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

Percentage Accurate: 99.9% → 99.9%

Time: 7.8s

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: 86.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{-25} \lor \neg \left(x \leq 1.1 \cdot 10^{-25}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -5.6e-25) (not (<= x 1.1e-25))) (- x (/ 2.0 x)) (- x y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -5.6e-25) || !(x <= 1.1e-25)) {
		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 ((x <= (-5.6d-25)) .or. (.not. (x <= 1.1d-25))) 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 ((x <= -5.6e-25) || !(x <= 1.1e-25)) {
		tmp = x - (2.0 / x);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -5.6e-25) or not (x <= 1.1e-25):
		tmp = x - (2.0 / x)
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -5.6e-25) || !(x <= 1.1e-25))
		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 ((x <= -5.6e-25) || ~((x <= 1.1e-25)))
		tmp = x - (2.0 / x);
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -5.6e-25], N[Not[LessEqual[x, 1.1e-25]], $MachinePrecision]], N[(x - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(x - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.59999999999999976e-25 or 1.1000000000000001e-25 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 95.2%

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

      1. associate-*r/ 95.2%

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

      2. metadata-eval 95.2%

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

   5. Simplified 95.2%

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

   if -5.59999999999999976e-25 < x < 1.1000000000000001e-25

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 84.2%

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

      1. neg-mul-1 84.2%

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

      2. unsub-neg 84.2%

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

   5. Simplified 84.2%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{-25} \lor \neg \left(x \leq 1.1 \cdot 10^{-25}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-24}:\\ \;\;\;\;x \cdot \left(1 - \frac{2}{x \cdot x}\right)\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-23}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{2}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.6e-24)
   (* x (- 1.0 (/ 2.0 (* x x))))
   (if (<= x 4.6e-23) (- x y) (- x (/ 2.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.6e-24) {
		tmp = x * (1.0 - (2.0 / (x * x)));
	} else if (x <= 4.6e-23) {
		tmp = x - y;
	} else {
		tmp = x - (2.0 / 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 <= (-2.6d-24)) then
        tmp = x * (1.0d0 - (2.0d0 / (x * x)))
    else if (x <= 4.6d-23) then
        tmp = x - y
    else
        tmp = x - (2.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.6e-24) {
		tmp = x * (1.0 - (2.0 / (x * x)));
	} else if (x <= 4.6e-23) {
		tmp = x - y;
	} else {
		tmp = x - (2.0 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.6e-24:
		tmp = x * (1.0 - (2.0 / (x * x)))
	elif x <= 4.6e-23:
		tmp = x - y
	else:
		tmp = x - (2.0 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.6e-24)
		tmp = Float64(x * Float64(1.0 - Float64(2.0 / Float64(x * x))));
	elseif (x <= 4.6e-23)
		tmp = Float64(x - y);
	else
		tmp = Float64(x - Float64(2.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.6e-24)
		tmp = x * (1.0 - (2.0 / (x * x)));
	elseif (x <= 4.6e-23)
		tmp = x - y;
	else
		tmp = x - (2.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.6e-24], N[(x * N[(1.0 - N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.6e-23], N[(x - y), $MachinePrecision], N[(x - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.6e-24

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 93.8%

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

      1. associate-*r/ 93.8%

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

      2. metadata-eval 93.8%

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

   5. Simplified 93.8%

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{2}{{x}^{2}}\right)} \]
   6. Step-by-step derivation

      1. unpow2 93.8%

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

   7. Applied egg-rr 93.8%

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

   if -2.6e-24 < x < 4.6000000000000002e-23

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 84.2%

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

      1. neg-mul-1 84.2%

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

      2. unsub-neg 84.2%

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

   5. Simplified 84.2%

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

   if 4.6000000000000002e-23 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 96.6%

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

      1. associate-*r/ 96.6%

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

      2. metadata-eval 96.6%

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

   5. Simplified 96.6%

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

Alternative 4: 86.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6000:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 0.0034:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6000.0) x (if (<= x 0.0034) (- x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6000.0) {
		tmp = x;
	} else if (x <= 0.0034) {
		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 <= (-6000.0d0)) then
        tmp = x
    else if (x <= 0.0034d0) 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 <= -6000.0) {
		tmp = x;
	} else if (x <= 0.0034) {
		tmp = x - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6000.0:
		tmp = x
	elif x <= 0.0034:
		tmp = x - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6000.0)
		tmp = x;
	elseif (x <= 0.0034)
		tmp = Float64(x - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6000.0)
		tmp = x;
	elseif (x <= 0.0034)
		tmp = x - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6000.0], x, If[LessEqual[x, 0.0034], N[(x - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -6e3 or 0.00339999999999999981 < 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 99.2%

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

   if -6e3 < x < 0.00339999999999999981

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 80.1%

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

      1. neg-mul-1 80.1%

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

      2. unsub-neg 80.1%

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

   5. Simplified 80.1%

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

Alternative 5: 78.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-66}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-134}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9e-66) x (if (<= x 9.5e-134) (- y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9e-66) {
		tmp = x;
	} else if (x <= 9.5e-134) {
		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 <= (-9d-66)) then
        tmp = x
    else if (x <= 9.5d-134) 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 <= -9e-66) {
		tmp = x;
	} else if (x <= 9.5e-134) {
		tmp = -y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9e-66:
		tmp = x
	elif x <= 9.5e-134:
		tmp = -y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9e-66)
		tmp = x;
	elseif (x <= 9.5e-134)
		tmp = Float64(-y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9e-66)
		tmp = x;
	elseif (x <= 9.5e-134)
		tmp = -y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9e-66], x, If[LessEqual[x, 9.5e-134], (-y), x]]

Derivation

1. Split input into 2 regimes

2. if x < -8.9999999999999995e-66 or 9.5000000000000008e-134 < 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 86.6%

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

   if -8.9999999999999995e-66 < x < 9.5000000000000008e-134

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 70.7%

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

      1. neg-mul-1 70.7%

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

   5. Simplified 70.7%

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

   1. clear-num 99.9%

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

   2. associate-/r/ 99.9%

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

   3. +-commutative 99.9%

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

   4. associate-/l* 99.9%

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

   5. fma-define 99.9%

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

   6. div-inv 99.9%

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

   7. metadata-eval 99.9%

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

4. Applied egg-rr 99.9%

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

   1. associate-*l/ 99.9%

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

   2. *-un-lft-identity 99.9%

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

   3. clear-num 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around 0 99.9%

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

8. Final simplification 99.9%

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

Alternative 7: 62.8% 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 59.7%

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

Reproduce

herbie shell --seed 2024191 
(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)))))