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

Percentage Accurate: 99.9% → 99.9%

Time: 6.3s

Alternatives: 4

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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(x + N[(-1.0 / N[(x * 0.5 + N[(1.0 / y), $MachinePrecision]), $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. clear-num 99.9%

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

   2. inv-pow 99.9%

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

   3. +-commutative 99.9%

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

   4. *-commutative 99.9%

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

   5. associate-/l* 99.9%

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

   6. fma-define 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in y around 0 99.9%

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

   1. +-commutative 99.9%

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

7. Simplified 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. unpow-1 99.9%

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

   3. +-commutative 99.9%

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

   4. *-commutative 99.9%

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

   5. fma-define 99.9%

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

9. Applied egg-rr 99.9%

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

    1. *-lft-identity 99.9%

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

11. Simplified 99.9%

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

12. Final simplification 99.9%

    \[\leadsto x + \frac{-1}{\mathsf{fma}\left(x, 0.5, \frac{1}{y}\right)} \]
13. Add Preprocessing

Alternative 2: 91.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{+80} \lor \neg \left(y \leq 3.6 \cdot 10^{+111}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5.4e+80) (not (<= y 3.6e+111))) (- x (/ 2.0 x)) (- x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5.4e+80) || !(y <= 3.6e+111)) {
		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 <= (-5.4d+80)) .or. (.not. (y <= 3.6d+111))) 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 <= -5.4e+80) || !(y <= 3.6e+111)) {
		tmp = x - (2.0 / x);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -5.4e+80) or not (y <= 3.6e+111):
		tmp = x - (2.0 / x)
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5.4e+80) || !(y <= 3.6e+111))
		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 <= -5.4e+80) || ~((y <= 3.6e+111)))
		tmp = x - (2.0 / x);
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5.4e+80], N[Not[LessEqual[y, 3.6e+111]], $MachinePrecision]], N[(x - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(x - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.39999999999999966e80 or 3.6000000000000002e111 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf 82.4%

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

   if -5.39999999999999966e80 < y < 3.6000000000000002e111

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 97.4%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{+80} \lor \neg \left(y \leq 3.6 \cdot 10^{+111}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 99.9%

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

3. Final simplification 99.9%

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

Alternative 4: 74.9% accurate, 3.7× 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 73.8%

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

4. Final simplification 73.8%

   \[\leadsto x - y \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024048 
(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)))))