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

Percentage Accurate: 99.9% → 99.9%

Time: 4.4s

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 99.9%

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

Alternative 2: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-71}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-70}:\\ \;\;\;\;x - y\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.1e-8)
   x
   (if (<= x -9.5e-71)
     (/ -2.0 x)
     (if (<= x 8.2e-70) (- x y) (if (<= x 2e-12) (/ -2.0 x) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.1e-8) {
		tmp = x;
	} else if (x <= -9.5e-71) {
		tmp = -2.0 / x;
	} else if (x <= 8.2e-70) {
		tmp = x - y;
	} else if (x <= 2e-12) {
		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 <= (-2.1d-8)) then
        tmp = x
    else if (x <= (-9.5d-71)) then
        tmp = (-2.0d0) / x
    else if (x <= 8.2d-70) then
        tmp = x - y
    else if (x <= 2d-12) 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 <= -2.1e-8) {
		tmp = x;
	} else if (x <= -9.5e-71) {
		tmp = -2.0 / x;
	} else if (x <= 8.2e-70) {
		tmp = x - y;
	} else if (x <= 2e-12) {
		tmp = -2.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.1e-8:
		tmp = x
	elif x <= -9.5e-71:
		tmp = -2.0 / x
	elif x <= 8.2e-70:
		tmp = x - y
	elif x <= 2e-12:
		tmp = -2.0 / x
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.1e-8)
		tmp = x;
	elseif (x <= -9.5e-71)
		tmp = Float64(-2.0 / x);
	elseif (x <= 8.2e-70)
		tmp = Float64(x - y);
	elseif (x <= 2e-12)
		tmp = Float64(-2.0 / x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.1e-8)
		tmp = x;
	elseif (x <= -9.5e-71)
		tmp = -2.0 / x;
	elseif (x <= 8.2e-70)
		tmp = x - y;
	elseif (x <= 2e-12)
		tmp = -2.0 / x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.1e-8], x, If[LessEqual[x, -9.5e-71], N[(-2.0 / x), $MachinePrecision], If[LessEqual[x, 8.2e-70], N[(x - y), $MachinePrecision], If[LessEqual[x, 2e-12], N[(-2.0 / x), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.09999999999999994e-8 or 1.99999999999999996e-12 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 98.2%

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

   if -2.09999999999999994e-8 < x < -9.4999999999999994e-71 or 8.19999999999999955e-70 < x < 1.99999999999999996e-12

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf 68.2%

      \[\leadsto \color{blue}{x - 2 \cdot \frac{1}{x}} \]
   4. Step-by-step derivation

      1. associate-*r/ 68.2%

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

      2. metadata-eval 68.2%

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

   5. Simplified 68.2%

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

   6. Taylor expanded in x around 0 68.2%

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

   if -9.4999999999999994e-71 < x < 8.19999999999999955e-70

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 85.9%

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

      1. neg-mul-1 85.9%

         \[\leadsto x + \color{blue}{\left(-y\right)} \]

      2. unsub-neg 85.9%

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

   5. Simplified 85.9%

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

Alternative 3: 90.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+40} \lor \neg \left(y \leq 3.5 \cdot 10^{+63}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.02e+40) (not (<= y 3.5e+63))) (- x (/ 2.0 x)) (- x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.02e+40) || !(y <= 3.5e+63)) {
		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 <= (-1.02d+40)) .or. (.not. (y <= 3.5d+63))) 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 <= -1.02e+40) || !(y <= 3.5e+63)) {
		tmp = x - (2.0 / x);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.02e+40) or not (y <= 3.5e+63):
		tmp = x - (2.0 / x)
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.02e+40) || !(y <= 3.5e+63))
		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 <= -1.02e+40) || ~((y <= 3.5e+63)))
		tmp = x - (2.0 / x);
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.02e+40], N[Not[LessEqual[y, 3.5e+63]], $MachinePrecision]], N[(x - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(x - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.02e40 or 3.50000000000000029e63 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 82.5%

      \[\leadsto \color{blue}{x - 2 \cdot \frac{1}{x}} \]
   4. Step-by-step derivation

      1. associate-*r/ 82.5%

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

      2. metadata-eval 82.5%

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

   5. Simplified 82.5%

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

   if -1.02e40 < y < 3.50000000000000029e63

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.3%

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

      1. neg-mul-1 99.3%

         \[\leadsto x + \color{blue}{\left(-y\right)} \]

      2. unsub-neg 99.3%

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

   5. Simplified 99.3%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+40} \lor \neg \left(y \leq 3.5 \cdot 10^{+63}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.006:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.006) x (if (<= x 1.4) (- x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.006) {
		tmp = x;
	} else if (x <= 1.4) {
		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 <= (-0.006d0)) then
        tmp = x
    else if (x <= 1.4d0) 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 <= -0.006) {
		tmp = x;
	} else if (x <= 1.4) {
		tmp = x - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.006:
		tmp = x
	elif x <= 1.4:
		tmp = x - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.006)
		tmp = x;
	elseif (x <= 1.4)
		tmp = Float64(x - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.006)
		tmp = x;
	elseif (x <= 1.4)
		tmp = x - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.006], x, If[LessEqual[x, 1.4], N[(x - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0060000000000000001 or 1.3999999999999999 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 99.0%

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

   if -0.0060000000000000001 < x < 1.3999999999999999

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 72.6%

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

      1. neg-mul-1 72.6%

         \[\leadsto x + \color{blue}{\left(-y\right)} \]

      2. unsub-neg 72.6%

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

   5. Simplified 72.6%

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

Alternative 5: 79.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-60}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-110}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.5e-60) x (if (<= x 2.2e-110) (- y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.5e-60) {
		tmp = x;
	} else if (x <= 2.2e-110) {
		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 <= (-1.5d-60)) then
        tmp = x
    else if (x <= 2.2d-110) 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 <= -1.5e-60) {
		tmp = x;
	} else if (x <= 2.2e-110) {
		tmp = -y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.5e-60:
		tmp = x
	elif x <= 2.2e-110:
		tmp = -y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.5e-60)
		tmp = x;
	elseif (x <= 2.2e-110)
		tmp = Float64(-y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.5e-60)
		tmp = x;
	elseif (x <= 2.2e-110)
		tmp = -y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.5e-60], x, If[LessEqual[x, 2.2e-110], (-y), x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.50000000000000009e-60 or 2.1999999999999999e-110 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 84.4%

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

   if -1.50000000000000009e-60 < x < 2.1999999999999999e-110

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 70.0%

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

      1. neg-mul-1 70.0%

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

   5. Simplified 70.0%

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

Alternative 6: 63.9% 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. Add Preprocessing

3. Taylor expanded in x around inf 58.7%

   \[\leadsto \color{blue}{x} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024170 
(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)))))