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

Percentage Accurate: 99.9% → 99.9%

Time: 9.1s

Alternatives: 5

Speedup: 1.0×

• 25.2% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - y \cdot \left(y \cdot x\right)\\ \mathbf{if}\;y \leq -2 \cdot 10^{+94}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 10^{+56}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 0.0 (* y (* y x)))))
   (if (<= y -2e+94) t_0 (if (<= y 1e+56) (* x (* y (- 1.0 y))) t_0))))

C

double code(double x, double y) {
	double t_0 = 0.0 - (y * (y * x));
	double tmp;
	if (y <= -2e+94) {
		tmp = t_0;
	} else if (y <= 1e+56) {
		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 = 0.0d0 - (y * (y * x))
    if (y <= (-2d+94)) then
        tmp = t_0
    else if (y <= 1d+56) 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 = 0.0 - (y * (y * x));
	double tmp;
	if (y <= -2e+94) {
		tmp = t_0;
	} else if (y <= 1e+56) {
		tmp = x * (y * (1.0 - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.0 - (y * (y * x))
	tmp = 0
	if y <= -2e+94:
		tmp = t_0
	elif y <= 1e+56:
		tmp = x * (y * (1.0 - y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.0 - Float64(y * Float64(y * x)))
	tmp = 0.0
	if (y <= -2e+94)
		tmp = t_0;
	elseif (y <= 1e+56)
		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 = 0.0 - (y * (y * x));
	tmp = 0.0;
	if (y <= -2e+94)
		tmp = t_0;
	elseif (y <= 1e+56)
		tmp = x * (y * (1.0 - y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.0 - N[(y * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2e+94], t$95$0, If[LessEqual[y, 1e+56], N[(x * N[(y * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2e94 or 1.00000000000000009e56 < 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 86.6%

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

   3. Simplified 86.6%

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

   6. Applied egg-rr 86.6%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. frac-times N/A

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

      4. times-frac N/A

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

      5. neg-mul-1 N/A

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

      6. remove-double-neg N/A

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

      7. frac-2neg N/A

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

      8. associate-*r/ N/A

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

      9. un-div-inv N/A

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

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

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

      11. clear-num N/A

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

      12. /-lowering-/.f64 N/A

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

      13. distribute-frac-neg N/A

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

      14. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \left(0 - \color{blue}{\frac{\left(-1 + y \cdot y\right) \cdot y}{y + 1}}\right)\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{\left(-1 + y \cdot y\right) \cdot y}{y + 1}\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(0, \left(\frac{y \cdot \left(-1 + y \cdot y\right)}{\color{blue}{y} + 1}\right)\right)\right)\right) \]

      17. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(0, \left(y \cdot \color{blue}{\frac{-1 + y \cdot y}{y + 1}}\right)\right)\right)\right) \]

   8. Applied egg-rr 86.6%

      \[\leadsto \color{blue}{\frac{x}{\frac{1}{0 - y \cdot \left(y + -1\right)}}} \]

   9. Taylor expanded in y around inf

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

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

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(-1, \color{blue}{\left({y}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

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

       3. *-lowering-*.f64 86.6%

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

   11. Simplified 86.6%

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

       1. associate-/r* N/A

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

       2. associate-/r/ N/A

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

       3. associate-*l/ N/A

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

       4. *-commutative N/A

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

       5. clear-num N/A

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

       6. inv-pow N/A

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

       7. sqr-pow N/A

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

       8. pow-prod-down N/A

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

       9. div-inv N/A

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

       10. metadata-eval N/A

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

       11. div-inv N/A

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

       12. metadata-eval N/A

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

       13. swap-sqr N/A

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

       14. metadata-eval N/A

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

       15. *-rgt-identity N/A

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

       16. pow-prod-down N/A

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

       17. sqr-pow N/A

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

       18. inv-pow N/A

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

       19. frac-2neg N/A

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

       20. neg-mul-1 N/A

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

       21. div-inv N/A

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

       22. distribute-frac-neg N/A

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

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

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

       24. clear-num N/A

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

       25. inv-pow N/A

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

       26. sqr-pow N/A

           \[\leadsto \mathsf{neg.f64}\left(\left(\frac{y \cdot x}{{\left(\frac{y}{-1}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{y}{-1}\right)}^{\left(\frac{-1}{2}\right)}}\right)\right) \]

   13. Applied egg-rr 99.9%

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

   if -2e94 < y < 1.00000000000000009e56

   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 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+94}:\\ \;\;\;\;0 - y \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;y \leq 10^{+56}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - y \cdot \left(y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 0.0 (* y (* y x)))))
   (if (<= y -1.0) t_0 (if (<= y 1.0) (* y x) t_0))))

C

double code(double x, double y) {
	double t_0 = 0.0 - (y * (y * x));
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = y * x;
	} 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 = 0.0d0 - (y * (y * x))
    if (y <= (-1.0d0)) then
        tmp = t_0
    else if (y <= 1.0d0) then
        tmp = y * x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.0 - (y * (y * x));
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = y * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.0 - (y * (y * x))
	tmp = 0
	if y <= -1.0:
		tmp = t_0
	elif y <= 1.0:
		tmp = y * x
	else:
		tmp = t_0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.0 - N[(y * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 1.0], N[(y * x), $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 90.2%

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

   3. Simplified 90.2%

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

   6. Applied egg-rr 90.1%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. frac-times N/A

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

      4. times-frac N/A

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

      5. neg-mul-1 N/A

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

      6. remove-double-neg N/A

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

      7. frac-2neg N/A

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

      8. associate-*r/ N/A

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

      9. un-div-inv N/A

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

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

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

      11. clear-num N/A

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

      12. /-lowering-/.f64 N/A

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

      13. distribute-frac-neg N/A

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

      14. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \left(0 - \color{blue}{\frac{\left(-1 + y \cdot y\right) \cdot y}{y + 1}}\right)\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{\left(-1 + y \cdot y\right) \cdot y}{y + 1}\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(0, \left(\frac{y \cdot \left(-1 + y \cdot y\right)}{\color{blue}{y} + 1}\right)\right)\right)\right) \]

      17. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(0, \left(y \cdot \color{blue}{\frac{-1 + y \cdot y}{y + 1}}\right)\right)\right)\right) \]

   8. Applied egg-rr 90.1%

      \[\leadsto \color{blue}{\frac{x}{\frac{1}{0 - y \cdot \left(y + -1\right)}}} \]

   9. Taylor expanded in y around inf

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

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

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(-1, \color{blue}{\left({y}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

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

       3. *-lowering-*.f64 89.2%

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

   11. Simplified 89.2%

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

       1. associate-/r* N/A

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

       2. associate-/r/ N/A

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

       3. associate-*l/ N/A

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

       4. *-commutative N/A

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

       5. clear-num N/A

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

       6. inv-pow N/A

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

       7. sqr-pow N/A

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

       8. pow-prod-down N/A

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

       9. div-inv N/A

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

       10. metadata-eval N/A

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

       11. div-inv N/A

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

       12. metadata-eval N/A

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

       13. swap-sqr N/A

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

       14. metadata-eval N/A

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

       15. *-rgt-identity N/A

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

       16. pow-prod-down N/A

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

       17. sqr-pow N/A

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

       18. inv-pow N/A

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

       19. frac-2neg N/A

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

       20. neg-mul-1 N/A

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

       21. div-inv N/A

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

       22. distribute-frac-neg N/A

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

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

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

       24. clear-num N/A

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

       25. inv-pow N/A

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

       26. sqr-pow N/A

           \[\leadsto \mathsf{neg.f64}\left(\left(\frac{y \cdot x}{{\left(\frac{y}{-1}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{y}{-1}\right)}^{\left(\frac{-1}{2}\right)}}\right)\right) \]

   13. Applied egg-rr 98.9%

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

   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 96.3%

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

   7. Simplified 96.3%

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

4. Final simplification 97.6%

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

Alternative 3: 63.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;0 - y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.0) (* y x) (- 0.0 (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.0) {
		tmp = y * x;
	} else {
		tmp = 0.0 - (y * x);
	}
	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 = y * x
    else
        tmp = 0.0d0 - (y * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.0) {
		tmp = y * x;
	} else {
		tmp = 0.0 - (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.0:
		tmp = y * x
	else:
		tmp = 0.0 - (y * x)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[y, 1.0], N[(y * x), $MachinePrecision], N[(0.0 - N[(y * x), $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. 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 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 73.7%

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

   7. Simplified 73.7%

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

   if 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 87.6%

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

   3. Simplified 87.6%

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

   6. Applied egg-rr 87.5%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. frac-times N/A

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

      4. times-frac N/A

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

      5. neg-mul-1 N/A

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

      6. remove-double-neg N/A

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

      7. frac-2neg N/A

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

      8. associate-*r/ N/A

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

      9. un-div-inv N/A

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

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

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

      11. clear-num N/A

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

      12. /-lowering-/.f64 N/A

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

      13. distribute-frac-neg N/A

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

      14. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \left(0 - \color{blue}{\frac{\left(-1 + y \cdot y\right) \cdot y}{y + 1}}\right)\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{\left(-1 + y \cdot y\right) \cdot y}{y + 1}\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(0, \left(\frac{y \cdot \left(-1 + y \cdot y\right)}{\color{blue}{y} + 1}\right)\right)\right)\right) \]

      17. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(0, \left(y \cdot \color{blue}{\frac{-1 + y \cdot y}{y + 1}}\right)\right)\right)\right) \]

   8. Applied egg-rr 87.6%

      \[\leadsto \color{blue}{\frac{x}{\frac{1}{0 - y \cdot \left(y + -1\right)}}} \]

   9. Taylor expanded in y around 0

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

       1. /-lowering-/.f64 0.8%

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

   11. Simplified 0.8%

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

       1. inv-pow N/A

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

       2. sqr-pow N/A

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

       3. pow-prod-down N/A

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

       4. *-rgt-identity N/A

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

       5. metadata-eval N/A

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

       6. swap-sqr N/A

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

       7. metadata-eval N/A

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

       8. div-inv N/A

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

       9. metadata-eval N/A

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

       10. div-inv N/A

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

       11. pow-prod-down N/A

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

       12. sqr-pow N/A

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

       13. inv-pow N/A

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

       14. clear-num N/A

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

       15. clear-num N/A

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

       16. associate-/r/ N/A

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

       17. frac-2neg N/A

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

       18. metadata-eval N/A

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

       19. remove-double-div N/A

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

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

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

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

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

       22. *-lowering-*.f64 24.6%

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

   13. Applied egg-rr 24.6%

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

4. Final simplification 62.0%

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

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(y * N[(x * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $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 95.2%

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

3. Simplified 95.2%

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

6. Applied egg-rr 95.1%

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

   1. *-commutative N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. associate-/r/ N/A

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

   5. *-commutative N/A

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

   6. div-inv N/A

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

   7. times-frac N/A

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

   8. +-commutative N/A

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

   9. metadata-eval N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. flip-- N/A

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

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

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

   14. sub-neg N/A

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

   15. metadata-eval N/A

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

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

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

   17. /-lowering-/.f64 N/A

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

   18. /-lowering-/.f64 N/A

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

   19. /-lowering-/.f64 99.7%

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

8. Applied egg-rr 99.7%

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

   1. associate-/r/ N/A

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

   2. clear-num N/A

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

   3. clear-num N/A

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

   4. associate-*l/ N/A

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

   5. *-commutative N/A

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

   6. *-rgt-identity N/A

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

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

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

   8. /-lowering-/.f64 99.7%

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

10. Applied egg-rr 99.7%

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

11. Taylor expanded in y around 0

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

13. Simplified 99.9%

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

Alternative 5: 56.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(y * x), $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 95.2%

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

3. Simplified 95.2%

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

5. Taylor expanded in y around 0

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

   1. *-lowering-*.f64 56.3%

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

7. Simplified 56.3%

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

8. Final simplification 56.3%

   \[\leadsto y \cdot x \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024158 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))