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

Percentage Accurate: 99.9% → 99.9%

Time: 11.8s

Alternatives: 6

Speedup: 1.0×

• 24.9% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+134} \lor \neg \left(y \leq 2 \cdot 10^{+144}\right):\\ \;\;\;\;0 - y \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5e+134) (not (<= y 2e+144)))
   (- 0.0 (* y (* x y)))
   (* x (* y (- 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5e+134) || !(y <= 2e+144)) {
		tmp = 0.0 - (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+134)) .or. (.not. (y <= 2d+144))) then
        tmp = 0.0d0 - (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+134) || !(y <= 2e+144)) {
		tmp = 0.0 - (y * (x * y));
	} else {
		tmp = x * (y * (1.0 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -5e+134) or not (y <= 2e+144):
		tmp = 0.0 - (y * (x * y))
	else:
		tmp = x * (y * (1.0 - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5e+134) || !(y <= 2e+144))
		tmp = Float64(0.0 - Float64(y * Float64(x * 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+134) || ~((y <= 2e+144)))
		tmp = 0.0 - (y * (x * y));
	else
		tmp = x * (y * (1.0 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5e+134], N[Not[LessEqual[y, 2e+144]], $MachinePrecision]], N[(0.0 - N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.99999999999999981e134 or 2.00000000000000005e144 < 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 79.3%

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

   3. Simplified 79.3%

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

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

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

      4. *-lft-identity N/A

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

      5. sub-neg N/A

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

      6. --lowering--.f64 N/A

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

      7. *-lowering-*.f64 79.3%

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

   6. Applied egg-rr 79.3%

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

   8. Applied egg-rr 82.2%

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

   9. Taylor expanded in y around inf

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

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \left({y}^{2}\right)\right)\right) \]

       2. unpow2 N/A

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

       3. *-lowering-*.f64 79.3%

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

   11. Simplified 79.3%

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

       1. associate-*r* N/A

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

       2. remove-double-div N/A

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

       3. un-div-inv N/A

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

       4. inv-pow N/A

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

       5. sqr-pow N/A

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

       6. pow-prod-down N/A

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

       7. remove-double-div N/A

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

       8. metadata-eval N/A

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

       9. distribute-neg-frac N/A

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

       10. associate-/l/ N/A

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

       11. inv-pow N/A

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

       12. pow-pow N/A

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

       13. neg-mul-1 N/A

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

       14. associate-*r/ N/A

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

       15. associate-*l/ N/A

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

       16. pow-prod-down N/A

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

       17. pow-prod-up N/A

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

       18. metadata-eval N/A

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

       19. metadata-eval N/A

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

       20. metadata-eval N/A

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

       21. metadata-eval N/A

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

       22. metadata-eval N/A

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

       23. unpow1 N/A

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

       24. frac-2neg N/A

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

       25. metadata-eval N/A

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

   13. Applied egg-rr 99.9%

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

   if -4.99999999999999981e134 < y < 2.00000000000000005e144

   1. Initial program 99.9%

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

      1. associate-*l* N/A

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

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

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

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

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

      4. --lowering--.f64 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+134} \lor \neg \left(y \leq 2 \cdot 10^{+144}\right):\\ \;\;\;\;0 - y \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;0 - y \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0))) (- 0.0 (* y (* x y))) (* x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = 0.0 - (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 = 0.0d0 - (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 = 0.0 - (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 = 0.0 - (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(0.0 - Float64(y * Float64(x * 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 = 0.0 - (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[(0.0 - N[(y * N[(x * y), $MachinePrecision]), $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* 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 87.9%

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

   3. Simplified 87.9%

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

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

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

      4. *-lft-identity N/A

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

      5. sub-neg N/A

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

      6. --lowering--.f64 N/A

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

      7. *-lowering-*.f64 87.9%

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

   6. Applied egg-rr 87.9%

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

   8. Applied egg-rr 89.5%

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

   9. Taylor expanded in y around inf

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

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \left({y}^{2}\right)\right)\right) \]

       2. unpow2 N/A

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

       3. *-lowering-*.f64 86.6%

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

   11. Simplified 86.6%

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

       1. associate-*r* N/A

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

       2. remove-double-div N/A

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

       3. un-div-inv N/A

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

       4. inv-pow N/A

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

       5. sqr-pow N/A

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

       6. pow-prod-down N/A

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

       7. remove-double-div N/A

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

       8. metadata-eval N/A

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

       9. distribute-neg-frac N/A

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

       10. associate-/l/ N/A

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

       11. inv-pow N/A

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

       12. pow-pow N/A

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

       13. neg-mul-1 N/A

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

       14. associate-*r/ N/A

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

       15. associate-*l/ N/A

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

       16. pow-prod-down N/A

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

       17. pow-prod-up N/A

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

       18. metadata-eval N/A

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

       19. metadata-eval N/A

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

       20. metadata-eval N/A

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

       21. metadata-eval N/A

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

       22. metadata-eval N/A

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

       23. unpow1 N/A

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

       24. frac-2neg N/A

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

       25. metadata-eval N/A

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

   13. Applied egg-rr 98.5%

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

   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* 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 98.1%

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

   7. Simplified 98.1%

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

4. Final simplification 98.3%

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

Alternative 3: 92.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;x \cdot \left(0 - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0))) (* x (- 0.0 (* y y))) (* x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = x * (0.0 - (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 * (0.0d0 - (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 * (0.0 - (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 * (0.0 - (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(0.0 - Float64(y * 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 * (0.0 - (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[(0.0 - N[(y * y), $MachinePrecision]), $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* 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 87.9%

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

   3. Simplified 87.9%

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

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

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

      4. *-lft-identity N/A

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

      5. sub-neg N/A

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

      6. --lowering--.f64 N/A

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

      7. *-lowering-*.f64 87.9%

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

   6. Applied egg-rr 87.9%

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

   8. Applied egg-rr 89.5%

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

   9. Taylor expanded in y around inf

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

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \left({y}^{2}\right)\right)\right) \]

       2. unpow2 N/A

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

       3. *-lowering-*.f64 86.6%

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

   11. Simplified 86.6%

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

   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* 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 98.1%

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

   7. Simplified 98.1%

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

4. Final simplification 92.5%

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

Alternative 4: 63.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 99.9%

      \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
   2. 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 96.4%

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

   3. Simplified 96.4%

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

   5. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 79.7%

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

   7. Simplified 79.7%

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

   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* 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 88.6%

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

   3. Simplified 88.6%

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

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

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

      4. *-lft-identity N/A

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

      5. sub-neg N/A

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

      6. --lowering--.f64 N/A

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

      7. *-lowering-*.f64 88.6%

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

   6. Applied egg-rr 88.6%

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

   8. Applied egg-rr 88.6%

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

   9. Taylor expanded in y around 0

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

   11. Simplified 0.7%

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

       1. frac-2neg N/A

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

       2. neg-mul-1 N/A

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

       3. div-inv N/A

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

       4. distribute-frac-neg N/A

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

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

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

       6. frac-2neg N/A

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

       7. metadata-eval N/A

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

       8. metadata-eval N/A

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

       9. frac-2neg N/A

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

       10. unpow1 N/A

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

       11. metadata-eval N/A

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

       12. metadata-eval N/A

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

       13. metadata-eval N/A

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

       14. pow-pow N/A

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

       15. pow-pow N/A

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

       16. inv-pow N/A

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

       17. frac-2neg N/A

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

       18. metadata-eval N/A

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

       19. remove-double-div N/A

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

       20. pow2 N/A

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

       21. sqr-neg N/A

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

       22. pow-prod-down N/A

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

       23. sqr-pow N/A

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

       24. inv-pow N/A

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

       25. div-inv N/A

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

       26. clear-num N/A

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

   13. Applied egg-rr 27.6%

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

4. Final simplification 64.7%

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

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

Alternative 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* 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 94.1%

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

3. Simplified 94.1%

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

5. Taylor expanded in y around 0

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

   1. *-lowering-*.f64 56.9%

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

7. Simplified 56.9%

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

Reproduce

herbie shell --seed 2024125 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))