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

Percentage Accurate: 69.2% → 99.8%

Time: 12.1s

Alternatives: 22

Speedup: 0.8×

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.7%

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

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. associate-+l+ N/A

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

   15. +-commutative N/A

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

   16. associate-+l+ N/A

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

   17. lower-+.f64 N/A

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

   18. lower-+.f64 N/A

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

   19. lower-/.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 85.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ t_1 := y + \left(1 + x\right)\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{t\_1} \cdot 1}{y + x}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-239}:\\ \;\;\;\;t\_0 \cdot \frac{y}{t\_1 \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{1 + \mathsf{fma}\left(x, 2 + \frac{1}{y}, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))) (t_1 (+ y (+ 1.0 x))))
   (if (<= x -1.4e+154)
     (/ (* (/ y t_1) 1.0) (+ y x))
     (if (<= x -4e-239)
       (* t_0 (/ y (* t_1 (+ y x))))
       (/ t_0 (+ 1.0 (fma x (+ 2.0 (/ 1.0 y)) y)))))))

C

double code(double x, double y) {
	double t_0 = x / (y + x);
	double t_1 = y + (1.0 + x);
	double tmp;
	if (x <= -1.4e+154) {
		tmp = ((y / t_1) * 1.0) / (y + x);
	} else if (x <= -4e-239) {
		tmp = t_0 * (y / (t_1 * (y + x)));
	} else {
		tmp = t_0 / (1.0 + fma(x, (2.0 + (1.0 / y)), y));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	t_1 = Float64(y + Float64(1.0 + x))
	tmp = 0.0
	if (x <= -1.4e+154)
		tmp = Float64(Float64(Float64(y / t_1) * 1.0) / Float64(y + x));
	elseif (x <= -4e-239)
		tmp = Float64(t_0 * Float64(y / Float64(t_1 * Float64(y + x))));
	else
		tmp = Float64(t_0 / Float64(1.0 + fma(x, Float64(2.0 + Float64(1.0 / y)), y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.4e+154], N[(N[(N[(y / t$95$1), $MachinePrecision] * 1.0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4e-239], N[(t$95$0 * N[(y / N[(t$95$1 * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(1.0 + N[(x * N[(2.0 + N[(1.0 / y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4e154

   1. Initial program 56.7%

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

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lower-+.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 87.0%

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

   if -1.4e154 < x < -4.0000000000000003e-239

   1. Initial program 75.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lower-+.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. lower-/.f64 97.9

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

   4. Applied rewrites 97.9%

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

   if -4.0000000000000003e-239 < x

   1. Initial program 68.4%

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

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lower-+.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. div-inv N/A

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

      10. clear-num N/A

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

      11. lift-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 99.4

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.4

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 77.7

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

   9. Applied rewrites 77.7%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{y + \left(1 + x\right)} \cdot 1}{y + x}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-239}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{\left(y + \left(1 + x\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{1 + \mathsf{fma}\left(x, 2 + \frac{1}{y}, y\right)}\\ \end{array} \]
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 2024228 
(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))))