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

Percentage Accurate: 99.9% → 99.9%

Time: 5.4s

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, 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. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 99.9

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

5. Applied rewrites 99.9%

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

7. Applied rewrites 99.9%

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

Alternative 2: 90.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{+103} \lor \neg \left(y \leq 2.15 \cdot 10^{+88}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.12e+103) (not (<= y 2.15e+88))) (- x (/ 2.0 x)) (- x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.12e+103) || !(y <= 2.15e+88)) {
		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 <= (-1.12d+103)) .or. (.not. (y <= 2.15d+88))) 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 <= -1.12e+103) || !(y <= 2.15e+88)) {
		tmp = x - (2.0 / x);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.12e+103) or not (y <= 2.15e+88):
		tmp = x - (2.0 / x)
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.12e+103) || !(y <= 2.15e+88))
		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 <= -1.12e+103) || ~((y <= 2.15e+88)))
		tmp = x - (2.0 / x);
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.12e+103], N[Not[LessEqual[y, 2.15e+88]], $MachinePrecision]], N[(x - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(x - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.12000000000000007e103 or 2.14999999999999987e88 < y

   1. Initial program 99.8%

      \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
   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 86.3

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

   5. Applied rewrites 86.3%

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

   if -1.12000000000000007e103 < y < 2.14999999999999987e88

   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. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y} \]

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 96.5

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

   5. Applied rewrites 96.5%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{+103} \lor \neg \left(y \leq 2.15 \cdot 10^{+88}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 99.9

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

5. Applied rewrites 99.9%

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

Alternative 4: 75.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.95 \cdot 10^{+152}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -3.95e+152) (/ -2.0 x) (- x y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.95e+152) {
		tmp = -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 <= (-3.95d+152)) then
        tmp = (-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 <= -3.95e+152) {
		tmp = -2.0 / x;
	} else {
		tmp = x - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.95e+152:
		tmp = -2.0 / x
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.95e+152)
		tmp = 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 <= -3.95e+152)
		tmp = -2.0 / x;
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.95e+152], N[(-2.0 / x), $MachinePrecision], N[(x - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.95e152

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 67.2

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

   5. Applied rewrites 67.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 56.7%

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

   if -3.95e152 < y

   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. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y} \]

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 80.2

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

   5. Applied rewrites 80.2%

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

Alternative 5: 74.3% 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. fp-cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y} \]

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower--.f64 73.6

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

5. Applied rewrites 73.6%

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

Alternative 6: 26.6% 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 27.2

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

5. Applied rewrites 27.2%

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

Reproduce

herbie shell --seed 2024320 
(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)))))