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

Percentage Accurate: 99.9% → 100.0%

Time: 5.3s

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: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. associate-/l* 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 87.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-46}:\\ \;\;\;\;x - y\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-23}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.1e-8)
   x
   (if (<= x 1.3e-46) (- x y) (if (<= x 2.15e-23) (/ -2.0 x) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.1e-8) {
		tmp = x;
	} else if (x <= 1.3e-46) {
		tmp = x - y;
	} else if (x <= 2.15e-23) {
		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 <= (-5.1d-8)) then
        tmp = x
    else if (x <= 1.3d-46) then
        tmp = x - y
    else if (x <= 2.15d-23) 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 <= -5.1e-8) {
		tmp = x;
	} else if (x <= 1.3e-46) {
		tmp = x - y;
	} else if (x <= 2.15e-23) {
		tmp = -2.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.1e-8:
		tmp = x
	elif x <= 1.3e-46:
		tmp = x - y
	elif x <= 2.15e-23:
		tmp = -2.0 / x
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.1e-8)
		tmp = x;
	elseif (x <= 1.3e-46)
		tmp = Float64(x - y);
	elseif (x <= 2.15e-23)
		tmp = Float64(-2.0 / x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.1e-8)
		tmp = x;
	elseif (x <= 1.3e-46)
		tmp = x - y;
	elseif (x <= 2.15e-23)
		tmp = -2.0 / x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.1e-8], x, If[LessEqual[x, 1.3e-46], N[(x - y), $MachinePrecision], If[LessEqual[x, 2.15e-23], N[(-2.0 / x), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.10000000000000001e-8 or 2.15000000000000001e-23 < x

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 94.8%

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

   5. Taylor expanded in x around inf 96.7%

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

   if -5.10000000000000001e-8 < x < 1.3000000000000001e-46

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 75.3%

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

   if 1.3000000000000001e-46 < x < 2.15000000000000001e-23

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around inf 78.2%

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

   5. Taylor expanded in x around 0 78.2%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-46}:\\ \;\;\;\;x - y\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-23}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 90.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.1 \cdot 10^{+126} \lor \neg \left(y \leq 1.65 \cdot 10^{+139}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5.1e+126) (not (<= y 1.65e+139))) (- x (/ 2.0 x)) (- x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5.1e+126) || !(y <= 1.65e+139)) {
		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 <= (-5.1d+126)) .or. (.not. (y <= 1.65d+139))) 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 <= -5.1e+126) || !(y <= 1.65e+139)) {
		tmp = x - (2.0 / x);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -5.1e+126) or not (y <= 1.65e+139):
		tmp = x - (2.0 / x)
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5.1e+126) || !(y <= 1.65e+139))
		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 <= -5.1e+126) || ~((y <= 1.65e+139)))
		tmp = x - (2.0 / x);
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5.1e+126], N[Not[LessEqual[y, 1.65e+139]], $MachinePrecision]], N[(x - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(x - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.1000000000000001e126 or 1.6500000000000001e139 < y

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 88.2%

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

   if -5.1000000000000001e126 < y < 1.6500000000000001e139

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 93.2%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.1 \cdot 10^{+126} \lor \neg \left(y \leq 1.65 \cdot 10^{+139}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]

Alternative 4: 87.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 0.00095:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.7e-8) x (if (<= x 0.00095) (- x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.7e-8) {
		tmp = x;
	} else if (x <= 0.00095) {
		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 <= (-4.7d-8)) then
        tmp = x
    else if (x <= 0.00095d0) 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 <= -4.7e-8) {
		tmp = x;
	} else if (x <= 0.00095) {
		tmp = x - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.7e-8:
		tmp = x
	elif x <= 0.00095:
		tmp = x - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.7e-8)
		tmp = x;
	elseif (x <= 0.00095)
		tmp = Float64(x - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.7e-8)
		tmp = x;
	elseif (x <= 0.00095)
		tmp = x - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.7e-8], x, If[LessEqual[x, 0.00095], N[(x - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -4.6999999999999997e-8 or 9.49999999999999998e-4 < x

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 97.1%

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

   5. Taylor expanded in x around inf 98.1%

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

   if -4.6999999999999997e-8 < x < 9.49999999999999998e-4

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 71.3%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 0.00095:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 63.4% 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. Step-by-step derivation

   1. *-commutative 99.9%

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

   2. associate-/l* 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in y around inf 60.6%

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

5. Taylor expanded in x around inf 59.5%

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

6. Final simplification 59.5%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023274 
(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)))))