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

Percentage Accurate: 99.9% → 99.9%

Time: 6.2s

Alternatives: 5

Speedup: N/A×

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

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

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 99.9%

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

   1. clear-num N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{1 + \frac{x \cdot y}{2}}{y}}}\right)\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 + \frac{x \cdot y}{2}}{y}\right)}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(1 + \frac{x \cdot y}{2}\right), \color{blue}{y}\right)\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{x \cdot y}{2}\right)\right), y\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(x \cdot y\right), 2\right)\right), y\right)\right)\right) \]

   6. *-lowering-*.f64 99.9%

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), 2\right)\right), y\right)\right)\right) \]

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot x + \frac{1}{y}\right)}\right)\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(1, \left(\frac{1}{y} + \color{blue}{\frac{1}{2} \cdot x}\right)\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{1}{y}\right), \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right)\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), \left(\color{blue}{\frac{1}{2}} \cdot x\right)\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), \left(x \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   5. *-lowering-*.f64 99.9%

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 2: 86.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00039:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-22}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-24}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.00039)
   x
   (if (<= x -9.5e-22) (/ -2.0 x) (if (<= x 6.5e-24) (- x y) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.00039) {
		tmp = x;
	} else if (x <= -9.5e-22) {
		tmp = -2.0 / x;
	} else if (x <= 6.5e-24) {
		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 <= (-0.00039d0)) then
        tmp = x
    else if (x <= (-9.5d-22)) then
        tmp = (-2.0d0) / x
    else if (x <= 6.5d-24) 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 <= -0.00039) {
		tmp = x;
	} else if (x <= -9.5e-22) {
		tmp = -2.0 / x;
	} else if (x <= 6.5e-24) {
		tmp = x - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.00039:
		tmp = x
	elif x <= -9.5e-22:
		tmp = -2.0 / x
	elif x <= 6.5e-24:
		tmp = x - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.00039)
		tmp = x;
	elseif (x <= -9.5e-22)
		tmp = Float64(-2.0 / x);
	elseif (x <= 6.5e-24)
		tmp = Float64(x - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.00039)
		tmp = x;
	elseif (x <= -9.5e-22)
		tmp = -2.0 / x;
	elseif (x <= 6.5e-24)
		tmp = x - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.00039], x, If[LessEqual[x, -9.5e-22], N[(-2.0 / x), $MachinePrecision], If[LessEqual[x, 6.5e-24], N[(x - y), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.89999999999999993e-4 or 6.5e-24 < 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

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

   5. Simplified 99.5%

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

   if -3.89999999999999993e-4 < x < -9.4999999999999994e-22

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{2}{x}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 99.7%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(2, \color{blue}{x}\right)\right) \]

   5. Simplified 99.7%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 85.7%

         \[\leadsto \mathsf{/.f64}\left(-2, \color{blue}{x}\right) \]

   8. Simplified 85.7%

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

   if -9.4999999999999994e-22 < x < 6.5e-24

   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

         \[\leadsto x + \left(\mathsf{neg}\left(y\right)\right) \]

      2. unsub-neg N/A

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

      3. --lowering--.f64 72.4%

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{y}\right) \]

   5. Simplified 72.4%

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

Alternative 3: 90.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{2}{x}\\ \mathbf{if}\;y \leq -3.3 \cdot 10^{+171}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+93}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ 2.0 x))))
   (if (<= y -3.3e+171) t_0 (if (<= y 6e+93) (- x y) t_0))))

C

double code(double x, double y) {
	double t_0 = x - (2.0 / x);
	double tmp;
	if (y <= -3.3e+171) {
		tmp = t_0;
	} else if (y <= 6e+93) {
		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 <= (-3.3d+171)) then
        tmp = t_0
    else if (y <= 6d+93) 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 <= -3.3e+171) {
		tmp = t_0;
	} else if (y <= 6e+93) {
		tmp = x - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x - (2.0 / x)
	tmp = 0
	if y <= -3.3e+171:
		tmp = t_0
	elif y <= 6e+93:
		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 <= -3.3e+171)
		tmp = t_0;
	elseif (y <= 6e+93)
		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 <= -3.3e+171)
		tmp = t_0;
	elseif (y <= 6e+93)
		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, -3.3e+171], t$95$0, If[LessEqual[y, 6e+93], N[(x - y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.29999999999999991e171 or 5.99999999999999957e93 < 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

      \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{2}{x}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 92.1%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(2, \color{blue}{x}\right)\right) \]

   5. Simplified 92.1%

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

   if -3.29999999999999991e171 < y < 5.99999999999999957e93

   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

         \[\leadsto x + \left(\mathsf{neg}\left(y\right)\right) \]

      2. unsub-neg N/A

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

      3. --lowering--.f64 96.4%

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{y}\right) \]

   5. Simplified 96.4%

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

Alternative 4: 86.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-24}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7.5e-6) x (if (<= x 6.5e-24) (- x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.5e-6) {
		tmp = x;
	} else if (x <= 6.5e-24) {
		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 <= (-7.5d-6)) then
        tmp = x
    else if (x <= 6.5d-24) 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 <= -7.5e-6) {
		tmp = x;
	} else if (x <= 6.5e-24) {
		tmp = x - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7.5e-6:
		tmp = x
	elif x <= 6.5e-24:
		tmp = x - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7.5e-6)
		tmp = x;
	elseif (x <= 6.5e-24)
		tmp = Float64(x - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.5e-6)
		tmp = x;
	elseif (x <= 6.5e-24)
		tmp = x - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7.5e-6], x, If[LessEqual[x, 6.5e-24], N[(x - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -7.50000000000000019e-6 or 6.5e-24 < 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

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

   5. Simplified 98.0%

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

   if -7.50000000000000019e-6 < x < 6.5e-24

   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

         \[\leadsto x + \left(\mathsf{neg}\left(y\right)\right) \]

      2. unsub-neg N/A

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

      3. --lowering--.f64 70.8%

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{y}\right) \]

   5. Simplified 70.8%

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

Alternative 5: 62.0% 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 99.9%

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

3. Taylor expanded in x around inf

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

5. Simplified 62.8%

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

Reproduce

herbie shell --seed 2024155 
(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)))))