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

Percentage Accurate: 99.7% → 99.7%

Time: 10.0s

Alternatives: 17

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: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(1.0 + N[(-1.0 / N[(x * 9.0), $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 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 61.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{-1}{x \cdot 9}\right) - \frac{y}{\sqrt{x} \cdot 3} \leq -5000:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (+ 1.0 (/ -1.0 (* x 9.0))) (/ y (* (sqrt x) 3.0))) -5000.0)
   (/ -0.1111111111111111 x)
   1.0))

C

double code(double x, double y) {
	double tmp;
	if (((1.0 + (-1.0 / (x * 9.0))) - (y / (sqrt(x) * 3.0))) <= -5000.0) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((1.0d0 + ((-1.0d0) / (x * 9.0d0))) - (y / (sqrt(x) * 3.0d0))) <= (-5000.0d0)) then
        tmp = (-0.1111111111111111d0) / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((1.0 + (-1.0 / (x * 9.0))) - (y / (Math.sqrt(x) * 3.0))) <= -5000.0) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((1.0 + (-1.0 / (x * 9.0))) - (y / (math.sqrt(x) * 3.0))) <= -5000.0:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(1.0 + Float64(-1.0 / Float64(x * 9.0))) - Float64(y / Float64(sqrt(x) * 3.0))) <= -5000.0)
		tmp = Float64(-0.1111111111111111 / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((1.0 + (-1.0 / (x * 9.0))) - (y / (sqrt(x) * 3.0))) <= -5000.0)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y / N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5000.0], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) (/.f64 y (*.f64 #s(literal 3 binary64) (sqrt.f64 x)))) < -5e3

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 95.1

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

   5. Applied rewrites 95.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 60.2%

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

   if -5e3 < (-.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) (/.f64 y (*.f64 #s(literal 3 binary64) (sqrt.f64 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 y around 0

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 58.5

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

   5. Applied rewrites 58.5%

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

   6. Taylor expanded in x around inf

      \[\leadsto 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 58.1%

      \[\leadsto 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{-1}{x \cdot 9}\right) - \frac{y}{\sqrt{x} \cdot 3} \leq -5000:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Final simplification 99.7%

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

Alternative 4: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - \frac{0.3333333333333333}{x \cdot 3}\right) - \frac{y}{\sqrt{x} \cdot 3} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (- (- 1.0 (/ 0.3333333333333333 (* x 3.0))) (/ y (* (sqrt x) 3.0))))

C

double code(double x, double y) {
	return (1.0 - (0.3333333333333333 / (x * 3.0))) - (y / (sqrt(x) * 3.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - (0.3333333333333333d0 / (x * 3.0d0))) - (y / (sqrt(x) * 3.0d0))
end function

Java

public static double code(double x, double y) {
	return (1.0 - (0.3333333333333333 / (x * 3.0))) - (y / (Math.sqrt(x) * 3.0));
}

Python

def code(x, y):
	return (1.0 - (0.3333333333333333 / (x * 3.0))) - (y / (math.sqrt(x) * 3.0))

Julia

function code(x, y)
	return Float64(Float64(1.0 - Float64(0.3333333333333333 / Float64(x * 3.0))) - Float64(y / Float64(sqrt(x) * 3.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - (0.3333333333333333 / (x * 3.0))) - (y / (sqrt(x) * 3.0));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(0.3333333333333333 / N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y / N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. lift-*.f64 N/A

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

   2. rem-square-sqrt N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. metadata-eval N/A

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

   6. swap-sqr N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. lift-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. associate-*r* N/A

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

   18. lift-sqrt.f64 N/A

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

   19. lift-sqrt.f64 N/A

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

   20. rem-square-sqrt N/A

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

   21. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. metadata-eval 99.6

      \[\leadsto \left(1 - \frac{\color{blue}{0.3333333333333333}}{x \cdot 3}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

6. Applied rewrites 99.6%

   \[\leadsto \left(1 - \color{blue}{\frac{0.3333333333333333}{x \cdot 3}}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

7. Final simplification 99.6%

   \[\leadsto \left(1 - \frac{0.3333333333333333}{x \cdot 3}\right) - \frac{y}{\sqrt{x} \cdot 3} \]
8. Add Preprocessing

Alternative 5: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

   6. lift-/.f64 N/A

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

   7. inv-pow N/A

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

   8. lift-*.f64 N/A

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

   9. unpow-prod-down N/A

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

   10. 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-rgt-neg-in N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lower--.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 6: 94.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+47}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x} \cdot 3}\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{+75}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{0.3333333333333333}{\sqrt{x}}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.6e+47)
   (- 1.0 (/ y (* (sqrt x) 3.0)))
   (if (<= y 5.7e+75)
     (+ 1.0 (/ 1.0 (* x -9.0)))
     (fma (- y) (/ 0.3333333333333333 (sqrt x)) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.6e+47) {
		tmp = 1.0 - (y / (sqrt(x) * 3.0));
	} else if (y <= 5.7e+75) {
		tmp = 1.0 + (1.0 / (x * -9.0));
	} else {
		tmp = fma(-y, (0.3333333333333333 / sqrt(x)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.6e+47)
		tmp = Float64(1.0 - Float64(y / Float64(sqrt(x) * 3.0)));
	elseif (y <= 5.7e+75)
		tmp = Float64(1.0 + Float64(1.0 / Float64(x * -9.0)));
	else
		tmp = fma(Float64(-y), Float64(0.3333333333333333 / sqrt(x)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.6e+47], N[(1.0 - N[(y / N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.7e+75], N[(1.0 + N[(1.0 / N[(x * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-y) * N[(0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.60000000000000008e47

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 91.5%

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

   if -3.60000000000000008e47 < y < 5.7000000000000004e75

   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 y around 0

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 97.9

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

   5. Applied rewrites 97.9%

      \[\leadsto \color{blue}{1 + \frac{-0.1111111111111111}{x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.0%

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

   if 5.7000000000000004e75 < 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 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around inf

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

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

      7. associate-*r* N/A

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

      8. rem-square-sqrt N/A

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

      9. unpow2 N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

   7. Applied rewrites 97.8%

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

   9. Applied rewrites 97.9%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+47}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x} \cdot 3}\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{+75}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{0.3333333333333333}{\sqrt{x}}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-y, \frac{0.3333333333333333}{\sqrt{x}}, 1\right)\\ \mathbf{if}\;y \leq -3.6 \cdot 10^{+47}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{+75}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (- y) (/ 0.3333333333333333 (sqrt x)) 1.0)))
   (if (<= y -3.6e+47)
     t_0
     (if (<= y 5.7e+75) (+ 1.0 (/ 1.0 (* x -9.0))) t_0))))

C

double code(double x, double y) {
	double t_0 = fma(-y, (0.3333333333333333 / sqrt(x)), 1.0);
	double tmp;
	if (y <= -3.6e+47) {
		tmp = t_0;
	} else if (y <= 5.7e+75) {
		tmp = 1.0 + (1.0 / (x * -9.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(-y), Float64(0.3333333333333333 / sqrt(x)), 1.0)
	tmp = 0.0
	if (y <= -3.6e+47)
		tmp = t_0;
	elseif (y <= 5.7e+75)
		tmp = Float64(1.0 + Float64(1.0 / Float64(x * -9.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[((-y) * N[(0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, -3.6e+47], t$95$0, If[LessEqual[y, 5.7e+75], N[(1.0 + N[(1.0 / N[(x * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.60000000000000008e47 or 5.7000000000000004e75 < 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 99.6

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

   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 inf

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

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

      7. associate-*r* N/A

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

      8. rem-square-sqrt N/A

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

      9. unpow2 N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

   7. Applied rewrites 94.2%

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

   9. Applied rewrites 94.3%

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

   if -3.60000000000000008e47 < y < 5.7000000000000004e75

   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 y around 0

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 97.9

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

   5. Applied rewrites 97.9%

      \[\leadsto \color{blue}{1 + \frac{-0.1111111111111111}{x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.0%

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

Alternative 8: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * y + 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. clear-num N/A

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

   6. associate-/r/ N/A

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

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

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

   8. distribute-frac-neg2 N/A

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

   9. lower-fma.f64 N/A

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

   10. distribute-frac-neg2 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/r* N/A

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

   13. distribute-neg-frac N/A

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

   14. lower-/.f64 N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval 99.7

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

   17. lift--.f64 N/A

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

   18. sub-neg N/A

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

   19. lower-+.f64 N/A

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

   20. lift-/.f64 N/A

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

   21. lift-*.f64 N/A

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

   22. *-commutative N/A

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

4. Applied rewrites 99.6%

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

Alternative 9: 94.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+47}:\\ \;\;\;\;1 + \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{+75}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\sqrt{x}}, -0.3333333333333333, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.6e+47)
   (+ 1.0 (/ (* y -0.3333333333333333) (sqrt x)))
   (if (<= y 5.7e+75)
     (+ 1.0 (/ 1.0 (* x -9.0)))
     (fma (/ y (sqrt x)) -0.3333333333333333 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.6e+47) {
		tmp = 1.0 + ((y * -0.3333333333333333) / sqrt(x));
	} else if (y <= 5.7e+75) {
		tmp = 1.0 + (1.0 / (x * -9.0));
	} else {
		tmp = fma((y / sqrt(x)), -0.3333333333333333, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.6e+47)
		tmp = Float64(1.0 + Float64(Float64(y * -0.3333333333333333) / sqrt(x)));
	elseif (y <= 5.7e+75)
		tmp = Float64(1.0 + Float64(1.0 / Float64(x * -9.0)));
	else
		tmp = fma(Float64(y / sqrt(x)), -0.3333333333333333, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.6e+47], N[(1.0 + N[(N[(y * -0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.7e+75], N[(1.0 + N[(1.0 / N[(x * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.60000000000000008e47

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      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 inf

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

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

      7. associate-*r* N/A

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

      8. rem-square-sqrt N/A

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

      9. unpow2 N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

   7. Applied rewrites 91.2%

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

   9. Applied rewrites 91.3%

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

   if -3.60000000000000008e47 < y < 5.7000000000000004e75

   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 y around 0

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 97.9

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

   5. Applied rewrites 97.9%

      \[\leadsto \color{blue}{1 + \frac{-0.1111111111111111}{x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.0%

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

   if 5.7000000000000004e75 < 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 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around inf

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

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

      7. associate-*r* N/A

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

      8. rem-square-sqrt N/A

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

      9. unpow2 N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

   7. Applied rewrites 97.8%

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

   9. Applied rewrites 97.8%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+47}:\\ \;\;\;\;1 + \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{+75}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\sqrt{x}}, -0.3333333333333333, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 94.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{y}{\sqrt{x}}, -0.3333333333333333, 1\right)\\ \mathbf{if}\;y \leq -3.6 \cdot 10^{+47}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{+75}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (/ y (sqrt x)) -0.3333333333333333 1.0)))
   (if (<= y -3.6e+47)
     t_0
     (if (<= y 5.7e+75) (+ 1.0 (/ 1.0 (* x -9.0))) t_0))))

C

double code(double x, double y) {
	double t_0 = fma((y / sqrt(x)), -0.3333333333333333, 1.0);
	double tmp;
	if (y <= -3.6e+47) {
		tmp = t_0;
	} else if (y <= 5.7e+75) {
		tmp = 1.0 + (1.0 / (x * -9.0));
	} 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 <= -3.6e+47)
		tmp = t_0;
	elseif (y <= 5.7e+75)
		tmp = Float64(1.0 + Float64(1.0 / Float64(x * -9.0)));
	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, -3.6e+47], t$95$0, If[LessEqual[y, 5.7e+75], N[(1.0 + N[(1.0 / N[(x * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.60000000000000008e47 or 5.7000000000000004e75 < 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 99.6

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

   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 inf

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

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

      7. associate-*r* N/A

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

      8. rem-square-sqrt N/A

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

      9. unpow2 N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

   7. Applied rewrites 94.2%

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

   9. Applied rewrites 94.2%

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

   if -3.60000000000000008e47 < y < 5.7000000000000004e75

   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 y around 0

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 97.9

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

   5. Applied rewrites 97.9%

      \[\leadsto \color{blue}{1 + \frac{-0.1111111111111111}{x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.0%

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

Alternative 11: 92.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{if}\;y \leq -1.4 \cdot 10^{+48}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{+104}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (* (sqrt x) -3.0))))
   (if (<= y -1.4e+48)
     t_0
     (if (<= y 9.6e+104) (+ 1.0 (/ 1.0 (* x -9.0))) t_0))))

C

double code(double x, double y) {
	double t_0 = y / (sqrt(x) * -3.0);
	double tmp;
	if (y <= -1.4e+48) {
		tmp = t_0;
	} else if (y <= 9.6e+104) {
		tmp = 1.0 + (1.0 / (x * -9.0));
	} 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 / (sqrt(x) * (-3.0d0))
    if (y <= (-1.4d+48)) then
        tmp = t_0
    else if (y <= 9.6d+104) then
        tmp = 1.0d0 + (1.0d0 / (x * (-9.0d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y / (Math.sqrt(x) * -3.0);
	double tmp;
	if (y <= -1.4e+48) {
		tmp = t_0;
	} else if (y <= 9.6e+104) {
		tmp = 1.0 + (1.0 / (x * -9.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (math.sqrt(x) * -3.0)
	tmp = 0
	if y <= -1.4e+48:
		tmp = t_0
	elif y <= 9.6e+104:
		tmp = 1.0 + (1.0 / (x * -9.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(sqrt(x) * -3.0))
	tmp = 0.0
	if (y <= -1.4e+48)
		tmp = t_0;
	elseif (y <= 9.6e+104)
		tmp = Float64(1.0 + Float64(1.0 / Float64(x * -9.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (sqrt(x) * -3.0);
	tmp = 0.0;
	if (y <= -1.4e+48)
		tmp = t_0;
	elseif (y <= 9.6e+104)
		tmp = 1.0 + (1.0 / (x * -9.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.4e+48], t$95$0, If[LessEqual[y, 9.6e+104], N[(1.0 + N[(1.0 / N[(x * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.40000000000000006e48 or 9.6e104 < 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

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

      2. rem-square-sqrt N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. metadata-eval N/A

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

      6. swap-sqr N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. lift-sqrt.f64 N/A

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

      19. lift-sqrt.f64 N/A

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

      20. rem-square-sqrt N/A

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

      21. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. rem-square-sqrt N/A

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

      7. unpow2 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. rem-square-sqrt N/A

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

      15. associate-*r* N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 90.0

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

   7. Applied rewrites 90.0%

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

   9. Applied rewrites 90.1%

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

   if -1.40000000000000006e48 < y < 9.6e104

   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 y around 0

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 96.6

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

   5. Applied rewrites 96.6%

      \[\leadsto \color{blue}{1 + \frac{-0.1111111111111111}{x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 96.7%

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

Alternative 12: 91.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{\sqrt{x}} \cdot -0.3333333333333333\\ \mathbf{if}\;y \leq -1.4 \cdot 10^{+48}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{+104}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ y (sqrt x)) -0.3333333333333333)))
   (if (<= y -1.4e+48)
     t_0
     (if (<= y 9.6e+104) (+ 1.0 (/ 1.0 (* x -9.0))) t_0))))

C

double code(double x, double y) {
	double t_0 = (y / sqrt(x)) * -0.3333333333333333;
	double tmp;
	if (y <= -1.4e+48) {
		tmp = t_0;
	} else if (y <= 9.6e+104) {
		tmp = 1.0 + (1.0 / (x * -9.0));
	} 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 / sqrt(x)) * (-0.3333333333333333d0)
    if (y <= (-1.4d+48)) then
        tmp = t_0
    else if (y <= 9.6d+104) then
        tmp = 1.0d0 + (1.0d0 / (x * (-9.0d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y / Math.sqrt(x)) * -0.3333333333333333;
	double tmp;
	if (y <= -1.4e+48) {
		tmp = t_0;
	} else if (y <= 9.6e+104) {
		tmp = 1.0 + (1.0 / (x * -9.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y / math.sqrt(x)) * -0.3333333333333333
	tmp = 0
	if y <= -1.4e+48:
		tmp = t_0
	elif y <= 9.6e+104:
		tmp = 1.0 + (1.0 / (x * -9.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y / sqrt(x)) * -0.3333333333333333)
	tmp = 0.0
	if (y <= -1.4e+48)
		tmp = t_0;
	elseif (y <= 9.6e+104)
		tmp = Float64(1.0 + Float64(1.0 / Float64(x * -9.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y / sqrt(x)) * -0.3333333333333333;
	tmp = 0.0;
	if (y <= -1.4e+48)
		tmp = t_0;
	elseif (y <= 9.6e+104)
		tmp = 1.0 + (1.0 / (x * -9.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]}, If[LessEqual[y, -1.4e+48], t$95$0, If[LessEqual[y, 9.6e+104], N[(1.0 + N[(1.0 / N[(x * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.40000000000000006e48 or 9.6e104 < 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

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

      2. rem-square-sqrt N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. metadata-eval N/A

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

      6. swap-sqr N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. lift-sqrt.f64 N/A

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

      19. lift-sqrt.f64 N/A

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

      20. rem-square-sqrt N/A

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

      21. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. rem-square-sqrt N/A

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

      7. unpow2 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. rem-square-sqrt N/A

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

      15. associate-*r* N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 90.0

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

   7. Applied rewrites 90.0%

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

   9. Applied rewrites 90.0%

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

   if -1.40000000000000006e48 < y < 9.6e104

   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 y around 0

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 96.6

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

   5. Applied rewrites 96.6%

      \[\leadsto \color{blue}{1 + \frac{-0.1111111111111111}{x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 96.7%

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

Alternative 13: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0004:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot \sqrt{x}, -0.3333333333333333, -0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x} \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.0004)
   (/ (fma (* y (sqrt x)) -0.3333333333333333 -0.1111111111111111) x)
   (- 1.0 (/ y (* (sqrt x) 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.0004) {
		tmp = fma((y * sqrt(x)), -0.3333333333333333, -0.1111111111111111) / x;
	} else {
		tmp = 1.0 - (y / (sqrt(x) * 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.0004)
		tmp = Float64(fma(Float64(y * sqrt(x)), -0.3333333333333333, -0.1111111111111111) / x);
	else
		tmp = Float64(1.0 - Float64(y / Float64(sqrt(x) * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.0004], N[(N[(N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333 + -0.1111111111111111), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - N[(y / N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000019e-4

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 98.6

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

   5. Applied rewrites 98.6%

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

   7. Applied rewrites 98.6%

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

   if 4.00000000000000019e-4 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 98.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0004:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot \sqrt{x}, -0.3333333333333333, -0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x} \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0004:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{x}, y \cdot -0.3333333333333333, -0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x} \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.0004)
   (/ (fma (sqrt x) (* y -0.3333333333333333) -0.1111111111111111) x)
   (- 1.0 (/ y (* (sqrt x) 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.0004) {
		tmp = fma(sqrt(x), (y * -0.3333333333333333), -0.1111111111111111) / x;
	} else {
		tmp = 1.0 - (y / (sqrt(x) * 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.0004)
		tmp = Float64(fma(sqrt(x), Float64(y * -0.3333333333333333), -0.1111111111111111) / x);
	else
		tmp = Float64(1.0 - Float64(y / Float64(sqrt(x) * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.0004], N[(N[(N[Sqrt[x], $MachinePrecision] * N[(y * -0.3333333333333333), $MachinePrecision] + -0.1111111111111111), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - N[(y / N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000019e-4

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if 4.00000000000000019e-4 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 98.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0004:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{x}, y \cdot -0.3333333333333333, -0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x} \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 62.1% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in y around 0

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

   1. sub-neg N/A

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

   2. lower-+.f64 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 59.6

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

5. Applied rewrites 59.6%

   \[\leadsto \color{blue}{1 + \frac{-0.1111111111111111}{x}} \]
6. Step-by-step derivation

7. Applied rewrites 59.7%

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

Alternative 16: 62.0% 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 y around 0

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

   1. sub-neg N/A

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

   2. lower-+.f64 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 59.6

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

5. Applied rewrites 59.6%

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

Alternative 17: 31.3% accurate, 49.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

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 y around 0

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

   1. sub-neg N/A

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

   2. lower-+.f64 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 59.6

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

5. Applied rewrites 59.6%

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

6. Taylor expanded in x around inf

   \[\leadsto 1 \]
7. Step-by-step derivation

8. Applied rewrites 30.0%

   \[\leadsto 1 \]
9. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024220 
(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)))))