 5.3d-41) then
        tmp = (-2.0d0) / x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.25e+35) {
		tmp = x;
	} else if (x <= 3.4e-93) {
		tmp = x - y;
	} else if (x <= 5.3e-41) {
		tmp = -2.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.25e+35:
		tmp = x
	elif x <= 3.4e-93:
		tmp = x - y
	elif x <= 5.3e-41:
		tmp = -2.0 / x
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.25e+35)
		tmp = x;
	elseif (x <= 3.4e-93)
		tmp = Float64(x - y);
	elseif (x <= 5.3e-41)
		tmp = Float64(-2.0 / x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.25e+35)
		tmp = x;
	elseif (x <= 3.4e-93)
		tmp = x - y;
	elseif (x <= 5.3e-41)
		tmp = -2.0 / x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.25e+35], x, If[LessEqual[x, 3.4e-93], N[(x - y), $MachinePrecision], If[LessEqual[x, 5.3e-41], N[(-2.0 / x), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.2499999999999998e35 or 5.29999999999999999e-41 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 98.0%

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

   if -2.2499999999999998e35 < x < 3.40000000000000001e-93

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 73.5%

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

      1. neg-mul-1 73.5%

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

      2. unsub-neg 73.5%

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

   5. Simplified 73.5%

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

   if 3.40000000000000001e-93 < x < 5.29999999999999999e-41

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf 76.4%

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

      1. associate-*r/ 76.4%

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

      2. metadata-eval 76.4%

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

   5. Simplified 76.4%

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

   6. Taylor expanded in x around 0 76.4%

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

Alternative 3: 90.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{+113} \lor \neg \left(y \leq 1.5 \cdot 10^{+66}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.65e+113) (not (<= y 1.5e+66))) (- x (/ 2.0 x)) (- x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.65e+113) || !(y <= 1.5e+66)) {
		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 <= (-2.65d+113)) .or. (.not. (y <= 1.5d+66))) 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 <= -2.65e+113) || !(y <= 1.5e+66)) {
		tmp = x - (2.0 / x);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.65e+113) or not (y <= 1.5e+66):
		tmp = x - (2.0 / x)
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.65e+113) || !(y <= 1.5e+66))
		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 <= -2.65e+113) || ~((y <= 1.5e+66)))
		tmp = x - (2.0 / x);
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.65e+113], N[Not[LessEqual[y, 1.5e+66]], $MachinePrecision]], N[(x - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(x - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.64999999999999984e113 or 1.50000000000000001e66 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf 83.8%

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

      1. associate-*r/ 83.8%

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

      2. metadata-eval 83.8%

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

   5. Simplified 83.8%

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

   if -2.64999999999999984e113 < y < 1.50000000000000001e66

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 97.6%

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

      1. neg-mul-1 97.6%

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

      2. unsub-neg 97.6%

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

   5. Simplified 97.6%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{+113} \lor \neg \left(y \leq 1.5 \cdot 10^{+66}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{+35}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{-16}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.25e+35) x (if (<= x 2.35e-16) (- x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.25e+35) {
		tmp = x;
	} else if (x <= 2.35e-16) {
		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 <= (-2.25d+35)) then
        tmp = x
    else if (x <= 2.35d-16) 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 <= -2.25e+35) {
		tmp = x;
	} else if (x <= 2.35e-16) {
		tmp = x - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.25e+35:
		tmp = x
	elif x <= 2.35e-16:
		tmp = x - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.25e+35)
		tmp = x;
	elseif (x <= 2.35e-16)
		tmp = Float64(x - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.25e+35)
		tmp = x;
	elseif (x <= 2.35e-16)
		tmp = x - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.25e+35], x, If[LessEqual[x, 2.35e-16], N[(x - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2499999999999998e35 or 2.35000000000000022e-16 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 99.9%

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

   if -2.2499999999999998e35 < x < 2.35000000000000022e-16

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 68.8%

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

      1. neg-mul-1 68.8%

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

      2. unsub-neg 68.8%

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

   5. Simplified 68.8%

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

Alternative 5: 78.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{-101}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-124}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.9e-101) x (if (<= x 5.6e-124) (- y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.9e-101) {
		tmp = x;
	} else if (x <= 5.6e-124) {
		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 <= (-4.9d-101)) then
        tmp = x
    else if (x <= 5.6d-124) 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 <= -4.9e-101) {
		tmp = x;
	} else if (x <= 5.6e-124) {
		tmp = -y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.9e-101:
		tmp = x
	elif x <= 5.6e-124:
		tmp = -y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.9e-101)
		tmp = x;
	elseif (x <= 5.6e-124)
		tmp = Float64(-y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.9e-101)
		tmp = x;
	elseif (x <= 5.6e-124)
		tmp = -y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.9e-101], x, If[LessEqual[x, 5.6e-124], (-y), x]]

Derivation

1. Split input into 2 regimes

2. if x < -4.9e-101 or 5.59999999999999996e-124 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 84.1%

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

   if -4.9e-101 < x < 5.59999999999999996e-124

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 64.5%

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

      1. neg-mul-1 64.5%

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

   5. Simplified 64.5%

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

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around inf 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. associate-/r* 99.9%

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

   3. *-inverses 99.9%

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

   4. fma-define 99.9%

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

5. Applied egg-rr 99.9%

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

   1. *-lft-identity 99.9%

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

   2. fma-undefine 99.9%

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

   3. *-commutative 99.9%

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

   4. fma-undefine 99.9%

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

7. Simplified 99.9%

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

   1. fma-undefine 99.9%

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

9. Applied egg-rr 99.9%

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

10. Final simplification 99.9%

    \[\leadsto x + \frac{1}{\frac{-1}{y} - x \cdot 0.5} \]
11. Add Preprocessing

Alternative 7: 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. Add Preprocessing

3. Taylor expanded in x around inf 61.7%

   \[\leadsto \color{blue}{x} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024130 
(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)))))