Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D

Percentage Accurate: 99.7% → 99.7%

Time: 6.0s

Alternatives: 13

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. inv-pow N/A

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

   6. lower-pow.f64 99.7

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

4. Applied rewrites 99.7%

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

Alternative 2: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 3: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 4: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Final simplification 99.7%

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

Alternative 5: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{\sqrt{x} \cdot 3} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (- (- 1.0 (/ 0.1111111111111111 x)) (/ y (* (sqrt x) 3.0))))

C

double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) - (y / (sqrt(x) * 3.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - (0.1111111111111111d0 / x)) - (y / (sqrt(x) * 3.0d0))
end function

Java

public static double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) - (y / (Math.sqrt(x) * 3.0));
}

Python

def code(x, y):
	return (1.0 - (0.1111111111111111 / x)) - (y / (math.sqrt(x) * 3.0))

Julia

function code(x, y)
	return Float64(Float64(1.0 - Float64(0.1111111111111111 / x)) - Float64(y / Float64(sqrt(x) * 3.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - (0.1111111111111111 / x)) - (y / (sqrt(x) * 3.0));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] - N[(y / N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. metadata-eval 99.7

      \[\leadsto \left(1 - \frac{\color{blue}{0.1111111111111111}}{x}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.7

       \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{\color{blue}{\sqrt{x} \cdot 3}} \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{\sqrt{x} \cdot 3}} \]
5. Add Preprocessing

Alternative 6: 93.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\sqrt{x}}, -0.3333333333333333, 1\right)\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+100}:\\ \;\;\;\;\frac{x - 0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.35e+27)
   (fma (/ y (sqrt x)) -0.3333333333333333 1.0)
   (if (<= y 9e+100)
     (/ (- x 0.1111111111111111) x)
     (- 1.0 (/ y (* 3.0 (sqrt x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.35e+27) {
		tmp = fma((y / sqrt(x)), -0.3333333333333333, 1.0);
	} else if (y <= 9e+100) {
		tmp = (x - 0.1111111111111111) / x;
	} else {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.35e+27)
		tmp = fma(Float64(y / sqrt(x)), -0.3333333333333333, 1.0);
	elseif (y <= 9e+100)
		tmp = Float64(Float64(x - 0.1111111111111111) / x);
	else
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.35e+27], N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision], If[LessEqual[y, 9e+100], N[(N[(x - 0.1111111111111111), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.3499999999999999e27

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} - \frac{y}{3 \cdot \sqrt{x}} \]
   4. Step-by-step derivation

   5. Applied rewrites 93.6%

      \[\leadsto \color{blue}{1} - \frac{y}{3 \cdot \sqrt{x}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 1 - \color{blue}{\frac{y}{3 \cdot \sqrt{x}}} \]

      2. lift-*.f64 N/A

         \[\leadsto 1 - \frac{y}{\color{blue}{3 \cdot \sqrt{x}}} \]

      3. *-commutative N/A

         \[\leadsto 1 - \frac{y}{\color{blue}{\sqrt{x} \cdot 3}} \]

      4. associate-/r* N/A

         \[\leadsto 1 - \color{blue}{\frac{\frac{y}{\sqrt{x}}}{3}} \]

      5. lower-/.f64 N/A

         \[\leadsto 1 - \color{blue}{\frac{\frac{y}{\sqrt{x}}}{3}} \]

      6. lower-/.f64 93.7

         \[\leadsto 1 - \frac{\color{blue}{\frac{y}{\sqrt{x}}}}{3} \]

   7. Applied rewrites 93.7%

      \[\leadsto 1 - \color{blue}{\frac{\frac{y}{\sqrt{x}}}{3}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 1 - \color{blue}{\frac{\frac{y}{\sqrt{x}}}{3}} \]

      2. lift-/.f64 N/A

         \[\leadsto 1 - \frac{\color{blue}{\frac{y}{\sqrt{x}}}}{3} \]

      3. associate-/r* N/A

         \[\leadsto 1 - \color{blue}{\frac{y}{\sqrt{x} \cdot 3}} \]

      4. *-commutative N/A

         \[\leadsto 1 - \frac{y}{\color{blue}{3 \cdot \sqrt{x}}} \]

      5. lift-*.f64 N/A

         \[\leadsto 1 - \frac{y}{\color{blue}{3 \cdot \sqrt{x}}} \]

      6. lift-/.f64 93.6

         \[\leadsto 1 - \color{blue}{\frac{y}{3 \cdot \sqrt{x}}} \]

      7. *-lft-identity N/A

         \[\leadsto 1 - \color{blue}{1 \cdot \frac{y}{3 \cdot \sqrt{x}}} \]

      8. lift-/.f64 N/A

         \[\leadsto 1 - 1 \cdot \color{blue}{\frac{y}{3 \cdot \sqrt{x}}} \]

      9. associate-*r/ N/A

         \[\leadsto 1 - \color{blue}{\frac{1 \cdot y}{3 \cdot \sqrt{x}}} \]

      10. lift-*.f64 N/A

          \[\leadsto 1 - \frac{1 \cdot y}{\color{blue}{3 \cdot \sqrt{x}}} \]

      11. times-frac N/A

          \[\leadsto 1 - \color{blue}{\frac{1}{3} \cdot \frac{y}{\sqrt{x}}} \]

      12. metadata-eval N/A

          \[\leadsto 1 - \color{blue}{\frac{1}{3}} \cdot \frac{y}{\sqrt{x}} \]

      13. metadata-eval N/A

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

      14. lift-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 93.6

          \[\leadsto 1 - \color{blue}{0.3333333333333333} \cdot \frac{y}{\sqrt{x}} \]

   9. Applied rewrites 93.6%

      \[\leadsto \color{blue}{1 - 0.3333333333333333 \cdot \frac{y}{\sqrt{x}}} \]
   10. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto \color{blue}{1 - \frac{1}{3} \cdot \frac{y}{\sqrt{x}}} \]

       2. lift-*.f64 N/A

          \[\leadsto 1 - \color{blue}{\frac{1}{3} \cdot \frac{y}{\sqrt{x}}} \]

       3. fp-cancel-sub-sign-inv N/A

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

       4. +-commutative N/A

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

       5. metadata-eval N/A

          \[\leadsto \color{blue}{\frac{-1}{3}} \cdot \frac{y}{\sqrt{x}} + 1 \]

       6. *-commutative N/A

          \[\leadsto \color{blue}{\frac{y}{\sqrt{x}} \cdot \frac{-1}{3}} + 1 \]

       7. lower-fma.f64 93.6

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\sqrt{x}}, -0.3333333333333333, 1\right)} \]

   11. Applied rewrites 93.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\sqrt{x}}, -0.3333333333333333, 1\right)} \]

   if -1.3499999999999999e27 < y < 9.00000000000000073e100

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. inv-pow N/A

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

      6. lower-pow.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333 \cdot \sqrt{x}, y, x\right)}{x}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{x - \frac{1}{9}}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 97.3%

       \[\leadsto \frac{x - 0.1111111111111111}{x} \]

   if 9.00000000000000073e100 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} - \frac{y}{3 \cdot \sqrt{x}} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.8%

      \[\leadsto \color{blue}{1} - \frac{y}{3 \cdot \sqrt{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\sqrt{x}}, -0.3333333333333333, 1\right)\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+100}:\\ \;\;\;\;\frac{x - 0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{x} \cdot y, -0.3333333333333333, -0.1111111111111111\right)}{x} - -1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{y}{\sqrt{x}}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2e+15)
   (- (/ (fma (* (sqrt x) y) -0.3333333333333333 -0.1111111111111111) x) -1.0)
   (- 1.0 (/ (/ y (sqrt x)) 3.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2e+15) {
		tmp = (fma((sqrt(x) * y), -0.3333333333333333, -0.1111111111111111) / x) - -1.0;
	} else {
		tmp = 1.0 - ((y / sqrt(x)) / 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2e+15)
		tmp = Float64(Float64(fma(Float64(sqrt(x) * y), -0.3333333333333333, -0.1111111111111111) / x) - -1.0);
	else
		tmp = Float64(1.0 - Float64(Float64(y / sqrt(x)) / 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2e+15], N[(N[(N[(N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] * -0.3333333333333333 + -0.1111111111111111), $MachinePrecision] / x), $MachinePrecision] - -1.0), $MachinePrecision], N[(1.0 - N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e15

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. inv-pow N/A

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

      6. lower-pow.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333 \cdot \sqrt{x}, y, x\right)}{x}} \]

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. metadata-eval N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. associate--l+ N/A

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

      6. +-commutative N/A

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

      7. *-lft-identity N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. metadata-eval N/A

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

      10. div-sub N/A

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

      11. *-lft-identity N/A

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

      12. associate-*l/ N/A

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

      13. mul-1-neg N/A

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

      14. distribute-rgt-neg-out N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 N/A

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

   10. Applied rewrites 99.5%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{x} \cdot y, -0.3333333333333333, -0.1111111111111111\right)}{x} - -1} \]

   if 2e15 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} - \frac{y}{3 \cdot \sqrt{x}} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{1} - \frac{y}{3 \cdot \sqrt{x}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 1 - \color{blue}{\frac{y}{3 \cdot \sqrt{x}}} \]

      2. lift-*.f64 N/A

         \[\leadsto 1 - \frac{y}{\color{blue}{3 \cdot \sqrt{x}}} \]

      3. *-commutative N/A

         \[\leadsto 1 - \frac{y}{\color{blue}{\sqrt{x} \cdot 3}} \]

      4. associate-/r* N/A

         \[\leadsto 1 - \color{blue}{\frac{\frac{y}{\sqrt{x}}}{3}} \]

      5. lower-/.f64 N/A

         \[\leadsto 1 - \color{blue}{\frac{\frac{y}{\sqrt{x}}}{3}} \]

      6. lower-/.f64 99.8

         \[\leadsto 1 - \frac{\color{blue}{\frac{y}{\sqrt{x}}}}{3} \]

   7. Applied rewrites 99.8%

      \[\leadsto 1 - \color{blue}{\frac{\frac{y}{\sqrt{x}}}{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{x} \cdot y, -0.3333333333333333, -0.1111111111111111\right)}{x} - -1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{y}{\sqrt{x}}}{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+51}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{x} \cdot y, -0.3333333333333333, -0.1111111111111111\right)}{x} - -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\sqrt{x}}, -0.3333333333333333, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 4e+51)
   (- (/ (fma (* (sqrt x) y) -0.3333333333333333 -0.1111111111111111) x) -1.0)
   (fma (/ y (sqrt x)) -0.3333333333333333 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4e+51) {
		tmp = (fma((sqrt(x) * y), -0.3333333333333333, -0.1111111111111111) / x) - -1.0;
	} else {
		tmp = fma((y / sqrt(x)), -0.3333333333333333, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4e+51)
		tmp = Float64(Float64(fma(Float64(sqrt(x) * y), -0.3333333333333333, -0.1111111111111111) / x) - -1.0);
	else
		tmp = fma(Float64(y / sqrt(x)), -0.3333333333333333, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4e+51], N[(N[(N[(N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] * -0.3333333333333333 + -0.1111111111111111), $MachinePrecision] / x), $MachinePrecision] - -1.0), $MachinePrecision], N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4e51

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. inv-pow N/A

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

      6. lower-pow.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333 \cdot \sqrt{x}, y, x\right)}{x}} \]

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. metadata-eval N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. associate--l+ N/A

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

      6. +-commutative N/A

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

      7. *-lft-identity N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. metadata-eval N/A

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

      10. div-sub N/A

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

      11. *-lft-identity N/A

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

      12. associate-*l/ N/A

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

      13. mul-1-neg N/A

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

      14. distribute-rgt-neg-out N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 N/A

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

   10. Applied rewrites 99.5%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{x} \cdot y, -0.3333333333333333, -0.1111111111111111\right)}{x} - -1} \]

   if 4e51 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} - \frac{y}{3 \cdot \sqrt{x}} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{1} - \frac{y}{3 \cdot \sqrt{x}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 1 - \color{blue}{\frac{y}{3 \cdot \sqrt{x}}} \]

      2. lift-*.f64 N/A

         \[\leadsto 1 - \frac{y}{\color{blue}{3 \cdot \sqrt{x}}} \]

      3. *-commutative N/A

         \[\leadsto 1 - \frac{y}{\color{blue}{\sqrt{x} \cdot 3}} \]

      4. associate-/r* N/A

         \[\leadsto 1 - \color{blue}{\frac{\frac{y}{\sqrt{x}}}{3}} \]

      5. lower-/.f64 N/A

         \[\leadsto 1 - \color{blue}{\frac{\frac{y}{\sqrt{x}}}{3}} \]

      6. lower-/.f64 99.8

         \[\leadsto 1 - \frac{\color{blue}{\frac{y}{\sqrt{x}}}}{3} \]

   7. Applied rewrites 99.8%

      \[\leadsto 1 - \color{blue}{\frac{\frac{y}{\sqrt{x}}}{3}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 1 - \color{blue}{\frac{\frac{y}{\sqrt{x}}}{3}} \]

      2. lift-/.f64 N/A

         \[\leadsto 1 - \frac{\color{blue}{\frac{y}{\sqrt{x}}}}{3} \]

      3. associate-/r* N/A

         \[\leadsto 1 - \color{blue}{\frac{y}{\sqrt{x} \cdot 3}} \]

      4. *-commutative N/A

         \[\leadsto 1 - \frac{y}{\color{blue}{3 \cdot \sqrt{x}}} \]

      5. lift-*.f64 N/A

         \[\leadsto 1 - \frac{y}{\color{blue}{3 \cdot \sqrt{x}}} \]

      6. lift-/.f64 99.8

         \[\leadsto 1 - \color{blue}{\frac{y}{3 \cdot \sqrt{x}}} \]

      7. *-lft-identity N/A

         \[\leadsto 1 - \color{blue}{1 \cdot \frac{y}{3 \cdot \sqrt{x}}} \]

      8. lift-/.f64 N/A

         \[\leadsto 1 - 1 \cdot \color{blue}{\frac{y}{3 \cdot \sqrt{x}}} \]

      9. associate-*r/ N/A

         \[\leadsto 1 - \color{blue}{\frac{1 \cdot y}{3 \cdot \sqrt{x}}} \]

      10. lift-*.f64 N/A

          \[\leadsto 1 - \frac{1 \cdot y}{\color{blue}{3 \cdot \sqrt{x}}} \]

      11. times-frac N/A

          \[\leadsto 1 - \color{blue}{\frac{1}{3} \cdot \frac{y}{\sqrt{x}}} \]

      12. metadata-eval N/A

          \[\leadsto 1 - \color{blue}{\frac{1}{3}} \cdot \frac{y}{\sqrt{x}} \]

      13. metadata-eval N/A

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

      14. lift-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 99.8

          \[\leadsto 1 - \color{blue}{0.3333333333333333} \cdot \frac{y}{\sqrt{x}} \]

   9. Applied rewrites 99.8%

      \[\leadsto \color{blue}{1 - 0.3333333333333333 \cdot \frac{y}{\sqrt{x}}} \]
   10. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto \color{blue}{1 - \frac{1}{3} \cdot \frac{y}{\sqrt{x}}} \]

       2. lift-*.f64 N/A

          \[\leadsto 1 - \color{blue}{\frac{1}{3} \cdot \frac{y}{\sqrt{x}}} \]

       3. fp-cancel-sub-sign-inv N/A

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

       4. +-commutative N/A

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

       5. metadata-eval N/A

          \[\leadsto \color{blue}{\frac{-1}{3}} \cdot \frac{y}{\sqrt{x}} + 1 \]

       6. *-commutative N/A

          \[\leadsto \color{blue}{\frac{y}{\sqrt{x}} \cdot \frac{-1}{3}} + 1 \]

       7. lower-fma.f64 99.8

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\sqrt{x}}, -0.3333333333333333, 1\right)} \]

   11. Applied rewrites 99.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\sqrt{x}}, -0.3333333333333333, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+51}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{x} \cdot y, -0.3333333333333333, -0.1111111111111111\right)}{x} - -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\sqrt{x}}, -0.3333333333333333, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+51}:\\ \;\;\;\;\frac{x - \mathsf{fma}\left(\sqrt{x} \cdot y, 0.3333333333333333, 0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\sqrt{x}}, -0.3333333333333333, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 4e+51)
   (/ (- x (fma (* (sqrt x) y) 0.3333333333333333 0.1111111111111111)) x)
   (fma (/ y (sqrt x)) -0.3333333333333333 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4e+51) {
		tmp = (x - fma((sqrt(x) * y), 0.3333333333333333, 0.1111111111111111)) / x;
	} else {
		tmp = fma((y / sqrt(x)), -0.3333333333333333, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4e+51)
		tmp = Float64(Float64(x - fma(Float64(sqrt(x) * y), 0.3333333333333333, 0.1111111111111111)) / x);
	else
		tmp = fma(Float64(y / sqrt(x)), -0.3333333333333333, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4e+51], N[(N[(x - N[(N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] * 0.3333333333333333 + 0.1111111111111111), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4e51

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x - \left(\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 99.5

         \[\leadsto \frac{x - \mathsf{fma}\left(\color{blue}{\sqrt{x}} \cdot y, 0.3333333333333333, 0.1111111111111111\right)}{x} \]

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{x - \mathsf{fma}\left(\sqrt{x} \cdot y, 0.3333333333333333, 0.1111111111111111\right)}{x}} \]

   if 4e51 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} - \frac{y}{3 \cdot \sqrt{x}} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{1} - \frac{y}{3 \cdot \sqrt{x}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 1 - \color{blue}{\frac{y}{3 \cdot \sqrt{x}}} \]

      2. lift-*.f64 N/A

         \[\leadsto 1 - \frac{y}{\color{blue}{3 \cdot \sqrt{x}}} \]

      3. *-commutative N/A

         \[\leadsto 1 - \frac{y}{\color{blue}{\sqrt{x} \cdot 3}} \]

      4. associate-/r* N/A

         \[\leadsto 1 - \color{blue}{\frac{\frac{y}{\sqrt{x}}}{3}} \]

      5. lower-/.f64 N/A

         \[\leadsto 1 - \color{blue}{\frac{\frac{y}{\sqrt{x}}}{3}} \]

      6. lower-/.f64 99.8

         \[\leadsto 1 - \frac{\color{blue}{\frac{y}{\sqrt{x}}}}{3} \]

   7. Applied rewrites 99.8%

      \[\leadsto 1 - \color{blue}{\frac{\frac{y}{\sqrt{x}}}{3}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 1 - \color{blue}{\frac{\frac{y}{\sqrt{x}}}{3}} \]

      2. lift-/.f64 N/A

         \[\leadsto 1 - \frac{\color{blue}{\frac{y}{\sqrt{x}}}}{3} \]

      3. associate-/r* N/A

         \[\leadsto 1 - \color{blue}{\frac{y}{\sqrt{x} \cdot 3}} \]

      4. *-commutative N/A

         \[\leadsto 1 - \frac{y}{\color{blue}{3 \cdot \sqrt{x}}} \]

      5. lift-*.f64 N/A

         \[\leadsto 1 - \frac{y}{\color{blue}{3 \cdot \sqrt{x}}} \]

      6. lift-/.f64 99.8

         \[\leadsto 1 - \color{blue}{\frac{y}{3 \cdot \sqrt{x}}} \]

      7. *-lft-identity N/A

         \[\leadsto 1 - \color{blue}{1 \cdot \frac{y}{3 \cdot \sqrt{x}}} \]

      8. lift-/.f64 N/A

         \[\leadsto 1 - 1 \cdot \color{blue}{\frac{y}{3 \cdot \sqrt{x}}} \]

      9. associate-*r/ N/A

         \[\leadsto 1 - \color{blue}{\frac{1 \cdot y}{3 \cdot \sqrt{x}}} \]

      10. lift-*.f64 N/A

          \[\leadsto 1 - \frac{1 \cdot y}{\color{blue}{3 \cdot \sqrt{x}}} \]

      11. times-frac N/A

          \[\leadsto 1 - \color{blue}{\frac{1}{3} \cdot \frac{y}{\sqrt{x}}} \]

      12. metadata-eval N/A

          \[\leadsto 1 - \color{blue}{\frac{1}{3}} \cdot \frac{y}{\sqrt{x}} \]

      13. metadata-eval N/A

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

      14. lift-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 99.8

          \[\leadsto 1 - \color{blue}{0.3333333333333333} \cdot \frac{y}{\sqrt{x}} \]

   9. Applied rewrites 99.8%

      \[\leadsto \color{blue}{1 - 0.3333333333333333 \cdot \frac{y}{\sqrt{x}}} \]
   10. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto \color{blue}{1 - \frac{1}{3} \cdot \frac{y}{\sqrt{x}}} \]

       2. lift-*.f64 N/A

          \[\leadsto 1 - \color{blue}{\frac{1}{3} \cdot \frac{y}{\sqrt{x}}} \]

       3. fp-cancel-sub-sign-inv N/A

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

       4. +-commutative N/A

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

       5. metadata-eval N/A

          \[\leadsto \color{blue}{\frac{-1}{3}} \cdot \frac{y}{\sqrt{x}} + 1 \]

       6. *-commutative N/A

          \[\leadsto \color{blue}{\frac{y}{\sqrt{x}} \cdot \frac{-1}{3}} + 1 \]

       7. lower-fma.f64 99.8

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\sqrt{x}}, -0.3333333333333333, 1\right)} \]

   11. Applied rewrites 99.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\sqrt{x}}, -0.3333333333333333, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 93.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+27} \lor \neg \left(y \leq 9 \cdot 10^{+100}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\sqrt{x}}, -0.3333333333333333, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.35e+27) (not (<= y 9e+100)))
   (fma (/ y (sqrt x)) -0.3333333333333333 1.0)
   (/ (- x 0.1111111111111111) x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.35e+27) || !(y <= 9e+100)) {
		tmp = fma((y / sqrt(x)), -0.3333333333333333, 1.0);
	} else {
		tmp = (x - 0.1111111111111111) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.35e+27) || !(y <= 9e+100))
		tmp = fma(Float64(y / sqrt(x)), -0.3333333333333333, 1.0);
	else
		tmp = Float64(Float64(x - 0.1111111111111111) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.35e+27], N[Not[LessEqual[y, 9e+100]], $MachinePrecision]], N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision], N[(N[(x - 0.1111111111111111), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3499999999999999e27 or 9.00000000000000073e100 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} - \frac{y}{3 \cdot \sqrt{x}} \]
   4. Step-by-step derivation

   5. Applied rewrites 96.0%

      \[\leadsto \color{blue}{1} - \frac{y}{3 \cdot \sqrt{x}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 1 - \color{blue}{\frac{y}{3 \cdot \sqrt{x}}} \]

      2. lift-*.f64 N/A

         \[\leadsto 1 - \frac{y}{\color{blue}{3 \cdot \sqrt{x}}} \]

      3. *-commutative N/A

         \[\leadsto 1 - \frac{y}{\color{blue}{\sqrt{x} \cdot 3}} \]

      4. associate-/r* N/A

         \[\leadsto 1 - \color{blue}{\frac{\frac{y}{\sqrt{x}}}{3}} \]

      5. lower-/.f64 N/A

         \[\leadsto 1 - \color{blue}{\frac{\frac{y}{\sqrt{x}}}{3}} \]

      6. lower-/.f64 96.0

         \[\leadsto 1 - \frac{\color{blue}{\frac{y}{\sqrt{x}}}}{3} \]

   7. Applied rewrites 96.0%

      \[\leadsto 1 - \color{blue}{\frac{\frac{y}{\sqrt{x}}}{3}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 1 - \color{blue}{\frac{\frac{y}{\sqrt{x}}}{3}} \]

      2. lift-/.f64 N/A

         \[\leadsto 1 - \frac{\color{blue}{\frac{y}{\sqrt{x}}}}{3} \]

      3. associate-/r* N/A

         \[\leadsto 1 - \color{blue}{\frac{y}{\sqrt{x} \cdot 3}} \]

      4. *-commutative N/A

         \[\leadsto 1 - \frac{y}{\color{blue}{3 \cdot \sqrt{x}}} \]

      5. lift-*.f64 N/A

         \[\leadsto 1 - \frac{y}{\color{blue}{3 \cdot \sqrt{x}}} \]

      6. lift-/.f64 96.0

         \[\leadsto 1 - \color{blue}{\frac{y}{3 \cdot \sqrt{x}}} \]

      7. *-lft-identity N/A

         \[\leadsto 1 - \color{blue}{1 \cdot \frac{y}{3 \cdot \sqrt{x}}} \]

      8. lift-/.f64 N/A

         \[\leadsto 1 - 1 \cdot \color{blue}{\frac{y}{3 \cdot \sqrt{x}}} \]

      9. associate-*r/ N/A

         \[\leadsto 1 - \color{blue}{\frac{1 \cdot y}{3 \cdot \sqrt{x}}} \]

      10. lift-*.f64 N/A

          \[\leadsto 1 - \frac{1 \cdot y}{\color{blue}{3 \cdot \sqrt{x}}} \]

      11. times-frac N/A

          \[\leadsto 1 - \color{blue}{\frac{1}{3} \cdot \frac{y}{\sqrt{x}}} \]

      12. metadata-eval N/A

          \[\leadsto 1 - \color{blue}{\frac{1}{3}} \cdot \frac{y}{\sqrt{x}} \]

      13. metadata-eval N/A

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

      14. lift-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 95.9

          \[\leadsto 1 - \color{blue}{0.3333333333333333} \cdot \frac{y}{\sqrt{x}} \]

   9. Applied rewrites 95.9%

      \[\leadsto \color{blue}{1 - 0.3333333333333333 \cdot \frac{y}{\sqrt{x}}} \]
   10. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto \color{blue}{1 - \frac{1}{3} \cdot \frac{y}{\sqrt{x}}} \]

       2. lift-*.f64 N/A

          \[\leadsto 1 - \color{blue}{\frac{1}{3} \cdot \frac{y}{\sqrt{x}}} \]

       3. fp-cancel-sub-sign-inv N/A

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

       4. +-commutative N/A

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

       5. metadata-eval N/A

          \[\leadsto \color{blue}{\frac{-1}{3}} \cdot \frac{y}{\sqrt{x}} + 1 \]

       6. *-commutative N/A

          \[\leadsto \color{blue}{\frac{y}{\sqrt{x}} \cdot \frac{-1}{3}} + 1 \]

       7. lower-fma.f64 95.9

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\sqrt{x}}, -0.3333333333333333, 1\right)} \]

   11. Applied rewrites 95.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\sqrt{x}}, -0.3333333333333333, 1\right)} \]

   if -1.3499999999999999e27 < y < 9.00000000000000073e100

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. inv-pow N/A

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

      6. lower-pow.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333 \cdot \sqrt{x}, y, x\right)}{x}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{x - \frac{1}{9}}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 97.3%

       \[\leadsto \frac{x - 0.1111111111111111}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+27} \lor \neg \left(y \leq 9 \cdot 10^{+100}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\sqrt{x}}, -0.3333333333333333, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 0.1111111111111111}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.3333333333333333, \sqrt{x} \cdot y, -0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\sqrt{x}}, -0.3333333333333333, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.112)
   (/ (fma -0.3333333333333333 (* (sqrt x) y) -0.1111111111111111) x)
   (fma (/ y (sqrt x)) -0.3333333333333333 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = fma(-0.3333333333333333, (sqrt(x) * y), -0.1111111111111111) / x;
	} else {
		tmp = fma((y / sqrt(x)), -0.3333333333333333, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.112)
		tmp = Float64(fma(-0.3333333333333333, Float64(sqrt(x) * y), -0.1111111111111111) / x);
	else
		tmp = fma(Float64(y / sqrt(x)), -0.3333333333333333, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.112], N[(N[(-0.3333333333333333 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] + -0.1111111111111111), $MachinePrecision] / x), $MachinePrecision], N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.112000000000000002

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)}{x}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 98.7

          \[\leadsto \frac{\mathsf{fma}\left(-0.3333333333333333, \color{blue}{\sqrt{x}} \cdot y, -0.1111111111111111\right)}{x} \]

   5. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.3333333333333333, \sqrt{x} \cdot y, -0.1111111111111111\right)}{x}} \]

   if 0.112000000000000002 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} - \frac{y}{3 \cdot \sqrt{x}} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{1} - \frac{y}{3 \cdot \sqrt{x}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 1 - \color{blue}{\frac{y}{3 \cdot \sqrt{x}}} \]

      2. lift-*.f64 N/A

         \[\leadsto 1 - \frac{y}{\color{blue}{3 \cdot \sqrt{x}}} \]

      3. *-commutative N/A

         \[\leadsto 1 - \frac{y}{\color{blue}{\sqrt{x} \cdot 3}} \]

      4. associate-/r* N/A

         \[\leadsto 1 - \color{blue}{\frac{\frac{y}{\sqrt{x}}}{3}} \]

      5. lower-/.f64 N/A

         \[\leadsto 1 - \color{blue}{\frac{\frac{y}{\sqrt{x}}}{3}} \]

      6. lower-/.f64 99.1

         \[\leadsto 1 - \frac{\color{blue}{\frac{y}{\sqrt{x}}}}{3} \]

   7. Applied rewrites 99.1%

      \[\leadsto 1 - \color{blue}{\frac{\frac{y}{\sqrt{x}}}{3}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 1 - \color{blue}{\frac{\frac{y}{\sqrt{x}}}{3}} \]

      2. lift-/.f64 N/A

         \[\leadsto 1 - \frac{\color{blue}{\frac{y}{\sqrt{x}}}}{3} \]

      3. associate-/r* N/A

         \[\leadsto 1 - \color{blue}{\frac{y}{\sqrt{x} \cdot 3}} \]

      4. *-commutative N/A

         \[\leadsto 1 - \frac{y}{\color{blue}{3 \cdot \sqrt{x}}} \]

      5. lift-*.f64 N/A

         \[\leadsto 1 - \frac{y}{\color{blue}{3 \cdot \sqrt{x}}} \]

      6. lift-/.f64 99.1

         \[\leadsto 1 - \color{blue}{\frac{y}{3 \cdot \sqrt{x}}} \]

      7. *-lft-identity N/A

         \[\leadsto 1 - \color{blue}{1 \cdot \frac{y}{3 \cdot \sqrt{x}}} \]

      8. lift-/.f64 N/A

         \[\leadsto 1 - 1 \cdot \color{blue}{\frac{y}{3 \cdot \sqrt{x}}} \]

      9. associate-*r/ N/A

         \[\leadsto 1 - \color{blue}{\frac{1 \cdot y}{3 \cdot \sqrt{x}}} \]

      10. lift-*.f64 N/A

          \[\leadsto 1 - \frac{1 \cdot y}{\color{blue}{3 \cdot \sqrt{x}}} \]

      11. times-frac N/A

          \[\leadsto 1 - \color{blue}{\frac{1}{3} \cdot \frac{y}{\sqrt{x}}} \]

      12. metadata-eval N/A

          \[\leadsto 1 - \color{blue}{\frac{1}{3}} \cdot \frac{y}{\sqrt{x}} \]

      13. metadata-eval N/A

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

      14. lift-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 99.1

          \[\leadsto 1 - \color{blue}{0.3333333333333333} \cdot \frac{y}{\sqrt{x}} \]

   9. Applied rewrites 99.1%

      \[\leadsto \color{blue}{1 - 0.3333333333333333 \cdot \frac{y}{\sqrt{x}}} \]
   10. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto \color{blue}{1 - \frac{1}{3} \cdot \frac{y}{\sqrt{x}}} \]

       2. lift-*.f64 N/A

          \[\leadsto 1 - \color{blue}{\frac{1}{3} \cdot \frac{y}{\sqrt{x}}} \]

       3. fp-cancel-sub-sign-inv N/A

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

       4. +-commutative N/A

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

       5. metadata-eval N/A

          \[\leadsto \color{blue}{\frac{-1}{3}} \cdot \frac{y}{\sqrt{x}} + 1 \]

       6. *-commutative N/A

          \[\leadsto \color{blue}{\frac{y}{\sqrt{x}} \cdot \frac{-1}{3}} + 1 \]

       7. lower-fma.f64 99.1

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\sqrt{x}}, -0.3333333333333333, 1\right)} \]

   11. Applied rewrites 99.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\sqrt{x}}, -0.3333333333333333, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 62.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{x - 0.1111111111111111}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (- x 0.1111111111111111) x))

C

double code(double x, double y) {
	return (x - 0.1111111111111111) / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x - 0.1111111111111111d0) / x
end function

Java

public static double code(double x, double y) {
	return (x - 0.1111111111111111) / x;
}

Python

def code(x, y):
	return (x - 0.1111111111111111) / x

Julia

function code(x, y)
	return Float64(Float64(x - 0.1111111111111111) / x)
end

MATLAB

function tmp = code(x, y)
	tmp = (x - 0.1111111111111111) / x;
end

Wolfram

code[x_, y_] := N[(N[(x - 0.1111111111111111), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. inv-pow N/A

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

   6. lower-pow.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 92.2%

   \[\leadsto \color{blue}{\frac{-0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333 \cdot \sqrt{x}, y, x\right)}{x}} \]

8. Taylor expanded in y around 0

   \[\leadsto \frac{x - \frac{1}{9}}{x} \]
9. Step-by-step derivation

10. Applied rewrites 60.7%

    \[\leadsto \frac{x - 0.1111111111111111}{x} \]

11. Final simplification 60.7%

    \[\leadsto \frac{x - 0.1111111111111111}{x} \]
12. Add Preprocessing

Alternative 13: 31.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \frac{-0.1111111111111111}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ -0.1111111111111111 x))

C

double code(double x, double y) {
	return -0.1111111111111111 / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (-0.1111111111111111d0) / x
end function

Java

public static double code(double x, double y) {
	return -0.1111111111111111 / x;
}

Python

def code(x, y):
	return -0.1111111111111111 / x

Julia

function code(x, y)
	return Float64(-0.1111111111111111 / x)
end

MATLAB

function tmp = code(x, y)
	tmp = -0.1111111111111111 / x;
end

Wolfram

code[x_, y_] := N[(-0.1111111111111111 / x), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)}{x}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 58.0

       \[\leadsto \frac{\mathsf{fma}\left(-0.3333333333333333, \color{blue}{\sqrt{x}} \cdot y, -0.1111111111111111\right)}{x} \]

5. Applied rewrites 58.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.3333333333333333, \sqrt{x} \cdot y, -0.1111111111111111\right)}{x}} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{\frac{-1}{9}}{x} \]
7. Step-by-step derivation

8. Applied rewrites 27.1%

   \[\leadsto \frac{-0.1111111111111111}{x} \]
9. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024329 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

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

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