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

Percentage Accurate: 99.7% → 99.6%

Time: 7.4s

Alternatives: 19

Speedup: 1.0×

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: 99.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return (1.0 - (1.0 / (9.0 * 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 - (1.0d0 / (9.0d0 * x))) - ((y / sqrt(x)) / 3.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(1.0 / N[(9.0 * x), $MachinePrecision]), $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. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return (1.0 - (1.0 / (9.0 * x))) - (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 / (9.0d0 * x))) - (y / (3.0d0 * sqrt(x)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

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

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

   12. lower-fma.f64 N/A

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

   13. neg-mul-1 N/A

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

   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. Final simplification 99.6%

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

Alternative 4: 95.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+65}:\\ \;\;\;\;1 - \frac{0.3333333333333333 \cdot y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+46}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9.2e+65)
   (- 1.0 (/ (* 0.3333333333333333 y) (sqrt x)))
   (if (<= y 2.5e+46)
     (- 1.0 (/ 0.1111111111111111 x))
     (- 1.0 (/ y (* 3.0 (sqrt x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9.2e+65) {
		tmp = 1.0 - ((0.3333333333333333 * y) / sqrt(x));
	} else if (y <= 2.5e+46) {
		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 <= (-9.2d+65)) then
        tmp = 1.0d0 - ((0.3333333333333333d0 * y) / sqrt(x))
    else if (y <= 2.5d+46) 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 <= -9.2e+65) {
		tmp = 1.0 - ((0.3333333333333333 * y) / Math.sqrt(x));
	} else if (y <= 2.5e+46) {
		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 <= -9.2e+65:
		tmp = 1.0 - ((0.3333333333333333 * y) / math.sqrt(x))
	elif y <= 2.5e+46:
		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 <= -9.2e+65)
		tmp = Float64(1.0 - Float64(Float64(0.3333333333333333 * y) / sqrt(x)));
	elseif (y <= 2.5e+46)
		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 <= -9.2e+65)
		tmp = 1.0 - ((0.3333333333333333 * y) / sqrt(x));
	elseif (y <= 2.5e+46)
		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, -9.2e+65], N[(1.0 - N[(N[(0.3333333333333333 * y), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.5e+46], 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 < -9.2e65

   1. Initial program 99.4%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   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. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 97.7%

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

   if -9.2e65 < y < 2.5000000000000001e46

   1. Initial program 99.9%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   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 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 98.5%

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

   10. Applied rewrites 98.5%

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

   if 2.5000000000000001e46 < 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 93.0%

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

Alternative 5: 95.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{if}\;y \leq -9.2 \cdot 10^{+65}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+46}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y (* 3.0 (sqrt x))))))
   (if (<= y -9.2e+65)
     t_0
     (if (<= y 2.5e+46) (- 1.0 (/ 0.1111111111111111 x)) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (y / (3.0 * sqrt(x)));
	double tmp;
	if (y <= -9.2e+65) {
		tmp = t_0;
	} else if (y <= 2.5e+46) {
		tmp = 1.0 - (0.1111111111111111 / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (y / (3.0d0 * sqrt(x)))
    if (y <= (-9.2d+65)) then
        tmp = t_0
    else if (y <= 2.5d+46) then
        tmp = 1.0d0 - (0.1111111111111111d0 / x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - (y / (3.0 * Math.sqrt(x)));
	double tmp;
	if (y <= -9.2e+65) {
		tmp = t_0;
	} else if (y <= 2.5e+46) {
		tmp = 1.0 - (0.1111111111111111 / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (y / (3.0 * math.sqrt(x)))
	tmp = 0
	if y <= -9.2e+65:
		tmp = t_0
	elif y <= 2.5e+46:
		tmp = 1.0 - (0.1111111111111111 / x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))))
	tmp = 0.0
	if (y <= -9.2e+65)
		tmp = t_0;
	elseif (y <= 2.5e+46)
		tmp = Float64(1.0 - Float64(0.1111111111111111 / x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (y / (3.0 * sqrt(x)));
	tmp = 0.0;
	if (y <= -9.2e+65)
		tmp = t_0;
	elseif (y <= 2.5e+46)
		tmp = 1.0 - (0.1111111111111111 / x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9.2e+65], t$95$0, If[LessEqual[y, 2.5e+46], N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.2e65 or 2.5000000000000001e46 < 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 95.3%

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

   if -9.2e65 < y < 2.5000000000000001e46

   1. Initial program 99.9%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   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 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 98.5%

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

   10. Applied rewrites 98.5%

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

Alternative 6: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+28}:\\ \;\;\;\;\frac{x - \mathsf{fma}\left(0.3333333333333333 \cdot \sqrt{x}, y, 0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{y}{\sqrt{x}}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1e+28)
   (/ (- x (fma (* 0.3333333333333333 (sqrt x)) y 0.1111111111111111)) x)
   (- 1.0 (/ (/ y (sqrt x)) 3.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1e+28) {
		tmp = (x - fma((0.3333333333333333 * sqrt(x)), y, 0.1111111111111111)) / x;
	} else {
		tmp = 1.0 - ((y / sqrt(x)) / 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1e+28)
		tmp = Float64(Float64(x - fma(Float64(0.3333333333333333 * sqrt(x)), y, 0.1111111111111111)) / x);
	else
		tmp = Float64(1.0 - Float64(Float64(y / sqrt(x)) / 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1e+28], N[(N[(x - N[(N[(0.3333333333333333 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * y + 0.1111111111111111), $MachinePrecision]), $MachinePrecision] / x), $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 < 9.99999999999999958e27

   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

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

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

   7. Applied rewrites 99.5%

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

   if 9.99999999999999958e27 < x

   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-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.8%

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

Alternative 7: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+19}:\\ \;\;\;\;1 - \frac{\mathsf{fma}\left(0.3333333333333333, \sqrt{x} \cdot y, 0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot y, \sqrt{\frac{1}{x}}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2e+19)
   (- 1.0 (/ (fma 0.3333333333333333 (* (sqrt x) y) 0.1111111111111111) x))
   (fma (* -0.3333333333333333 y) (sqrt (/ 1.0 x)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2e+19) {
		tmp = 1.0 - (fma(0.3333333333333333, (sqrt(x) * y), 0.1111111111111111) / x);
	} else {
		tmp = fma((-0.3333333333333333 * y), sqrt((1.0 / x)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2e+19)
		tmp = Float64(1.0 - Float64(fma(0.3333333333333333, Float64(sqrt(x) * y), 0.1111111111111111) / x));
	else
		tmp = fma(Float64(-0.3333333333333333 * y), sqrt(Float64(1.0 / x)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2e+19], N[(1.0 - N[(N[(0.3333333333333333 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] + 0.1111111111111111), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(-0.3333333333333333 * y), $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e19

   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

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

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

   7. Applied rewrites 99.5%

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

   if 2e19 < x

   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

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

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

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

      8. lower-fma.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333 \cdot y, \sqrt{\frac{1}{x}}, 1\right)} \]
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: 95.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{y}{\sqrt{x}}, -0.3333333333333333, 1\right)\\ \mathbf{if}\;y \leq -9.2 \cdot 10^{+65}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+46}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (/ y (sqrt x)) -0.3333333333333333 1.0)))
   (if (<= y -9.2e+65)
     t_0
     (if (<= y 2.5e+46) (- 1.0 (/ 0.1111111111111111 x)) t_0))))

C

double code(double x, double y) {
	double t_0 = fma((y / sqrt(x)), -0.3333333333333333, 1.0);
	double tmp;
	if (y <= -9.2e+65) {
		tmp = t_0;
	} else if (y <= 2.5e+46) {
		tmp = 1.0 - (0.1111111111111111 / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(y / sqrt(x)), -0.3333333333333333, 1.0)
	tmp = 0.0
	if (y <= -9.2e+65)
		tmp = t_0;
	elseif (y <= 2.5e+46)
		tmp = Float64(1.0 - Float64(0.1111111111111111 / x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision]}, If[LessEqual[y, -9.2e+65], t$95$0, If[LessEqual[y, 2.5e+46], N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.2e65 or 2.5000000000000001e46 < 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 95.3%

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

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval 95.2

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

   7. Applied rewrites 95.2%

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

   if -9.2e65 < y < 2.5000000000000001e46

   1. Initial program 99.9%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   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 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 98.5%

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

   10. Applied rewrites 98.5%

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

Alternative 10: 92.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+79}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+123}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{-3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.2e+79)
   (/ (* -0.3333333333333333 y) (sqrt x))
   (if (<= y 1.55e+123)
     (- 1.0 (/ 0.1111111111111111 x))
     (/ y (* -3.0 (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.2e+79) {
		tmp = (-0.3333333333333333 * y) / sqrt(x);
	} else if (y <= 1.55e+123) {
		tmp = 1.0 - (0.1111111111111111 / x);
	} else {
		tmp = 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 <= (-2.2d+79)) then
        tmp = ((-0.3333333333333333d0) * y) / sqrt(x)
    else if (y <= 1.55d+123) then
        tmp = 1.0d0 - (0.1111111111111111d0 / x)
    else
        tmp = y / ((-3.0d0) * sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.2e+79) {
		tmp = (-0.3333333333333333 * y) / Math.sqrt(x);
	} else if (y <= 1.55e+123) {
		tmp = 1.0 - (0.1111111111111111 / x);
	} else {
		tmp = y / (-3.0 * Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.2e+79:
		tmp = (-0.3333333333333333 * y) / math.sqrt(x)
	elif y <= 1.55e+123:
		tmp = 1.0 - (0.1111111111111111 / x)
	else:
		tmp = y / (-3.0 * math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.2e+79)
		tmp = Float64(Float64(-0.3333333333333333 * y) / sqrt(x));
	elseif (y <= 1.55e+123)
		tmp = Float64(1.0 - Float64(0.1111111111111111 / x));
	else
		tmp = Float64(y / Float64(-3.0 * sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.2e+79)
		tmp = (-0.3333333333333333 * y) / sqrt(x);
	elseif (y <= 1.55e+123)
		tmp = 1.0 - (0.1111111111111111 / x);
	else
		tmp = y / (-3.0 * sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.2e+79], N[(N[(-0.3333333333333333 * y), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.55e+123], N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(y / N[(-3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.1999999999999999e79

   1. Initial program 99.4%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   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-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. 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. Taylor expanded in x around 0

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

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

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

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

      7. metadata-eval N/A

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

      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 77.1

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

   7. Applied rewrites 77.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 95.1%

       \[\leadsto \left(-0.3333333333333333 \cdot y\right) \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 95.2%

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

   if -2.1999999999999999e79 < y < 1.55000000000000003e123

   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 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 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 92.2%

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

   10. Applied rewrites 92.2%

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

   if 1.55000000000000003e123 < y

   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-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

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

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

      7. metadata-eval N/A

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

      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 73.9

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

   7. Applied rewrites 73.9%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 96.8%

       \[\leadsto \left(-0.3333333333333333 \cdot y\right) \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 97.0%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+79}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+123}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{-3 \cdot \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{-3 \cdot \sqrt{x}}\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{+79}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+123}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (* -3.0 (sqrt x)))))
   (if (<= y -2.2e+79)
     t_0
     (if (<= y 1.55e+123) (- 1.0 (/ 0.1111111111111111 x)) t_0))))

C

double code(double x, double y) {
	double t_0 = y / (-3.0 * sqrt(x));
	double tmp;
	if (y <= -2.2e+79) {
		tmp = t_0;
	} else if (y <= 1.55e+123) {
		tmp = 1.0 - (0.1111111111111111 / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / ((-3.0d0) * sqrt(x))
    if (y <= (-2.2d+79)) then
        tmp = t_0
    else if (y <= 1.55d+123) then
        tmp = 1.0d0 - (0.1111111111111111d0 / x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y / (-3.0 * Math.sqrt(x));
	double tmp;
	if (y <= -2.2e+79) {
		tmp = t_0;
	} else if (y <= 1.55e+123) {
		tmp = 1.0 - (0.1111111111111111 / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (-3.0 * math.sqrt(x))
	tmp = 0
	if y <= -2.2e+79:
		tmp = t_0
	elif y <= 1.55e+123:
		tmp = 1.0 - (0.1111111111111111 / x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(-3.0 * sqrt(x)))
	tmp = 0.0
	if (y <= -2.2e+79)
		tmp = t_0;
	elseif (y <= 1.55e+123)
		tmp = Float64(1.0 - Float64(0.1111111111111111 / x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (-3.0 * sqrt(x));
	tmp = 0.0;
	if (y <= -2.2e+79)
		tmp = t_0;
	elseif (y <= 1.55e+123)
		tmp = 1.0 - (0.1111111111111111 / x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(-3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.2e+79], t$95$0, If[LessEqual[y, 1.55e+123], N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.1999999999999999e79 or 1.55000000000000003e123 < y

   1. Initial program 99.5%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   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-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. 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. Taylor expanded in x around 0

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

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

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

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

      7. metadata-eval N/A

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

      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 75.7

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

   7. Applied rewrites 75.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 95.8%

       \[\leadsto \left(-0.3333333333333333 \cdot y\right) \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 95.9%

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

   if -2.1999999999999999e79 < y < 1.55000000000000003e123

   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 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 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 92.2%

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

   10. Applied rewrites 92.2%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+79}:\\ \;\;\;\;\frac{y}{-3 \cdot \sqrt{x}}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+123}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{-3 \cdot \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 91.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+79}:\\ \;\;\;\;\frac{-0.3333333333333333}{\sqrt{x}} \cdot y\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+123}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.2e+79)
   (* (/ -0.3333333333333333 (sqrt x)) y)
   (if (<= y 1.55e+123)
     (- 1.0 (/ 0.1111111111111111 x))
     (* -0.3333333333333333 (/ y (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.2e+79) {
		tmp = (-0.3333333333333333 / sqrt(x)) * y;
	} else if (y <= 1.55e+123) {
		tmp = 1.0 - (0.1111111111111111 / x);
	} else {
		tmp = -0.3333333333333333 * (y / 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 <= (-2.2d+79)) then
        tmp = ((-0.3333333333333333d0) / sqrt(x)) * y
    else if (y <= 1.55d+123) then
        tmp = 1.0d0 - (0.1111111111111111d0 / x)
    else
        tmp = (-0.3333333333333333d0) * (y / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.2e+79) {
		tmp = (-0.3333333333333333 / Math.sqrt(x)) * y;
	} else if (y <= 1.55e+123) {
		tmp = 1.0 - (0.1111111111111111 / x);
	} else {
		tmp = -0.3333333333333333 * (y / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.2e+79:
		tmp = (-0.3333333333333333 / math.sqrt(x)) * y
	elif y <= 1.55e+123:
		tmp = 1.0 - (0.1111111111111111 / x)
	else:
		tmp = -0.3333333333333333 * (y / math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.2e+79)
		tmp = Float64(Float64(-0.3333333333333333 / sqrt(x)) * y);
	elseif (y <= 1.55e+123)
		tmp = Float64(1.0 - Float64(0.1111111111111111 / x));
	else
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.2e+79)
		tmp = (-0.3333333333333333 / sqrt(x)) * y;
	elseif (y <= 1.55e+123)
		tmp = 1.0 - (0.1111111111111111 / x);
	else
		tmp = -0.3333333333333333 * (y / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.2e+79], N[(N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1.55e+123], N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.1999999999999999e79

   1. Initial program 99.4%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   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-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. 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. Taylor expanded in x around 0

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

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

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

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

      7. metadata-eval N/A

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

      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 77.1

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

   7. Applied rewrites 77.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 95.1%

       \[\leadsto \left(-0.3333333333333333 \cdot y\right) \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 95.0%

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

   if -2.1999999999999999e79 < y < 1.55000000000000003e123

   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 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 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 92.2%

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

   10. Applied rewrites 92.2%

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

   if 1.55000000000000003e123 < y

   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-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

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

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

      7. metadata-eval N/A

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

      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 73.9

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

   7. Applied rewrites 73.9%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 96.8%

       \[\leadsto \left(-0.3333333333333333 \cdot y\right) \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 96.7%

       \[\leadsto \color{blue}{\frac{y}{\sqrt{x}} \cdot -0.3333333333333333} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+79}:\\ \;\;\;\;\frac{-0.3333333333333333}{\sqrt{x}} \cdot y\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+123}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 91.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.3333333333333333}{\sqrt{x}} \cdot y\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{+79}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+123}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ -0.3333333333333333 (sqrt x)) y)))
   (if (<= y -2.2e+79)
     t_0
     (if (<= y 1.55e+123) (- 1.0 (/ 0.1111111111111111 x)) t_0))))

C

double code(double x, double y) {
	double t_0 = (-0.3333333333333333 / sqrt(x)) * y;
	double tmp;
	if (y <= -2.2e+79) {
		tmp = t_0;
	} else if (y <= 1.55e+123) {
		tmp = 1.0 - (0.1111111111111111 / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-0.3333333333333333d0) / sqrt(x)) * y
    if (y <= (-2.2d+79)) then
        tmp = t_0
    else if (y <= 1.55d+123) then
        tmp = 1.0d0 - (0.1111111111111111d0 / x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (-0.3333333333333333 / Math.sqrt(x)) * y;
	double tmp;
	if (y <= -2.2e+79) {
		tmp = t_0;
	} else if (y <= 1.55e+123) {
		tmp = 1.0 - (0.1111111111111111 / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (-0.3333333333333333 / math.sqrt(x)) * y
	tmp = 0
	if y <= -2.2e+79:
		tmp = t_0
	elif y <= 1.55e+123:
		tmp = 1.0 - (0.1111111111111111 / x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(-0.3333333333333333 / sqrt(x)) * y)
	tmp = 0.0
	if (y <= -2.2e+79)
		tmp = t_0;
	elseif (y <= 1.55e+123)
		tmp = Float64(1.0 - Float64(0.1111111111111111 / x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (-0.3333333333333333 / sqrt(x)) * y;
	tmp = 0.0;
	if (y <= -2.2e+79)
		tmp = t_0;
	elseif (y <= 1.55e+123)
		tmp = 1.0 - (0.1111111111111111 / x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -2.2e+79], t$95$0, If[LessEqual[y, 1.55e+123], N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.1999999999999999e79 or 1.55000000000000003e123 < y

   1. Initial program 99.5%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   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-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. 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. Taylor expanded in x around 0

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

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

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

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

      7. metadata-eval N/A

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

      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 75.7

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

   7. Applied rewrites 75.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 95.8%

       \[\leadsto \left(-0.3333333333333333 \cdot y\right) \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 95.7%

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

   if -2.1999999999999999e79 < y < 1.55000000000000003e123

   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 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 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 92.2%

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

   10. Applied rewrites 92.2%

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

Alternative 14: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.3333333333333333 \cdot \sqrt{x}, y, -0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot y, \sqrt{\frac{1}{x}}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.11)
   (/ (fma (* -0.3333333333333333 (sqrt x)) y -0.1111111111111111) x)
   (fma (* -0.3333333333333333 y) (sqrt (/ 1.0 x)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = fma((-0.3333333333333333 * sqrt(x)), y, -0.1111111111111111) / x;
	} else {
		tmp = fma((-0.3333333333333333 * y), sqrt((1.0 / x)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.11)
		tmp = Float64(fma(Float64(-0.3333333333333333 * sqrt(x)), y, -0.1111111111111111) / x);
	else
		tmp = fma(Float64(-0.3333333333333333 * y), sqrt(Float64(1.0 / x)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.11], N[(N[(N[(-0.3333333333333333 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * y + -0.1111111111111111), $MachinePrecision] / x), $MachinePrecision], N[(N[(-0.3333333333333333 * y), $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   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-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. 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. Taylor expanded in x around 0

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

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

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

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

      7. metadata-eval N/A

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

      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 98.6

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

   7. Applied rewrites 98.6%

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

   9. Applied rewrites 98.6%

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

   if 0.110000000000000001 < 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

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

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

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

      8. lower-fma.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 99.2

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

   5. Applied rewrites 99.2%

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

Alternative 15: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.3333333333333333, \sqrt{x} \cdot y, -0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot y, \sqrt{\frac{1}{x}}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.11)
   (/ (fma -0.3333333333333333 (* (sqrt x) y) -0.1111111111111111) x)
   (fma (* -0.3333333333333333 y) (sqrt (/ 1.0 x)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = fma(-0.3333333333333333, (sqrt(x) * y), -0.1111111111111111) / x;
	} else {
		tmp = fma((-0.3333333333333333 * y), sqrt((1.0 / x)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.11)
		tmp = Float64(fma(-0.3333333333333333, Float64(sqrt(x) * y), -0.1111111111111111) / x);
	else
		tmp = fma(Float64(-0.3333333333333333 * y), sqrt(Float64(1.0 / x)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.11], N[(N[(-0.3333333333333333 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] + -0.1111111111111111), $MachinePrecision] / x), $MachinePrecision], N[(N[(-0.3333333333333333 * y), $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   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. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. metadata-eval N/A

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

      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 98.6

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

   5. Applied rewrites 98.6%

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

   if 0.110000000000000001 < 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

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

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

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

      8. lower-fma.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 99.2

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

   5. Applied rewrites 99.2%

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

Alternative 16: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.3333333333333333, \sqrt{x} \cdot y, -0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.11)
   (/ (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 <= 0.11) {
		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 <= 0.11)
		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, 0.11], 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 < 0.110000000000000001

   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. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. metadata-eval N/A

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

      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 98.6

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

   5. Applied rewrites 98.6%

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

   if 0.110000000000000001 < 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 99.2%

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

Alternative 17: 64.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.1 \cdot 10^{+153}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 0.1111111111111111}{x \cdot x} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.1e+153)
   (- 1.0 (/ 0.1111111111111111 x))
   (* (/ (- x 0.1111111111111111) (* x x)) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.1e+153) {
		tmp = 1.0 - (0.1111111111111111 / x);
	} else {
		tmp = ((x - 0.1111111111111111) / (x * x)) * 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 <= 5.1d+153) then
        tmp = 1.0d0 - (0.1111111111111111d0 / x)
    else
        tmp = ((x - 0.1111111111111111d0) / (x * x)) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5.1e+153) {
		tmp = 1.0 - (0.1111111111111111 / x);
	} else {
		tmp = ((x - 0.1111111111111111) / (x * x)) * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.1e+153:
		tmp = 1.0 - (0.1111111111111111 / x)
	else:
		tmp = ((x - 0.1111111111111111) / (x * x)) * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.1e+153)
		tmp = Float64(1.0 - Float64(0.1111111111111111 / x));
	else
		tmp = Float64(Float64(Float64(x - 0.1111111111111111) / Float64(x * x)) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.1e+153)
		tmp = 1.0 - (0.1111111111111111 / x);
	else
		tmp = ((x - 0.1111111111111111) / (x * x)) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.1e+153], N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - 0.1111111111111111), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.10000000000000035e153

   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

      \[\leadsto \color{blue}{\frac{x - \left(\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)}{x}} \]
   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 95.9

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 71.5%

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

   10. Applied rewrites 71.6%

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

   if 5.10000000000000035e153 < 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 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 71.2

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

   5. Applied rewrites 71.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 3.4%

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

   10. Applied rewrites 18.4%

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

   12. Applied rewrites 19.0%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.1 \cdot 10^{+153}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 0.1111111111111111}{x \cdot x} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 62.5% 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. 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 93.0

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

5. Applied rewrites 93.0%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 63.6%

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

10. Applied rewrites 63.6%

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

Alternative 19: 32.0% 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. 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-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in x around 0

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

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

   4. +-commutative N/A

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

   5. distribute-neg-in N/A

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

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

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

   7. metadata-eval N/A

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

   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 60.0

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

7. Applied rewrites 60.0%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 32.2%

    \[\leadsto \frac{-0.1111111111111111}{x} \]
11. 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 2024296 
(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)))))