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

Percentage Accurate: 99.9% → 99.9%

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: 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. Final simplification 100.0%

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

Alternative 2: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0069:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-163}:\\ \;\;\;\;x - y\\ \mathbf{elif}\;x \leq 5.9 \cdot 10^{-147}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-33}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.0069)
   x
   (if (<= x 1.8e-163)
     (- x y)
     (if (<= x 5.9e-147) (/ -2.0 x) (if (<= x 7.2e-33) (- x y) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.0069) {
		tmp = x;
	} else if (x <= 1.8e-163) {
		tmp = x - y;
	} else if (x <= 5.9e-147) {
		tmp = -2.0 / x;
	} else if (x <= 7.2e-33) {
		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 <= (-0.0069d0)) then
        tmp = x
    else if (x <= 1.8d-163) then
        tmp = x - y
    else if (x <= 5.9d-147) then
        tmp = (-2.0d0) / x
    else if (x <= 7.2d-33) 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 <= -0.0069) {
		tmp = x;
	} else if (x <= 1.8e-163) {
		tmp = x - y;
	} else if (x <= 5.9e-147) {
		tmp = -2.0 / x;
	} else if (x <= 7.2e-33) {
		tmp = x - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.0069:
		tmp = x
	elif x <= 1.8e-163:
		tmp = x - y
	elif x <= 5.9e-147:
		tmp = -2.0 / x
	elif x <= 7.2e-33:
		tmp = x - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.0069)
		tmp = x;
	elseif (x <= 1.8e-163)
		tmp = Float64(x - y);
	elseif (x <= 5.9e-147)
		tmp = Float64(-2.0 / x);
	elseif (x <= 7.2e-33)
		tmp = Float64(x - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.0069)
		tmp = x;
	elseif (x <= 1.8e-163)
		tmp = x - y;
	elseif (x <= 5.9e-147)
		tmp = -2.0 / x;
	elseif (x <= 7.2e-33)
		tmp = x - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.0069], x, If[LessEqual[x, 1.8e-163], N[(x - y), $MachinePrecision], If[LessEqual[x, 5.9e-147], N[(-2.0 / x), $MachinePrecision], If[LessEqual[x, 7.2e-33], N[(x - y), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0068999999999999999 or 7.20000000000000068e-33 < 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. Taylor expanded in y around inf 96.4%

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

   5. Taylor expanded in x around inf 98.5%

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

   if -0.0068999999999999999 < x < 1.7999999999999999e-163 or 5.9000000000000003e-147 < x < 7.20000000000000068e-33

   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. Taylor expanded in y around 0 77.6%

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

   if 1.7999999999999999e-163 < x < 5.9000000000000003e-147

   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. Taylor expanded in y around inf 82.5%

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

   5. Taylor expanded in x around 0 82.5%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0069:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-163}:\\ \;\;\;\;x - y\\ \mathbf{elif}\;x \leq 5.9 \cdot 10^{-147}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-33}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 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* 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. Final simplification 100.0%

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

Alternative 4: 90.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+65} \lor \neg \left(y \leq 2.4 \cdot 10^{+82}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.6e+65) (not (<= y 2.4e+82))) (- x (/ 2.0 x)) (- x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.6e+65) || !(y <= 2.4e+82)) {
		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 <= (-3.6d+65)) .or. (.not. (y <= 2.4d+82))) 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 <= -3.6e+65) || !(y <= 2.4e+82)) {
		tmp = x - (2.0 / x);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.6e+65) or not (y <= 2.4e+82):
		tmp = x - (2.0 / x)
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.6e+65) || !(y <= 2.4e+82))
		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 <= -3.6e+65) || ~((y <= 2.4e+82)))
		tmp = x - (2.0 / x);
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.6e+65], N[Not[LessEqual[y, 2.4e+82]], $MachinePrecision]], N[(x - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(x - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.59999999999999978e65 or 2.39999999999999998e82 < y

   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. Taylor expanded in y around inf 81.6%

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

   if -3.59999999999999978e65 < y < 2.39999999999999998e82

   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. Taylor expanded in y around 0 97.7%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+65} \lor \neg \left(y \leq 2.4 \cdot 10^{+82}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]

Alternative 5: 87.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0069:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-33}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.0069) x (if (<= x 5.1e-33) (- x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.0069) {
		tmp = x;
	} else if (x <= 5.1e-33) {
		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 <= (-0.0069d0)) then
        tmp = x
    else if (x <= 5.1d-33) 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 <= -0.0069) {
		tmp = x;
	} else if (x <= 5.1e-33) {
		tmp = x - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.0069:
		tmp = x
	elif x <= 5.1e-33:
		tmp = x - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.0069)
		tmp = x;
	elseif (x <= 5.1e-33)
		tmp = Float64(x - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.0069)
		tmp = x;
	elseif (x <= 5.1e-33)
		tmp = x - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.0069], x, If[LessEqual[x, 5.1e-33], N[(x - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0068999999999999999 or 5.10000000000000008e-33 < 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. Taylor expanded in y around inf 96.4%

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

   5. Taylor expanded in x around inf 98.5%

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

   if -0.0068999999999999999 < x < 5.10000000000000008e-33

   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. Taylor expanded in y around 0 74.1%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0069:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-33}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

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* 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. Taylor expanded in y around inf 62.3%

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

5. Taylor expanded in x around inf 64.3%

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

6. Final simplification 64.3%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023310 
(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)))))