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

Percentage Accurate: 99.9% → 99.9%

Time: 8.4s

Alternatives: 4

Speedup: 1.0×

• 26.1% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot \left(0 - y\right)\right)\\ \mathbf{if}\;y \leq -2.5 \cdot 10^{+72}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+60}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x (- 0.0 y)))))
   (if (<= y -2.5e+72) t_0 (if (<= y 1.15e+60) (* x (* y (- 1.0 y))) t_0))))

C

double code(double x, double y) {
	double t_0 = y * (x * (0.0 - y));
	double tmp;
	if (y <= -2.5e+72) {
		tmp = t_0;
	} else if (y <= 1.15e+60) {
		tmp = x * (y * (1.0 - 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 * (0.0d0 - y))
    if (y <= (-2.5d+72)) then
        tmp = t_0
    else if (y <= 1.15d+60) then
        tmp = x * (y * (1.0d0 - 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 * (0.0 - y));
	double tmp;
	if (y <= -2.5e+72) {
		tmp = t_0;
	} else if (y <= 1.15e+60) {
		tmp = x * (y * (1.0 - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (x * (0.0 - y))
	tmp = 0
	if y <= -2.5e+72:
		tmp = t_0
	elif y <= 1.15e+60:
		tmp = x * (y * (1.0 - y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * Float64(0.0 - y)))
	tmp = 0.0
	if (y <= -2.5e+72)
		tmp = t_0;
	elseif (y <= 1.15e+60)
		tmp = Float64(x * Float64(y * Float64(1.0 - y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (x * (0.0 - y));
	tmp = 0.0;
	if (y <= -2.5e+72)
		tmp = t_0;
	elseif (y <= 1.15e+60)
		tmp = x * (y * (1.0 - y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * N[(0.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.5e+72], t$95$0, If[LessEqual[y, 1.15e+60], N[(x * N[(y * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.49999999999999996e72 or 1.15000000000000008e60 < y

   1. Initial program 99.9%

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

      1. associate-*l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot \left(1 - y\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 - y\right)}\right)\right) \]

      4. --lowering--.f64 83.9%

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

   3. Simplified 83.9%

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

   6. Applied egg-rr 83.8%

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

      1. frac-times N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. frac-2neg N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. neg-mul-1 N/A

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

      9. times-frac N/A

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

      10. clear-num N/A

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

      11. inv-pow N/A

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

      12. inv-pow N/A

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

      13. pow-prod-down N/A

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

      14. *-commutative N/A

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

      15. inv-pow N/A

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

      16. /-lowering-/.f64 N/A

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

      17. associate-*r/ N/A

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \color{blue}{\left(\frac{-1}{y}\right)}\right) \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 99.8%

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

   11. Simplified 99.8%

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

       1. /-rgt-identity N/A

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

       2. clear-num N/A

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

       3. associate-/l/ N/A

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

       4. associate-/r* N/A

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

       5. frac-2neg N/A

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

       6. metadata-eval N/A

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

       7. remove-double-div N/A

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

       8. distribute-neg-frac N/A

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

       9. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{y}{\frac{1}{x \cdot y}}\right)\right) \]

       10. div-inv N/A

           \[\leadsto \mathsf{neg.f64}\left(\left(y \cdot \frac{1}{\frac{1}{x \cdot y}}\right)\right) \]

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

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{\frac{1}{x \cdot y}}\right)\right)\right) \]

       12. clear-num N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{x \cdot y}{1}\right)\right)\right) \]

       13. /-rgt-identity N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(y, \left(x \cdot y\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(y, \left(y \cdot x\right)\right)\right) \]

       15. *-lowering-*.f64 99.9%

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, x\right)\right)\right) \]

   13. Applied egg-rr 99.9%

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

   if -2.49999999999999996e72 < y < 1.15000000000000008e60

   1. Initial program 99.9%

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

      1. associate-*l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot \left(1 - y\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 - y\right)}\right)\right) \]

      4. --lowering--.f64 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot \left(1 - y\right)\right)} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+72}:\\ \;\;\;\;y \cdot \left(x \cdot \left(0 - y\right)\right)\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+60}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \left(0 - y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot \left(0 - y\right)\right)\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x (- 0.0 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 * (0.0 - 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 * (0.0d0 - 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 * (0.0 - 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 * (0.0 - 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(y * Float64(x * Float64(0.0 - 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 * (0.0 - 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 * N[(0.0 - y), $MachinePrecision]), $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. Step-by-step derivation

      1. associate-*l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot \left(1 - y\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 - y\right)}\right)\right) \]

      4. --lowering--.f64 87.3%

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

   3. Simplified 87.3%

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

   6. Applied egg-rr 87.2%

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

      1. frac-times N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. frac-2neg N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. neg-mul-1 N/A

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

      9. times-frac N/A

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

      10. clear-num N/A

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

      11. inv-pow N/A

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

      12. inv-pow N/A

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

      13. pow-prod-down N/A

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

      14. *-commutative N/A

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

      15. inv-pow N/A

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

      16. /-lowering-/.f64 N/A

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

      17. associate-*r/ N/A

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \color{blue}{\left(\frac{-1}{y}\right)}\right) \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 97.9%

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

   11. Simplified 97.9%

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

       1. /-rgt-identity N/A

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

       2. clear-num N/A

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

       3. associate-/l/ N/A

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

       4. associate-/r* N/A

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

       5. frac-2neg N/A

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

       6. metadata-eval N/A

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

       7. remove-double-div N/A

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

       8. distribute-neg-frac N/A

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

       9. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{y}{\frac{1}{x \cdot y}}\right)\right) \]

       10. div-inv N/A

           \[\leadsto \mathsf{neg.f64}\left(\left(y \cdot \frac{1}{\frac{1}{x \cdot y}}\right)\right) \]

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

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{\frac{1}{x \cdot y}}\right)\right)\right) \]

       12. clear-num N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{x \cdot y}{1}\right)\right)\right) \]

       13. /-rgt-identity N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(y, \left(x \cdot y\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(y, \left(y \cdot x\right)\right)\right) \]

       15. *-lowering-*.f64 98.0%

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, x\right)\right)\right) \]

   13. Applied egg-rr 98.0%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

      1. associate-*l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot \left(1 - y\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 - y\right)}\right)\right) \]

      4. --lowering--.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 96.7%

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

   7. Simplified 96.7%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;y \cdot \left(x \cdot \left(0 - y\right)\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \left(0 - y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 4: 56.8% 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. Step-by-step derivation

   1. associate-*l* N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot \left(1 - y\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 - y\right)}\right)\right) \]

   4. --lowering--.f64 93.4%

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

3. Simplified 93.4%

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

5. Taylor expanded in y around 0

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

   1. *-lowering-*.f64 56.0%

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

7. Simplified 56.0%

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

Reproduce

herbie shell --seed 2024152 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))