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

Percentage Accurate: 99.9% → 99.9%

Time: 33.6s

Alternatives: 6

Speedup: N/A×

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, 1.3× speedup

Math

\[\begin{array}{l} \\ x - \frac{y}{\mathsf{fma}\left(y \cdot 0.5, x, 1\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

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

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

   5. div-inv N/A

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

   6. *-lowering-*.f64 N/A

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

   7. metadata-eval 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 85.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{-21}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.26 \cdot 10^{-107}:\\ \;\;\;\;x - y\\ \mathbf{elif}\;x \leq 0.0045:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9.2e-21)
   x
   (if (<= x 2.26e-107) (- x y) (if (<= x 0.0045) (/ -2.0 x) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9.2e-21) {
		tmp = x;
	} else if (x <= 2.26e-107) {
		tmp = x - y;
	} else if (x <= 0.0045) {
		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 <= (-9.2d-21)) then
        tmp = x
    else if (x <= 2.26d-107) then
        tmp = x - y
    else if (x <= 0.0045d0) 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 <= -9.2e-21) {
		tmp = x;
	} else if (x <= 2.26e-107) {
		tmp = x - y;
	} else if (x <= 0.0045) {
		tmp = -2.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9.2e-21:
		tmp = x
	elif x <= 2.26e-107:
		tmp = x - y
	elif x <= 0.0045:
		tmp = -2.0 / x
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9.2e-21)
		tmp = x;
	elseif (x <= 2.26e-107)
		tmp = Float64(x - y);
	elseif (x <= 0.0045)
		tmp = Float64(-2.0 / x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9.2e-21)
		tmp = x;
	elseif (x <= 2.26e-107)
		tmp = x - y;
	elseif (x <= 0.0045)
		tmp = -2.0 / x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9.2e-21], x, If[LessEqual[x, 2.26e-107], N[(x - y), $MachinePrecision], If[LessEqual[x, 0.0045], N[(-2.0 / x), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.19999999999999998e-21 or 0.00449999999999999966 < 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 97.6%

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

   if -9.19999999999999998e-21 < x < 2.25999999999999995e-107

   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

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 76.8

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

   5. Simplified 76.8%

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

   if 2.25999999999999995e-107 < x < 0.00449999999999999966

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 61.3

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

   5. Simplified 61.3%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 58.9

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

   8. Simplified 58.9%

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

Alternative 3: 90.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{2}{x}\\ \mathbf{if}\;y \leq -8 \cdot 10^{+109}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{+55}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ 2.0 x))))
   (if (<= y -8e+109) t_0 (if (<= y 7.6e+55) (- x y) t_0))))

C

double code(double x, double y) {
	double t_0 = x - (2.0 / x);
	double tmp;
	if (y <= -8e+109) {
		tmp = t_0;
	} else if (y <= 7.6e+55) {
		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 <= (-8d+109)) then
        tmp = t_0
    else if (y <= 7.6d+55) 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 <= -8e+109) {
		tmp = t_0;
	} else if (y <= 7.6e+55) {
		tmp = x - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x - (2.0 / x)
	tmp = 0
	if y <= -8e+109:
		tmp = t_0
	elif y <= 7.6e+55:
		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 <= -8e+109)
		tmp = t_0;
	elseif (y <= 7.6e+55)
		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 <= -8e+109)
		tmp = t_0;
	elseif (y <= 7.6e+55)
		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, -8e+109], t$95$0, If[LessEqual[y, 7.6e+55], N[(x - y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.99999999999999985e109 or 7.5999999999999999e55 < 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

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

      1. /-lowering-/.f64 84.4

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

   5. Simplified 84.4%

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

   if -7.99999999999999985e109 < y < 7.5999999999999999e55

   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

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 97.0

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

   5. Simplified 97.0%

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

Alternative 4: 87.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{-21}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-11}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9.2e-21) x (if (<= x 3.8e-11) (- x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9.2e-21) {
		tmp = x;
	} else if (x <= 3.8e-11) {
		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 <= (-9.2d-21)) then
        tmp = x
    else if (x <= 3.8d-11) 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 <= -9.2e-21) {
		tmp = x;
	} else if (x <= 3.8e-11) {
		tmp = x - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9.2e-21:
		tmp = x
	elif x <= 3.8e-11:
		tmp = x - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9.2e-21)
		tmp = x;
	elseif (x <= 3.8e-11)
		tmp = Float64(x - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9.2e-21)
		tmp = x;
	elseif (x <= 3.8e-11)
		tmp = x - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9.2e-21], x, If[LessEqual[x, 3.8e-11], N[(x - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -9.19999999999999998e-21 or 3.7999999999999998e-11 < 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 96.1%

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

   if -9.19999999999999998e-21 < x < 3.7999999999999998e-11

   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

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 71.0

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

   5. Simplified 71.0%

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

Alternative 5: 78.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-133}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-82}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9.5e-133) x (if (<= x 3.2e-82) (- y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9.5e-133) {
		tmp = x;
	} else if (x <= 3.2e-82) {
		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 <= (-9.5d-133)) then
        tmp = x
    else if (x <= 3.2d-82) 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 <= -9.5e-133) {
		tmp = x;
	} else if (x <= 3.2e-82) {
		tmp = -y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9.5e-133:
		tmp = x
	elif x <= 3.2e-82:
		tmp = -y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9.5e-133)
		tmp = x;
	elseif (x <= 3.2e-82)
		tmp = Float64(-y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9.5e-133)
		tmp = x;
	elseif (x <= 3.2e-82)
		tmp = -y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9.5e-133], x, If[LessEqual[x, 3.2e-82], (-y), x]]

Derivation

1. Split input into 2 regimes

2. if x < -9.4999999999999992e-133 or 3.2000000000000001e-82 < 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 86.4%

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

   if -9.4999999999999992e-133 < x < 3.2000000000000001e-82

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(y\right)} \]

      2. neg-lowering-neg.f64 60.3

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

   5. Simplified 60.3%

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

Alternative 6: 63.7% accurate, 34.0× 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

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

5. Simplified 62.0%

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

Reproduce

herbie shell --seed 2024205 
(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)))))