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

Percentage Accurate: 99.9% → 99.9%

Time: 6.0s

Alternatives: 6

Speedup: 1.0×

• 25.1% 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, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+49} \lor \neg \left(y \leq 1.05 \cdot 10^{+71}\right):\\ \;\;\;\;y \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5e+49) (not (<= y 1.05e+71)))
   (* y (* x (- y)))
   (* x (* y (- 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5e+49) || !(y <= 1.05e+71)) {
		tmp = y * (x * -y);
	} else {
		tmp = x * (y * (1.0 - 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 <= (-5d+49)) .or. (.not. (y <= 1.05d+71))) then
        tmp = y * (x * -y)
    else
        tmp = x * (y * (1.0d0 - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -5e+49) || !(y <= 1.05e+71)) {
		tmp = y * (x * -y);
	} else {
		tmp = x * (y * (1.0 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -5e+49) or not (y <= 1.05e+71):
		tmp = y * (x * -y)
	else:
		tmp = x * (y * (1.0 - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5e+49) || !(y <= 1.05e+71))
		tmp = Float64(y * Float64(x * Float64(-y)));
	else
		tmp = Float64(x * Float64(y * Float64(1.0 - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -5e+49) || ~((y <= 1.05e+71)))
		tmp = y * (x * -y);
	else
		tmp = x * (y * (1.0 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5e+49], N[Not[LessEqual[y, 1.05e+71]], $MachinePrecision]], N[(y * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.0000000000000004e49 or 1.04999999999999995e71 < y

   1. Initial program 99.8%

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

      1. associate-*l* 85.6%

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

   3. Simplified 85.6%

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

      1. associate-*r* 99.8%

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

      2. flip-- 85.7%

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

      3. associate-*r/ 77.8%

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

      4. metadata-eval 77.8%

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

      5. pow2 77.8%

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

      6. +-commutative 77.8%

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

   5. Applied egg-rr 77.8%

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

      1. associate-/l* 85.7%

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

   7. Simplified 85.7%

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

      1. div-inv 85.7%

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

      2. clear-num 85.7%

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

      3. metadata-eval 85.7%

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

      4. unpow2 85.7%

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

      5. +-commutative 85.7%

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

      6. flip-- 99.8%

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

      7. associate-*r* 85.6%

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

      8. *-commutative 85.6%

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

      9. associate-*r* 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in y around inf 99.8%

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

       1. neg-mul-1 99.8%

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

       2. distribute-rgt-neg-in 99.8%

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

   12. Simplified 99.8%

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

   if -5.0000000000000004e49 < y < 1.04999999999999995e71

   1. Initial program 99.9%

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

      1. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+49} \lor \neg \left(y \leq 1.05 \cdot 10^{+71}\right):\\ \;\;\;\;y \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \end{array} \]

Alternative 2: 98.0% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0))) (* y (* x (- y))) (* x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = y * (x * -y);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = y * (x * -y)
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 1.0):
		tmp = y * (x * -y)
	else:
		tmp = x * y
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(y * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]

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* 88.8%

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

   3. Simplified 88.8%

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

      1. associate-*r* 99.8%

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

      2. flip-- 88.7%

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

      3. associate-*r/ 81.9%

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

      4. metadata-eval 81.9%

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

      5. pow2 81.9%

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

      6. +-commutative 81.9%

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

   5. Applied egg-rr 81.9%

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

      1. associate-/l* 88.7%

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

   7. Simplified 88.7%

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

      1. div-inv 88.7%

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

      2. clear-num 88.7%

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

      3. metadata-eval 88.7%

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

      4. unpow2 88.7%

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

      5. +-commutative 88.7%

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

      6. flip-- 99.8%

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

      7. associate-*r* 88.8%

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

      8. *-commutative 88.8%

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

      9. associate-*r* 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in y around inf 97.9%

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

       1. neg-mul-1 97.9%

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

       2. distribute-rgt-neg-in 97.9%

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

   12. Simplified 97.9%

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

   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* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 97.5%

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

4. Final simplification 97.7%

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

Alternative 3: 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* 94.8%

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

3. Simplified 94.8%

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

   1. associate-*r* 99.9%

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

   2. flip-- 94.7%

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

   3. associate-*r/ 91.6%

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

   4. metadata-eval 91.6%

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

   5. pow2 91.6%

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

   6. +-commutative 91.6%

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

5. Applied egg-rr 91.6%

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

   1. associate-/l* 94.7%

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

7. Simplified 94.7%

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

   1. div-inv 94.7%

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

   2. clear-num 94.7%

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

   3. metadata-eval 94.7%

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

   4. unpow2 94.7%

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

   5. +-commutative 94.7%

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

   6. flip-- 99.9%

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

   7. associate-*r* 94.8%

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

   8. *-commutative 94.8%

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

   9. associate-*r* 99.9%

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

9. Applied egg-rr 99.9%

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

10. Final simplification 99.9%

    \[\leadsto y \cdot \left(x \cdot \left(1 - y\right)\right) \]

Alternative 4: 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. Final simplification 99.9%

   \[\leadsto \left(1 - y\right) \cdot \left(x \cdot y\right) \]

Alternative 5: 63.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.0d0) then
        tmp = x * y
    else
        tmp = x * -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 * -y;
	}
	return tmp;
}

Python

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

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.0)
		tmp = Float64(x * y);
	else
		tmp = Float64(x * Float64(-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 * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.0], N[(x * y), $MachinePrecision], N[(x * (-y)), $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* 96.1%

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

   3. Simplified 96.1%

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

   4. Taylor expanded in y around 0 75.2%

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

   if 1 < y

   1. Initial program 99.9%

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

      1. associate-*l* 90.2%

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

   3. Simplified 90.2%

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

      1. associate-*r* 99.9%

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

      2. flip-- 90.2%

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

      3. associate-*r/ 84.1%

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

      4. metadata-eval 84.1%

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

      5. pow2 84.1%

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

      6. +-commutative 84.1%

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

   5. Applied egg-rr 84.1%

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

      1. associate-*l* 77.5%

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

      2. associate-/l* 80.4%

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

      3. sub-neg 80.4%

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

      4. distribute-lft-in 80.4%

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

      5. *-rgt-identity 80.4%

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

      6. distribute-rgt-neg-in 80.4%

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

      7. unpow2 80.4%

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

      8. cube-mult 80.4%

         \[\leadsto \frac{x}{\frac{y + 1}{y + \left(-\color{blue}{{y}^{3}}\right)}} \]

      9. unsub-neg 80.4%

         \[\leadsto \frac{x}{\frac{y + 1}{\color{blue}{y - {y}^{3}}}} \]

   7. Simplified 80.4%

      \[\leadsto \color{blue}{\frac{x}{\frac{y + 1}{y - {y}^{3}}}} \]

   8. Taylor expanded in y around 0 0.7%

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

      1. frac-2neg 0.7%

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

      2. div-inv 0.7%

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

      3. neg-mul-1 0.7%

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

      4. div-inv 0.7%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 55.3%

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

      7. frac-times 55.3%

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

      8. metadata-eval 55.3%

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

      9. metadata-eval 55.3%

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

      10. frac-times 55.3%

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

      11. sqrt-unprod 35.6%

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

      12. add-sqr-sqrt 35.6%

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

      13. remove-double-div 35.6%

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

   10. Applied egg-rr 35.6%

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

4. Final simplification 66.2%

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

Alternative 6: 56.3% 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* 94.8%

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

3. Simplified 94.8%

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

4. Taylor expanded in y around 0 58.3%

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

5. Final simplification 58.3%

   \[\leadsto x \cdot y \]

Reproduce

herbie shell --seed 2023318 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))