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

Percentage Accurate: 99.9% → 99.9%

Time: 6.6s

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, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(x - N[Power[N[(0.5 * x + N[Power[y, -1.0], $MachinePrecision]), $MachinePrecision], -1.0], $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. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 99.9

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-/.f64 N/A

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

   8. clear-num N/A

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

   9. associate-/r/ N/A

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

   10. lower-fma.f64 N/A

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

   11. metadata-eval 99.9

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-/.f64 100.0

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

7. Applied rewrites 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 90.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+122} \lor \neg \left(y \leq 6.5 \cdot 10^{+154}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.2e+122) (not (<= y 6.5e+154))) (- x (/ 2.0 x)) (- x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.2e+122) || !(y <= 6.5e+154)) {
		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 <= (-2.2d+122)) .or. (.not. (y <= 6.5d+154))) 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 <= -2.2e+122) || !(y <= 6.5e+154)) {
		tmp = x - (2.0 / x);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.2e+122) or not (y <= 6.5e+154):
		tmp = x - (2.0 / x)
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.2e+122) || !(y <= 6.5e+154))
		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 <= -2.2e+122) || ~((y <= 6.5e+154)))
		tmp = x - (2.0 / x);
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.2e+122], N[Not[LessEqual[y, 6.5e+154]], $MachinePrecision]], N[(x - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(x - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.1999999999999999e122 or 6.5000000000000005e154 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 88.1

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

   5. Applied rewrites 88.1%

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

   if -2.1999999999999999e122 < y < 6.5000000000000005e154

   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

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

      2. unsub-neg N/A

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

      3. lower--.f64 93.3

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

   5. Applied rewrites 93.3%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+122} \lor \neg \left(y \leq 6.5 \cdot 10^{+154}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 1.3× speedup

Math

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

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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. div-inv N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

Alternative 4: 78.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+212} \lor \neg \left(y \leq 6.5 \cdot 10^{+239}\right):\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -7e+212) (not (<= y 6.5e+239))) (/ -2.0 x) (- x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -7e+212) || !(y <= 6.5e+239)) {
		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 <= (-7d+212)) .or. (.not. (y <= 6.5d+239))) 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 <= -7e+212) || !(y <= 6.5e+239)) {
		tmp = -2.0 / x;
	} else {
		tmp = x - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -7e+212) or not (y <= 6.5e+239):
		tmp = -2.0 / x
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -7e+212) || !(y <= 6.5e+239))
		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 <= -7e+212) || ~((y <= 6.5e+239)))
		tmp = -2.0 / x;
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -7e+212], N[Not[LessEqual[y, 6.5e+239]], $MachinePrecision]], N[(-2.0 / x), $MachinePrecision], N[(x - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.99999999999999974e212 or 6.5e239 < y

   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. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. distribute-neg-frac N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 84.2

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.6%

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

   if -6.99999999999999974e212 < y < 6.5e239

   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

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

      2. unsub-neg N/A

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

      3. lower--.f64 85.2

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

   5. Applied rewrites 85.2%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+212} \lor \neg \left(y \leq 6.5 \cdot 10^{+239}\right):\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.6% 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. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 74.7

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

5. Applied rewrites 74.7%

   \[\leadsto \color{blue}{x - y} \]
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 27.3

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

5. Applied rewrites 27.3%

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

Reproduce

herbie shell --seed 2024324 
(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)))))