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

Percentage Accurate: 99.9% → 99.9%

Time: 4.4s

Alternatives: 5

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: 90.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+165} \lor \neg \left(y \leq 1.6 \cdot 10^{+72}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.9e+165) (not (<= y 1.6e+72))) (- x (/ 2.0 x)) (- x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.9e+165) || !(y <= 1.6e+72)) {
		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 <= (-2.9d+165)) .or. (.not. (y <= 1.6d+72))) 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 <= -2.9e+165) || !(y <= 1.6e+72)) {
		tmp = x - (2.0 / x);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.9e+165) or not (y <= 1.6e+72):
		tmp = x - (2.0 / x)
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.9e+165) || !(y <= 1.6e+72))
		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 <= -2.9e+165) || ~((y <= 1.6e+72)))
		tmp = x - (2.0 / x);
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.9e+165], N[Not[LessEqual[y, 1.6e+72]], $MachinePrecision]], N[(x - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(x - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.90000000000000006e165 or 1.6000000000000001e72 < 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 90.3%

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

   if -2.90000000000000006e165 < y < 1.6000000000000001e72

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 96.2%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+165} \lor \neg \left(y \leq 1.6 \cdot 10^{+72}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-7}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.6e-7) x (if (<= x 1.4) (- x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.6e-7) {
		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 <= (-2.6d-7)) 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 <= -2.6e-7) {
		tmp = x;
	} else if (x <= 1.4) {
		tmp = x - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.6e-7:
		tmp = x
	elif x <= 1.4:
		tmp = x - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.6e-7)
		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 <= -2.6e-7)
		tmp = x;
	elseif (x <= 1.4)
		tmp = x - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.6e-7], x, If[LessEqual[x, 1.4], N[(x - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.59999999999999999e-7 or 1.3999999999999999 < 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 98.8%

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

   if -2.59999999999999999e-7 < x < 1.3999999999999999

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 79.0%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-7}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-105}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.3e-51) x (if (<= x 3.2e-105) (- y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.3e-51) {
		tmp = x;
	} else if (x <= 3.2e-105) {
		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 <= (-4.3d-51)) then
        tmp = x
    else if (x <= 3.2d-105) 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 <= -4.3e-51) {
		tmp = x;
	} else if (x <= 3.2e-105) {
		tmp = -y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.3e-51:
		tmp = x
	elif x <= 3.2e-105:
		tmp = -y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.3e-51)
		tmp = x;
	elseif (x <= 3.2e-105)
		tmp = Float64(-y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.3e-51)
		tmp = x;
	elseif (x <= 3.2e-105)
		tmp = -y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.3e-51], x, If[LessEqual[x, 3.2e-105], (-y), x]]

Derivation

1. Split input into 2 regimes

2. if x < -4.2999999999999997e-51 or 3.19999999999999981e-105 < 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 88.4%

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

   if -4.2999999999999997e-51 < x < 3.19999999999999981e-105

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 71.1%

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

      1. neg-mul-1 71.1%

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

   5. Simplified 71.1%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-105}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.0% 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 60.0%

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

4. Final simplification 60.0%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(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)))))