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

Percentage Accurate: 99.9% → 99.9%

Time: 6.7s

Alternatives: 5

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, 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 2: 91.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{2}{x}\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{+117}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 10^{+102}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ 2.0 x))))
   (if (<= y -1.8e+117) t_0 (if (<= y 1e+102) (- x y) t_0))))

C

double code(double x, double y) {
	double t_0 = x - (2.0 / x);
	double tmp;
	if (y <= -1.8e+117) {
		tmp = t_0;
	} else if (y <= 1e+102) {
		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 <= (-1.8d+117)) then
        tmp = t_0
    else if (y <= 1d+102) 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 <= -1.8e+117) {
		tmp = t_0;
	} else if (y <= 1e+102) {
		tmp = x - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x - (2.0 / x)
	tmp = 0
	if y <= -1.8e+117:
		tmp = t_0
	elif y <= 1e+102:
		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 <= -1.8e+117)
		tmp = t_0;
	elseif (y <= 1e+102)
		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 <= -1.8e+117)
		tmp = t_0;
	elseif (y <= 1e+102)
		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, -1.8e+117], t$95$0, If[LessEqual[y, 1e+102], N[(x - y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.80000000000000006e117 or 9.99999999999999977e101 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto x - \color{blue}{\frac{2}{x}} \]
   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 -1.80000000000000006e117 < y < 9.99999999999999977e101

   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 94.9

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

   5. Applied rewrites 94.9%

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

Alternative 3: 78.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{+187}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+132}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.75e+187) (/ -2.0 x) (if (<= y 2.5e+132) (- x y) (/ -2.0 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.75e+187) {
		tmp = -2.0 / x;
	} else if (y <= 2.5e+132) {
		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 <= (-1.75d+187)) then
        tmp = (-2.0d0) / x
    else if (y <= 2.5d+132) 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 <= -1.75e+187) {
		tmp = -2.0 / x;
	} else if (y <= 2.5e+132) {
		tmp = x - y;
	} else {
		tmp = -2.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.75e+187:
		tmp = -2.0 / x
	elif y <= 2.5e+132:
		tmp = x - y
	else:
		tmp = -2.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.75e+187)
		tmp = Float64(-2.0 / x);
	elseif (y <= 2.5e+132)
		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 <= -1.75e+187)
		tmp = -2.0 / x;
	elseif (y <= 2.5e+132)
		tmp = x - y;
	else
		tmp = -2.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.75e+187], N[(-2.0 / x), $MachinePrecision], If[LessEqual[y, 2.5e+132], N[(x - y), $MachinePrecision], N[(-2.0 / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.7499999999999999e187 or 2.5000000000000001e132 < y

   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. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 - 2 \cdot \frac{1}{{x}^{2}}\right)} \]
   6. 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 85.8

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

   7. Applied rewrites 85.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 33.0%

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

   if -1.7499999999999999e187 < y < 2.5000000000000001e132

   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 91.4

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

   5. Applied rewrites 91.4%

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

Alternative 4: 74.8% 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 72.6

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

5. Applied rewrites 72.6%

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

Alternative 5: 26.7% 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 26.9

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

5. Applied rewrites 26.9%

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

Reproduce

herbie shell --seed 2024294 
(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)))))