Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A

Percentage Accurate: 69.2% → 99.8%

Time: 12.5s

Alternatives: 24

Speedup: 1.6×

Specification

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{\frac{\frac{y}{\left(y + x\right) + 1}}{y + x} \cdot x}{y + x} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (/ (* (/ (/ y (+ (+ y x) 1.0)) (+ y x)) x) (+ y x)))

C

assert(x < y);
double code(double x, double y) {
	return (((y / ((y + x) + 1.0)) / (y + x)) * x) / (y + x);
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (((y / ((y + x) + 1.0d0)) / (y + x)) * x) / (y + x)
end function

Java

assert x < y;
public static double code(double x, double y) {
	return (((y / ((y + x) + 1.0)) / (y + x)) * x) / (y + x);
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return (((y / ((y + x) + 1.0)) / (y + x)) * x) / (y + x)

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(Float64(Float64(Float64(y / Float64(Float64(y + x) + 1.0)) / Float64(y + x)) * x) / Float64(y + x))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = (((y / ((y + x) + 1.0)) / (y + x)) * x) / (y + x);
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(N[(N[(N[(y / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.4%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. clear-num N/A

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

   6. frac-times N/A

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

   7. *-commutative N/A

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

   8. *-lft-identity N/A

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

   9. div-inv N/A

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

   10. clear-num N/A

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

   11. lift-/.f64 N/A

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

   12. div-inv N/A

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

   13. clear-num N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 99.8

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

   \[\leadsto \frac{\frac{\frac{y}{\left(y + x\right) + 1}}{y + x} \cdot x}{y + x} \]
8. Add Preprocessing

Alternative 2: 95.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+159}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot x}{\left(\left(1 + x\right) + y\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{\mathsf{fma}\left(2 + \frac{1}{y}, x, 1 + y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 9.5e+159)
   (/ (* (/ y (+ y x)) x) (* (+ (+ 1.0 x) y) (+ y x)))
   (/ (/ x (+ y x)) (fma (+ 2.0 (/ 1.0 y)) x (+ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 9.5e+159) {
		tmp = ((y / (y + x)) * x) / (((1.0 + x) + y) * (y + x));
	} else {
		tmp = (x / (y + x)) / fma((2.0 + (1.0 / y)), x, (1.0 + y));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 9.5e+159)
		tmp = Float64(Float64(Float64(y / Float64(y + x)) * x) / Float64(Float64(Float64(1.0 + x) + y) * Float64(y + x)));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / fma(Float64(2.0 + Float64(1.0 / y)), x, Float64(1.0 + y)));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 9.5e+159], N[(N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / N[(N[(N[(1.0 + x), $MachinePrecision] + y), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + N[(1.0 / y), $MachinePrecision]), $MachinePrecision] * x + N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.5000000000000003e159

   1. Initial program 70.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 96.1

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. associate-+r+ N/A

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

      21. lower-+.f64 N/A

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

      22. lower-+.f64 96.1

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 96.1

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

   4. Applied rewrites 96.1%

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

   if 9.5000000000000003e159 < y

   1. Initial program 63.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. *-lft-identity N/A

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

      9. div-inv N/A

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

      10. clear-num N/A

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

      11. lift-/.f64 N/A

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

      12. div-inv N/A

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

      13. clear-num N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 99.9

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

   6. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l/ N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. associate-+l+ N/A

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

      14. +-commutative N/A

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

      15. associate-+l+ N/A

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

      16. +-commutative N/A

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

      17. lift-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. associate-+l+ N/A

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

       4. *-commutative N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\mathsf{fma}\left(2 + \frac{1}{y}, x, 1 + y\right)}} \]

       7. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\mathsf{fma}\left(\color{blue}{\frac{1}{y} + 2}, x, 1 + y\right)} \]

       8. lower-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\mathsf{fma}\left(\color{blue}{\frac{1}{y} + 2}, x, 1 + y\right)} \]

       9. lower-/.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\mathsf{fma}\left(\color{blue}{\frac{1}{y}} + 2, x, 1 + y\right)} \]

       10. lower-+.f64 93.4

           \[\leadsto \frac{\frac{x}{y + x}}{\mathsf{fma}\left(\frac{1}{y} + 2, x, \color{blue}{1 + y}\right)} \]

   11. Applied rewrites 93.4%

       \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\mathsf{fma}\left(\frac{1}{y} + 2, x, 1 + y\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+159}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot x}{\left(\left(1 + x\right) + y\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{\mathsf{fma}\left(2 + \frac{1}{y}, x, 1 + y\right)}\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x)))))

C

double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Fortran

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

Java

public static double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Python

def code(x, y):
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)))

Julia

function code(x, y)
	return Float64(Float64(Float64(x / Float64(Float64(y + 1.0) + x)) / Float64(y + x)) / Float64(1.0 / Float64(y / Float64(y + x))))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
end

Wolfram

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

Reproduce

herbie shell --seed 2024230 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (/ (/ x (+ (+ y 1) x)) (+ y x)) (/ 1 (/ y (+ y x)))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))