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

Percentage Accurate: 99.7% → 99.7%

Time: 6.5s

Alternatives: 13

Speedup: 0.4×

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, 0.9× 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]

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. inv-pow N/A

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

   5. frac-2neg N/A

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

   6. lower-/.f64 N/A

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

   7. neg-mul-1 N/A

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

   8. inv-pow N/A

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

   9. un-div-inv N/A

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

   10. lower-/.f64 N/A

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

   11. metadata-eval 99.7

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

4. Applied rewrites 99.7%

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

Alternative 2: 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 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

   6. lift-/.f64 N/A

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

   7. inv-pow N/A

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

   8. lift-*.f64 N/A

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

   9. unpow-prod-down N/A

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

   10. distribute-lft-neg-in N/A

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

   11. lower-fma.f64 N/A

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

   12. neg-mul-1 N/A

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

   13. inv-pow N/A

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

   14. un-div-inv N/A

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

   15. lower-/.f64 N/A

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

   16. metadata-eval N/A

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

   17. lower--.f64 99.6

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 4: 94.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+34}:\\ \;\;\;\;1 - \frac{y \cdot 0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+68}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.2e+34)
   (- 1.0 (/ (* y 0.3333333333333333) (sqrt x)))
   (if (<= y 3.1e+68)
     (- 1.0 (/ 0.1111111111111111 x))
     (- 1.0 (/ y (* 3.0 (sqrt x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.2e+34) {
		tmp = 1.0 - ((y * 0.3333333333333333) / sqrt(x));
	} else if (y <= 3.1e+68) {
		tmp = 1.0 - (0.1111111111111111 / x);
	} else {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.2d+34)) then
        tmp = 1.0d0 - ((y * 0.3333333333333333d0) / sqrt(x))
    else if (y <= 3.1d+68) then
        tmp = 1.0d0 - (0.1111111111111111d0 / x)
    else
        tmp = 1.0d0 - (y / (3.0d0 * sqrt(x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.2e+34) {
		tmp = 1.0 - ((y * 0.3333333333333333) / Math.sqrt(x));
	} else if (y <= 3.1e+68) {
		tmp = 1.0 - (0.1111111111111111 / x);
	} else {
		tmp = 1.0 - (y / (3.0 * Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.2e+34:
		tmp = 1.0 - ((y * 0.3333333333333333) / math.sqrt(x))
	elif y <= 3.1e+68:
		tmp = 1.0 - (0.1111111111111111 / x)
	else:
		tmp = 1.0 - (y / (3.0 * math.sqrt(x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.2e+34)
		tmp = Float64(1.0 - Float64(Float64(y * 0.3333333333333333) / sqrt(x)));
	elseif (y <= 3.1e+68)
		tmp = Float64(1.0 - Float64(0.1111111111111111 / x));
	else
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.2e+34)
		tmp = 1.0 - ((y * 0.3333333333333333) / sqrt(x));
	elseif (y <= 3.1e+68)
		tmp = 1.0 - (0.1111111111111111 / x);
	else
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.2e+34], N[(1.0 - N[(N[(y * 0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.1e+68], N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $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 < -1.19999999999999993e34

   1. Initial program 99.3%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   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 85.2%

      \[\leadsto \color{blue}{1} - \frac{y}{3 \cdot \sqrt{x}} \]
   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. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 85.3

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

   7. Applied rewrites 85.3%

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

   if -1.19999999999999993e34 < y < 3.0999999999999998e68

   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. inv-pow N/A

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

      5. frac-2neg N/A

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

      6. lower-/.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. inv-pow N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x - \left(\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)}{x}} \]
   6. 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 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 98.0%

       \[\leadsto \frac{x - 0.1111111111111111}{x} \]
   11. Step-by-step derivation

   12. Applied rewrites 98.0%

       \[\leadsto 1 - \color{blue}{\frac{0.1111111111111111}{x}} \]

   if 3.0999999999999998e68 < y

   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 inf

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

   5. Applied rewrites 99.7%

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

Alternative 5: 94.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\sqrt{x}}, -0.3333333333333333, 1\right)\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+68}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.2e+34)
   (fma (/ y (sqrt x)) -0.3333333333333333 1.0)
   (if (<= y 3.1e+68)
     (- 1.0 (/ 0.1111111111111111 x))
     (- 1.0 (/ y (* 3.0 (sqrt x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.2e+34) {
		tmp = fma((y / sqrt(x)), -0.3333333333333333, 1.0);
	} else if (y <= 3.1e+68) {
		tmp = 1.0 - (0.1111111111111111 / x);
	} else {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.2e+34)
		tmp = fma(Float64(y / sqrt(x)), -0.3333333333333333, 1.0);
	elseif (y <= 3.1e+68)
		tmp = Float64(1.0 - Float64(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, -1.2e+34], N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision], If[LessEqual[y, 3.1e+68], N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $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 < -1.19999999999999993e34

   1. Initial program 99.3%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   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 85.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac N/A

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

      5. neg-mul-1 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. metadata-eval N/A

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

      9. associate-/l* N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. +-commutative N/A

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

   7. Applied rewrites 85.2%

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

   if -1.19999999999999993e34 < y < 3.0999999999999998e68

   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. inv-pow N/A

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

      5. frac-2neg N/A

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

      6. lower-/.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. inv-pow N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x - \left(\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)}{x}} \]
   6. 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 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 98.0%

       \[\leadsto \frac{x - 0.1111111111111111}{x} \]
   11. Step-by-step derivation

   12. Applied rewrites 98.0%

       \[\leadsto 1 - \color{blue}{\frac{0.1111111111111111}{x}} \]

   if 3.0999999999999998e68 < y

   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 inf

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

   5. Applied rewrites 99.7%

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

Alternative 6: 99.6% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 5e+15)
   (- 1.0 (/ (fma 0.3333333333333333 (* (sqrt x) y) 0.1111111111111111) x))
   (- 1.0 (/ y (* 3.0 (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5e+15) {
		tmp = 1.0 - (fma(0.3333333333333333, (sqrt(x) * y), 0.1111111111111111) / x);
	} else {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5e+15)
		tmp = Float64(1.0 - Float64(fma(0.3333333333333333, Float64(sqrt(x) * y), 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[x, 5e+15], N[(1.0 - N[(N[(0.3333333333333333 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] + 0.1111111111111111), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5e15

   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. inv-pow N/A

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

      5. frac-2neg N/A

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

      6. lower-/.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. inv-pow N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

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

   7. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{x - \mathsf{fma}\left(\sqrt{x} \cdot y, 0.3333333333333333, 0.1111111111111111\right)}{x}} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.5%

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

   if 5e15 < 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%

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

Alternative 7: 99.6% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 5e+15)
   (/ (- x (fma (* (sqrt x) y) 0.3333333333333333 0.1111111111111111)) x)
   (- 1.0 (/ y (* 3.0 (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5e+15) {
		tmp = (x - fma((sqrt(x) * y), 0.3333333333333333, 0.1111111111111111)) / x;
	} else {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5e+15)
		tmp = Float64(Float64(x - fma(Float64(sqrt(x) * y), 0.3333333333333333, 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[x, 5e+15], N[(N[(x - N[(N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] * 0.3333333333333333 + 0.1111111111111111), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5e15

   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 5e15 < 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%

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

Alternative 8: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.3333333333333333, \frac{y}{\sqrt{x}}, 1 - \frac{0.1111111111111111}{x}\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (fma -0.3333333333333333 (/ y (sqrt x)) (- 1.0 (/ 0.1111111111111111 x))))

C

double code(double x, double y) {
	return fma(-0.3333333333333333, (y / sqrt(x)), (1.0 - (0.1111111111111111 / x)));
}

Julia

function code(x, y)
	return fma(-0.3333333333333333, Float64(y / sqrt(x)), Float64(1.0 - Float64(0.1111111111111111 / x)))
end

Wolfram

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

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. distribute-neg-frac N/A

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

   6. neg-mul-1 N/A

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

   7. lift-*.f64 N/A

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

   8. times-frac N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. lower-/.f64 99.7

       \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \color{blue}{\frac{y}{\sqrt{x}}}, 1 - \frac{1}{x \cdot 9}\right) \]

   16. lift-/.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. associate-/r* N/A

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

   20. metadata-eval N/A

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

   21. metadata-eval N/A

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

   22. lower-/.f64 N/A

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

   23. metadata-eval 99.6

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

4. Applied rewrites 99.6%

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

Alternative 9: 94.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+34} \lor \neg \left(y \leq 3.1 \cdot 10^{+68}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\sqrt{x}}, -0.3333333333333333, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.2e+34) (not (<= y 3.1e+68)))
   (fma (/ y (sqrt x)) -0.3333333333333333 1.0)
   (- 1.0 (/ 0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.2e+34) || !(y <= 3.1e+68)) {
		tmp = fma((y / sqrt(x)), -0.3333333333333333, 1.0);
	} else {
		tmp = 1.0 - (0.1111111111111111 / x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.2e+34) || !(y <= 3.1e+68))
		tmp = fma(Float64(y / sqrt(x)), -0.3333333333333333, 1.0);
	else
		tmp = Float64(1.0 - Float64(0.1111111111111111 / x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.2e+34], N[Not[LessEqual[y, 3.1e+68]], $MachinePrecision]], N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision], N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.19999999999999993e34 or 3.0999999999999998e68 < y

   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 92.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac N/A

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

      5. neg-mul-1 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. metadata-eval N/A

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

      9. associate-/l* N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. +-commutative N/A

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

   7. Applied rewrites 91.9%

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

   if -1.19999999999999993e34 < y < 3.0999999999999998e68

   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. inv-pow N/A

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

      5. frac-2neg N/A

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

      6. lower-/.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. inv-pow N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x - \left(\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)}{x}} \]
   6. 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 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 98.0%

       \[\leadsto \frac{x - 0.1111111111111111}{x} \]
   11. Step-by-step derivation

   12. Applied rewrites 98.0%

       \[\leadsto 1 - \color{blue}{\frac{0.1111111111111111}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+34} \lor \neg \left(y \leq 3.1 \cdot 10^{+68}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\sqrt{x}}, -0.3333333333333333, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 94.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+34} \lor \neg \left(y \leq 3.1 \cdot 10^{+68}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.2e+34) (not (<= y 3.1e+68)))
   (fma (/ -0.3333333333333333 (sqrt x)) y 1.0)
   (- 1.0 (/ 0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.2e+34) || !(y <= 3.1e+68)) {
		tmp = fma((-0.3333333333333333 / sqrt(x)), y, 1.0);
	} else {
		tmp = 1.0 - (0.1111111111111111 / x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.2e+34) || !(y <= 3.1e+68))
		tmp = fma(Float64(-0.3333333333333333 / sqrt(x)), y, 1.0);
	else
		tmp = Float64(1.0 - Float64(0.1111111111111111 / x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.2e+34], N[Not[LessEqual[y, 3.1e+68]], $MachinePrecision]], N[(N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.19999999999999993e34 or 3.0999999999999998e68 < y

   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 92.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac N/A

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

      5. neg-mul-1 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. metadata-eval N/A

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

      9. associate-/l* N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 91.8

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

   7. Applied rewrites 91.8%

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

   if -1.19999999999999993e34 < y < 3.0999999999999998e68

   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. inv-pow N/A

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

      5. frac-2neg N/A

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

      6. lower-/.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. inv-pow N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x - \left(\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)}{x}} \]
   6. 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 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 98.0%

       \[\leadsto \frac{x - 0.1111111111111111}{x} \]
   11. Step-by-step derivation

   12. Applied rewrites 98.0%

       \[\leadsto 1 - \color{blue}{\frac{0.1111111111111111}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+34} \lor \neg \left(y \leq 3.1 \cdot 10^{+68}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.115:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.3333333333333333, \sqrt{x} \cdot y, -0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.115)
   (/ (fma -0.3333333333333333 (* (sqrt x) y) -0.1111111111111111) x)
   (- 1.0 (/ y (* 3.0 (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.115) {
		tmp = fma(-0.3333333333333333, (sqrt(x) * y), -0.1111111111111111) / x;
	} else {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.115)
		tmp = Float64(fma(-0.3333333333333333, Float64(sqrt(x) * y), -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[x, 0.115], N[(N[(-0.3333333333333333 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] + -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 2 regimes

2. if x < 0.115000000000000005

   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 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. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-neg-in N/A

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

      8. metadata-eval N/A

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

      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 97.8

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

   5. Applied rewrites 97.8%

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

   if 0.115000000000000005 < 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 98.4%

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

Alternative 12: 63.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{0.1111111111111111}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return 1.0 - (0.1111111111111111 / x);
}

Python

def code(x, y):
	return 1.0 - (0.1111111111111111 / x)

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(1.0 - N[(0.1111111111111111 / x), $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. inv-pow N/A

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

   5. frac-2neg N/A

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

   6. lower-/.f64 N/A

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

   7. neg-mul-1 N/A

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

   8. inv-pow N/A

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

   9. un-div-inv N/A

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

   10. lower-/.f64 N/A

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

   11. metadata-eval 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{x - \left(\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)}{x}} \]
6. 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 94.1

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

7. Applied rewrites 94.1%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 67.1%

    \[\leadsto \frac{x - 0.1111111111111111}{x} \]
11. Step-by-step derivation

12. Applied rewrites 67.1%

    \[\leadsto 1 - \color{blue}{\frac{0.1111111111111111}{x}} \]
13. Add Preprocessing

Alternative 13: 32.6% 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. mul-1-neg N/A

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

   2. distribute-neg-frac N/A

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

   3. lower-/.f64 N/A

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

   4. distribute-neg-in N/A

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

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

   7. distribute-lft-neg-in N/A

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

   8. metadata-eval N/A

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

   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 59.3

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

5. Applied rewrites 59.3%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 33.0%

   \[\leadsto \frac{-0.1111111111111111}{x} \]
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 2024318 
(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)))))