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

Percentage Accurate: 99.9% → 99.9%

Time: 3.3s

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. Final simplification 100.0%

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

Alternative 2: 85.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+17}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-95}:\\ \;\;\;\;x - y\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.3e+17)
   x
   (if (<= x 1.2e-95) (- x y) (if (<= x 1.2e-5) (/ -2.0 x) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.3e+17) {
		tmp = x;
	} else if (x <= 1.2e-95) {
		tmp = x - y;
	} else if (x <= 1.2e-5) {
		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 <= (-1.3d+17)) then
        tmp = x
    else if (x <= 1.2d-95) then
        tmp = x - y
    else if (x <= 1.2d-5) 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 <= -1.3e+17) {
		tmp = x;
	} else if (x <= 1.2e-95) {
		tmp = x - y;
	} else if (x <= 1.2e-5) {
		tmp = -2.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.3e+17:
		tmp = x
	elif x <= 1.2e-95:
		tmp = x - y
	elif x <= 1.2e-5:
		tmp = -2.0 / x
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.3e+17)
		tmp = x;
	elseif (x <= 1.2e-95)
		tmp = Float64(x - y);
	elseif (x <= 1.2e-5)
		tmp = Float64(-2.0 / x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.3e+17)
		tmp = x;
	elseif (x <= 1.2e-95)
		tmp = x - y;
	elseif (x <= 1.2e-5)
		tmp = -2.0 / x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.3e+17], x, If[LessEqual[x, 1.2e-95], N[(x - y), $MachinePrecision], If[LessEqual[x, 1.2e-5], N[(-2.0 / x), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3e17 or 1.2e-5 < x

   1. Initial program 100.0%

      \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x - \frac{y}{1 + \frac{x}{\frac{2}{y}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 98.7%

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

   6. Taylor expanded in x around inf 98.5%

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

   if -1.3e17 < x < 1.2e-95

   1. Initial program 99.9%

      \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
   2. Step-by-step derivation

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x - \frac{y}{1 + \frac{x}{\frac{2}{y}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 79.4%

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

   if 1.2e-95 < x < 1.2e-5

   1. Initial program 99.9%

      \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
   2. Step-by-step derivation

      1. associate-/l* 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x - \frac{y}{1 + \frac{x}{\frac{2}{y}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 69.6%

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

   6. Taylor expanded in x around 0 69.6%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+17}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-95}:\\ \;\;\;\;x - y\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+55} \lor \neg \left(y \leq 6.8 \cdot 10^{+136}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.1e+55) (not (<= y 6.8e+136))) (- x (/ 2.0 x)) (- x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.1e+55) || !(y <= 6.8e+136)) {
		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 <= (-3.1d+55)) .or. (.not. (y <= 6.8d+136))) 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 <= -3.1e+55) || !(y <= 6.8e+136)) {
		tmp = x - (2.0 / x);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.1e+55) or not (y <= 6.8e+136):
		tmp = x - (2.0 / x)
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.1e+55) || !(y <= 6.8e+136))
		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 <= -3.1e+55) || ~((y <= 6.8e+136)))
		tmp = x - (2.0 / x);
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.1e+55], N[Not[LessEqual[y, 6.8e+136]], $MachinePrecision]], N[(x - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(x - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.09999999999999994e55 or 6.79999999999999993e136 < y

   1. Initial program 99.9%

      \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
   2. Step-by-step derivation

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x - \frac{y}{1 + \frac{x}{\frac{2}{y}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 90.1%

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

   if -3.09999999999999994e55 < y < 6.79999999999999993e136

   1. Initial program 100.0%

      \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x - \frac{y}{1 + \frac{x}{\frac{2}{y}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 96.6%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+55} \lor \neg \left(y \leq 6.8 \cdot 10^{+136}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+17}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.3e+17) x (if (<= x 1.45) (- x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.3e+17) {
		tmp = x;
	} else if (x <= 1.45) {
		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 <= (-1.3d+17)) then
        tmp = x
    else if (x <= 1.45d0) 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 <= -1.3e+17) {
		tmp = x;
	} else if (x <= 1.45) {
		tmp = x - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.3e+17:
		tmp = x
	elif x <= 1.45:
		tmp = x - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.3e+17)
		tmp = x;
	elseif (x <= 1.45)
		tmp = Float64(x - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.3e+17)
		tmp = x;
	elseif (x <= 1.45)
		tmp = x - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.3e+17], x, If[LessEqual[x, 1.45], N[(x - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3e17 or 1.44999999999999996 < x

   1. Initial program 100.0%

      \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x - \frac{y}{1 + \frac{x}{\frac{2}{y}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 99.4%

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

   6. Taylor expanded in x around inf 98.9%

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

   if -1.3e17 < x < 1.44999999999999996

   1. Initial program 99.9%

      \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
   2. Step-by-step derivation

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x - \frac{y}{1 + \frac{x}{\frac{2}{y}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 72.2%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+17}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \frac{y}{1 + \frac{x}{\frac{2}{y}}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- x (/ y (+ 1.0 (/ x (/ 2.0 y))))))

C

double code(double x, double y) {
	return x - (y / (1.0 + (x / (2.0 / y))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x - (y / (1.0d0 + (x / (2.0d0 / y))))
end function

Java

public static double code(double x, double y) {
	return x - (y / (1.0 + (x / (2.0 / y))));
}

Python

def code(x, y):
	return x - (y / (1.0 + (x / (2.0 / y))))

Julia

function code(x, y)
	return Float64(x - Float64(y / Float64(1.0 + Float64(x / Float64(2.0 / y)))))
end

MATLAB

function tmp = code(x, y)
	tmp = x - (y / (1.0 + (x / (2.0 / y))));
end

Wolfram

code[x_, y_] := N[(x - N[(y / N[(1.0 + N[(x / N[(2.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
2. Step-by-step derivation

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{x - \frac{y}{1 + \frac{x}{\frac{2}{y}}}} \]
4. Add Preprocessing

5. Final simplification 99.9%

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

Alternative 6: 62.5% 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. Step-by-step derivation

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{x - \frac{y}{1 + \frac{x}{\frac{2}{y}}}} \]
4. Add Preprocessing

5. Taylor expanded in y around inf 64.8%

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

6. Taylor expanded in x around inf 63.1%

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

7. Final simplification 63.1%

   \[\leadsto x \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(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)))))