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

Percentage Accurate: 99.9% → 99.9%

Time: 6.6s

Alternatives: 6

Speedup: 1.3×

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, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(x - N[(y / N[(N[(x / N[(2.0 / y), $MachinePrecision]), $MachinePrecision] + 1.0), $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. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 90.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{2}{x}\\ \mathbf{if}\;y \leq -4.5 \cdot 10^{+88}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+173}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ 2.0 x))))
   (if (<= y -4.5e+88) t_0 (if (<= y 5.5e+173) (- x y) t_0))))

C

double code(double x, double y) {
	double t_0 = x - (2.0 / x);
	double tmp;
	if (y <= -4.5e+88) {
		tmp = t_0;
	} else if (y <= 5.5e+173) {
		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 <= (-4.5d+88)) then
        tmp = t_0
    else if (y <= 5.5d+173) 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 <= -4.5e+88) {
		tmp = t_0;
	} else if (y <= 5.5e+173) {
		tmp = x - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x - (2.0 / x)
	tmp = 0
	if y <= -4.5e+88:
		tmp = t_0
	elif y <= 5.5e+173:
		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 <= -4.5e+88)
		tmp = t_0;
	elseif (y <= 5.5e+173)
		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 <= -4.5e+88)
		tmp = t_0;
	elseif (y <= 5.5e+173)
		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, -4.5e+88], t$95$0, If[LessEqual[y, 5.5e+173], N[(x - y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.5e88 or 5.50000000000000049e173 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 83.7

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

   5. Applied rewrites 83.7%

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

   if -4.5e88 < y < 5.50000000000000049e173

   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. lower--.f64 97.3

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

   5. Applied rewrites 97.3%

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

Alternative 3: 99.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(x - N[(y / N[(N[(y * x), $MachinePrecision] * 0.5 + 1.0), $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. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. div-inv N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

Alternative 4: 79.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+232}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+174}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3e+232) (/ -2.0 x) (if (<= y 1.2e+174) (- x y) (/ -2.0 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3e+232) {
		tmp = -2.0 / x;
	} else if (y <= 1.2e+174) {
		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 <= (-3d+232)) then
        tmp = (-2.0d0) / x
    else if (y <= 1.2d+174) 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 <= -3e+232) {
		tmp = -2.0 / x;
	} else if (y <= 1.2e+174) {
		tmp = x - y;
	} else {
		tmp = -2.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3e+232:
		tmp = -2.0 / x
	elif y <= 1.2e+174:
		tmp = x - y
	else:
		tmp = -2.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3e+232)
		tmp = Float64(-2.0 / x);
	elseif (y <= 1.2e+174)
		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 <= -3e+232)
		tmp = -2.0 / x;
	elseif (y <= 1.2e+174)
		tmp = x - y;
	else
		tmp = -2.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3e+232], N[(-2.0 / x), $MachinePrecision], If[LessEqual[y, 1.2e+174], N[(x - y), $MachinePrecision], N[(-2.0 / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.00000000000000003e232 or 1.1999999999999999e174 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(2 \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(2 \cdot \frac{1}{{x}^{2}}\right)\right) + 1\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot x + 1 \cdot x} \]

      4. *-lft-identity N/A

         \[\leadsto \left(\mathsf{neg}\left(2 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot x + \color{blue}{x} \]

      5. lower-fma.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\frac{\color{blue}{2}}{{x}^{2}}\right), x, x\right) \]

      8. distribute-neg-frac N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(2\right)}{{x}^{2}}}, x, x\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(2\right)}{{x}^{2}}}, x, x\right) \]

      10. metadata-eval N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 73.8

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

   5. Applied rewrites 73.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 47.4%

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

   if -3.00000000000000003e232 < y < 1.1999999999999999e174

   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. lower--.f64 88.6

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

   5. Applied rewrites 88.6%

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

Alternative 5: 75.1% accurate, 8.5× 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 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. lower--.f64 76.4

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

5. Applied rewrites 76.4%

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

Alternative 6: 26.3% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ -y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- y))

C

double code(double x, double y) {
	return -y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -y
end function

Java

public static double code(double x, double y) {
	return -y;
}

Python

def code(x, y):
	return -y

Julia

function code(x, y)
	return Float64(-y)
end

MATLAB

function tmp = code(x, y)
	tmp = -y;
end

Wolfram

code[x_, y_] := (-y)

Derivation

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. lower-neg.f64 30.0

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

5. Applied rewrites 30.0%

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

Reproduce

herbie shell --seed 2024248 
(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)))))