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

Percentage Accurate: 99.9% → 99.9%

Time: 31.2s

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, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift-/.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. div-inv N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 2: 90.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{2}{x}\\ \mathbf{if}\;y \leq -1 \cdot 10^{+79}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.44 \cdot 10^{+159}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ 2.0 x))))
   (if (<= y -1e+79) t_0 (if (<= y 1.44e+159) (- x y) t_0))))

C

double code(double x, double y) {
	double t_0 = x - (2.0 / x);
	double tmp;
	if (y <= -1e+79) {
		tmp = t_0;
	} else if (y <= 1.44e+159) {
		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 <= (-1d+79)) then
        tmp = t_0
    else if (y <= 1.44d+159) 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 <= -1e+79) {
		tmp = t_0;
	} else if (y <= 1.44e+159) {
		tmp = x - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x - (2.0 / x)
	tmp = 0
	if y <= -1e+79:
		tmp = t_0
	elif y <= 1.44e+159:
		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 <= -1e+79)
		tmp = t_0;
	elseif (y <= 1.44e+159)
		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 <= -1e+79)
		tmp = t_0;
	elseif (y <= 1.44e+159)
		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, -1e+79], t$95$0, If[LessEqual[y, 1.44e+159], N[(x - y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.99999999999999967e78 or 1.43999999999999995e159 < y

   1. Initial program 99.8%

      \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
   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 85.3

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

   5. Simplified 85.3%

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

   if -9.99999999999999967e78 < y < 1.43999999999999995e159

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-fma.f64 N/A

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

      12. distribute-lft-neg-in N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 84.6

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

   5. Simplified 84.6%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot y} \]
   7. 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 95.3

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

   8. Simplified 95.3%

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

Alternative 3: 86.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5:\\ \;\;\;\;x + -1\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-6}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.5) (+ x -1.0) (if (<= x 5.4e-6) (- x y) (+ x -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.5) {
		tmp = x + -1.0;
	} else if (x <= 5.4e-6) {
		tmp = x - y;
	} else {
		tmp = x + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.5d0)) then
        tmp = x + (-1.0d0)
    else if (x <= 5.4d-6) then
        tmp = x - y
    else
        tmp = x + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.5) {
		tmp = x + -1.0;
	} else if (x <= 5.4e-6) {
		tmp = x - y;
	} else {
		tmp = x + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.5:
		tmp = x + -1.0
	elif x <= 5.4e-6:
		tmp = x - y
	else:
		tmp = x + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.5)
		tmp = Float64(x + -1.0);
	elseif (x <= 5.4e-6)
		tmp = Float64(x - y);
	else
		tmp = Float64(x + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.5)
		tmp = x + -1.0;
	elseif (x <= 5.4e-6)
		tmp = x - y;
	else
		tmp = x + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.5], N[(x + -1.0), $MachinePrecision], If[LessEqual[x, 5.4e-6], N[(x - y), $MachinePrecision], N[(x + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5 or 5.39999999999999997e-6 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

         \[\leadsto x + \color{blue}{-1} \]

      3. lower-+.f64 95.2

         \[\leadsto \color{blue}{x + -1} \]

   5. Simplified 95.2%

      \[\leadsto \color{blue}{x + -1} \]

   if -2.5 < x < 5.39999999999999997e-6

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-fma.f64 N/A

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

      12. distribute-lft-neg-in N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 74.6

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

   5. Simplified 74.6%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot y} \]
   7. 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 80.0

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

   8. Simplified 80.0%

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

Alternative 4: 73.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-68}:\\ \;\;\;\;x + -1\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-26}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;x + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.4e-68) (+ x -1.0) (if (<= x 4.7e-26) (- y) (+ x -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.4e-68) {
		tmp = x + -1.0;
	} else if (x <= 4.7e-26) {
		tmp = -y;
	} else {
		tmp = x + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4.4d-68)) then
        tmp = x + (-1.0d0)
    else if (x <= 4.7d-26) then
        tmp = -y
    else
        tmp = x + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.4e-68) {
		tmp = x + -1.0;
	} else if (x <= 4.7e-26) {
		tmp = -y;
	} else {
		tmp = x + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.4e-68:
		tmp = x + -1.0
	elif x <= 4.7e-26:
		tmp = -y
	else:
		tmp = x + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.4e-68)
		tmp = Float64(x + -1.0);
	elseif (x <= 4.7e-26)
		tmp = Float64(-y);
	else
		tmp = Float64(x + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.4e-68)
		tmp = x + -1.0;
	elseif (x <= 4.7e-26)
		tmp = -y;
	else
		tmp = x + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.4e-68], N[(x + -1.0), $MachinePrecision], If[LessEqual[x, 4.7e-26], (-y), N[(x + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.40000000000000005e-68 or 4.69999999999999989e-26 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

         \[\leadsto x + \color{blue}{-1} \]

      3. lower-+.f64 85.6

         \[\leadsto \color{blue}{x + -1} \]

   5. Simplified 85.6%

      \[\leadsto \color{blue}{x + -1} \]

   if -4.40000000000000005e-68 < x < 4.69999999999999989e-26

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-fma.f64 N/A

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

      12. distribute-lft-neg-in N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 75.6

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

   5. Simplified 75.6%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 62.9

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

   8. Simplified 62.9%

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

Alternative 5: 26.3% 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 y around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. associate-*r* N/A

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

   5. associate-*r* N/A

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

   6. unpow2 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. metadata-eval N/A

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

   10. distribute-lft-neg-in N/A

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

   11. lower-fma.f64 N/A

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

   12. distribute-lft-neg-in N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 62.4

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

5. Simplified 62.4%

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

6. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 30.7

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

8. Simplified 30.7%

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

Alternative 6: 3.7% accurate, 34.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -2.0)

C

double code(double x, double y) {
	return -2.0;
}

Fortran

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

Java

public static double code(double x, double y) {
	return -2.0;
}

Python

def code(x, y):
	return -2.0

Julia

function code(x, y)
	return -2.0
end

MATLAB

function tmp = code(x, y)
	tmp = -2.0;
end

Wolfram

code[x_, y_] := -2.0

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around inf

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

5. Simplified 47.8%

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

6. Taylor expanded in x around 0

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

8. Simplified 4.0%

   \[\leadsto \color{blue}{-2} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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)))))