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

Percentage Accurate: 99.9% → 99.9%

Time: 5.6s

Alternatives: 5

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. Step-by-step derivation

   1. clear-num 99.9%

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

   2. inv-pow 99.9%

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

   3. *-commutative 99.9%

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

3. Applied egg-rr 99.9%

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

   1. clear-num 99.9%

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

   2. inv-pow 99.9%

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

   3. unpow-1 99.9%

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

5. Applied egg-rr 99.9%

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

6. Taylor expanded in y around 0 99.9%

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

   1. *-commutative 99.9%

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

8. Simplified 99.9%

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

   1. add-log-exp 67.8%

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

   2. *-un-lft-identity 67.8%

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

   3. log-prod 67.8%

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

   4. metadata-eval 67.8%

      \[\leadsto x - \left(\color{blue}{0} + \log \left(e^{{\left(\frac{1}{y} + x \cdot 0.5\right)}^{-1}}\right)\right) \]

   5. add-log-exp 99.9%

      \[\leadsto x - \left(0 + \color{blue}{{\left(\frac{1}{y} + x \cdot 0.5\right)}^{-1}}\right) \]

   6. unpow-1 99.9%

      \[\leadsto x - \left(0 + \color{blue}{\frac{1}{\frac{1}{y} + x \cdot 0.5}}\right) \]

   7. +-commutative 99.9%

      \[\leadsto x - \left(0 + \frac{1}{\color{blue}{x \cdot 0.5 + \frac{1}{y}}}\right) \]

   8. fma-def 99.9%

      \[\leadsto x - \left(0 + \frac{1}{\color{blue}{\mathsf{fma}\left(x, 0.5, \frac{1}{y}\right)}}\right) \]

10. Applied egg-rr 99.9%

    \[\leadsto x - \color{blue}{\left(0 + \frac{1}{\mathsf{fma}\left(x, 0.5, \frac{1}{y}\right)}\right)} \]
11. Step-by-step derivation

    1. +-lft-identity 99.9%

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

12. Simplified 99.9%

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

13. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
2. Step-by-step derivation

   1. clear-num 99.9%

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

   2. inv-pow 99.9%

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

   3. *-commutative 99.9%

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

3. Applied egg-rr 99.9%

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

   1. unpow-1 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

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. Final simplification 99.9%

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

Alternative 4: 90.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+161} \lor \neg \left(y \leq 3 \cdot 10^{+57}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.1e+161) (not (<= y 3e+57))) (- x (/ 2.0 x)) (- x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.1e+161) || !(y <= 3e+57)) {
		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 <= (-2.1d+161)) .or. (.not. (y <= 3d+57))) 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 <= -2.1e+161) || !(y <= 3e+57)) {
		tmp = x - (2.0 / x);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.1e+161) or not (y <= 3e+57):
		tmp = x - (2.0 / x)
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.1e+161) || !(y <= 3e+57))
		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 <= -2.1e+161) || ~((y <= 3e+57)))
		tmp = x - (2.0 / x);
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.1e+161], N[Not[LessEqual[y, 3e+57]], $MachinePrecision]], N[(x - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(x - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.1e161 or 3e57 < y

   1. Initial program 99.8%

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

   2. Taylor expanded in y around inf 83.0%

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

   if -2.1e161 < y < 3e57

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 94.5%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+161} \lor \neg \left(y \leq 3 \cdot 10^{+57}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]

Alternative 5: 75.7% 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. Taylor expanded in y around 0 75.2%

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

3. Final simplification 75.2%

   \[\leadsto x - y \]

Reproduce

herbie shell --seed 2023257 
(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)))))