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

Percentage Accurate: 99.7% → 99.6%

Time: 9.9s

Alternatives: 14

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. distribute-neg-frac N/A

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

   6. neg-mul-1 N/A

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

   7. lift-*.f64 N/A

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

   8. times-frac N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. lower-/.f64 99.7

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

   16. lift--.f64 N/A

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

   17. sub-neg N/A

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

   18. lower-+.f64 N/A

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

   19. lift-/.f64 N/A

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

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. associate-/r* N/A

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

   23. metadata-eval N/A

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

   24. metadata-eval N/A

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

4. Applied rewrites 99.7%

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

Alternative 2: 61.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{-1}{x \cdot 9}\right) - \frac{y}{\sqrt{x} \cdot 3} \leq -0.05:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (+ 1.0 (/ -1.0 (* x 9.0))) (/ y (* (sqrt x) 3.0))) -0.05)
   (/ -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))) <= -0.05) {
		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))) <= (-0.05d0)) 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))) <= -0.05) {
		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))) <= -0.05:
		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))) <= -0.05)
		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))) <= -0.05)
		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], -0.05], 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)))) < -0.050000000000000003

   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 93.9

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 61.2%

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

   if -0.050000000000000003 < (-.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 62.1

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

   5. Applied rewrites 62.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 61.7%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{-1}{x \cdot 9}\right) - \frac{y}{\sqrt{x} \cdot 3} \leq -0.05:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, 1\right)\\ \mathbf{elif}\;y \leq 1.26 \cdot 10^{+29}:\\ \;\;\;\;1 + \frac{\frac{1}{x}}{-9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x} \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1e+73)
   (fma (/ -0.3333333333333333 (sqrt x)) y 1.0)
   (if (<= y 1.26e+29)
     (+ 1.0 (/ (/ 1.0 x) -9.0))
     (- 1.0 (/ y (* (sqrt x) 3.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1e+73) {
		tmp = fma((-0.3333333333333333 / sqrt(x)), y, 1.0);
	} else if (y <= 1.26e+29) {
		tmp = 1.0 + ((1.0 / x) / -9.0);
	} else {
		tmp = 1.0 - (y / (sqrt(x) * 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1e+73)
		tmp = fma(Float64(-0.3333333333333333 / sqrt(x)), y, 1.0);
	elseif (y <= 1.26e+29)
		tmp = Float64(1.0 + Float64(Float64(1.0 / x) / -9.0));
	else
		tmp = Float64(1.0 - Float64(y / Float64(sqrt(x) * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1e+73], N[(N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision], If[LessEqual[y, 1.26e+29], N[(1.0 + N[(N[(1.0 / x), $MachinePrecision] / -9.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y / N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.99999999999999983e72

   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

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

      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 99.7%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.4%

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

   if -9.99999999999999983e72 < y < 1.26e29

   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 99.0

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

   5. Applied rewrites 99.0%

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

   7. Applied rewrites 99.1%

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

   if 1.26e29 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 93.1%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, 1\right)\\ \mathbf{elif}\;y \leq 1.26 \cdot 10^{+29}:\\ \;\;\;\;1 + \frac{\frac{1}{x}}{-9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x} \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.3333333333333333, y \cdot \sqrt{x}, x + -0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{\sqrt{x}}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.5e+15)
   (/ (fma -0.3333333333333333 (* y (sqrt x)) (+ x -0.1111111111111111)) x)
   (fma -0.3333333333333333 (/ y (sqrt x)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.5e+15) {
		tmp = fma(-0.3333333333333333, (y * sqrt(x)), (x + -0.1111111111111111)) / x;
	} else {
		tmp = fma(-0.3333333333333333, (y / sqrt(x)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.5e+15)
		tmp = Float64(fma(-0.3333333333333333, Float64(y * sqrt(x)), Float64(x + -0.1111111111111111)) / x);
	else
		tmp = fma(-0.3333333333333333, Float64(y / sqrt(x)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.5e+15], N[(N[(-0.3333333333333333 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(x + -0.1111111111111111), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5e15

   1. Initial program 99.5%

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

   4. Applied rewrites 47.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{\left(x \cdot -9\right) \cdot \left(x \cdot \left(x \cdot 81\right)\right)} + 1}{\mathsf{fma}\left(\frac{0.1111111111111111}{x}, 1 + \frac{0.1111111111111111}{x}, 1\right)}} - \frac{y}{3 \cdot \sqrt{x}} \]

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 47.4%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate--r+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 99.4

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

   10. Applied rewrites 99.4%

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

   if 2.5e15 < x

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. neg-mul-1 N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 99.8

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

      18. lower-+.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. associate-/r* N/A

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

      23. metadata-eval N/A

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

      24. metadata-eval N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.3333333333333333, y \cdot \sqrt{x}, x + -0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{\sqrt{x}}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, 1\right)\\ \mathbf{elif}\;y \leq 1.26 \cdot 10^{+29}:\\ \;\;\;\;1 + \frac{\frac{1}{x}}{-9}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{\sqrt{x}}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1e+73)
   (fma (/ -0.3333333333333333 (sqrt x)) y 1.0)
   (if (<= y 1.26e+29)
     (+ 1.0 (/ (/ 1.0 x) -9.0))
     (fma -0.3333333333333333 (/ y (sqrt x)) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1e+73) {
		tmp = fma((-0.3333333333333333 / sqrt(x)), y, 1.0);
	} else if (y <= 1.26e+29) {
		tmp = 1.0 + ((1.0 / x) / -9.0);
	} else {
		tmp = fma(-0.3333333333333333, (y / sqrt(x)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1e+73)
		tmp = fma(Float64(-0.3333333333333333 / sqrt(x)), y, 1.0);
	elseif (y <= 1.26e+29)
		tmp = Float64(1.0 + Float64(Float64(1.0 / x) / -9.0));
	else
		tmp = fma(-0.3333333333333333, Float64(y / sqrt(x)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1e+73], N[(N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision], If[LessEqual[y, 1.26e+29], N[(1.0 + N[(N[(1.0 / x), $MachinePrecision] / -9.0), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.99999999999999983e72

   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

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

      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 99.7%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.4%

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

   if -9.99999999999999983e72 < y < 1.26e29

   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 99.0

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

   5. Applied rewrites 99.0%

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

   7. Applied rewrites 99.1%

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

   if 1.26e29 < y

   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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. neg-mul-1 N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 99.6

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

      18. lower-+.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. associate-/r* N/A

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

      23. metadata-eval N/A

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

      24. metadata-eval N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 93.0%

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

Alternative 6: 94.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.3333333333333333, \frac{y}{\sqrt{x}}, 1\right)\\ \mathbf{if}\;y \leq -1 \cdot 10^{+73}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.26 \cdot 10^{+29}:\\ \;\;\;\;1 + \frac{\frac{1}{x}}{-9}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma -0.3333333333333333 (/ y (sqrt x)) 1.0)))
   (if (<= y -1e+73) t_0 (if (<= y 1.26e+29) (+ 1.0 (/ (/ 1.0 x) -9.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = fma(-0.3333333333333333, (y / sqrt(x)), 1.0);
	double tmp;
	if (y <= -1e+73) {
		tmp = t_0;
	} else if (y <= 1.26e+29) {
		tmp = 1.0 + ((1.0 / x) / -9.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(-0.3333333333333333, Float64(y / sqrt(x)), 1.0)
	tmp = 0.0
	if (y <= -1e+73)
		tmp = t_0;
	elseif (y <= 1.26e+29)
		tmp = Float64(1.0 + Float64(Float64(1.0 / x) / -9.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, -1e+73], t$95$0, If[LessEqual[y, 1.26e+29], N[(1.0 + N[(N[(1.0 / x), $MachinePrecision] / -9.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.99999999999999983e72 or 1.26e29 < 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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. neg-mul-1 N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 99.6

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

      18. lower-+.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. associate-/r* N/A

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

      23. metadata-eval N/A

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

      24. metadata-eval N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 96.1%

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

   if -9.99999999999999983e72 < y < 1.26e29

   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 99.0

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

   5. Applied rewrites 99.0%

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

   7. Applied rewrites 99.1%

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

Alternative 7: 92.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+74}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+50}:\\ \;\;\;\;1 + \frac{\frac{1}{x}}{-9}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.2e+74)
   (* y (/ -0.3333333333333333 (sqrt x)))
   (if (<= y 8.6e+50) (+ 1.0 (/ (/ 1.0 x) -9.0)) (/ y (* (sqrt x) -3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.2e+74) {
		tmp = y * (-0.3333333333333333 / sqrt(x));
	} else if (y <= 8.6e+50) {
		tmp = 1.0 + ((1.0 / x) / -9.0);
	} else {
		tmp = y / (sqrt(x) * -3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.2d+74)) then
        tmp = y * ((-0.3333333333333333d0) / sqrt(x))
    else if (y <= 8.6d+50) then
        tmp = 1.0d0 + ((1.0d0 / x) / (-9.0d0))
    else
        tmp = y / (sqrt(x) * (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.2e+74) {
		tmp = y * (-0.3333333333333333 / Math.sqrt(x));
	} else if (y <= 8.6e+50) {
		tmp = 1.0 + ((1.0 / x) / -9.0);
	} else {
		tmp = y / (Math.sqrt(x) * -3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.2e+74:
		tmp = y * (-0.3333333333333333 / math.sqrt(x))
	elif y <= 8.6e+50:
		tmp = 1.0 + ((1.0 / x) / -9.0)
	else:
		tmp = y / (math.sqrt(x) * -3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.2e+74)
		tmp = Float64(y * Float64(-0.3333333333333333 / sqrt(x)));
	elseif (y <= 8.6e+50)
		tmp = Float64(1.0 + Float64(Float64(1.0 / x) / -9.0));
	else
		tmp = Float64(y / Float64(sqrt(x) * -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.2e+74)
		tmp = y * (-0.3333333333333333 / sqrt(x));
	elseif (y <= 8.6e+50)
		tmp = 1.0 + ((1.0 / x) / -9.0);
	else
		tmp = y / (sqrt(x) * -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.2e+74], N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.6e+50], N[(1.0 + N[(N[(1.0 / x), $MachinePrecision] / -9.0), $MachinePrecision]), $MachinePrecision], N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.1999999999999998e74

   1. Initial program 99.5%

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

   4. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\frac{\frac{1}{\left(x \cdot -9\right) \cdot \left(x \cdot \left(x \cdot 81\right)\right)} + 1}{\mathsf{fma}\left(\frac{0.1111111111111111}{x}, 1 + \frac{0.1111111111111111}{x}, 1\right)}} - \frac{y}{3 \cdot \sqrt{x}} \]

   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

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. rem-square-sqrt N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      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. lower-*.f64 98.8

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

   7. Applied rewrites 98.8%

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

   9. Applied rewrites 99.1%

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

   if -4.1999999999999998e74 < y < 8.5999999999999994e50

   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 98.4

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

   5. Applied rewrites 98.4%

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

   7. Applied rewrites 98.5%

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

   if 8.5999999999999994e50 < y

   1. Initial program 99.6%

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

   4. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{\left(x \cdot -9\right) \cdot \left(x \cdot \left(x \cdot 81\right)\right)} + 1}{\mathsf{fma}\left(\frac{0.1111111111111111}{x}, 1 + \frac{0.1111111111111111}{x}, 1\right)}} - \frac{y}{3 \cdot \sqrt{x}} \]

   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

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. rem-square-sqrt N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      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. lower-*.f64 85.2

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

   7. Applied rewrites 85.2%

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

   9. Applied rewrites 85.2%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+74}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+50}:\\ \;\;\;\;1 + \frac{\frac{1}{x}}{-9}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+74}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+50}:\\ \;\;\;\;1 + \frac{\frac{1}{x}}{-9}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.2e+74)
   (* y (/ -0.3333333333333333 (sqrt x)))
   (if (<= y 8.6e+50)
     (+ 1.0 (/ (/ 1.0 x) -9.0))
     (* -0.3333333333333333 (/ y (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.2e+74) {
		tmp = y * (-0.3333333333333333 / sqrt(x));
	} else if (y <= 8.6e+50) {
		tmp = 1.0 + ((1.0 / x) / -9.0);
	} else {
		tmp = -0.3333333333333333 * (y / sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.2d+74)) then
        tmp = y * ((-0.3333333333333333d0) / sqrt(x))
    else if (y <= 8.6d+50) then
        tmp = 1.0d0 + ((1.0d0 / x) / (-9.0d0))
    else
        tmp = (-0.3333333333333333d0) * (y / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.2e+74) {
		tmp = y * (-0.3333333333333333 / Math.sqrt(x));
	} else if (y <= 8.6e+50) {
		tmp = 1.0 + ((1.0 / x) / -9.0);
	} else {
		tmp = -0.3333333333333333 * (y / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.2e+74:
		tmp = y * (-0.3333333333333333 / math.sqrt(x))
	elif y <= 8.6e+50:
		tmp = 1.0 + ((1.0 / x) / -9.0)
	else:
		tmp = -0.3333333333333333 * (y / math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.2e+74)
		tmp = Float64(y * Float64(-0.3333333333333333 / sqrt(x)));
	elseif (y <= 8.6e+50)
		tmp = Float64(1.0 + Float64(Float64(1.0 / x) / -9.0));
	else
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.2e+74)
		tmp = y * (-0.3333333333333333 / sqrt(x));
	elseif (y <= 8.6e+50)
		tmp = 1.0 + ((1.0 / x) / -9.0);
	else
		tmp = -0.3333333333333333 * (y / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.2e+74], N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.6e+50], N[(1.0 + N[(N[(1.0 / x), $MachinePrecision] / -9.0), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.1999999999999998e74

   1. Initial program 99.5%

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

   4. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\frac{\frac{1}{\left(x \cdot -9\right) \cdot \left(x \cdot \left(x \cdot 81\right)\right)} + 1}{\mathsf{fma}\left(\frac{0.1111111111111111}{x}, 1 + \frac{0.1111111111111111}{x}, 1\right)}} - \frac{y}{3 \cdot \sqrt{x}} \]

   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

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. rem-square-sqrt N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      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. lower-*.f64 98.8

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

   7. Applied rewrites 98.8%

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

   9. Applied rewrites 99.1%

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

   if -4.1999999999999998e74 < y < 8.5999999999999994e50

   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 98.4

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

   5. Applied rewrites 98.4%

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

   7. Applied rewrites 98.5%

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

   if 8.5999999999999994e50 < y

   1. Initial program 99.6%

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

   4. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{\left(x \cdot -9\right) \cdot \left(x \cdot \left(x \cdot 81\right)\right)} + 1}{\mathsf{fma}\left(\frac{0.1111111111111111}{x}, 1 + \frac{0.1111111111111111}{x}, 1\right)}} - \frac{y}{3 \cdot \sqrt{x}} \]

   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

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. rem-square-sqrt N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      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. lower-*.f64 85.2

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

   7. Applied rewrites 85.2%

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

   9. Applied rewrites 85.2%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+74}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+50}:\\ \;\;\;\;1 + \frac{\frac{1}{x}}{-9}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{if}\;y \leq -4.2 \cdot 10^{+74}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+50}:\\ \;\;\;\;1 + \frac{\frac{1}{x}}{-9}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* -0.3333333333333333 (/ y (sqrt x)))))
   (if (<= y -4.2e+74)
     t_0
     (if (<= y 8.6e+50) (+ 1.0 (/ (/ 1.0 x) -9.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = -0.3333333333333333 * (y / sqrt(x));
	double tmp;
	if (y <= -4.2e+74) {
		tmp = t_0;
	} else if (y <= 8.6e+50) {
		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 = (-0.3333333333333333d0) * (y / sqrt(x))
    if (y <= (-4.2d+74)) then
        tmp = t_0
    else if (y <= 8.6d+50) 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 = -0.3333333333333333 * (y / Math.sqrt(x));
	double tmp;
	if (y <= -4.2e+74) {
		tmp = t_0;
	} else if (y <= 8.6e+50) {
		tmp = 1.0 + ((1.0 / x) / -9.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -0.3333333333333333 * (y / math.sqrt(x))
	tmp = 0
	if y <= -4.2e+74:
		tmp = t_0
	elif y <= 8.6e+50:
		tmp = 1.0 + ((1.0 / x) / -9.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(-0.3333333333333333 * Float64(y / sqrt(x)))
	tmp = 0.0
	if (y <= -4.2e+74)
		tmp = t_0;
	elseif (y <= 8.6e+50)
		tmp = Float64(1.0 + Float64(Float64(1.0 / x) / -9.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = -0.3333333333333333 * (y / sqrt(x));
	tmp = 0.0;
	if (y <= -4.2e+74)
		tmp = t_0;
	elseif (y <= 8.6e+50)
		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[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.2e+74], t$95$0, If[LessEqual[y, 8.6e+50], N[(1.0 + N[(N[(1.0 / x), $MachinePrecision] / -9.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.1999999999999998e74 or 8.5999999999999994e50 < y

   1. Initial program 99.5%

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

   4. Applied rewrites 71.5%

      \[\leadsto \color{blue}{\frac{\frac{1}{\left(x \cdot -9\right) \cdot \left(x \cdot \left(x \cdot 81\right)\right)} + 1}{\mathsf{fma}\left(\frac{0.1111111111111111}{x}, 1 + \frac{0.1111111111111111}{x}, 1\right)}} - \frac{y}{3 \cdot \sqrt{x}} \]

   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

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. rem-square-sqrt N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      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. lower-*.f64 92.0

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

   7. Applied rewrites 92.0%

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

   9. Applied rewrites 92.1%

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

   if -4.1999999999999998e74 < y < 8.5999999999999994e50

   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 98.4

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

   5. Applied rewrites 98.4%

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

   7. Applied rewrites 98.5%

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

Alternative 10: 98.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   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 98.8

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

   5. Applied rewrites 98.8%

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

   7. Applied rewrites 98.8%

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

   if 0.110000000000000001 < x

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. neg-mul-1 N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 99.8

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

      18. lower-+.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. associate-/r* N/A

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

      23. metadata-eval N/A

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

      24. metadata-eval N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 98.6%

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

4. Final simplification 98.7%

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

Alternative 11: 98.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   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 98.8

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

   5. Applied rewrites 98.8%

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

   if 0.110000000000000001 < x

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. neg-mul-1 N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 99.8

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

      18. lower-+.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. associate-/r* N/A

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

      23. metadata-eval N/A

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

      24. metadata-eval N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 98.6%

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

4. Final simplification 98.7%

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

Alternative 12: 62.2% accurate, 1.9× speedup

Math

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

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(Float64(1.0 / 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[(N[(1.0 / x), $MachinePrecision] / -9.0), $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. 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 61.8

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

5. Applied rewrites 61.8%

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

7. Applied rewrites 61.8%

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

Alternative 13: 62.1% 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 61.8

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

5. Applied rewrites 61.8%

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

Alternative 14: 31.4% 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 61.8

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

5. Applied rewrites 61.8%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 34.0%

   \[\leadsto 1 \]
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 2024221 
(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)))))