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

Percentage Accurate: 99.9% → 99.9%

Time: 5.0s

Alternatives: 4

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 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 91.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+84} \lor \neg \left(y \leq 4.3 \cdot 10^{+145}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5.5e+84) (not (<= y 4.3e+145))) (- x (/ 2.0 x)) (- x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5.5e+84) || !(y <= 4.3e+145)) {
		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 <= (-5.5d+84)) .or. (.not. (y <= 4.3d+145))) 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 <= -5.5e+84) || !(y <= 4.3e+145)) {
		tmp = x - (2.0 / x);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -5.5e+84) or not (y <= 4.3e+145):
		tmp = x - (2.0 / x)
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5.5e+84) || !(y <= 4.3e+145))
		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 <= -5.5e+84) || ~((y <= 4.3e+145)))
		tmp = x - (2.0 / x);
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5.5e+84], N[Not[LessEqual[y, 4.3e+145]], $MachinePrecision]], N[(x - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(x - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.5000000000000004e84 or 4.29999999999999998e145 < 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 91.2%

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

   if -5.5000000000000004e84 < y < 4.29999999999999998e145

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 96.0%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+84} \lor \neg \left(y \leq 4.3 \cdot 10^{+145}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{1}{\frac{-1}{y} - x \cdot 0.5} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (/ 1.0 (- (/ -1.0 y) (* x 0.5)))))

C

double code(double x, double y) {
	return x + (1.0 / ((-1.0 / y) - (x * 0.5)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + (1.0d0 / (((-1.0d0) / y) - (x * 0.5d0)))
end function

Java

public static double code(double x, double y) {
	return x + (1.0 / ((-1.0 / y) - (x * 0.5)));
}

Python

def code(x, y):
	return x + (1.0 / ((-1.0 / y) - (x * 0.5)))

Julia

function code(x, y)
	return Float64(x + Float64(1.0 / Float64(Float64(-1.0 / y) - Float64(x * 0.5))))
end

MATLAB

function tmp = code(x, y)
	tmp = x + (1.0 / ((-1.0 / y) - (x * 0.5)));
end

Wolfram

code[x_, y_] := N[(x + N[(1.0 / N[(N[(-1.0 / y), $MachinePrecision] - N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. clear-num 99.9%

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

   2. associate-/r/ 99.9%

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

   3. +-commutative 99.9%

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

   4. *-commutative 99.9%

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

   5. associate-/l* 99.9%

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

   6. fma-define 99.9%

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

4. Applied egg-rr 99.9%

   \[\leadsto x - \color{blue}{\frac{1}{\mathsf{fma}\left(y, \frac{x}{2}, 1\right)} \cdot y} \]
5. Step-by-step derivation

   1. associate-*l/ 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. clear-num 99.9%

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

   4. div-inv 99.9%

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

   5. metadata-eval 99.9%

      \[\leadsto x - \frac{1}{\frac{\mathsf{fma}\left(y, x \cdot \color{blue}{0.5}, 1\right)}{y}} \]

6. Applied egg-rr 99.9%

   \[\leadsto x - \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y, x \cdot 0.5, 1\right)}{y}}} \]

7. Taylor expanded in y around inf 99.9%

   \[\leadsto x - \frac{1}{\color{blue}{0.5 \cdot x + \frac{1}{y}}} \]

8. Final simplification 99.9%

   \[\leadsto x + \frac{1}{\frac{-1}{y} - x \cdot 0.5} \]
9. Add Preprocessing

Alternative 4: 74.3% 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. Add Preprocessing

3. Taylor expanded in y around 0 75.1%

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

Reproduce

herbie shell --seed 2024086 
(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)))))