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

Percentage Accurate: 99.9% → 100.0%

Time: 3.7s

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 99.9%

   \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
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. Final simplification 100.0%

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

Alternative 2: 85.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.072:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{-51}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-183}:\\ \;\;\;\;x - y\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{-197}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-19}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.072)
   x
   (if (<= x -2.15e-51)
     (/ -2.0 x)
     (if (<= x -7e-183)
       (- x y)
       (if (<= x -2.5e-197) (/ -2.0 x) (if (<= x 8e-19) (- x y) x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.072) {
		tmp = x;
	} else if (x <= -2.15e-51) {
		tmp = -2.0 / x;
	} else if (x <= -7e-183) {
		tmp = x - y;
	} else if (x <= -2.5e-197) {
		tmp = -2.0 / x;
	} else if (x <= 8e-19) {
		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 <= (-0.072d0)) then
        tmp = x
    else if (x <= (-2.15d-51)) then
        tmp = (-2.0d0) / x
    else if (x <= (-7d-183)) then
        tmp = x - y
    else if (x <= (-2.5d-197)) then
        tmp = (-2.0d0) / x
    else if (x <= 8d-19) 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 <= -0.072) {
		tmp = x;
	} else if (x <= -2.15e-51) {
		tmp = -2.0 / x;
	} else if (x <= -7e-183) {
		tmp = x - y;
	} else if (x <= -2.5e-197) {
		tmp = -2.0 / x;
	} else if (x <= 8e-19) {
		tmp = x - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.072:
		tmp = x
	elif x <= -2.15e-51:
		tmp = -2.0 / x
	elif x <= -7e-183:
		tmp = x - y
	elif x <= -2.5e-197:
		tmp = -2.0 / x
	elif x <= 8e-19:
		tmp = x - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.072)
		tmp = x;
	elseif (x <= -2.15e-51)
		tmp = Float64(-2.0 / x);
	elseif (x <= -7e-183)
		tmp = Float64(x - y);
	elseif (x <= -2.5e-197)
		tmp = Float64(-2.0 / x);
	elseif (x <= 8e-19)
		tmp = Float64(x - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.072)
		tmp = x;
	elseif (x <= -2.15e-51)
		tmp = -2.0 / x;
	elseif (x <= -7e-183)
		tmp = x - y;
	elseif (x <= -2.5e-197)
		tmp = -2.0 / x;
	elseif (x <= 8e-19)
		tmp = x - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.072], x, If[LessEqual[x, -2.15e-51], N[(-2.0 / x), $MachinePrecision], If[LessEqual[x, -7e-183], N[(x - y), $MachinePrecision], If[LessEqual[x, -2.5e-197], N[(-2.0 / x), $MachinePrecision], If[LessEqual[x, 8e-19], N[(x - y), $MachinePrecision], x]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0719999999999999946 or 7.9999999999999998e-19 < x

   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 inf 100.0%

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

   if -0.0719999999999999946 < x < -2.1499999999999999e-51 or -6.99999999999999983e-183 < x < -2.5000000000000001e-197

   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 81.2%

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

   5. Taylor expanded in x around 0 76.0%

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

   if -2.1499999999999999e-51 < x < -6.99999999999999983e-183 or -2.5000000000000001e-197 < x < 7.9999999999999998e-19

   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 78.2%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.072:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{-51}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-183}:\\ \;\;\;\;x - y\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{-197}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-19}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 89.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+154} \lor \neg \left(y \leq 1.75 \cdot 10^{+45}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.5e+154) (not (<= y 1.75e+45))) (- x (/ 2.0 x)) (- x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.5e+154) || !(y <= 1.75e+45)) {
		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 <= (-1.5d+154)) .or. (.not. (y <= 1.75d+45))) 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 <= -1.5e+154) || !(y <= 1.75e+45)) {
		tmp = x - (2.0 / x);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.5e+154) or not (y <= 1.75e+45):
		tmp = x - (2.0 / x)
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.5e+154) || !(y <= 1.75e+45))
		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 <= -1.5e+154) || ~((y <= 1.75e+45)))
		tmp = x - (2.0 / x);
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.5e+154], N[Not[LessEqual[y, 1.75e+45]], $MachinePrecision]], N[(x - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(x - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.50000000000000013e154 or 1.75000000000000011e45 < y

   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* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 89.1%

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

   if -1.50000000000000013e154 < y < 1.75000000000000011e45

   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 95.1%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+154} \lor \neg \left(y \leq 1.75 \cdot 10^{+45}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]

Alternative 4: 86.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+16}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-19}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.7e+16) x (if (<= x 8e-19) (- x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.7e+16) {
		tmp = x;
	} else if (x <= 8e-19) {
		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 <= (-1.7d+16)) then
        tmp = x
    else if (x <= 8d-19) 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 <= -1.7e+16) {
		tmp = x;
	} else if (x <= 8e-19) {
		tmp = x - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.7e+16:
		tmp = x
	elif x <= 8e-19:
		tmp = x - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.7e+16)
		tmp = x;
	elseif (x <= 8e-19)
		tmp = Float64(x - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.7e+16)
		tmp = x;
	elseif (x <= 8e-19)
		tmp = x - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.7e+16], x, If[LessEqual[x, 8e-19], N[(x - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7e16 or 7.9999999999999998e-19 < x

   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 inf 100.0%

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

   if -1.7e16 < x < 7.9999999999999998e-19

   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* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 68.7%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+16}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-19}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 77.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.95 \cdot 10^{-99}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-65}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.95e-99) x (if (<= x 9.2e-65) (- y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.95e-99) {
		tmp = x;
	} else if (x <= 9.2e-65) {
		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 <= (-2.95d-99)) then
        tmp = x
    else if (x <= 9.2d-65) 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 <= -2.95e-99) {
		tmp = x;
	} else if (x <= 9.2e-65) {
		tmp = -y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.95e-99:
		tmp = x
	elif x <= 9.2e-65:
		tmp = -y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.95e-99)
		tmp = x;
	elseif (x <= 9.2e-65)
		tmp = Float64(-y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.95e-99)
		tmp = x;
	elseif (x <= 9.2e-65)
		tmp = -y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.95e-99], x, If[LessEqual[x, 9.2e-65], (-y), x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.95e-99 or 9.1999999999999999e-65 < x

   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 inf 88.5%

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

   if -2.95e-99 < x < 9.1999999999999999e-65

   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 x around 0 63.4%

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

      1. neg-mul-1 63.4%

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

   6. Simplified 63.4%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.95 \cdot 10^{-99}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-65}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 62.7% 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. *-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 x around inf 63.4%

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

5. Final simplification 63.4%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023271 
(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)))))