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

Percentage Accurate: 99.9% → 99.9%

Time: 5.9s

Alternatives: 6

Speedup: 1.0×

• 25.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+29} \lor \neg \left(y \leq 2000000000000\right):\\ \;\;\;\;y \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -6.2e+29) (not (<= y 2000000000000.0)))
   (* y (* x (- y)))
   (* x (* y (- 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -6.2e+29) || !(y <= 2000000000000.0)) {
		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 <= (-6.2d+29)) .or. (.not. (y <= 2000000000000.0d0))) 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 <= -6.2e+29) || !(y <= 2000000000000.0)) {
		tmp = y * (x * -y);
	} else {
		tmp = x * (y * (1.0 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -6.2e+29) or not (y <= 2000000000000.0):
		tmp = y * (x * -y)
	else:
		tmp = x * (y * (1.0 - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -6.2e+29) || !(y <= 2000000000000.0))
		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 <= -6.2e+29) || ~((y <= 2000000000000.0)))
		tmp = y * (x * -y);
	else
		tmp = x * (y * (1.0 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -6.2e+29], N[Not[LessEqual[y, 2000000000000.0]], $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 < -6.1999999999999998e29 or 2e12 < y

   1. Initial program 99.9%

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

      1. associate-*l* 87.1%

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

   3. Simplified 87.1%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot \left(1 - y\right)\right)} \]
   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-- 87.1%

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

      3. associate-*r/ 80.4%

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

      4. metadata-eval 80.4%

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

      5. pow2 80.4%

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

      6. +-commutative 80.4%

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

   5. Applied egg-rr 80.4%

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

      1. associate-/l* 87.0%

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

   7. Simplified 87.0%

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

   8. Taylor expanded in y around inf 99.8%

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

      1. div-inv 99.8%

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

      2. frac-2neg 99.8%

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

      3. metadata-eval 99.8%

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

      4. remove-double-div 99.9%

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

   10. Applied egg-rr 99.9%

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

   if -6.1999999999999998e29 < y < 2e12

   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%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+29} \lor \neg \left(y \leq 2000000000000\right):\\ \;\;\;\;y \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \end{array} \]

Alternative 3: 92.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.0))
		tmp = Float64(x * Float64(y * 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 = x * (y * -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[(x * N[(y * (-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* 87.6%

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

   3. Simplified 87.6%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot \left(1 - y\right)\right)} \]
   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-- 87.6%

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

      3. associate-*r/ 81.2%

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

      4. metadata-eval 81.2%

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

      5. pow2 81.2%

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

      6. +-commutative 81.2%

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

   5. Applied egg-rr 81.2%

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

      1. associate-/l* 87.5%

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

   7. Simplified 87.5%

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

   8. Taylor expanded in y around inf 98.7%

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

      1. div-inv 98.7%

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

      2. clear-num 98.8%

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

      3. associate-*l* 86.5%

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

      4. clear-num 86.4%

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

      5. frac-2neg 86.4%

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

      6. metadata-eval 86.4%

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

      7. remove-double-div 86.5%

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

   10. Applied egg-rr 86.5%

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

   if -1 < y < 1

   1. Initial program 100.0%

      \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
   2. 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 99.7%

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

4. Final simplification 92.3%

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

Alternative 4: 97.8% 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* 87.6%

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

   3. Simplified 87.6%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot \left(1 - y\right)\right)} \]
   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-- 87.6%

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

      3. associate-*r/ 81.2%

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

      4. metadata-eval 81.2%

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

      5. pow2 81.2%

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

      6. +-commutative 81.2%

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

   5. Applied egg-rr 81.2%

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

      1. associate-/l* 87.5%

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

   7. Simplified 87.5%

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

   8. Taylor expanded in y around inf 98.7%

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

      1. div-inv 98.7%

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

      2. frac-2neg 98.7%

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

      3. metadata-eval 98.7%

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

      4. remove-double-div 98.8%

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

   10. Applied egg-rr 98.8%

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

   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* 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 99.7%

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

4. Final simplification 99.2%

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

Alternative 5: 63.6% 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* 96.0%

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

   3. Simplified 96.0%

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

   4. Taylor expanded in y around 0 75.9%

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

   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* 84.9%

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

   3. Simplified 84.9%

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

   4. Taylor expanded in y around 0 57.0%

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

      1. +-commutative 57.0%

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

      2. mul-1-neg 57.0%

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

      3. unsub-neg 57.0%

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

      4. distribute-lft-out-- 84.9%

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

   6. Simplified 84.9%

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

      1. sub-neg 84.9%

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

      2. sub-neg 84.9%

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

      3. *-un-lft-identity 84.9%

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

      4. unpow2 84.9%

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

      5. distribute-rgt-out-- 84.9%

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

      6. associate-*r* 99.8%

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

      7. flip-- 84.9%

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

      8. metadata-eval 84.9%

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

      9. unpow2 84.9%

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

      10. +-commutative 84.9%

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

      11. clear-num 84.9%

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

      12. div-inv 84.8%

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

      13. *-commutative 84.8%

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

      14. associate-/l* 84.9%

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

      15. clear-num 84.9%

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

      16. metadata-eval 84.9%

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

      17. unpow2 84.9%

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

      18. +-commutative 84.9%

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

      19. flip-- 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in y around 0 0.7%

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

       1. frac-2neg 0.7%

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

       2. metadata-eval 0.7%

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

       3. associate-/r/ 0.7%

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

       4. div-inv 0.7%

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

       5. metadata-eval 0.7%

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

       6. *-commutative 0.7%

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

       7. neg-mul-1 0.7%

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

       8. add-sqr-sqrt 0.0%

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

       9. sqrt-unprod 48.0%

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

       10. sqr-neg 48.0%

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

       11. sqrt-unprod 32.1%

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

       12. add-sqr-sqrt 32.1%

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

   11. Applied egg-rr 32.1%

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

4. Final simplification 64.2%

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

Alternative 6: 56.5% 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* 93.1%

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

3. Simplified 93.1%

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

4. Taylor expanded in y around 0 55.9%

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

5. Final simplification 55.9%

   \[\leadsto x \cdot y \]

Reproduce

herbie shell --seed 2023310 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))