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

Percentage Accurate: 99.9% → 99.9%

Time: 8.3s

Alternatives: 6

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, 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 100.0%

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

Alternative 2: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{-86} \lor \neg \left(x \leq 1.3 \cdot 10^{-44}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -9.2e-86) (not (<= x 1.3e-44))) (- x (/ 2.0 x)) (- x y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -9.2e-86) || !(x <= 1.3e-44)) {
		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 ((x <= (-9.2d-86)) .or. (.not. (x <= 1.3d-44))) 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 ((x <= -9.2e-86) || !(x <= 1.3e-44)) {
		tmp = x - (2.0 / x);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -9.2e-86) or not (x <= 1.3e-44):
		tmp = x - (2.0 / x)
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -9.2e-86) || !(x <= 1.3e-44))
		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 ((x <= -9.2e-86) || ~((x <= 1.3e-44)))
		tmp = x - (2.0 / x);
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -9.2e-86], N[Not[LessEqual[x, 1.3e-44]], $MachinePrecision]], N[(x - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(x - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.19999999999999985e-86 or 1.2999999999999999e-44 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 93.8%

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

      1. associate-*r/ 93.8%

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

      2. metadata-eval 93.8%

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

   5. Simplified 93.8%

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

   if -9.19999999999999985e-86 < x < 1.2999999999999999e-44

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 86.8%

      \[\leadsto \color{blue}{x + -1 \cdot y} \]
   4. Step-by-step derivation

      1. neg-mul-1 86.8%

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

      2. unsub-neg 86.8%

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

   5. Simplified 86.8%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{-86} \lor \neg \left(x \leq 1.3 \cdot 10^{-44}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-19}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 0.000125:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.15e-19) x (if (<= x 0.000125) (- x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.15e-19) {
		tmp = x;
	} else if (x <= 0.000125) {
		tmp = x - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.15d-19)) then
        tmp = x
    else if (x <= 0.000125d0) then
        tmp = x - y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.15e-19) {
		tmp = x;
	} else if (x <= 0.000125) {
		tmp = x - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.15e-19:
		tmp = x
	elif x <= 0.000125:
		tmp = x - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.15e-19)
		tmp = x;
	elseif (x <= 0.000125)
		tmp = Float64(x - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.15e-19)
		tmp = x;
	elseif (x <= 0.000125)
		tmp = x - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.15e-19], x, If[LessEqual[x, 0.000125], N[(x - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1499999999999999e-19 or 1.25e-4 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 96.3%

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

   if -1.1499999999999999e-19 < x < 1.25e-4

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 78.7%

      \[\leadsto \color{blue}{x + -1 \cdot y} \]
   4. Step-by-step derivation

      1. neg-mul-1 78.7%

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

      2. unsub-neg 78.7%

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

   5. Simplified 78.7%

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

Alternative 4: 78.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-122}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-58}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7.5e-122) x (if (<= x 1.9e-58) (- y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.5e-122) {
		tmp = x;
	} else if (x <= 1.9e-58) {
		tmp = -y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7.5d-122)) then
        tmp = x
    else if (x <= 1.9d-58) then
        tmp = -y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -7.5e-122) {
		tmp = x;
	} else if (x <= 1.9e-58) {
		tmp = -y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7.5e-122:
		tmp = x
	elif x <= 1.9e-58:
		tmp = -y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7.5e-122)
		tmp = x;
	elseif (x <= 1.9e-58)
		tmp = Float64(-y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.5e-122)
		tmp = x;
	elseif (x <= 1.9e-58)
		tmp = -y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7.5e-122], x, If[LessEqual[x, 1.9e-58], (-y), x]]

Derivation

1. Split input into 2 regimes

2. if x < -7.4999999999999998e-122 or 1.8999999999999999e-58 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 87.9%

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

   if -7.4999999999999998e-122 < x < 1.8999999999999999e-58

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 74.1%

      \[\leadsto \color{blue}{-1 \cdot y} \]
   4. Step-by-step derivation

      1. neg-mul-1 74.1%

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

   5. Simplified 74.1%

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

Alternative 5: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{-1}{\frac{1}{y} + x \cdot 0.5} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + ((-1.0d0) / ((1.0d0 / y) + (x * 0.5d0)))
end function

Java

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

Python

def code(x, y):
	return x + (-1.0 / ((1.0 / y) + (x * 0.5)))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = x + (-1.0 / ((1.0 / y) + (x * 0.5)));
end

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
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. associate-/l* 99.9%

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

   5. fma-define 99.9%

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

   6. div-inv 99.9%

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

   7. metadata-eval 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around 0 99.9%

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

   1. sub-neg 99.9%

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

   2. unpow-1 99.9%

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

   3. distribute-neg-frac 99.9%

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

   4. metadata-eval 99.9%

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

   5. *-commutative 99.9%

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

   6. fma-define 99.9%

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

7. Applied egg-rr 99.9%

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

   1. metadata-eval 99.9%

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

   2. distribute-neg-frac 99.9%

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

   3. unsub-neg 99.9%

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

9. Simplified 99.9%

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

    1. fma-undefine 99.9%

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

    2. +-commutative 99.9%

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

11. Applied egg-rr 99.9%

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

12. Final simplification 99.9%

    \[\leadsto x + \frac{-1}{\frac{1}{y} + x \cdot 0.5} \]
13. Add Preprocessing

Alternative 6: 62.8% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around inf 61.8%

   \[\leadsto \color{blue}{x} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024170 
(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)))))