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

Percentage Accurate: 99.9% → 99.9%

Time: 5.1s

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: 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. Final simplification 99.9%

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

Alternative 2: 89.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{+95} \lor \neg \left(y \leq 2.8 \cdot 10^{+120}\right) \land \left(y \leq 6.8 \cdot 10^{+187} \lor \neg \left(y \leq 5 \cdot 10^{+235}\right)\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5.6e+95)
         (and (not (<= y 2.8e+120)) (or (<= y 6.8e+187) (not (<= y 5e+235)))))
   (- x (/ 2.0 x))
   (- x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5.6e+95) || (!(y <= 2.8e+120) && ((y <= 6.8e+187) || !(y <= 5e+235)))) {
		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 <= (-5.6d+95)) .or. (.not. (y <= 2.8d+120)) .and. (y <= 6.8d+187) .or. (.not. (y <= 5d+235))) 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 <= -5.6e+95) || (!(y <= 2.8e+120) && ((y <= 6.8e+187) || !(y <= 5e+235)))) {
		tmp = x - (2.0 / x);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -5.6e+95) or (not (y <= 2.8e+120) and ((y <= 6.8e+187) or not (y <= 5e+235))):
		tmp = x - (2.0 / x)
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5.6e+95) || (!(y <= 2.8e+120) && ((y <= 6.8e+187) || !(y <= 5e+235))))
		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 <= -5.6e+95) || (~((y <= 2.8e+120)) && ((y <= 6.8e+187) || ~((y <= 5e+235)))))
		tmp = x - (2.0 / x);
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5.6e+95], And[N[Not[LessEqual[y, 2.8e+120]], $MachinePrecision], Or[LessEqual[y, 6.8e+187], N[Not[LessEqual[y, 5e+235]], $MachinePrecision]]]], N[(x - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(x - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.5999999999999995e95 or 2.8000000000000001e120 < y < 6.7999999999999999e187 or 5.00000000000000027e235 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 82.9%

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

   if -5.5999999999999995e95 < y < 2.8000000000000001e120 or 6.7999999999999999e187 < y < 5.00000000000000027e235

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 95.9%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{+95} \lor \neg \left(y \leq 2.8 \cdot 10^{+120}\right) \land \left(y \leq 6.8 \cdot 10^{+187} \lor \neg \left(y \leq 5 \cdot 10^{+235}\right)\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]

Alternative 3: 87.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.7e-8) x (if (<= x 5.8e-7) (- x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.7e-8) {
		tmp = x;
	} else if (x <= 5.8e-7) {
		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 <= (-2.7d-8)) then
        tmp = x
    else if (x <= 5.8d-7) 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 <= -2.7e-8) {
		tmp = x;
	} else if (x <= 5.8e-7) {
		tmp = x - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.7e-8:
		tmp = x
	elif x <= 5.8e-7:
		tmp = x - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.7e-8)
		tmp = x;
	elseif (x <= 5.8e-7)
		tmp = Float64(x - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.7e-8)
		tmp = x;
	elseif (x <= 5.8e-7)
		tmp = x - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.7e-8], x, If[LessEqual[x, 5.8e-7], N[(x - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.70000000000000002e-8 or 5.7999999999999995e-7 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 97.2%

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

   if -2.70000000000000002e-8 < x < 5.7999999999999995e-7

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 77.6%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 78.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-138}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-76}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.3e-138) x (if (<= x 2.4e-76) (- y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.3e-138) {
		tmp = x;
	} else if (x <= 2.4e-76) {
		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 <= (-1.3d-138)) then
        tmp = x
    else if (x <= 2.4d-76) 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 <= -1.3e-138) {
		tmp = x;
	} else if (x <= 2.4e-76) {
		tmp = -y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.3e-138:
		tmp = x
	elif x <= 2.4e-76:
		tmp = -y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.3e-138)
		tmp = x;
	elseif (x <= 2.4e-76)
		tmp = Float64(-y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.3e-138)
		tmp = x;
	elseif (x <= 2.4e-76)
		tmp = -y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.3e-138], x, If[LessEqual[x, 2.4e-76], (-y), x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3e-138 or 2.40000000000000013e-76 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 84.9%

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

   if -1.3e-138 < x < 2.40000000000000013e-76

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 75.5%

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

      1. neg-mul-1 75.5%

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

   4. Simplified 75.5%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-138}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-76}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 63.2% 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. Taylor expanded in x around inf 59.0%

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

3. Final simplification 59.0%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023230 
(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)))))