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

Percentage Accurate: 99.9% → 99.9%

Time: 7.3s

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 100.0%

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

Alternative 2: 90.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+42} \lor \neg \left(y \leq 1.66 \cdot 10^{+109}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.6e+42) (not (<= y 1.66e+109))) (- x (/ 2.0 x)) (- x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.6e+42) || !(y <= 1.66e+109)) {
		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 <= (-4.6d+42)) .or. (.not. (y <= 1.66d+109))) 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 <= -4.6e+42) || !(y <= 1.66e+109)) {
		tmp = x - (2.0 / x);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.6e+42) or not (y <= 1.66e+109):
		tmp = x - (2.0 / x)
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.6e+42) || !(y <= 1.66e+109))
		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 <= -4.6e+42) || ~((y <= 1.66e+109)))
		tmp = x - (2.0 / x);
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.6e+42], N[Not[LessEqual[y, 1.66e+109]], $MachinePrecision]], N[(x - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(x - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.6e42 or 1.6599999999999999e109 < 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 84.8%

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

      1. associate-*r/ 84.8%

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

      2. metadata-eval 84.8%

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

   5. Simplified 84.8%

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

   if -4.6e42 < y < 1.6599999999999999e109

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 98.2%

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

      1. neg-mul-1 98.2%

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

      2. unsub-neg 98.2%

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

   5. Simplified 98.2%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+42} \lor \neg \left(y \leq 1.66 \cdot 10^{+109}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-12}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -8.8e-5) x (if (<= x 2e-12) (- x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -8.8e-5) {
		tmp = x;
	} else if (x <= 2e-12) {
		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 <= (-8.8d-5)) then
        tmp = x
    else if (x <= 2d-12) 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 <= -8.8e-5) {
		tmp = x;
	} else if (x <= 2e-12) {
		tmp = x - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -8.8e-5:
		tmp = x
	elif x <= 2e-12:
		tmp = x - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -8.8e-5)
		tmp = x;
	elseif (x <= 2e-12)
		tmp = Float64(x - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8.8e-5)
		tmp = x;
	elseif (x <= 2e-12)
		tmp = x - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -8.8e-5], x, If[LessEqual[x, 2e-12], N[(x - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -8.7999999999999998e-5 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 96.5%

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

   if -8.7999999999999998e-5 < x < 1.99999999999999996e-12

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 70.9%

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

      1. neg-mul-1 70.9%

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

      2. unsub-neg 70.9%

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

   5. Simplified 70.9%

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

Alternative 4: 78.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-135}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-149}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.65e-135) x (if (<= x 4.5e-149) (- y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.65e-135) {
		tmp = x;
	} else if (x <= 4.5e-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 <= (-1.65d-135)) then
        tmp = x
    else if (x <= 4.5d-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 <= -1.65e-135) {
		tmp = x;
	} else if (x <= 4.5e-149) {
		tmp = -y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.65e-135:
		tmp = x
	elif x <= 4.5e-149:
		tmp = -y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.65e-135)
		tmp = x;
	elseif (x <= 4.5e-149)
		tmp = Float64(-y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.65e-135)
		tmp = x;
	elseif (x <= 4.5e-149)
		tmp = -y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.65e-135], x, If[LessEqual[x, 4.5e-149], (-y), x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.65e-135 or 4.4999999999999998e-149 < 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 82.9%

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

   if -1.65e-135 < x < 4.4999999999999998e-149

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 71.0%

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

      1. neg-mul-1 71.0%

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

   5. Simplified 71.0%

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

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

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. Taylor expanded in y around inf 99.9%

   \[\leadsto x - \frac{y}{\color{blue}{y \cdot \left(0.5 \cdot x + \frac{1}{y}\right)}} \]
4. Step-by-step derivation

   1. *-un-lft-identity 99.9%

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

   2. associate-/r* 99.9%

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

   3. *-inverses 99.9%

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

   4. fma-define 99.9%

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

5. Applied egg-rr 99.9%

   \[\leadsto x - \color{blue}{1 \cdot \frac{1}{\mathsf{fma}\left(0.5, x, \frac{1}{y}\right)}} \]
6. Step-by-step derivation

   1. *-lft-identity 99.9%

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

   2. fma-undefine 99.9%

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

   3. *-commutative 99.9%

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

   4. fma-undefine 99.9%

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

7. Simplified 99.9%

   \[\leadsto x - \color{blue}{\frac{1}{\mathsf{fma}\left(x, 0.5, \frac{1}{y}\right)}} \]
8. Step-by-step derivation

   1. fma-undefine 99.9%

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

   2. +-commutative 99.9%

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

9. Applied egg-rr 99.9%

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

10. Final simplification 99.9%

    \[\leadsto x + \frac{-1}{\frac{1}{y} + x \cdot 0.5} \]
11. 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 100.0%

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

3. Taylor expanded in x around inf 64.0%

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

Reproduce

herbie shell --seed 2024172 
(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)))))