Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B

Percentage Accurate: 99.4% → 99.4%

Time: 1.5s

Alternatives: 12

Speedup: 1.0×

Specification

\[\mathsf{TRUE}\left(\right)\]

Math

\[\begin{array}{l} \\ \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))

C

double code(double x, double y) {
	return (3.0 * sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (3.0d0 * sqrt(x)) * ((y + (1.0d0 / (x * 9.0d0))) - 1.0d0)
end function

Java

public static double code(double x, double y) {
	return (3.0 * Math.sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
}

Python

def code(x, y):
	return (3.0 * math.sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0)

Julia

function code(x, y)
	return Float64(Float64(3.0 * sqrt(x)) * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) - 1.0))
end

MATLAB

function tmp = code(x, y)
	tmp = (3.0 * sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
end

Wolfram

code[x_, y_] := N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))

C

double code(double x, double y) {
	return (3.0 * sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (3.0d0 * sqrt(x)) * ((y + (1.0d0 / (x * 9.0d0))) - 1.0d0)
end function

Java

public static double code(double x, double y) {
	return (3.0 * Math.sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
}

Python

def code(x, y):
	return (3.0 * math.sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0)

Julia

function code(x, y)
	return Float64(Float64(3.0 * sqrt(x)) * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) - 1.0))
end

MATLAB

function tmp = code(x, y)
	tmp = (3.0 * sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
end

Wolfram

code[x_, y_] := N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))

C

double code(double x, double y) {
	return (3.0 * sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (3.0d0 * sqrt(x)) * ((y + (1.0d0 / (x * 9.0d0))) - 1.0d0)
end function

Java

public static double code(double x, double y) {
	return (3.0 * Math.sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
}

Python

def code(x, y):
	return (3.0 * math.sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0)

Julia

function code(x, y)
	return Float64(Float64(3.0 * sqrt(x)) * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) - 1.0))
end

MATLAB

function tmp = code(x, y)
	tmp = (3.0 * sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
end

Wolfram

code[x_, y_] := N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 51.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{x \cdot 9}\\ \mathbf{if}\;x \leq 0.00122:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(\left(y + t\_0\right) - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* x 9.0))))
   (if (<= x 0.00122)
     (* (* 3.0 (sqrt x)) t_0)
     (* (sqrt x) (- (+ y t_0) 1.0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 / (x * 9.0);
	double tmp;
	if (x <= 0.00122) {
		tmp = (3.0 * sqrt(x)) * t_0;
	} else {
		tmp = sqrt(x) * ((y + t_0) - 1.0);
	}
	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) :: tmp
    t_0 = 1.0d0 / (x * 9.0d0)
    if (x <= 0.00122d0) then
        tmp = (3.0d0 * sqrt(x)) * t_0
    else
        tmp = sqrt(x) * ((y + t_0) - 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 / (x * 9.0);
	double tmp;
	if (x <= 0.00122) {
		tmp = (3.0 * Math.sqrt(x)) * t_0;
	} else {
		tmp = Math.sqrt(x) * ((y + t_0) - 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 / (x * 9.0)
	tmp = 0
	if x <= 0.00122:
		tmp = (3.0 * math.sqrt(x)) * t_0
	else:
		tmp = math.sqrt(x) * ((y + t_0) - 1.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 / Float64(x * 9.0))
	tmp = 0.0
	if (x <= 0.00122)
		tmp = Float64(Float64(3.0 * sqrt(x)) * t_0);
	else
		tmp = Float64(sqrt(x) * Float64(Float64(y + t_0) - 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 / (x * 9.0);
	tmp = 0.0;
	if (x <= 0.00122)
		tmp = (3.0 * sqrt(x)) * t_0;
	else
		tmp = sqrt(x) * ((y + t_0) - 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.00122], N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(y + t$95$0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00121999999999999995

   1. Initial program 99.3%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\frac{\frac{1}{9}}{x}} \]

   5. Applied rewrites 73.7%

      \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\frac{1}{x \cdot 9}} \]

   if 0.00121999999999999995 < x

   1. Initial program 99.6%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(3 \cdot \sqrt{x}\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   5. Applied rewrites 27.6%

      \[\leadsto \color{blue}{\sqrt{x}} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 47.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{x \cdot 9}\\ \mathbf{if}\;x \leq 0.00122:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y + t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* x 9.0))))
   (if (<= x 0.00122) (* (* 3.0 (sqrt x)) t_0) (* (sqrt x) (+ y t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 / (x * 9.0);
	double tmp;
	if (x <= 0.00122) {
		tmp = (3.0 * sqrt(x)) * t_0;
	} else {
		tmp = sqrt(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) :: tmp
    t_0 = 1.0d0 / (x * 9.0d0)
    if (x <= 0.00122d0) then
        tmp = (3.0d0 * sqrt(x)) * t_0
    else
        tmp = sqrt(x) * (y + t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 / (x * 9.0);
	double tmp;
	if (x <= 0.00122) {
		tmp = (3.0 * Math.sqrt(x)) * t_0;
	} else {
		tmp = Math.sqrt(x) * (y + t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 / (x * 9.0)
	tmp = 0
	if x <= 0.00122:
		tmp = (3.0 * math.sqrt(x)) * t_0
	else:
		tmp = math.sqrt(x) * (y + t_0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 / Float64(x * 9.0))
	tmp = 0.0
	if (x <= 0.00122)
		tmp = Float64(Float64(3.0 * sqrt(x)) * t_0);
	else
		tmp = Float64(sqrt(x) * Float64(y + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 / (x * 9.0);
	tmp = 0.0;
	if (x <= 0.00122)
		tmp = (3.0 * sqrt(x)) * t_0;
	else
		tmp = sqrt(x) * (y + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.00122], N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(y + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00121999999999999995

   1. Initial program 99.3%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\frac{\frac{1}{9}}{x}} \]

   5. Applied rewrites 73.7%

      \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\frac{1}{x \cdot 9}} \]

   if 0.00121999999999999995 < x

   1. Initial program 99.6%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\frac{\frac{1}{9} + x \cdot \left(y - 1\right)}{x}} \]

   5. Applied rewrites 53.3%

      \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(y + \frac{1}{x \cdot 9}\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(3 \cdot \sqrt{x}\right)} \cdot \left(y + \frac{1}{x \cdot 9}\right) \]

   8. Applied rewrites 20.4%

      \[\leadsto \color{blue}{\sqrt{x}} \cdot \left(y + \frac{1}{x \cdot 9}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 3 \cdot \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* 3.0 (* (sqrt x) (- (+ y (/ 1.0 (* x 9.0))) 1.0))))

C

double code(double x, double y) {
	return 3.0 * (sqrt(x) * ((y + (1.0 / (x * 9.0))) - 1.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 3.0d0 * (sqrt(x) * ((y + (1.0d0 / (x * 9.0d0))) - 1.0d0))
end function

Java

public static double code(double x, double y) {
	return 3.0 * (Math.sqrt(x) * ((y + (1.0 / (x * 9.0))) - 1.0));
}

Python

def code(x, y):
	return 3.0 * (math.sqrt(x) * ((y + (1.0 / (x * 9.0))) - 1.0))

Julia

function code(x, y)
	return Float64(3.0 * Float64(sqrt(x) * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) - 1.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = 3.0 * (sqrt(x) * ((y + (1.0 / (x * 9.0))) - 1.0));
end

Wolfram

code[x_, y_] := N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 3.4%

   \[\leadsto \color{blue}{3 \cdot \sqrt{x}} \]

6. Taylor expanded in x around 0

   \[\leadsto 3 \cdot \sqrt{x} \]

8. Applied rewrites 99.3%

   \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \color{blue}{\left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)}\right) \]
9. Add Preprocessing

Alternative 5: 74.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(3 \cdot \sqrt{x}\right) \cdot \left(y + \frac{1}{x \cdot 9}\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (* 3.0 (sqrt x)) (+ y (/ 1.0 (* x 9.0)))))

C

double code(double x, double y) {
	return (3.0 * sqrt(x)) * (y + (1.0 / (x * 9.0)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (3.0d0 * sqrt(x)) * (y + (1.0d0 / (x * 9.0d0)))
end function

Java

public static double code(double x, double y) {
	return (3.0 * Math.sqrt(x)) * (y + (1.0 / (x * 9.0)));
}

Python

def code(x, y):
	return (3.0 * math.sqrt(x)) * (y + (1.0 / (x * 9.0)))

Julia

function code(x, y)
	return Float64(Float64(3.0 * sqrt(x)) * Float64(y + Float64(1.0 / Float64(x * 9.0))))
end

MATLAB

function tmp = code(x, y)
	tmp = (3.0 * sqrt(x)) * (y + (1.0 / (x * 9.0)));
end

Wolfram

code[x_, y_] := N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\frac{\frac{1}{9} + x \cdot \left(y - 1\right)}{x}} \]

5. Applied rewrites 76.0%

   \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(y + \frac{1}{x \cdot 9}\right)} \]
6. Add Preprocessing

Alternative 6: 19.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{x} \cdot \left(y + \frac{1}{x \cdot 9}\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (sqrt x) (+ y (/ 1.0 (* x 9.0)))))

C

double code(double x, double y) {
	return sqrt(x) * (y + (1.0 / (x * 9.0)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt(x) * (y + (1.0d0 / (x * 9.0d0)))
end function

Java

public static double code(double x, double y) {
	return Math.sqrt(x) * (y + (1.0 / (x * 9.0)));
}

Python

def code(x, y):
	return math.sqrt(x) * (y + (1.0 / (x * 9.0)))

Julia

function code(x, y)
	return Float64(sqrt(x) * Float64(y + Float64(1.0 / Float64(x * 9.0))))
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt(x) * (y + (1.0 / (x * 9.0)));
end

Wolfram

code[x_, y_] := N[(N[Sqrt[x], $MachinePrecision] * N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\frac{\frac{1}{9} + x \cdot \left(y - 1\right)}{x}} \]

5. Applied rewrites 76.0%

   \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(y + \frac{1}{x \cdot 9}\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(3 \cdot \sqrt{x}\right)} \cdot \left(y + \frac{1}{x \cdot 9}\right) \]

8. Applied rewrites 19.0%

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

Alternative 7: 8.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{x} \cdot \frac{1}{x \cdot 9} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (sqrt x) (/ 1.0 (* x 9.0))))

C

double code(double x, double y) {
	return sqrt(x) * (1.0 / (x * 9.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt(x) * (1.0d0 / (x * 9.0d0))
end function

Java

public static double code(double x, double y) {
	return Math.sqrt(x) * (1.0 / (x * 9.0));
}

Python

def code(x, y):
	return math.sqrt(x) * (1.0 / (x * 9.0))

Julia

function code(x, y)
	return Float64(sqrt(x) * Float64(1.0 / Float64(x * 9.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt(x) * (1.0 / (x * 9.0));
end

Wolfram

code[x_, y_] := N[(N[Sqrt[x], $MachinePrecision] * N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\frac{\frac{1}{9}}{x}} \]

5. Applied rewrites 38.7%

   \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\frac{1}{x \cdot 9}} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(3 \cdot \sqrt{x}\right)} \cdot \frac{1}{x \cdot 9} \]

8. Applied rewrites 8.5%

   \[\leadsto \color{blue}{\sqrt{x}} \cdot \frac{1}{x \cdot 9} \]
9. Add Preprocessing

Alternative 8: 5.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ y + \frac{1}{x \cdot 9} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ y (/ 1.0 (* x 9.0))))

C

double code(double x, double y) {
	return y + (1.0 / (x * 9.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y + (1.0d0 / (x * 9.0d0))
end function

Java

public static double code(double x, double y) {
	return y + (1.0 / (x * 9.0));
}

Python

def code(x, y):
	return y + (1.0 / (x * 9.0))

Julia

function code(x, y)
	return Float64(y + Float64(1.0 / Float64(x * 9.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = y + (1.0 / (x * 9.0));
end

Wolfram

code[x_, y_] := N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 3.4%

   \[\leadsto \color{blue}{3 \cdot \sqrt{x}} \]

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{3 \cdot \left(\sqrt{x} \cdot \left(y - 1\right)\right)} \]

8. Applied rewrites 5.9%

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

Alternative 9: 4.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot 9} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 (* x 9.0)))

C

double code(double x, double y) {
	return 1.0 / (x * 9.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / (x * 9.0d0)
end function

Java

public static double code(double x, double y) {
	return 1.0 / (x * 9.0);
}

Python

def code(x, y):
	return 1.0 / (x * 9.0)

Julia

function code(x, y)
	return Float64(1.0 / Float64(x * 9.0))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / (x * 9.0);
end

Wolfram

code[x_, y_] := N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 3.4%

   \[\leadsto \color{blue}{3 \cdot \sqrt{x}} \]

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{3 \cdot \left(\sqrt{x} \cdot \left(y - 1\right)\right)} \]

8. Applied rewrites 5.9%

   \[\leadsto \color{blue}{y + \frac{1}{x \cdot 9}} \]

9. Taylor expanded in x around inf

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

11. Applied rewrites 4.1%

    \[\leadsto \frac{1}{\color{blue}{x \cdot 9}} \]
12. Add Preprocessing

Alternative 10: 3.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ 3 \cdot \sqrt{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* 3.0 (sqrt x)))

C

double code(double x, double y) {
	return 3.0 * sqrt(x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 3.0d0 * sqrt(x)
end function

Java

public static double code(double x, double y) {
	return 3.0 * Math.sqrt(x);
}

Python

def code(x, y):
	return 3.0 * math.sqrt(x)

Julia

function code(x, y)
	return Float64(3.0 * sqrt(x))
end

MATLAB

function tmp = code(x, y)
	tmp = 3.0 * sqrt(x);
end

Wolfram

code[x_, y_] := N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 3.4%

   \[\leadsto \color{blue}{3 \cdot \sqrt{x}} \]
6. Add Preprocessing

Alternative 11: 3.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (sqrt x))

C

double code(double x, double y) {
	return sqrt(x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt(x)
end function

Java

public static double code(double x, double y) {
	return Math.sqrt(x);
}

Python

def code(x, y):
	return math.sqrt(x)

Julia

function code(x, y)
	return sqrt(x)
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt(x);
end

Wolfram

code[x_, y_] := N[Sqrt[x], $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 3.4%

   \[\leadsto \color{blue}{3 \cdot \sqrt{x}} \]

6. Taylor expanded in x around 0

   \[\leadsto 3 \cdot \color{blue}{\sqrt{x}} \]

8. Applied rewrites 3.4%

   \[\leadsto \sqrt{x} \]
9. Add Preprocessing

Alternative 12: 3.0% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ x \cdot 9 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x 9.0))

C

double code(double x, double y) {
	return x * 9.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * 9.0d0
end function

Java

public static double code(double x, double y) {
	return x * 9.0;
}

Python

def code(x, y):
	return x * 9.0

Julia

function code(x, y)
	return Float64(x * 9.0)
end

MATLAB

function tmp = code(x, y)
	tmp = x * 9.0;
end

Wolfram

code[x_, y_] := N[(x * 9.0), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 3.4%

   \[\leadsto \color{blue}{3 \cdot \sqrt{x}} \]

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{3 \cdot \left(\sqrt{x} \cdot \left(y - 1\right)\right)} \]

8. Applied rewrites 5.9%

   \[\leadsto \color{blue}{y + \frac{1}{x \cdot 9}} \]

9. Taylor expanded in x around inf

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

11. Applied rewrites 4.1%

    \[\leadsto \frac{1}{\color{blue}{x \cdot 9}} \]

12. Taylor expanded in x around 0

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

14. Applied rewrites 3.1%

    \[\leadsto x \cdot \color{blue}{9} \]
15. Add Preprocessing

Developer Target 1: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ 3 \cdot \left(y \cdot \sqrt{x} + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \sqrt{x}\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* 3.0 (+ (* y (sqrt x)) (* (- (/ 1.0 (* x 9.0)) 1.0) (sqrt x)))))

C

double code(double x, double y) {
	return 3.0 * ((y * sqrt(x)) + (((1.0 / (x * 9.0)) - 1.0) * sqrt(x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 3.0d0 * ((y * sqrt(x)) + (((1.0d0 / (x * 9.0d0)) - 1.0d0) * sqrt(x)))
end function

Java

public static double code(double x, double y) {
	return 3.0 * ((y * Math.sqrt(x)) + (((1.0 / (x * 9.0)) - 1.0) * Math.sqrt(x)));
}

Python

def code(x, y):
	return 3.0 * ((y * math.sqrt(x)) + (((1.0 / (x * 9.0)) - 1.0) * math.sqrt(x)))

Julia

function code(x, y)
	return Float64(3.0 * Float64(Float64(y * sqrt(x)) + Float64(Float64(Float64(1.0 / Float64(x * 9.0)) - 1.0) * sqrt(x))))
end

MATLAB

function tmp = code(x, y)
	tmp = 3.0 * ((y * sqrt(x)) + (((1.0 / (x * 9.0)) - 1.0) * sqrt(x)));
end

Wolfram

code[x_, y_] := N[(3.0 * N[(N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024321 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64
  :pre (TRUE)

  :alt
  (! :herbie-platform default (* 3 (+ (* y (sqrt x)) (* (- (/ 1 (* x 9)) 1) (sqrt x)))))

  (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))