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

Percentage Accurate: 99.9% → 100.0%

Time: 3.8s

Alternatives: 3

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 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. Final simplification 100.0%

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

Alternative 2: 91.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+154} \lor \neg \left(y \leq 1.25 \cdot 10^{+109}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.2e+154) (not (<= y 1.25e+109))) (- x (/ 2.0 x)) (- x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.2e+154) || !(y <= 1.25e+109)) {
		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 <= (-2.2d+154)) .or. (.not. (y <= 1.25d+109))) 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 <= -2.2e+154) || !(y <= 1.25e+109)) {
		tmp = x - (2.0 / x);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.2e+154) or not (y <= 1.25e+109):
		tmp = x - (2.0 / x)
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.2e+154) || !(y <= 1.25e+109))
		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 <= -2.2e+154) || ~((y <= 1.25e+109)))
		tmp = x - (2.0 / x);
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.2e+154], N[Not[LessEqual[y, 1.25e+109]], $MachinePrecision]], N[(x - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(x - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.2000000000000001e154 or 1.25e109 < y

   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* 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 87.6%

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

   if -2.2000000000000001e154 < y < 1.25e109

   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 97.5%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+154} \lor \neg \left(y \leq 1.25 \cdot 10^{+109}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]

Alternative 3: 74.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ x - y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- x y))

C

double code(double x, double y) {
	return x - y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x - y
end function

Java

public static double code(double x, double y) {
	return x - y;
}

Python

def code(x, y):
	return x - y

Julia

function code(x, y)
	return Float64(x - y)
end

MATLAB

function tmp = code(x, y)
	tmp = x - y;
end

Wolfram

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

Derivation

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 79.8%

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

5. Final simplification 79.8%

   \[\leadsto x - y \]

Reproduce

herbie shell --seed 2023268 
(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)))))