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

Percentage Accurate: 69.2% → 77.0%

Time: 1.3s

Alternatives: 5

Speedup: 0.5×

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: 77.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y\right) \cdot \left(x + y\right)\\ t_1 := \left(x + y\right) + 1\\ t_2 := \frac{x \cdot y}{t\_0 \cdot t\_1}\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\left(x + y\right) \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (+ x y) (+ x y)))
        (t_1 (+ (+ x y) 1.0))
        (t_2 (/ (* x y) (* t_0 t_1))))
   (if (<= t_2 5e-9) t_2 (/ t_1 (* (+ x y) t_0)))))

C

double code(double x, double y) {
	double t_0 = (x + y) * (x + y);
	double t_1 = (x + y) + 1.0;
	double t_2 = (x * y) / (t_0 * t_1);
	double tmp;
	if (t_2 <= 5e-9) {
		tmp = t_2;
	} else {
		tmp = t_1 / ((x + y) * t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (x + y) * (x + y)
    t_1 = (x + y) + 1.0d0
    t_2 = (x * y) / (t_0 * t_1)
    if (t_2 <= 5d-9) then
        tmp = t_2
    else
        tmp = t_1 / ((x + y) * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x + y) * (x + y);
	double t_1 = (x + y) + 1.0;
	double t_2 = (x * y) / (t_0 * t_1);
	double tmp;
	if (t_2 <= 5e-9) {
		tmp = t_2;
	} else {
		tmp = t_1 / ((x + y) * t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x + y) * (x + y)
	t_1 = (x + y) + 1.0
	t_2 = (x * y) / (t_0 * t_1)
	tmp = 0
	if t_2 <= 5e-9:
		tmp = t_2
	else:
		tmp = t_1 / ((x + y) * t_0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) * Float64(x + y))
	t_1 = Float64(Float64(x + y) + 1.0)
	t_2 = Float64(Float64(x * y) / Float64(t_0 * t_1))
	tmp = 0.0
	if (t_2 <= 5e-9)
		tmp = t_2;
	else
		tmp = Float64(t_1 / Float64(Float64(x + y) * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x + y) * (x + y);
	t_1 = (x + y) + 1.0;
	t_2 = (x * y) / (t_0 * t_1);
	tmp = 0.0;
	if (t_2 <= 5e-9)
		tmp = t_2;
	else
		tmp = t_1 / ((x + y) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] / N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 5e-9], t$95$2, N[(t$95$1 / N[(N[(x + y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 (+.f64 x y) (+.f64 x y)) (+.f64 (+.f64 x y) #s(literal 1 binary64)))) < 5.0000000000000001e-9

   1. Initial program 83.2%

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

   if 5.0000000000000001e-9 < (/.f64 (*.f64 x y) (*.f64 (*.f64 (+.f64 x y) (+.f64 x y)) (+.f64 (+.f64 x y) #s(literal 1 binary64))))

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 46.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 46.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 46.5%

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

Alternative 2: 45.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

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 + y) * (x + y)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 43.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 37.0%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 43.5%

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

Alternative 3: 39.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

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 + y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 43.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 37.0%

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

Alternative 4: 3.6% accurate, 6.5× 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 70.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 3.7%

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

Alternative 5: 2.8% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ x + y \end{array} \]

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 70.5%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(-1 \cdot \left(x \cdot \left(-1 \cdot \frac{\left(2 \cdot \left(y \cdot \left(1 + y\right)\right) + {y}^{2}\right) \cdot \left(\frac{1}{y \cdot {\left(1 + y\right)}^{2}} + 2 \cdot \frac{1}{{y}^{2} \cdot \left(1 + y\right)}\right)}{{y}^{2} \cdot \left(1 + y\right)} + \left(3 \cdot \frac{1}{{y}^{2} \cdot {\left(1 + y\right)}^{2}} + \frac{1}{{y}^{3} \cdot {\left(1 + y\right)}^{2}}\right)\right)\right) - \left(2 \cdot \frac{1}{{y}^{2} \cdot \left(1 + y\right)} + \frac{1}{y \cdot {\left(1 + y\right)}^{2}}\right)\right) + \frac{1}{y \cdot \left(1 + y\right)}\right)} \]

5. Applied rewrites 2.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 2.7%

   \[\leadsto x + \color{blue}{y} \]
9. 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 2024313 
(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))))