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

Percentage Accurate: 99.9% → 99.9%

Time: 5.1s

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 100.0%

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

Alternative 2: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-38}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-92}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-58}:\\ \;\;\;\;x - y\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-15}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.4e-38)
   x
   (if (<= x -5.8e-92)
     (/ -2.0 x)
     (if (<= x 2.25e-58) (- x y) (if (<= x 6e-15) (/ -2.0 x) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.4e-38) {
		tmp = x;
	} else if (x <= -5.8e-92) {
		tmp = -2.0 / x;
	} else if (x <= 2.25e-58) {
		tmp = x - y;
	} else if (x <= 6e-15) {
		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 <= (-3.4d-38)) then
        tmp = x
    else if (x <= (-5.8d-92)) then
        tmp = (-2.0d0) / x
    else if (x <= 2.25d-58) then
        tmp = x - y
    else if (x <= 6d-15) 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 <= -3.4e-38) {
		tmp = x;
	} else if (x <= -5.8e-92) {
		tmp = -2.0 / x;
	} else if (x <= 2.25e-58) {
		tmp = x - y;
	} else if (x <= 6e-15) {
		tmp = -2.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.4e-38:
		tmp = x
	elif x <= -5.8e-92:
		tmp = -2.0 / x
	elif x <= 2.25e-58:
		tmp = x - y
	elif x <= 6e-15:
		tmp = -2.0 / x
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.4e-38)
		tmp = x;
	elseif (x <= -5.8e-92)
		tmp = Float64(-2.0 / x);
	elseif (x <= 2.25e-58)
		tmp = Float64(x - y);
	elseif (x <= 6e-15)
		tmp = Float64(-2.0 / x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.4e-38)
		tmp = x;
	elseif (x <= -5.8e-92)
		tmp = -2.0 / x;
	elseif (x <= 2.25e-58)
		tmp = x - y;
	elseif (x <= 6e-15)
		tmp = -2.0 / x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.4e-38], x, If[LessEqual[x, -5.8e-92], N[(-2.0 / x), $MachinePrecision], If[LessEqual[x, 2.25e-58], N[(x - y), $MachinePrecision], If[LessEqual[x, 6e-15], N[(-2.0 / x), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.4000000000000002e-38 or 6e-15 < 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 97.7%

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

   if -3.4000000000000002e-38 < x < -5.79999999999999969e-92 or 2.2500000000000001e-58 < x < 6e-15

   1. Initial program 99.8%

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

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

   if -5.79999999999999969e-92 < x < 2.2500000000000001e-58

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 83.4%

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

      1. neg-mul-1 83.4%

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

      2. unsub-neg 83.4%

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

   5. Simplified 83.4%

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

Alternative 3: 91.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+117} \lor \neg \left(y \leq 1.68 \cdot 10^{+137}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -6.8e+117) (not (<= y 1.68e+137))) (- x (/ 2.0 x)) (- x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -6.8e+117) || !(y <= 1.68e+137)) {
		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 <= (-6.8d+117)) .or. (.not. (y <= 1.68d+137))) 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 <= -6.8e+117) || !(y <= 1.68e+137)) {
		tmp = x - (2.0 / x);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -6.8e+117) or not (y <= 1.68e+137):
		tmp = x - (2.0 / x)
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -6.8e+117) || !(y <= 1.68e+137))
		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 <= -6.8e+117) || ~((y <= 1.68e+137)))
		tmp = x - (2.0 / x);
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -6.8e+117], N[Not[LessEqual[y, 1.68e+137]], $MachinePrecision]], N[(x - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(x - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.8000000000000002e117 or 1.6799999999999999e137 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 89.6%

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

      1. associate-*r/ 89.6%

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

      2. metadata-eval 89.6%

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

   5. Simplified 89.6%

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

   if -6.8000000000000002e117 < y < 1.6799999999999999e137

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 92.6%

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

      1. neg-mul-1 92.6%

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

      2. unsub-neg 92.6%

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

   5. Simplified 92.6%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+117} \lor \neg \left(y \leq 1.68 \cdot 10^{+137}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6700:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-30}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6700.0) x (if (<= x 5.8e-30) (- x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6700.0) {
		tmp = x;
	} else if (x <= 5.8e-30) {
		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 <= (-6700.0d0)) then
        tmp = x
    else if (x <= 5.8d-30) 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 <= -6700.0) {
		tmp = x;
	} else if (x <= 5.8e-30) {
		tmp = x - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6700.0:
		tmp = x
	elif x <= 5.8e-30:
		tmp = x - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6700.0)
		tmp = x;
	elseif (x <= 5.8e-30)
		tmp = Float64(x - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6700.0)
		tmp = x;
	elseif (x <= 5.8e-30)
		tmp = x - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6700.0], x, If[LessEqual[x, 5.8e-30], N[(x - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -6700 or 5.79999999999999978e-30 < 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 98.4%

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

   if -6700 < x < 5.79999999999999978e-30

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 72.1%

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

      1. neg-mul-1 72.1%

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

      2. unsub-neg 72.1%

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

   5. Simplified 72.1%

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

Alternative 5: 78.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{-41}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-83}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9.2e-41) x (if (<= x 1.3e-83) (- y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9.2e-41) {
		tmp = x;
	} else if (x <= 1.3e-83) {
		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 <= (-9.2d-41)) then
        tmp = x
    else if (x <= 1.3d-83) 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 <= -9.2e-41) {
		tmp = x;
	} else if (x <= 1.3e-83) {
		tmp = -y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9.2e-41:
		tmp = x
	elif x <= 1.3e-83:
		tmp = -y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9.2e-41)
		tmp = x;
	elseif (x <= 1.3e-83)
		tmp = Float64(-y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9.2e-41)
		tmp = x;
	elseif (x <= 1.3e-83)
		tmp = -y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9.2e-41], x, If[LessEqual[x, 1.3e-83], (-y), x]]

Derivation

1. Split input into 2 regimes

2. if x < -9.20000000000000041e-41 or 1.30000000000000004e-83 < 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 91.6%

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

   if -9.20000000000000041e-41 < x < 1.30000000000000004e-83

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 66.0%

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

      1. neg-mul-1 66.0%

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

   5. Simplified 66.0%

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

Alternative 6: 62.9% 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 100.0%

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

3. Taylor expanded in x around inf 59.3%

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

Reproduce

herbie shell --seed 2024165 
(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)))))