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

Percentage Accurate: 99.9% → 99.9%

Time: 5.9s

Alternatives: 7

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, 0.1× speedup

Math

\[\begin{array}{l} \\ x - \frac{y}{\mathsf{fma}\left(\frac{x}{2}, y, 1\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. associate-*l/ 99.9%

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

   3. fma-def 99.9%

      \[\leadsto x - \frac{y}{\color{blue}{\mathsf{fma}\left(\frac{x}{2}, y, 1\right)}} \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(\frac{x}{2}, y, 1\right)}} \]

4. Final simplification 99.9%

   \[\leadsto x - \frac{y}{\mathsf{fma}\left(\frac{x}{2}, y, 1\right)} \]

Alternative 2: 86.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-61}:\\ \;\;\;\;x - y\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-45}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.6e-5)
   x
   (if (<= x 9.2e-61)
     (- x y)
     (if (<= x 7.8e-45) (/ -2.0 x) (if (<= x 1.4) (- x y) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.6e-5) {
		tmp = x;
	} else if (x <= 9.2e-61) {
		tmp = x - y;
	} else if (x <= 7.8e-45) {
		tmp = -2.0 / 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 <= (-3.6d-5)) then
        tmp = x
    else if (x <= 9.2d-61) then
        tmp = x - y
    else if (x <= 7.8d-45) then
        tmp = (-2.0d0) / 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 <= -3.6e-5) {
		tmp = x;
	} else if (x <= 9.2e-61) {
		tmp = x - y;
	} else if (x <= 7.8e-45) {
		tmp = -2.0 / x;
	} else if (x <= 1.4) {
		tmp = x - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.6e-5:
		tmp = x
	elif x <= 9.2e-61:
		tmp = x - y
	elif x <= 7.8e-45:
		tmp = -2.0 / x
	elif x <= 1.4:
		tmp = x - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.6e-5)
		tmp = x;
	elseif (x <= 9.2e-61)
		tmp = Float64(x - y);
	elseif (x <= 7.8e-45)
		tmp = Float64(-2.0 / 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 <= -3.6e-5)
		tmp = x;
	elseif (x <= 9.2e-61)
		tmp = x - y;
	elseif (x <= 7.8e-45)
		tmp = -2.0 / x;
	elseif (x <= 1.4)
		tmp = x - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.6e-5], x, If[LessEqual[x, 9.2e-61], N[(x - y), $MachinePrecision], If[LessEqual[x, 7.8e-45], N[(-2.0 / x), $MachinePrecision], If[LessEqual[x, 1.4], N[(x - y), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.60000000000000009e-5 or 1.3999999999999999 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 97.8%

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

   if -3.60000000000000009e-5 < x < 9.19999999999999967e-61 or 7.7999999999999999e-45 < x < 1.3999999999999999

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 73.0%

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

   if 9.19999999999999967e-61 < x < 7.7999999999999999e-45

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-61}:\\ \;\;\;\;x - y\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-45}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

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 4: 90.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{+50} \lor \neg \left(y \leq 2.5 \cdot 10^{+88}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -8.8e+50) (not (<= y 2.5e+88))) (- x (/ 2.0 x)) (- x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -8.8e+50) || !(y <= 2.5e+88)) {
		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 <= (-8.8d+50)) .or. (.not. (y <= 2.5d+88))) 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 <= -8.8e+50) || !(y <= 2.5e+88)) {
		tmp = x - (2.0 / x);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -8.8e+50) or not (y <= 2.5e+88):
		tmp = x - (2.0 / x)
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -8.8e+50) || !(y <= 2.5e+88))
		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 <= -8.8e+50) || ~((y <= 2.5e+88)))
		tmp = x - (2.0 / x);
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -8.8e+50], N[Not[LessEqual[y, 2.5e+88]], $MachinePrecision]], N[(x - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(x - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.80000000000000067e50 or 2.49999999999999999e88 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 83.8%

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

   if -8.80000000000000067e50 < y < 2.49999999999999999e88

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 96.8%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{+50} \lor \neg \left(y \leq 2.5 \cdot 10^{+88}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]

Alternative 5: 86.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.36:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.6e-5) x (if (<= x 1.36) (- x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.6e-5) {
		tmp = x;
	} else if (x <= 1.36) {
		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 <= (-3.6d-5)) then
        tmp = x
    else if (x <= 1.36d0) 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 <= -3.6e-5) {
		tmp = x;
	} else if (x <= 1.36) {
		tmp = x - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.6e-5:
		tmp = x
	elif x <= 1.36:
		tmp = x - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.6e-5)
		tmp = x;
	elseif (x <= 1.36)
		tmp = Float64(x - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.6e-5)
		tmp = x;
	elseif (x <= 1.36)
		tmp = x - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.6e-5], x, If[LessEqual[x, 1.36], N[(x - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -3.60000000000000009e-5 or 1.3600000000000001 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 97.8%

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

   if -3.60000000000000009e-5 < x < 1.3600000000000001

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 70.4%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.36:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 77.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-50}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-149}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1e-50) x (if (<= x 5.2e-149) (- y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1e-50) {
		tmp = x;
	} else if (x <= 5.2e-149) {
		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 <= (-1d-50)) then
        tmp = x
    else if (x <= 5.2d-149) 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 <= -1e-50) {
		tmp = x;
	} else if (x <= 5.2e-149) {
		tmp = -y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1e-50:
		tmp = x
	elif x <= 5.2e-149:
		tmp = -y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1e-50)
		tmp = x;
	elseif (x <= 5.2e-149)
		tmp = Float64(-y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1e-50)
		tmp = x;
	elseif (x <= 5.2e-149)
		tmp = -y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1e-50], x, If[LessEqual[x, 5.2e-149], (-y), x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.00000000000000001e-50 or 5.19999999999999998e-149 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 84.3%

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

   if -1.00000000000000001e-50 < x < 5.19999999999999998e-149

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 57.1%

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

      1. neg-mul-1 57.1%

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

   4. Simplified 57.1%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-50}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-149}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 62.6% 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 59.3%

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

3. Final simplification 59.3%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023185 
(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)))))