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

Percentage Accurate: 99.7% → 99.7%

Time: 11.1s

Alternatives: 21

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 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]

Derivation

1. Initial program 99.7%

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

3. Final simplification 99.7%

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

Alternative 2: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.3 \cdot 10^{+42}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+83}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5.3e+42)
   (+ 1.0 (* y (/ -0.3333333333333333 (sqrt x))))
   (if (<= y 4.2e+83)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (- 1.0 (/ y (* 3.0 (sqrt x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5.3e+42) {
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	} else if (y <= 4.2e+83) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-5.3d+42)) then
        tmp = 1.0d0 + (y * ((-0.3333333333333333d0) / sqrt(x)))
    else if (y <= 4.2d+83) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = 1.0d0 - (y / (3.0d0 * sqrt(x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -5.3e+42) {
		tmp = 1.0 + (y * (-0.3333333333333333 / Math.sqrt(x)));
	} else if (y <= 4.2e+83) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 - (y / (3.0 * Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5.3e+42:
		tmp = 1.0 + (y * (-0.3333333333333333 / math.sqrt(x)))
	elif y <= 4.2e+83:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = 1.0 - (y / (3.0 * math.sqrt(x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5.3e+42)
		tmp = Float64(1.0 + Float64(y * Float64(-0.3333333333333333 / sqrt(x))));
	elseif (y <= 4.2e+83)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -5.3e+42)
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	elseif (y <= 4.2e+83)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5.3e+42], N[(1.0 + N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.2e+83], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.30000000000000028e42

   1. Initial program 99.5%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.6%

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

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{1}{\color{blue}{\frac{\sqrt{x} \cdot -3}{y}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{1}{\sqrt{x} \cdot -3} \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{\sqrt{x} \cdot -3}\right), \color{blue}{y}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{-3 \cdot \sqrt{x}}\right), y\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{1}{-3}}{\sqrt{x}}\right), y\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{-1}{3}}{\sqrt{x}}\right), y\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{1}{9} \cdot -3}{\sqrt{x}}\right), y\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{\left(\mathsf{neg}\left(\frac{-1}{9}\right)\right) \cdot -3}{\sqrt{x}}\right), y\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\frac{-1}{9}\right)\right) \cdot -3\right), \left(\sqrt{x}\right)\right), y\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{9} \cdot -3\right), \left(\sqrt{x}\right)\right), y\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, \left(\sqrt{x}\right)\right), y\right)\right) \]

      12. sqrt-lowering-sqrt.f64 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{sqrt.f64}\left(x\right)\right), y\right)\right) \]

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{sqrt.f64}\left(x\right)\right), y\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 91.2%

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

   if -5.30000000000000028e42 < y < 4.20000000000000005e83

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

      7. /-lowering-/.f64 98.5%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

   7. Simplified 98.5%

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

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{x}{\frac{-1}{9}}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \color{blue}{\frac{-1}{9}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\frac{1}{9}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{-1}{9}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\frac{-1}{9}\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \frac{1}{9}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \frac{1}{9}\right)\right)\right) \]

      8. div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{x}}{9}\right)\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x \cdot 9}}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{-1}{\color{blue}{x} \cdot 9}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot 9\right)}\right)\right) \]

      13. *-lowering-*.f64 98.6%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{9}\right)\right)\right) \]

   9. Applied egg-rr 98.6%

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

   if 4.20000000000000005e83 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(3, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 95.2%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.3 \cdot 10^{+42}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+83}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+42}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+89}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y}{\frac{\sqrt{x}}{-0.3333333333333333}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.4e+42)
   (+ 1.0 (* y (/ -0.3333333333333333 (sqrt x))))
   (if (<= y 1.95e+89)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (+ 1.0 (/ y (/ (sqrt x) -0.3333333333333333))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.4e+42) {
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	} else if (y <= 1.95e+89) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 + (y / (sqrt(x) / -0.3333333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.4d+42)) then
        tmp = 1.0d0 + (y * ((-0.3333333333333333d0) / sqrt(x)))
    else if (y <= 1.95d+89) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = 1.0d0 + (y / (sqrt(x) / (-0.3333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.4e+42) {
		tmp = 1.0 + (y * (-0.3333333333333333 / Math.sqrt(x)));
	} else if (y <= 1.95e+89) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 + (y / (Math.sqrt(x) / -0.3333333333333333));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.4e+42:
		tmp = 1.0 + (y * (-0.3333333333333333 / math.sqrt(x)))
	elif y <= 1.95e+89:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = 1.0 + (y / (math.sqrt(x) / -0.3333333333333333))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.4e+42)
		tmp = Float64(1.0 + Float64(y * Float64(-0.3333333333333333 / sqrt(x))));
	elseif (y <= 1.95e+89)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(1.0 + Float64(y / Float64(sqrt(x) / -0.3333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.4e+42)
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	elseif (y <= 1.95e+89)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = 1.0 + (y / (sqrt(x) / -0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.4e+42], N[(1.0 + N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.95e+89], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y / N[(N[Sqrt[x], $MachinePrecision] / -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.4e42

   1. Initial program 99.5%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.6%

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

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{1}{\color{blue}{\frac{\sqrt{x} \cdot -3}{y}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{1}{\sqrt{x} \cdot -3} \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{\sqrt{x} \cdot -3}\right), \color{blue}{y}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{-3 \cdot \sqrt{x}}\right), y\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{1}{-3}}{\sqrt{x}}\right), y\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{-1}{3}}{\sqrt{x}}\right), y\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{1}{9} \cdot -3}{\sqrt{x}}\right), y\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{\left(\mathsf{neg}\left(\frac{-1}{9}\right)\right) \cdot -3}{\sqrt{x}}\right), y\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\frac{-1}{9}\right)\right) \cdot -3\right), \left(\sqrt{x}\right)\right), y\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{9} \cdot -3\right), \left(\sqrt{x}\right)\right), y\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, \left(\sqrt{x}\right)\right), y\right)\right) \]

      12. sqrt-lowering-sqrt.f64 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{sqrt.f64}\left(x\right)\right), y\right)\right) \]

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{sqrt.f64}\left(x\right)\right), y\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 91.2%

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

   if -1.4e42 < y < 1.95000000000000005e89

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

      7. /-lowering-/.f64 97.8%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

   7. Simplified 97.8%

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

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{x}{\frac{-1}{9}}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \color{blue}{\frac{-1}{9}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\frac{1}{9}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{-1}{9}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\frac{-1}{9}\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \frac{1}{9}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \frac{1}{9}\right)\right)\right) \]

      8. div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{x}}{9}\right)\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x \cdot 9}}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{-1}{\color{blue}{x} \cdot 9}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot 9\right)}\right)\right) \]

      13. *-lowering-*.f64 97.9%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{9}\right)\right)\right) \]

   9. Applied egg-rr 97.9%

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

   if 1.95000000000000005e89 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(3, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 96.8%

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

      1. sub-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. neg-mul-1 N/A

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

      4. times-frac N/A

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

      5. metadata-eval N/A

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

      6. associate-/l* N/A

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

      7. associate-*l/ N/A

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{\frac{-1}{3}}{\sqrt{x}} \cdot y\right), \color{blue}{1}\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{\frac{-1}{3} \cdot y}{\sqrt{x}}\right), 1\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{-1}{3} \cdot \frac{y}{\sqrt{x}}\right), 1\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{-1}{3} \cdot \frac{y}{\sqrt{x}}\right), 1\right) \]

      13. times-frac N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{-1 \cdot y}{3 \cdot \sqrt{x}}\right), 1\right) \]

      14. neg-mul-1 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{\mathsf{neg}\left(y\right)}{3 \cdot \sqrt{x}}\right), 1\right) \]

      15. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right), 1\right) \]

      16. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{y}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}\right), 1\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)\right), 1\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \left(\mathsf{neg}\left(3\right)\right)\right)\right), 1\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot -3\right)\right), 1\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \frac{1}{\frac{-1}{3}}\right)\right), 1\right) \]

      22. div-inv N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, \left(\frac{\sqrt{x}}{\frac{-1}{3}}\right)\right), 1\right) \]

      23. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(\sqrt{x}\right), \frac{-1}{3}\right)\right), 1\right) \]

      24. sqrt-lowering-sqrt.f64 96.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), \frac{-1}{3}\right)\right), 1\right) \]

   7. Applied egg-rr 96.8%

      \[\leadsto \color{blue}{\frac{y}{\frac{\sqrt{x}}{-0.3333333333333333}} + 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+42}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+89}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y}{\frac{\sqrt{x}}{-0.3333333333333333}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{if}\;y \leq -6.6 \cdot 10^{+42}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+89}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* y (/ -0.3333333333333333 (sqrt x))))))
   (if (<= y -6.6e+42)
     t_0
     (if (<= y 1.95e+89) (+ 1.0 (/ -1.0 (* x 9.0))) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	double tmp;
	if (y <= -6.6e+42) {
		tmp = t_0;
	} else if (y <= 1.95e+89) {
		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 = 1.0d0 + (y * ((-0.3333333333333333d0) / sqrt(x)))
    if (y <= (-6.6d+42)) then
        tmp = t_0
    else if (y <= 1.95d+89) 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 = 1.0 + (y * (-0.3333333333333333 / Math.sqrt(x)));
	double tmp;
	if (y <= -6.6e+42) {
		tmp = t_0;
	} else if (y <= 1.95e+89) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (y * (-0.3333333333333333 / math.sqrt(x)))
	tmp = 0
	if y <= -6.6e+42:
		tmp = t_0
	elif y <= 1.95e+89:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(y * Float64(-0.3333333333333333 / sqrt(x))))
	tmp = 0.0
	if (y <= -6.6e+42)
		tmp = t_0;
	elseif (y <= 1.95e+89)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	tmp = 0.0;
	if (y <= -6.6e+42)
		tmp = t_0;
	elseif (y <= 1.95e+89)
		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[(1.0 + N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.6e+42], t$95$0, If[LessEqual[y, 1.95e+89], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.5999999999999998e42 or 1.95000000000000005e89 < y

   1. Initial program 99.6%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.6%

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

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{1}{\color{blue}{\frac{\sqrt{x} \cdot -3}{y}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{1}{\sqrt{x} \cdot -3} \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{\sqrt{x} \cdot -3}\right), \color{blue}{y}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{-3 \cdot \sqrt{x}}\right), y\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{1}{-3}}{\sqrt{x}}\right), y\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{-1}{3}}{\sqrt{x}}\right), y\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{1}{9} \cdot -3}{\sqrt{x}}\right), y\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{\left(\mathsf{neg}\left(\frac{-1}{9}\right)\right) \cdot -3}{\sqrt{x}}\right), y\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\frac{-1}{9}\right)\right) \cdot -3\right), \left(\sqrt{x}\right)\right), y\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{9} \cdot -3\right), \left(\sqrt{x}\right)\right), y\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, \left(\sqrt{x}\right)\right), y\right)\right) \]

      12. sqrt-lowering-sqrt.f64 99.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{sqrt.f64}\left(x\right)\right), y\right)\right) \]

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{sqrt.f64}\left(x\right)\right), y\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 93.7%

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

   if -6.5999999999999998e42 < y < 1.95000000000000005e89

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

      7. /-lowering-/.f64 97.8%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

   7. Simplified 97.8%

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

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{x}{\frac{-1}{9}}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \color{blue}{\frac{-1}{9}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\frac{1}{9}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{-1}{9}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\frac{-1}{9}\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \frac{1}{9}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \frac{1}{9}\right)\right)\right) \]

      8. div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{x}}{9}\right)\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x \cdot 9}}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{-1}{\color{blue}{x} \cdot 9}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot 9\right)}\right)\right) \]

      13. *-lowering-*.f64 97.9%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{9}\right)\right)\right) \]

   9. Applied egg-rr 97.9%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{+42}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+89}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot -0.3333333333333333\right) \cdot \sqrt{\frac{1}{x}}\\ \mathbf{if}\;y \leq -1.4 \cdot 10^{+53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+92}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y -0.3333333333333333) (sqrt (/ 1.0 x)))))
   (if (<= y -1.4e+53)
     t_0
     (if (<= y 1.45e+92) (+ 1.0 (/ -1.0 (* x 9.0))) t_0))))

C

double code(double x, double y) {
	double t_0 = (y * -0.3333333333333333) * sqrt((1.0 / x));
	double tmp;
	if (y <= -1.4e+53) {
		tmp = t_0;
	} else if (y <= 1.45e+92) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y * (-0.3333333333333333d0)) * sqrt((1.0d0 / x))
    if (y <= (-1.4d+53)) then
        tmp = t_0
    else if (y <= 1.45d+92) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * -0.3333333333333333) * Math.sqrt((1.0 / x));
	double tmp;
	if (y <= -1.4e+53) {
		tmp = t_0;
	} else if (y <= 1.45e+92) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * -0.3333333333333333) * math.sqrt((1.0 / x))
	tmp = 0
	if y <= -1.4e+53:
		tmp = t_0
	elif y <= 1.45e+92:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * -0.3333333333333333) * sqrt(Float64(1.0 / x)))
	tmp = 0.0
	if (y <= -1.4e+53)
		tmp = t_0;
	elseif (y <= 1.45e+92)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * -0.3333333333333333) * sqrt((1.0 / x));
	tmp = 0.0;
	if (y <= -1.4e+53)
		tmp = t_0;
	elseif (y <= 1.45e+92)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * -0.3333333333333333), $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.4e+53], t$95$0, If[LessEqual[y, 1.45e+92], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4e53 or 1.45e92 < y

   1. Initial program 99.6%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.6%

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

   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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}}\right), \color{blue}{\left(\frac{-1}{3} \cdot y\right)}\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right), \left(\color{blue}{\frac{-1}{3}} \cdot y\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\frac{-1}{3} \cdot y\right)\right) \]

      7. *-lowering-*.f64 91.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\frac{-1}{3}, \color{blue}{y}\right)\right) \]

   7. Simplified 91.2%

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

   if -1.4e53 < y < 1.45e92

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

      7. /-lowering-/.f64 96.7%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

   7. Simplified 96.7%

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

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{x}{\frac{-1}{9}}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \color{blue}{\frac{-1}{9}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\frac{1}{9}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{-1}{9}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\frac{-1}{9}\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \frac{1}{9}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \frac{1}{9}\right)\right)\right) \]

      8. div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{x}}{9}\right)\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x \cdot 9}}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{-1}{\color{blue}{x} \cdot 9}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot 9\right)}\right)\right) \]

      13. *-lowering-*.f64 96.8%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{9}\right)\right)\right) \]

   9. Applied egg-rr 96.8%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+53}:\\ \;\;\;\;\left(y \cdot -0.3333333333333333\right) \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+92}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -0.3333333333333333\right) \cdot \sqrt{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{-0.1111111111111111 + \sqrt{x} \cdot \left(y \cdot -0.3333333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.8e-11)
   (/ (+ -0.1111111111111111 (* (sqrt x) (* y -0.3333333333333333))) x)
   (+ 1.0 (* y (/ -0.3333333333333333 (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.8e-11) {
		tmp = (-0.1111111111111111 + (sqrt(x) * (y * -0.3333333333333333))) / x;
	} else {
		tmp = 1.0 + (y * (-0.3333333333333333 / 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 (x <= 1.8d-11) then
        tmp = ((-0.1111111111111111d0) + (sqrt(x) * (y * (-0.3333333333333333d0)))) / x
    else
        tmp = 1.0d0 + (y * ((-0.3333333333333333d0) / sqrt(x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.8e-11) {
		tmp = (-0.1111111111111111 + (Math.sqrt(x) * (y * -0.3333333333333333))) / x;
	} else {
		tmp = 1.0 + (y * (-0.3333333333333333 / Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.8e-11:
		tmp = (-0.1111111111111111 + (math.sqrt(x) * (y * -0.3333333333333333))) / x
	else:
		tmp = 1.0 + (y * (-0.3333333333333333 / math.sqrt(x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.8e-11)
		tmp = Float64(Float64(-0.1111111111111111 + Float64(sqrt(x) * Float64(y * -0.3333333333333333))) / x);
	else
		tmp = Float64(1.0 + Float64(y * Float64(-0.3333333333333333 / sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.8e-11)
		tmp = (-0.1111111111111111 + (sqrt(x) * (y * -0.3333333333333333))) / x;
	else
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.8e-11], N[(N[(-0.1111111111111111 + N[(N[Sqrt[x], $MachinePrecision] * N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(1.0 + N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.79999999999999992e-11

   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. associate-/r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{1}{x}}{9}\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(3, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{x}\right), 9\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(3, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right) \]

      3. /-lowering-/.f64 99.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), 9\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(3, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right) \]

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)\right)\right), \color{blue}{x}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)\right)\right), x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{9} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{9} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \left(\sqrt{x} \cdot y\right)\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{9} + \frac{-1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right), x\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{9}, \left(\frac{-1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)\right), x\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{9}, \left(\left(\frac{-1}{3} \cdot \sqrt{x}\right) \cdot y\right)\right), x\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{9}, \left(\left(\sqrt{x} \cdot \frac{-1}{3}\right) \cdot y\right)\right), x\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{9}, \left(\sqrt{x} \cdot \left(\frac{-1}{3} \cdot y\right)\right)\right), x\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{9}, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \left(\frac{-1}{3} \cdot y\right)\right)\right), x\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{9}, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\frac{-1}{3} \cdot y\right)\right)\right), x\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{9}, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(y \cdot \frac{-1}{3}\right)\right)\right), x\right) \]

      15. *-lowering-*.f64 99.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{9}, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{*.f64}\left(y, \frac{-1}{3}\right)\right)\right), x\right) \]

   7. Simplified 99.6%

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

   if 1.79999999999999992e-11 < x

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.8%

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

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{1}{\color{blue}{\frac{\sqrt{x} \cdot -3}{y}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{1}{\sqrt{x} \cdot -3} \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{\sqrt{x} \cdot -3}\right), \color{blue}{y}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{-3 \cdot \sqrt{x}}\right), y\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{1}{-3}}{\sqrt{x}}\right), y\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{-1}{3}}{\sqrt{x}}\right), y\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{1}{9} \cdot -3}{\sqrt{x}}\right), y\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{\left(\mathsf{neg}\left(\frac{-1}{9}\right)\right) \cdot -3}{\sqrt{x}}\right), y\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\frac{-1}{9}\right)\right) \cdot -3\right), \left(\sqrt{x}\right)\right), y\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{9} \cdot -3\right), \left(\sqrt{x}\right)\right), y\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, \left(\sqrt{x}\right)\right), y\right)\right) \]

      12. sqrt-lowering-sqrt.f64 99.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{sqrt.f64}\left(x\right)\right), y\right)\right) \]

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{sqrt.f64}\left(x\right)\right), y\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 98.3%

      \[\leadsto \color{blue}{1} + \frac{-0.3333333333333333}{\sqrt{x}} \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{-0.1111111111111111 + \sqrt{x} \cdot \left(y \cdot -0.3333333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{-0.1111111111111111 + -0.3333333333333333 \cdot \left(y \cdot \sqrt{x}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.1e-5)
   (/ (+ -0.1111111111111111 (* -0.3333333333333333 (* y (sqrt x)))) x)
   (+ 1.0 (* y (/ -0.3333333333333333 (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.1e-5) {
		tmp = (-0.1111111111111111 + (-0.3333333333333333 * (y * sqrt(x)))) / x;
	} else {
		tmp = 1.0 + (y * (-0.3333333333333333 / 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 (x <= 1.1d-5) then
        tmp = ((-0.1111111111111111d0) + ((-0.3333333333333333d0) * (y * sqrt(x)))) / x
    else
        tmp = 1.0d0 + (y * ((-0.3333333333333333d0) / sqrt(x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.1e-5) {
		tmp = (-0.1111111111111111 + (-0.3333333333333333 * (y * Math.sqrt(x)))) / x;
	} else {
		tmp = 1.0 + (y * (-0.3333333333333333 / Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.1e-5:
		tmp = (-0.1111111111111111 + (-0.3333333333333333 * (y * math.sqrt(x)))) / x
	else:
		tmp = 1.0 + (y * (-0.3333333333333333 / math.sqrt(x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.1e-5)
		tmp = Float64(Float64(-0.1111111111111111 + Float64(-0.3333333333333333 * Float64(y * sqrt(x)))) / x);
	else
		tmp = Float64(1.0 + Float64(y * Float64(-0.3333333333333333 / sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.1e-5)
		tmp = (-0.1111111111111111 + (-0.3333333333333333 * (y * sqrt(x)))) / x;
	else
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.1e-5], N[(N[(-0.1111111111111111 + N[(-0.3333333333333333 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(1.0 + N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1e-5

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-neg-in N/A

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

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)\right)\right), \color{blue}{x}\right) \]

      10. distribute-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)\right)\right), x\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{9} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)\right)\right), x\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{9}, \left(\mathsf{neg}\left(\frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{9}, \left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \left(\sqrt{x} \cdot y\right)\right)\right), x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{9}, \left(\frac{-1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)\right), x\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{9}, \mathsf{*.f64}\left(\frac{-1}{3}, \left(\sqrt{x} \cdot y\right)\right)\right), x\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{9}, \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left(\sqrt{x}\right), y\right)\right)\right), x\right) \]

      17. sqrt-lowering-sqrt.f64 99.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{9}, \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), y\right)\right)\right), x\right) \]

   7. Simplified 99.5%

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

   if 1.1e-5 < x

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.8%

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

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{1}{\color{blue}{\frac{\sqrt{x} \cdot -3}{y}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{1}{\sqrt{x} \cdot -3} \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{\sqrt{x} \cdot -3}\right), \color{blue}{y}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{-3 \cdot \sqrt{x}}\right), y\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{1}{-3}}{\sqrt{x}}\right), y\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{-1}{3}}{\sqrt{x}}\right), y\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{1}{9} \cdot -3}{\sqrt{x}}\right), y\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{\left(\mathsf{neg}\left(\frac{-1}{9}\right)\right) \cdot -3}{\sqrt{x}}\right), y\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\frac{-1}{9}\right)\right) \cdot -3\right), \left(\sqrt{x}\right)\right), y\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{9} \cdot -3\right), \left(\sqrt{x}\right)\right), y\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, \left(\sqrt{x}\right)\right), y\right)\right) \]

      12. sqrt-lowering-sqrt.f64 99.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{sqrt.f64}\left(x\right)\right), y\right)\right) \]

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{sqrt.f64}\left(x\right)\right), y\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 98.2%

      \[\leadsto \color{blue}{1} + \frac{-0.3333333333333333}{\sqrt{x}} \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{-0.1111111111111111 + -0.3333333333333333 \cdot \left(y \cdot \sqrt{x}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg N/A

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

   6. associate-/r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

   7. distribute-neg-frac N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

   11. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

   16. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

   17. metadata-eval 99.7%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

3. Simplified 99.7%

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

Alternative 9: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 + ((-0.1111111111111111d0) / x)) + (y * ((-0.3333333333333333d0) / sqrt(x)))
end function

Java

public static double code(double x, double y) {
	return (1.0 + (-0.1111111111111111 / x)) + (y * (-0.3333333333333333 / Math.sqrt(x)));
}

Python

def code(x, y):
	return (1.0 + (-0.1111111111111111 / x)) + (y * (-0.3333333333333333 / math.sqrt(x)))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = (1.0 + (-0.1111111111111111 / x)) + (y * (-0.3333333333333333 / sqrt(x)));
end

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg N/A

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

   6. associate-/r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

   7. distribute-neg-frac N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

   11. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

   16. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

   17. metadata-eval 99.7%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

3. Simplified 99.7%

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

   1. clear-num N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{1}{\color{blue}{\frac{\sqrt{x} \cdot -3}{y}}}\right)\right) \]

   2. associate-/r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{1}{\sqrt{x} \cdot -3} \cdot \color{blue}{y}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{\sqrt{x} \cdot -3}\right), \color{blue}{y}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{-3 \cdot \sqrt{x}}\right), y\right)\right) \]

   5. associate-/r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{1}{-3}}{\sqrt{x}}\right), y\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{-1}{3}}{\sqrt{x}}\right), y\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{1}{9} \cdot -3}{\sqrt{x}}\right), y\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\left(\frac{\left(\mathsf{neg}\left(\frac{-1}{9}\right)\right) \cdot -3}{\sqrt{x}}\right), y\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\frac{-1}{9}\right)\right) \cdot -3\right), \left(\sqrt{x}\right)\right), y\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{9} \cdot -3\right), \left(\sqrt{x}\right)\right), y\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, \left(\sqrt{x}\right)\right), y\right)\right) \]

   12. sqrt-lowering-sqrt.f64 99.7%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{sqrt.f64}\left(x\right)\right), y\right)\right) \]

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

   \[\leadsto \left(1 + \frac{-0.1111111111111111}{x}\right) + y \cdot \frac{-0.3333333333333333}{\sqrt{x}} \]
8. Add Preprocessing

Alternative 10: 68.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{-0.0013717421124828531}{x \cdot \left(x \cdot x\right)}\\ \mathbf{if}\;y \leq -3.2 \cdot 10^{+139}:\\ \;\;\;\;\left(1 + \frac{-0.1111111111111111}{x}\right) \cdot t\_0\\ \mathbf{elif}\;y \leq 3.15 \cdot 10^{+153}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{-0.1111111111111111 + \frac{0.0013717421124828531}{x \cdot x}}{x}\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ -0.0013717421124828531 (* x (* x x))))))
   (if (<= y -3.2e+139)
     (* (+ 1.0 (/ -0.1111111111111111 x)) t_0)
     (if (<= y 3.15e+153)
       (+ 1.0 (/ -1.0 (* x 9.0)))
       (*
        (+ 1.0 (/ (+ -0.1111111111111111 (/ 0.0013717421124828531 (* x x))) x))
        t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (-0.0013717421124828531 / (x * (x * x)));
	double tmp;
	if (y <= -3.2e+139) {
		tmp = (1.0 + (-0.1111111111111111 / x)) * t_0;
	} else if (y <= 3.15e+153) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (1.0 + ((-0.1111111111111111 + (0.0013717421124828531 / (x * x))) / x)) * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((-0.0013717421124828531d0) / (x * (x * x)))
    if (y <= (-3.2d+139)) then
        tmp = (1.0d0 + ((-0.1111111111111111d0) / x)) * t_0
    else if (y <= 3.15d+153) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = (1.0d0 + (((-0.1111111111111111d0) + (0.0013717421124828531d0 / (x * x))) / x)) * t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (-0.0013717421124828531 / (x * (x * x)));
	double tmp;
	if (y <= -3.2e+139) {
		tmp = (1.0 + (-0.1111111111111111 / x)) * t_0;
	} else if (y <= 3.15e+153) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (1.0 + ((-0.1111111111111111 + (0.0013717421124828531 / (x * x))) / x)) * t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (-0.0013717421124828531 / (x * (x * x)))
	tmp = 0
	if y <= -3.2e+139:
		tmp = (1.0 + (-0.1111111111111111 / x)) * t_0
	elif y <= 3.15e+153:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = (1.0 + ((-0.1111111111111111 + (0.0013717421124828531 / (x * x))) / x)) * t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(-0.0013717421124828531 / Float64(x * Float64(x * x))))
	tmp = 0.0
	if (y <= -3.2e+139)
		tmp = Float64(Float64(1.0 + Float64(-0.1111111111111111 / x)) * t_0);
	elseif (y <= 3.15e+153)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(-0.1111111111111111 + Float64(0.0013717421124828531 / Float64(x * x))) / x)) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (-0.0013717421124828531 / (x * (x * x)));
	tmp = 0.0;
	if (y <= -3.2e+139)
		tmp = (1.0 + (-0.1111111111111111 / x)) * t_0;
	elseif (y <= 3.15e+153)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = (1.0 + ((-0.1111111111111111 + (0.0013717421124828531 / (x * x))) / x)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(-0.0013717421124828531 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.2e+139], N[(N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y, 3.15e+153], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(-0.1111111111111111 + N[(0.0013717421124828531 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.2000000000000001e139

   1. Initial program 99.6%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

      7. /-lowering-/.f64 2.5%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

   7. Simplified 2.5%

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

   9. Applied egg-rr 2.4%

      \[\leadsto \color{blue}{\left(1 + \frac{-0.0013717421124828531}{x \cdot \left(x \cdot x\right)}\right) \cdot \frac{1}{1 + \frac{\frac{-0.1111111111111111}{x} - 1}{\frac{x}{-0.1111111111111111}}}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \color{blue}{\left(1 - \frac{1}{9} \cdot \frac{1}{x}\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right)\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right)\right) \]

       5. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right)\right) \]

       7. /-lowering-/.f64 27.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right)\right) \]

   12. Simplified 27.1%

       \[\leadsto \left(1 + \frac{-0.0013717421124828531}{x \cdot \left(x \cdot x\right)}\right) \cdot \color{blue}{\left(1 + \frac{-0.1111111111111111}{x}\right)} \]

   if -3.2000000000000001e139 < y < 3.1500000000000001e153

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

      7. /-lowering-/.f64 83.0%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

   7. Simplified 83.0%

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

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{x}{\frac{-1}{9}}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \color{blue}{\frac{-1}{9}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\frac{1}{9}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{-1}{9}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\frac{-1}{9}\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \frac{1}{9}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \frac{1}{9}\right)\right)\right) \]

      8. div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{x}}{9}\right)\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x \cdot 9}}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{-1}{\color{blue}{x} \cdot 9}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot 9\right)}\right)\right) \]

      13. *-lowering-*.f64 83.1%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{9}\right)\right)\right) \]

   9. Applied egg-rr 83.1%

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

   if 3.1500000000000001e153 < y

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

      7. /-lowering-/.f64 3.9%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

   7. Simplified 3.9%

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

   9. Applied egg-rr 4.5%

      \[\leadsto \color{blue}{\left(1 + \frac{-0.0013717421124828531}{x \cdot \left(x \cdot x\right)}\right) \cdot \frac{1}{1 + \frac{\frac{-0.1111111111111111}{x} - 1}{\frac{x}{-0.1111111111111111}}}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \color{blue}{\left(\left(1 + \frac{\frac{1}{729}}{{x}^{3}}\right) - \frac{1}{9} \cdot \frac{1}{x}\right)}\right) \]
   11. Step-by-step derivation

       1. associate--l+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(1 + \color{blue}{\left(\frac{\frac{1}{729}}{{x}^{3}} - \frac{1}{9} \cdot \frac{1}{x}\right)}\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{1}{729}}{{x}^{3}} - \frac{1}{9} \cdot \frac{1}{x}\right)}\right)\right) \]

       3. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{729}}{{x}^{3}} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right)\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right) + \color{blue}{\frac{\frac{1}{729}}{{x}^{3}}}\right)\right)\right) \]

       5. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\left(0 - \frac{1}{9} \cdot \frac{1}{x}\right) + \frac{\color{blue}{\frac{1}{729}}}{{x}^{3}}\right)\right)\right) \]

       6. associate--r- N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(0 - \color{blue}{\left(\frac{1}{9} \cdot \frac{1}{x} - \frac{\frac{1}{729}}{{x}^{3}}\right)}\right)\right)\right) \]

       7. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(0 - \left(\frac{\frac{1}{9} \cdot 1}{x} - \frac{\color{blue}{\frac{1}{729}}}{{x}^{3}}\right)\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(0 - \left(\frac{\frac{1}{9}}{x} - \frac{\frac{1}{729}}{{x}^{3}}\right)\right)\right)\right) \]

       9. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(0 - \left(\frac{\frac{1}{9}}{x} - \frac{\frac{1}{729}}{\left(x \cdot x\right) \cdot \color{blue}{x}}\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(0 - \left(\frac{\frac{1}{9}}{x} - \frac{\frac{1}{729}}{{x}^{2} \cdot x}\right)\right)\right)\right) \]

       11. associate-/r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(0 - \left(\frac{\frac{1}{9}}{x} - \frac{\frac{\frac{1}{729}}{{x}^{2}}}{\color{blue}{x}}\right)\right)\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(0 - \left(\frac{\frac{1}{9}}{x} - \frac{\frac{\frac{1}{729} \cdot 1}{{x}^{2}}}{x}\right)\right)\right)\right) \]

       13. associate-*r/ N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(0 - \left(\frac{\frac{1}{9}}{x} - \frac{\frac{1}{729} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right)\right) \]

       14. div-sub N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(0 - \frac{\frac{1}{9} - \frac{1}{729} \cdot \frac{1}{{x}^{2}}}{\color{blue}{x}}\right)\right)\right) \]

       15. neg-sub0 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} - \frac{1}{729} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right)\right) \]

       16. mul-1-neg N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(-1 \cdot \color{blue}{\frac{\frac{1}{9} - \frac{1}{729} \cdot \frac{1}{{x}^{2}}}{x}}\right)\right)\right) \]

       17. associate-*r/ N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\frac{-1 \cdot \left(\frac{1}{9} - \frac{1}{729} \cdot \frac{1}{{x}^{2}}\right)}{\color{blue}{x}}\right)\right)\right) \]

       18. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(-1 \cdot \left(\frac{1}{9} - \frac{1}{729} \cdot \frac{1}{{x}^{2}}\right)\right), \color{blue}{x}\right)\right)\right) \]

   12. Simplified 31.6%

       \[\leadsto \left(1 + \frac{-0.0013717421124828531}{x \cdot \left(x \cdot x\right)}\right) \cdot \color{blue}{\left(1 + \frac{-0.1111111111111111 + \frac{0.0013717421124828531}{x \cdot x}}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+139}:\\ \;\;\;\;\left(1 + \frac{-0.1111111111111111}{x}\right) \cdot \left(1 + \frac{-0.0013717421124828531}{x \cdot \left(x \cdot x\right)}\right)\\ \mathbf{elif}\;y \leq 3.15 \cdot 10^{+153}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{-0.1111111111111111 + \frac{0.0013717421124828531}{x \cdot x}}{x}\right) \cdot \left(1 + \frac{-0.0013717421124828531}{x \cdot \left(x \cdot x\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.4% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{-0.0013717421124828531}{x \cdot \left(x \cdot x\right)}\\ \mathbf{if}\;y \leq -3.4 \cdot 10^{+139}:\\ \;\;\;\;\left(1 + \frac{-0.1111111111111111}{x}\right) \cdot t\_0\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+150}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ -0.0013717421124828531 (* x (* x x))))))
   (if (<= y -3.4e+139)
     (* (+ 1.0 (/ -0.1111111111111111 x)) t_0)
     (if (<= y 8.6e+150) (+ 1.0 (/ -1.0 (* x 9.0))) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (-0.0013717421124828531 / (x * (x * x)));
	double tmp;
	if (y <= -3.4e+139) {
		tmp = (1.0 + (-0.1111111111111111 / x)) * t_0;
	} else if (y <= 8.6e+150) {
		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 = 1.0d0 + ((-0.0013717421124828531d0) / (x * (x * x)))
    if (y <= (-3.4d+139)) then
        tmp = (1.0d0 + ((-0.1111111111111111d0) / x)) * t_0
    else if (y <= 8.6d+150) 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 = 1.0 + (-0.0013717421124828531 / (x * (x * x)));
	double tmp;
	if (y <= -3.4e+139) {
		tmp = (1.0 + (-0.1111111111111111 / x)) * t_0;
	} else if (y <= 8.6e+150) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (-0.0013717421124828531 / (x * (x * x)))
	tmp = 0
	if y <= -3.4e+139:
		tmp = (1.0 + (-0.1111111111111111 / x)) * t_0
	elif y <= 8.6e+150:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(-0.0013717421124828531 / Float64(x * Float64(x * x))))
	tmp = 0.0
	if (y <= -3.4e+139)
		tmp = Float64(Float64(1.0 + Float64(-0.1111111111111111 / x)) * t_0);
	elseif (y <= 8.6e+150)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (-0.0013717421124828531 / (x * (x * x)));
	tmp = 0.0;
	if (y <= -3.4e+139)
		tmp = (1.0 + (-0.1111111111111111 / x)) * t_0;
	elseif (y <= 8.6e+150)
		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[(1.0 + N[(-0.0013717421124828531 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.4e+139], N[(N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y, 8.6e+150], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.4000000000000002e139

   1. Initial program 99.6%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

      7. /-lowering-/.f64 2.5%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

   7. Simplified 2.5%

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

   9. Applied egg-rr 2.4%

      \[\leadsto \color{blue}{\left(1 + \frac{-0.0013717421124828531}{x \cdot \left(x \cdot x\right)}\right) \cdot \frac{1}{1 + \frac{\frac{-0.1111111111111111}{x} - 1}{\frac{x}{-0.1111111111111111}}}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \color{blue}{\left(1 - \frac{1}{9} \cdot \frac{1}{x}\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right)\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right)\right) \]

       5. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right)\right) \]

       7. /-lowering-/.f64 27.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right)\right) \]

   12. Simplified 27.1%

       \[\leadsto \left(1 + \frac{-0.0013717421124828531}{x \cdot \left(x \cdot x\right)}\right) \cdot \color{blue}{\left(1 + \frac{-0.1111111111111111}{x}\right)} \]

   if -3.4000000000000002e139 < y < 8.59999999999999994e150

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

      7. /-lowering-/.f64 83.0%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

   7. Simplified 83.0%

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

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{x}{\frac{-1}{9}}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \color{blue}{\frac{-1}{9}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\frac{1}{9}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{-1}{9}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\frac{-1}{9}\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \frac{1}{9}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \frac{1}{9}\right)\right)\right) \]

      8. div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{x}}{9}\right)\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x \cdot 9}}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{-1}{\color{blue}{x} \cdot 9}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot 9\right)}\right)\right) \]

      13. *-lowering-*.f64 83.1%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{9}\right)\right)\right) \]

   9. Applied egg-rr 83.1%

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

   if 8.59999999999999994e150 < y

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

      7. /-lowering-/.f64 3.9%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

   7. Simplified 3.9%

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

   9. Applied egg-rr 4.5%

      \[\leadsto \color{blue}{\left(1 + \frac{-0.0013717421124828531}{x \cdot \left(x \cdot x\right)}\right) \cdot \frac{1}{1 + \frac{\frac{-0.1111111111111111}{x} - 1}{\frac{x}{-0.1111111111111111}}}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \color{blue}{1}\right) \]
   11. Step-by-step derivation

   12. Simplified 29.3%

       \[\leadsto \left(1 + \frac{-0.0013717421124828531}{x \cdot \left(x \cdot x\right)}\right) \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+139}:\\ \;\;\;\;\left(1 + \frac{-0.1111111111111111}{x}\right) \cdot \left(1 + \frac{-0.0013717421124828531}{x \cdot \left(x \cdot x\right)}\right)\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+150}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.0013717421124828531}{x \cdot \left(x \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.8% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+139}:\\ \;\;\;\;\frac{1 - \frac{0.012345679012345678}{x \cdot x}}{1 + \frac{-0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+152}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.0013717421124828531}{x \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.4e+139)
   (/
    (- 1.0 (/ 0.012345679012345678 (* x x)))
    (+ 1.0 (/ -0.1111111111111111 x)))
   (if (<= y 3.2e+152)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (+ 1.0 (/ -0.0013717421124828531 (* x (* x x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.4e+139) {
		tmp = (1.0 - (0.012345679012345678 / (x * x))) / (1.0 + (-0.1111111111111111 / x));
	} else if (y <= 3.2e+152) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 + (-0.0013717421124828531 / (x * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.4d+139)) then
        tmp = (1.0d0 - (0.012345679012345678d0 / (x * x))) / (1.0d0 + ((-0.1111111111111111d0) / x))
    else if (y <= 3.2d+152) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = 1.0d0 + ((-0.0013717421124828531d0) / (x * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.4e+139) {
		tmp = (1.0 - (0.012345679012345678 / (x * x))) / (1.0 + (-0.1111111111111111 / x));
	} else if (y <= 3.2e+152) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 + (-0.0013717421124828531 / (x * (x * x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.4e+139:
		tmp = (1.0 - (0.012345679012345678 / (x * x))) / (1.0 + (-0.1111111111111111 / x))
	elif y <= 3.2e+152:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = 1.0 + (-0.0013717421124828531 / (x * (x * x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.4e+139)
		tmp = Float64(Float64(1.0 - Float64(0.012345679012345678 / Float64(x * x))) / Float64(1.0 + Float64(-0.1111111111111111 / x)));
	elseif (y <= 3.2e+152)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(1.0 + Float64(-0.0013717421124828531 / Float64(x * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.4e+139)
		tmp = (1.0 - (0.012345679012345678 / (x * x))) / (1.0 + (-0.1111111111111111 / x));
	elseif (y <= 3.2e+152)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = 1.0 + (-0.0013717421124828531 / (x * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.4e+139], N[(N[(1.0 - N[(0.012345679012345678 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.2e+152], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.0013717421124828531 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.4000000000000002e139

   1. Initial program 99.6%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

      7. /-lowering-/.f64 2.5%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

   7. Simplified 2.5%

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

   9. Applied egg-rr 24.3%

      \[\leadsto \color{blue}{\frac{1 - \frac{0.012345679012345678}{x \cdot x}}{1 + \frac{-0.1111111111111111}{x}}} \]

   if -3.4000000000000002e139 < y < 3.20000000000000005e152

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

      7. /-lowering-/.f64 83.0%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

   7. Simplified 83.0%

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

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{x}{\frac{-1}{9}}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \color{blue}{\frac{-1}{9}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\frac{1}{9}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{-1}{9}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\frac{-1}{9}\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \frac{1}{9}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \frac{1}{9}\right)\right)\right) \]

      8. div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{x}}{9}\right)\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x \cdot 9}}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{-1}{\color{blue}{x} \cdot 9}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot 9\right)}\right)\right) \]

      13. *-lowering-*.f64 83.1%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{9}\right)\right)\right) \]

   9. Applied egg-rr 83.1%

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

   if 3.20000000000000005e152 < y

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

      7. /-lowering-/.f64 3.9%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

   7. Simplified 3.9%

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

   9. Applied egg-rr 4.5%

      \[\leadsto \color{blue}{\left(1 + \frac{-0.0013717421124828531}{x \cdot \left(x \cdot x\right)}\right) \cdot \frac{1}{1 + \frac{\frac{-0.1111111111111111}{x} - 1}{\frac{x}{-0.1111111111111111}}}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \color{blue}{1}\right) \]
   11. Step-by-step derivation

   12. Simplified 29.3%

       \[\leadsto \left(1 + \frac{-0.0013717421124828531}{x \cdot \left(x \cdot x\right)}\right) \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+139}:\\ \;\;\;\;\frac{1 - \frac{0.012345679012345678}{x \cdot x}}{1 + \frac{-0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+152}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.0013717421124828531}{x \cdot \left(x \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 67.5% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{-0.012345679012345678}{x \cdot x}}{1 + \frac{-0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+150}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.0013717421124828531}{x \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.7e+144)
   (/ (/ -0.012345679012345678 (* x x)) (+ 1.0 (/ -0.1111111111111111 x)))
   (if (<= y 4.8e+150)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (+ 1.0 (/ -0.0013717421124828531 (* x (* x x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.7e+144) {
		tmp = (-0.012345679012345678 / (x * x)) / (1.0 + (-0.1111111111111111 / x));
	} else if (y <= 4.8e+150) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 + (-0.0013717421124828531 / (x * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.7d+144)) then
        tmp = ((-0.012345679012345678d0) / (x * x)) / (1.0d0 + ((-0.1111111111111111d0) / x))
    else if (y <= 4.8d+150) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = 1.0d0 + ((-0.0013717421124828531d0) / (x * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.7e+144) {
		tmp = (-0.012345679012345678 / (x * x)) / (1.0 + (-0.1111111111111111 / x));
	} else if (y <= 4.8e+150) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 + (-0.0013717421124828531 / (x * (x * x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.7e+144:
		tmp = (-0.012345679012345678 / (x * x)) / (1.0 + (-0.1111111111111111 / x))
	elif y <= 4.8e+150:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = 1.0 + (-0.0013717421124828531 / (x * (x * x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.7e+144)
		tmp = Float64(Float64(-0.012345679012345678 / Float64(x * x)) / Float64(1.0 + Float64(-0.1111111111111111 / x)));
	elseif (y <= 4.8e+150)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(1.0 + Float64(-0.0013717421124828531 / Float64(x * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.7e+144)
		tmp = (-0.012345679012345678 / (x * x)) / (1.0 + (-0.1111111111111111 / x));
	elseif (y <= 4.8e+150)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = 1.0 + (-0.0013717421124828531 / (x * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.7e+144], N[(N[(-0.012345679012345678 / N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.8e+150], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.0013717421124828531 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.7e144

   1. Initial program 99.6%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

      7. /-lowering-/.f64 2.3%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

   7. Simplified 2.3%

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

   9. Applied egg-rr 26.0%

      \[\leadsto \color{blue}{\frac{1 - \frac{0.012345679012345678}{x \cdot x}}{1 + \frac{-0.1111111111111111}{x}}} \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{-1}{81}}{{x}^{2}}\right)}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{81}, \left({x}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{81}, \left(x \cdot x\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right)\right) \]

       3. *-lowering-*.f64 24.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{81}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right)\right) \]

   12. Simplified 24.7%

       \[\leadsto \frac{\color{blue}{\frac{-0.012345679012345678}{x \cdot x}}}{1 + \frac{-0.1111111111111111}{x}} \]

   if -1.7e144 < y < 4.80000000000000005e150

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

      7. /-lowering-/.f64 81.7%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

   7. Simplified 81.7%

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

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{x}{\frac{-1}{9}}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \color{blue}{\frac{-1}{9}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\frac{1}{9}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{-1}{9}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\frac{-1}{9}\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \frac{1}{9}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \frac{1}{9}\right)\right)\right) \]

      8. div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{x}}{9}\right)\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x \cdot 9}}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{-1}{\color{blue}{x} \cdot 9}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot 9\right)}\right)\right) \]

      13. *-lowering-*.f64 81.8%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{9}\right)\right)\right) \]

   9. Applied egg-rr 81.8%

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

   if 4.80000000000000005e150 < y

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

      7. /-lowering-/.f64 3.9%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

   7. Simplified 3.9%

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

   9. Applied egg-rr 4.5%

      \[\leadsto \color{blue}{\left(1 + \frac{-0.0013717421124828531}{x \cdot \left(x \cdot x\right)}\right) \cdot \frac{1}{1 + \frac{\frac{-0.1111111111111111}{x} - 1}{\frac{x}{-0.1111111111111111}}}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \color{blue}{1}\right) \]
   11. Step-by-step derivation

   12. Simplified 29.3%

       \[\leadsto \left(1 + \frac{-0.0013717421124828531}{x \cdot \left(x \cdot x\right)}\right) \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{-0.012345679012345678}{x \cdot x}}{1 + \frac{-0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+150}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.0013717421124828531}{x \cdot \left(x \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 67.4% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+144}:\\ \;\;\;\;\frac{1 - \frac{0.012345679012345678}{x \cdot x}}{\frac{-0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+152}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.0013717421124828531}{x \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.7e+144)
   (/ (- 1.0 (/ 0.012345679012345678 (* x x))) (/ -0.1111111111111111 x))
   (if (<= y 2e+152)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (+ 1.0 (/ -0.0013717421124828531 (* x (* x x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.7e+144) {
		tmp = (1.0 - (0.012345679012345678 / (x * x))) / (-0.1111111111111111 / x);
	} else if (y <= 2e+152) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 + (-0.0013717421124828531 / (x * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.7d+144)) then
        tmp = (1.0d0 - (0.012345679012345678d0 / (x * x))) / ((-0.1111111111111111d0) / x)
    else if (y <= 2d+152) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = 1.0d0 + ((-0.0013717421124828531d0) / (x * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.7e+144) {
		tmp = (1.0 - (0.012345679012345678 / (x * x))) / (-0.1111111111111111 / x);
	} else if (y <= 2e+152) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 + (-0.0013717421124828531 / (x * (x * x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.7e+144:
		tmp = (1.0 - (0.012345679012345678 / (x * x))) / (-0.1111111111111111 / x)
	elif y <= 2e+152:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = 1.0 + (-0.0013717421124828531 / (x * (x * x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.7e+144)
		tmp = Float64(Float64(1.0 - Float64(0.012345679012345678 / Float64(x * x))) / Float64(-0.1111111111111111 / x));
	elseif (y <= 2e+152)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(1.0 + Float64(-0.0013717421124828531 / Float64(x * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.7e+144)
		tmp = (1.0 - (0.012345679012345678 / (x * x))) / (-0.1111111111111111 / x);
	elseif (y <= 2e+152)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = 1.0 + (-0.0013717421124828531 / (x * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.7e+144], N[(N[(1.0 - N[(0.012345679012345678 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e+152], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.0013717421124828531 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.7e144

   1. Initial program 99.6%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

      7. /-lowering-/.f64 2.3%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

   7. Simplified 2.3%

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

   9. Applied egg-rr 26.0%

      \[\leadsto \color{blue}{\frac{1 - \frac{0.012345679012345678}{x \cdot x}}{1 + \frac{-0.1111111111111111}{x}}} \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \color{blue}{\left(\frac{\frac{-1}{9}}{x}\right)}\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 24.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

   12. Simplified 24.1%

       \[\leadsto \frac{1 - \frac{0.012345679012345678}{x \cdot x}}{\color{blue}{\frac{-0.1111111111111111}{x}}} \]

   if -1.7e144 < y < 2.0000000000000001e152

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

      7. /-lowering-/.f64 81.7%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

   7. Simplified 81.7%

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

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{x}{\frac{-1}{9}}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \color{blue}{\frac{-1}{9}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\frac{1}{9}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{-1}{9}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\frac{-1}{9}\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \frac{1}{9}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \frac{1}{9}\right)\right)\right) \]

      8. div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{x}}{9}\right)\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x \cdot 9}}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{-1}{\color{blue}{x} \cdot 9}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot 9\right)}\right)\right) \]

      13. *-lowering-*.f64 81.8%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{9}\right)\right)\right) \]

   9. Applied egg-rr 81.8%

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

   if 2.0000000000000001e152 < y

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

      7. /-lowering-/.f64 3.9%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

   7. Simplified 3.9%

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

   9. Applied egg-rr 4.5%

      \[\leadsto \color{blue}{\left(1 + \frac{-0.0013717421124828531}{x \cdot \left(x \cdot x\right)}\right) \cdot \frac{1}{1 + \frac{\frac{-0.1111111111111111}{x} - 1}{\frac{x}{-0.1111111111111111}}}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \color{blue}{1}\right) \]
   11. Step-by-step derivation

   12. Simplified 29.3%

       \[\leadsto \left(1 + \frac{-0.0013717421124828531}{x \cdot \left(x \cdot x\right)}\right) \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+144}:\\ \;\;\;\;\frac{1 - \frac{0.012345679012345678}{x \cdot x}}{\frac{-0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+152}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.0013717421124828531}{x \cdot \left(x \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 65.5% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{+150}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.0013717421124828531}{x \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.15e+150)
   (+ 1.0 (/ -1.0 (* x 9.0)))
   (+ 1.0 (/ -0.0013717421124828531 (* x (* x x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.15e+150) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 + (-0.0013717421124828531 / (x * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.15d+150) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = 1.0d0 + ((-0.0013717421124828531d0) / (x * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.15e+150) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 + (-0.0013717421124828531 / (x * (x * x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.15e+150:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = 1.0 + (-0.0013717421124828531 / (x * (x * x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.15e+150)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(1.0 + Float64(-0.0013717421124828531 / Float64(x * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.15e+150)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = 1.0 + (-0.0013717421124828531 / (x * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.15e+150], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.0013717421124828531 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.15000000000000001e150

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

      7. /-lowering-/.f64 69.7%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

   7. Simplified 69.7%

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

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{x}{\frac{-1}{9}}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \color{blue}{\frac{-1}{9}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\frac{1}{9}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{-1}{9}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\frac{-1}{9}\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \frac{1}{9}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \frac{1}{9}\right)\right)\right) \]

      8. div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{x}}{9}\right)\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x \cdot 9}}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{-1}{\color{blue}{x} \cdot 9}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot 9\right)}\right)\right) \]

      13. *-lowering-*.f64 69.8%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{9}\right)\right)\right) \]

   9. Applied egg-rr 69.8%

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

   if 1.15000000000000001e150 < y

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

      7. /-lowering-/.f64 3.9%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

   7. Simplified 3.9%

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

   9. Applied egg-rr 4.5%

      \[\leadsto \color{blue}{\left(1 + \frac{-0.0013717421124828531}{x \cdot \left(x \cdot x\right)}\right) \cdot \frac{1}{1 + \frac{\frac{-0.1111111111111111}{x} - 1}{\frac{x}{-0.1111111111111111}}}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \color{blue}{1}\right) \]
   11. Step-by-step derivation

   12. Simplified 29.3%

       \[\leadsto \left(1 + \frac{-0.0013717421124828531}{x \cdot \left(x \cdot x\right)}\right) \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{+150}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.0013717421124828531}{x \cdot \left(x \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 65.2% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{+148}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.012345679012345678}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.1e+148)
   (+ 1.0 (/ -1.0 (* x 9.0)))
   (/ -0.012345679012345678 (* x x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.1e+148) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = -0.012345679012345678 / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.1d+148) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = (-0.012345679012345678d0) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.1e+148) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = -0.012345679012345678 / (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.1e+148:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = -0.012345679012345678 / (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.1e+148)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(-0.012345679012345678 / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.1e+148)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = -0.012345679012345678 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.1e+148], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.012345679012345678 / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.0999999999999999e148

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

      7. /-lowering-/.f64 69.7%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

   7. Simplified 69.7%

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

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{x}{\frac{-1}{9}}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \color{blue}{\frac{-1}{9}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\frac{1}{9}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{-1}{9}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\frac{-1}{9}\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \frac{1}{9}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x} \cdot \frac{1}{9}\right)\right)\right) \]

      8. div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{x}}{9}\right)\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x \cdot 9}}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{-1}{\color{blue}{x} \cdot 9}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot 9\right)}\right)\right) \]

      13. *-lowering-*.f64 69.8%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{9}\right)\right)\right) \]

   9. Applied egg-rr 69.8%

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

   if 1.0999999999999999e148 < y

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

      7. /-lowering-/.f64 3.9%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

   7. Simplified 3.9%

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

   9. Applied egg-rr 0.6%

      \[\leadsto \color{blue}{\frac{1 - \frac{0.012345679012345678}{x \cdot x}}{1 + \frac{-0.1111111111111111}{x}}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \color{blue}{1}\right) \]
   11. Step-by-step derivation

   12. Simplified 26.4%

       \[\leadsto \frac{1 - \frac{0.012345679012345678}{x \cdot x}}{\color{blue}{1}} \]

   13. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{\frac{-1}{81}}{{x}^{2}}} \]
   14. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-1}{81}, \color{blue}{\left({x}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-1}{81}, \left(x \cdot \color{blue}{x}\right)\right) \]

       3. *-lowering-*.f64 27.1%

          \[\leadsto \mathsf{/.f64}\left(\frac{-1}{81}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   15. Simplified 27.1%

       \[\leadsto \color{blue}{\frac{-0.012345679012345678}{x \cdot x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 65.1% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{+149}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.012345679012345678}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.05e+149)
   (+ 1.0 (/ -0.1111111111111111 x))
   (/ -0.012345679012345678 (* x x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.05e+149) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = -0.012345679012345678 / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.05d+149) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = (-0.012345679012345678d0) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.05e+149) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = -0.012345679012345678 / (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.05e+149:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = -0.012345679012345678 / (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.05e+149)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(-0.012345679012345678 / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.05e+149)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = -0.012345679012345678 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.05e+149], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(-0.012345679012345678 / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.0500000000000001e149

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

      7. /-lowering-/.f64 69.7%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

   7. Simplified 69.7%

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

   if 1.0500000000000001e149 < y

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

      7. /-lowering-/.f64 3.9%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

   7. Simplified 3.9%

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

   9. Applied egg-rr 0.6%

      \[\leadsto \color{blue}{\frac{1 - \frac{0.012345679012345678}{x \cdot x}}{1 + \frac{-0.1111111111111111}{x}}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \color{blue}{1}\right) \]
   11. Step-by-step derivation

   12. Simplified 26.4%

       \[\leadsto \frac{1 - \frac{0.012345679012345678}{x \cdot x}}{\color{blue}{1}} \]

   13. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{\frac{-1}{81}}{{x}^{2}}} \]
   14. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-1}{81}, \color{blue}{\left({x}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-1}{81}, \left(x \cdot \color{blue}{x}\right)\right) \]

       3. *-lowering-*.f64 27.1%

          \[\leadsto \mathsf{/.f64}\left(\frac{-1}{81}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   15. Simplified 27.1%

       \[\leadsto \color{blue}{\frac{-0.012345679012345678}{x \cdot x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 61.6% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{-7}:\\ \;\;\;\;-0.1111111111111111 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 6.8e-7) (* -0.1111111111111111 (/ 1.0 x)) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6.8e-7) {
		tmp = -0.1111111111111111 * (1.0 / 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 (x <= 6.8d-7) then
        tmp = (-0.1111111111111111d0) * (1.0d0 / x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 6.8e-7) {
		tmp = -0.1111111111111111 * (1.0 / x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 6.8e-7:
		tmp = -0.1111111111111111 * (1.0 / x)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 6.8e-7)
		tmp = Float64(-0.1111111111111111 * Float64(1.0 / x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 6.8e-7)
		tmp = -0.1111111111111111 * (1.0 / x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 6.8e-7], N[(-0.1111111111111111 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 6.79999999999999948e-7

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

      7. /-lowering-/.f64 57.9%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

   7. Simplified 57.9%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 57.9%

         \[\leadsto \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right) \]

   10. Simplified 57.9%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\frac{-1}{9}}\right) \]

       4. /-lowering-/.f64 57.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \frac{-1}{9}\right) \]

   12. Applied egg-rr 57.9%

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

   if 6.79999999999999948e-7 < x

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

      7. /-lowering-/.f64 62.1%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

   7. Simplified 62.1%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 60.6%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{-7}:\\ \;\;\;\;-0.1111111111111111 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 61.7% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 6.8e-7) (/ -0.1111111111111111 x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6.8e-7) {
		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 (x <= 6.8d-7) 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 (x <= 6.8e-7) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 6.8e-7:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 6.8e-7)
		tmp = Float64(-0.1111111111111111 / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 6.8e-7)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 6.8e-7], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 6.79999999999999948e-7

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

      7. /-lowering-/.f64 57.9%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

   7. Simplified 57.9%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 57.9%

         \[\leadsto \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right) \]

   10. Simplified 57.9%

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

   if 6.79999999999999948e-7 < x

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

      7. /-lowering-/.f64 62.1%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

   7. Simplified 62.1%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 60.6%

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

Alternative 20: 62.6% accurate, 22.6× 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. Step-by-step derivation

   1. sub-neg N/A

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

   6. associate-/r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

   7. distribute-neg-frac N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

   11. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

   16. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

   17. metadata-eval 99.7%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

3. Simplified 99.7%

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

5. Taylor expanded in y around 0

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

   1. sub-neg N/A

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

   5. distribute-neg-frac N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

   7. /-lowering-/.f64 59.9%

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

7. Simplified 59.9%

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

Alternative 21: 32.4% accurate, 113.0× 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. Step-by-step derivation

   1. sub-neg N/A

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

   6. associate-/r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

   7. distribute-neg-frac N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

   11. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

   16. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

   17. metadata-eval 99.7%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

3. Simplified 99.7%

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

5. Taylor expanded in y around 0

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

   1. sub-neg N/A

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

   5. distribute-neg-frac N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

   7. /-lowering-/.f64 59.9%

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

7. Simplified 59.9%

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

8. Taylor expanded in x around inf

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

10. Simplified 29.9%

    \[\leadsto \color{blue}{1} \]
11. Add Preprocessing

Developer Target 1: 99.7% accurate, 1.0× 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 2024138 
(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)))))