Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2

Percentage Accurate: 99.9% → 99.9%

Time: 9.3s

Alternatives: 8

Speedup: 1.0×

• 25.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot y\right) \cdot \left(1 - y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (* x y) (- 1.0 y)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot y\right) \cdot \left(1 - y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (* x y) (- 1.0 y)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (* y (fma x (- y) x)))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

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

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

   5. sub-neg N/A

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

   6. +-commutative N/A

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

   7. distribute-lft-in N/A

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

   8. *-rgt-identity N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

   10. neg-lowering-neg.f64 99.9

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

   \[\leadsto y \cdot \mathsf{fma}\left(x, -y, x\right) \]
6. Add Preprocessing

Alternative 2: 97.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (* y (* x y)))))
   (if (<= y -1.0) t_0 (if (<= y 1.0) (* x y) t_0))))

C

double code(double x, double y) {
	double t_0 = -(y * (x * y));
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		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 = -(y * (x * y))
    if (y <= (-1.0d0)) then
        tmp = t_0
    else if (y <= 1.0d0) 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 = -(y * (x * y));
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = x * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -(y * (x * y))
	tmp = 0
	if y <= -1.0:
		tmp = t_0
	elif y <= 1.0:
		tmp = x * y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(-Float64(y * Float64(x * y)))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = Float64(x * y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = -(y * (x * y));
	tmp = 0.0;
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = x * y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = (-N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 1.0], N[(x * y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 99.8%

      \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 96.6

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

   5. Simplified 96.6%

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

   if -1 < y < 1

   1. Initial program 100.0%

      \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 97.9

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

   5. Simplified 97.9%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;-y \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;-y \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot y\right) \cdot \left(-x\right)\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y y) (- x))))
   (if (<= y -1.0) t_0 (if (<= y 1.0) (* x y) t_0))))

C

double code(double x, double y) {
	double t_0 = (y * y) * -x;
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		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 = (y * y) * -x
    if (y <= (-1.0d0)) then
        tmp = t_0
    else if (y <= 1.0d0) 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 = (y * y) * -x;
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = x * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * y) * -x
	tmp = 0
	if y <= -1.0:
		tmp = t_0
	elif y <= 1.0:
		tmp = x * y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) * Float64(-x))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = Float64(x * y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * y) * -x;
	tmp = 0.0;
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = x * y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * (-x)), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 1.0], N[(x * y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 99.8%

      \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. neg-lowering-neg.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

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

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

      4. unpow2 N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. *-lowering-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. neg-lowering-neg.f64 86.8

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

   7. Simplified 86.8%

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

   if -1 < y < 1

   1. Initial program 100.0%

      \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 97.9

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

   5. Simplified 97.9%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(-x\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{+66}:\\ \;\;\;\;x \cdot \left(y - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-y \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1e+66) (* x (- y (* y y))) (- (* y (* x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1e+66) {
		tmp = x * (y - (y * y));
	} else {
		tmp = -(y * (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 <= 1d+66) then
        tmp = x * (y - (y * y))
    else
        tmp = -(y * (x * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1e+66) {
		tmp = x * (y - (y * y));
	} else {
		tmp = -(y * (x * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1e+66:
		tmp = x * (y - (y * y))
	else:
		tmp = -(y * (x * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1e+66)
		tmp = Float64(x * Float64(y - Float64(y * y)));
	else
		tmp = Float64(-Float64(y * Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1e+66)
		tmp = x * (y - (y * y));
	else
		tmp = -(y * (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1e+66], N[(x * N[(y - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if y < 9.99999999999999945e65

   1. Initial program 99.9%

      \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. distribute-lft-out-- N/A

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

      3. *-rgt-identity N/A

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

      4. unpow2 N/A

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 98.1

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

   5. Simplified 98.1%

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

   if 9.99999999999999945e65 < y

   1. Initial program 99.9%

      \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 99.9

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

   5. Simplified 99.9%

      \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{\left(-y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+66}:\\ \;\;\;\;x \cdot \left(y - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-y \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-2 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.0) (* x y) (* x (- -2.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.0) {
		tmp = x * y;
	} else {
		tmp = x * (-2.0 - 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 <= 1.0d0) then
        tmp = x * y
    else
        tmp = x * ((-2.0d0) - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.0) {
		tmp = x * y;
	} else {
		tmp = x * (-2.0 - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.0:
		tmp = x * y
	else:
		tmp = x * (-2.0 - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.0)
		tmp = Float64(x * y);
	else
		tmp = Float64(x * Float64(-2.0 - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.0)
		tmp = x * y;
	else
		tmp = x * (-2.0 - y);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 99.9%

      \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 74.6

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

   5. Simplified 74.6%

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

   if 1 < y

   1. Initial program 99.7%

      \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 74.4%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

      16. unsub-neg N/A

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

      17. --lowering--.f64 74.4

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

   7. Simplified 74.4%

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

   8. Taylor expanded in y around inf

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

   10. Simplified 84.3%

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

   11. Taylor expanded in y around 0

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

       1. sub-neg N/A

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

       2. +-commutative N/A

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

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

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

       4. metadata-eval N/A

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

       5. *-commutative N/A

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

       6. mul-1-neg N/A

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

       7. distribute-rgt-neg-in N/A

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

       8. mul-1-neg N/A

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

       9. distribute-lft-out N/A

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

       10. *-lowering-*.f64 N/A

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

       11. mul-1-neg N/A

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

       12. unsub-neg N/A

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

       13. --lowering--.f64 28.9

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

   13. Simplified 28.9%

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

Alternative 6: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot y\right) \cdot \left(1 - y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (* x y) (- 1.0 y)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 7: 56.9% accurate, 2.3× speedup

Math

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

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%

   \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. *-lowering-*.f64 55.3

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

5. Simplified 55.3%

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

Alternative 8: 4.7% accurate, 4.7× 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 Float64(-x)
end

MATLAB

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

Wolfram

code[x_, y_] := (-x)

Derivation

1. Initial program 99.9%

   \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
2. Add Preprocessing

4. Applied egg-rr 87.2%

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

5. Taylor expanded in y around inf

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

   1. /-lowering-/.f64 55.5

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

7. Simplified 55.5%

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

8. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 4.8

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

10. Simplified 4.8%

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

Reproduce

herbie shell --seed 2024199 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))