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

Percentage Accurate: 99.9% → 99.9%

Time: 5.6s

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 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, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -260000:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-76}:\\ \;\;\;\;x - y\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-36}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -260000.0)
   x
   (if (<= x 1.45e-76) (- x y) (if (<= x 6e-36) (/ -2.0 x) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -260000.0) {
		tmp = x;
	} else if (x <= 1.45e-76) {
		tmp = x - y;
	} else if (x <= 6e-36) {
		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 <= (-260000.0d0)) then
        tmp = x
    else if (x <= 1.45d-76) then
        tmp = x - y
    else if (x <= 6d-36) 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 <= -260000.0) {
		tmp = x;
	} else if (x <= 1.45e-76) {
		tmp = x - y;
	} else if (x <= 6e-36) {
		tmp = -2.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -260000.0:
		tmp = x
	elif x <= 1.45e-76:
		tmp = x - y
	elif x <= 6e-36:
		tmp = -2.0 / x
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -260000.0)
		tmp = x;
	elseif (x <= 1.45e-76)
		tmp = Float64(x - y);
	elseif (x <= 6e-36)
		tmp = Float64(-2.0 / x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -260000.0)
		tmp = x;
	elseif (x <= 1.45e-76)
		tmp = x - y;
	elseif (x <= 6e-36)
		tmp = -2.0 / x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -260000.0], x, If[LessEqual[x, 1.45e-76], N[(x - y), $MachinePrecision], If[LessEqual[x, 6e-36], N[(-2.0 / x), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.6e5 or 6.0000000000000003e-36 < 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 97.0%

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

   if -2.6e5 < x < 1.4500000000000001e-76

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 80.3%

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

      1. neg-mul-1 80.3%

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

      2. unsub-neg 80.3%

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

   5. Simplified 80.3%

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

   if 1.4500000000000001e-76 < x < 6.0000000000000003e-36

   1. Initial program 99.5%

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

   3. Taylor expanded in y around inf 94.2%

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

      1. associate-*r/ 94.2%

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

      2. metadata-eval 94.2%

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

   5. Simplified 94.2%

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

   6. Taylor expanded in x around 0 94.2%

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

Alternative 3: 91.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+129} \lor \neg \left(y \leq 3.1 \cdot 10^{+95}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3e+129) (not (<= y 3.1e+95))) (- x (/ 2.0 x)) (- x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3e+129) || !(y <= 3.1e+95)) {
		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 <= (-3d+129)) .or. (.not. (y <= 3.1d+95))) 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 <= -3e+129) || !(y <= 3.1e+95)) {
		tmp = x - (2.0 / x);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3e+129) or not (y <= 3.1e+95):
		tmp = x - (2.0 / x)
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3e+129) || !(y <= 3.1e+95))
		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 <= -3e+129) || ~((y <= 3.1e+95)))
		tmp = x - (2.0 / x);
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3e+129], N[Not[LessEqual[y, 3.1e+95]], $MachinePrecision]], N[(x - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(x - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.0000000000000003e129 or 3.1000000000000003e95 < 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 86.6%

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

      1. associate-*r/ 86.6%

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

      2. metadata-eval 86.6%

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

   5. Simplified 86.6%

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

   if -3.0000000000000003e129 < y < 3.1000000000000003e95

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 96.0%

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

      1. neg-mul-1 96.0%

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

      2. unsub-neg 96.0%

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

   5. Simplified 96.0%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+129} \lor \neg \left(y \leq 3.1 \cdot 10^{+95}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -260000:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 0.0031:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -260000.0) x (if (<= x 0.0031) (- x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -260000.0) {
		tmp = x;
	} else if (x <= 0.0031) {
		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 <= (-260000.0d0)) then
        tmp = x
    else if (x <= 0.0031d0) 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 <= -260000.0) {
		tmp = x;
	} else if (x <= 0.0031) {
		tmp = x - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -260000.0:
		tmp = x
	elif x <= 0.0031:
		tmp = x - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -260000.0)
		tmp = x;
	elseif (x <= 0.0031)
		tmp = Float64(x - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -260000.0)
		tmp = x;
	elseif (x <= 0.0031)
		tmp = x - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -260000.0], x, If[LessEqual[x, 0.0031], N[(x - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.6e5 or 0.00309999999999999989 < 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 -2.6e5 < x < 0.00309999999999999989

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 75.8%

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

      1. neg-mul-1 75.8%

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

      2. unsub-neg 75.8%

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

   5. Simplified 75.8%

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

Alternative 5: 78.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-129}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-87}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.4e-129) x (if (<= x 5.8e-87) (- y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.4e-129) {
		tmp = x;
	} else if (x <= 5.8e-87) {
		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 <= (-1.4d-129)) then
        tmp = x
    else if (x <= 5.8d-87) 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 <= -1.4e-129) {
		tmp = x;
	} else if (x <= 5.8e-87) {
		tmp = -y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.4e-129:
		tmp = x
	elif x <= 5.8e-87:
		tmp = -y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.4e-129)
		tmp = x;
	elseif (x <= 5.8e-87)
		tmp = Float64(-y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.4e-129)
		tmp = x;
	elseif (x <= 5.8e-87)
		tmp = -y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.4e-129], x, If[LessEqual[x, 5.8e-87], (-y), x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4e-129 or 5.7999999999999998e-87 < 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 89.4%

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

   if -1.4e-129 < x < 5.7999999999999998e-87

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 62.5%

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

      1. neg-mul-1 62.5%

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

   5. Simplified 62.5%

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

Alternative 6: 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 64.3%

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

Reproduce

herbie shell --seed 2024184 
(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)))))