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

Percentage Accurate: 99.9% → 99.9%

Time: 33.2s

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. Final simplification 99.9%

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

Alternative 2: 90.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{2}{x}\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{+173}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{+125}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ 2.0 x))))
   (if (<= y -6.2e+173) t_0 (if (<= y 4.9e+125) (- x y) t_0))))

C

double code(double x, double y) {
	double t_0 = x - (2.0 / x);
	double tmp;
	if (y <= -6.2e+173) {
		tmp = t_0;
	} else if (y <= 4.9e+125) {
		tmp = x - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - (2.0d0 / x)
    if (y <= (-6.2d+173)) then
        tmp = t_0
    else if (y <= 4.9d+125) then
        tmp = x - y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x - (2.0 / x);
	double tmp;
	if (y <= -6.2e+173) {
		tmp = t_0;
	} else if (y <= 4.9e+125) {
		tmp = x - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x - (2.0 / x)
	tmp = 0
	if y <= -6.2e+173:
		tmp = t_0
	elif y <= 4.9e+125:
		tmp = x - y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x - Float64(2.0 / x))
	tmp = 0.0
	if (y <= -6.2e+173)
		tmp = t_0;
	elseif (y <= 4.9e+125)
		tmp = Float64(x - y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x - (2.0 / x);
	tmp = 0.0;
	if (y <= -6.2e+173)
		tmp = t_0;
	elseif (y <= 4.9e+125)
		tmp = x - y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.2e+173], t$95$0, If[LessEqual[y, 4.9e+125], N[(x - y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.2e173 or 4.90000000000000016e125 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{2}{x}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 89.5%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(2, \color{blue}{x}\right)\right) \]

   5. Simplified 89.5%

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

   if -6.2e173 < y < 4.90000000000000016e125

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto x + \left(\mathsf{neg}\left(y\right)\right) \]

      2. unsub-neg N/A

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

      3. --lowering--.f64 95.7%

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{y}\right) \]

   5. Simplified 95.7%

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

Alternative 3: 78.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+175}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{+221}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.2e+175) (/ -2.0 x) (if (<= y 4.9e+221) (- x y) (/ -2.0 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.2e+175) {
		tmp = -2.0 / x;
	} else if (y <= 4.9e+221) {
		tmp = x - y;
	} else {
		tmp = -2.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.2d+175)) then
        tmp = (-2.0d0) / x
    else if (y <= 4.9d+221) then
        tmp = x - y
    else
        tmp = (-2.0d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.2e+175) {
		tmp = -2.0 / x;
	} else if (y <= 4.9e+221) {
		tmp = x - y;
	} else {
		tmp = -2.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.2e+175:
		tmp = -2.0 / x
	elif y <= 4.9e+221:
		tmp = x - y
	else:
		tmp = -2.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.2e+175)
		tmp = Float64(-2.0 / x);
	elseif (y <= 4.9e+221)
		tmp = Float64(x - y);
	else
		tmp = Float64(-2.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.2e+175)
		tmp = -2.0 / x;
	elseif (y <= 4.9e+221)
		tmp = x - y;
	else
		tmp = -2.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.2e+175], N[(-2.0 / x), $MachinePrecision], If[LessEqual[y, 4.9e+221], N[(x - y), $MachinePrecision], N[(-2.0 / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.1999999999999998e175 or 4.8999999999999999e221 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{2}{x}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 96.0%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(2, \color{blue}{x}\right)\right) \]

   5. Simplified 96.0%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 56.9%

         \[\leadsto \mathsf{/.f64}\left(-2, \color{blue}{x}\right) \]

   8. Simplified 56.9%

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

   if -4.1999999999999998e175 < y < 4.8999999999999999e221

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto x + \left(\mathsf{neg}\left(y\right)\right) \]

      2. unsub-neg N/A

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

      3. --lowering--.f64 92.7%

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{y}\right) \]

   5. Simplified 92.7%

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

Alternative 4: 86.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-26}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.8e-17) x (if (<= x 6e-26) (- x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.8e-17) {
		tmp = x;
	} else if (x <= 6e-26) {
		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 <= (-3.8d-17)) then
        tmp = x
    else if (x <= 6d-26) 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 <= -3.8e-17) {
		tmp = x;
	} else if (x <= 6e-26) {
		tmp = x - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.8e-17:
		tmp = x
	elif x <= 6e-26:
		tmp = x - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.8e-17)
		tmp = x;
	elseif (x <= 6e-26)
		tmp = Float64(x - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.8e-17)
		tmp = x;
	elseif (x <= 6e-26)
		tmp = x - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.8e-17], x, If[LessEqual[x, 6e-26], N[(x - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -3.8000000000000001e-17 or 6.00000000000000023e-26 < 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

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

   5. Simplified 95.1%

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

   if -3.8000000000000001e-17 < x < 6.00000000000000023e-26

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto x + \left(\mathsf{neg}\left(y\right)\right) \]

      2. unsub-neg N/A

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

      3. --lowering--.f64 73.7%

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{y}\right) \]

   5. Simplified 73.7%

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

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + (y / ((-1.0d0) - (x / (2.0d0 / y))))
end function

Java

public static double code(double x, double y) {
	return x + (y / (-1.0 - (x / (2.0 / y))));
}

Python

def code(x, y):
	return x + (y / (-1.0 - (x / (2.0 / y))))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = x + (y / (-1.0 - (x / (2.0 / y))));
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-/l* N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\frac{y}{2}}\right)\right)\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(x \cdot \frac{1}{\color{blue}{\frac{2}{y}}}\right)\right)\right)\right) \]

   3. un-div-inv N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(\frac{x}{\color{blue}{\frac{2}{y}}}\right)\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{2}{y}\right)}\right)\right)\right)\right) \]

   5. /-lowering-/.f64 99.9%

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(2, \color{blue}{y}\right)\right)\right)\right)\right) \]

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 6: 62.5% 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

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

5. Simplified 62.9%

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

Reproduce

herbie shell --seed 2024152 
(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)))))