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

Percentage Accurate: 99.7% → 99.7%

Time: 12.6s

Alternatives: 25

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. 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 99.7%

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

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 94.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+42}:\\ \;\;\;\;1 + {x}^{-0.5} \cdot \frac{y}{-3}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+16}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{y}{\sqrt{x}}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.5e+42)
   (+ 1.0 (* (pow x -0.5) (/ y -3.0)))
   (if (<= y 1.5e+16)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (- 1.0 (/ (/ y (sqrt x)) 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.5e+42) {
		tmp = 1.0 + (pow(x, -0.5) * (y / -3.0));
	} else if (y <= 1.5e+16) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 - ((y / sqrt(x)) / 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-7.5d+42)) then
        tmp = 1.0d0 + ((x ** (-0.5d0)) * (y / (-3.0d0)))
    else if (y <= 1.5d+16) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = 1.0d0 - ((y / sqrt(x)) / 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7.5e+42) {
		tmp = 1.0 + (Math.pow(x, -0.5) * (y / -3.0));
	} else if (y <= 1.5e+16) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 - ((y / Math.sqrt(x)) / 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.5e+42:
		tmp = 1.0 + (math.pow(x, -0.5) * (y / -3.0))
	elif y <= 1.5e+16:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = 1.0 - ((y / math.sqrt(x)) / 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.5e+42)
		tmp = Float64(1.0 + Float64((x ^ -0.5) * Float64(y / -3.0)));
	elseif (y <= 1.5e+16)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(1.0 - Float64(Float64(y / sqrt(x)) / 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.5e+42)
		tmp = 1.0 + ((x ^ -0.5) * (y / -3.0));
	elseif (y <= 1.5e+16)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = 1.0 - ((y / sqrt(x)) / 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.5e+42], N[(1.0 + N[(N[Power[x, -0.5], $MachinePrecision] * N[(y / -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+16], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.50000000000000041e42

   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 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 92.1%

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

   7. Simplified 92.1%

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. div-inv N/A

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

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

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

      5. pow1/2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({\left(\frac{1}{x}\right)}^{\frac{1}{2}}\right), \left(\frac{\color{blue}{y}}{-3}\right)\right)\right) \]

      6. inv-pow N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({\left({x}^{-1}\right)}^{\frac{1}{2}}\right), \left(\frac{y}{-3}\right)\right)\right) \]

      7. pow-pow N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{\left(-1 \cdot \frac{1}{2}\right)}\right), \left(\frac{\color{blue}{y}}{-3}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \left(-1 \cdot \frac{1}{2}\right)\right), \left(\frac{\color{blue}{y}}{-3}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \frac{-1}{2}\right), \left(\frac{y}{-3}\right)\right)\right) \]

      10. /-lowering-/.f64 92.2%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \frac{-1}{2}\right), \mathsf{/.f64}\left(y, \color{blue}{-3}\right)\right)\right) \]

   9. Applied egg-rr 92.2%

      \[\leadsto 1 + \color{blue}{{x}^{-0.5} \cdot \frac{y}{-3}} \]

   if -7.50000000000000041e42 < y < 1.5e16

   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 98.0%

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

   7. Simplified 98.0%

      \[\leadsto \color{blue}{1 + \frac{-0.1111111111111111}{x}} \]
   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. clear-num N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

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

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

      13. associate-/r/ N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 98.1%

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

   9. Applied egg-rr 98.1%

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

   if 1.5e16 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 94.6%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. sqrt-lowering-sqrt.f64 94.7%

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

   7. Applied egg-rr 94.7%

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

Alternative 3: 94.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+44}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+16}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{y}{\sqrt{x}}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.05e+44)
   (- 1.0 (/ y (* 3.0 (sqrt x))))
   (if (<= y 1.5e+16)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (- 1.0 (/ (/ y (sqrt x)) 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.05e+44) {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	} else if (y <= 1.5e+16) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 - ((y / sqrt(x)) / 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.05d+44)) then
        tmp = 1.0d0 - (y / (3.0d0 * sqrt(x)))
    else if (y <= 1.5d+16) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = 1.0d0 - ((y / sqrt(x)) / 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.05e+44) {
		tmp = 1.0 - (y / (3.0 * Math.sqrt(x)));
	} else if (y <= 1.5e+16) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 - ((y / Math.sqrt(x)) / 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.05e+44:
		tmp = 1.0 - (y / (3.0 * math.sqrt(x)))
	elif y <= 1.5e+16:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = 1.0 - ((y / math.sqrt(x)) / 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.05e+44)
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	elseif (y <= 1.5e+16)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(1.0 - Float64(Float64(y / sqrt(x)) / 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.05e+44)
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	elseif (y <= 1.5e+16)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = 1.0 - ((y / sqrt(x)) / 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.05e+44], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+16], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.04999999999999993e44

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 92.1%

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

   if -1.04999999999999993e44 < y < 1.5e16

   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 98.0%

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

   7. Simplified 98.0%

      \[\leadsto \color{blue}{1 + \frac{-0.1111111111111111}{x}} \]
   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. clear-num N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

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

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

      13. associate-/r/ N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 98.1%

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

   9. Applied egg-rr 98.1%

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

   if 1.5e16 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 94.6%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. sqrt-lowering-sqrt.f64 94.7%

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

   7. Applied egg-rr 94.7%

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

Alternative 4: 94.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+42}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+16}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x}} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.5e+42)
   (- 1.0 (/ y (* 3.0 (sqrt x))))
   (if (<= y 1.5e+16)
     (+ 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 <= -8.5e+42) {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	} else if (y <= 1.5e+16) {
		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 <= (-8.5d+42)) then
        tmp = 1.0d0 - (y / (3.0d0 * sqrt(x)))
    else if (y <= 1.5d+16) 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 <= -8.5e+42) {
		tmp = 1.0 - (y / (3.0 * Math.sqrt(x)));
	} else if (y <= 1.5e+16) {
		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 <= -8.5e+42:
		tmp = 1.0 - (y / (3.0 * math.sqrt(x)))
	elif y <= 1.5e+16:
		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 <= -8.5e+42)
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	elseif (y <= 1.5e+16)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(1.0 - Float64(Float64(y / sqrt(x)) * 0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8.5e+42)
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	elseif (y <= 1.5e+16)
		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, -8.5e+42], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+16], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.5000000000000003e42

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 92.1%

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

   if -8.5000000000000003e42 < y < 1.5e16

   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 98.0%

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

   7. Simplified 98.0%

      \[\leadsto \color{blue}{1 + \frac{-0.1111111111111111}{x}} \]
   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. clear-num N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

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

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

      13. associate-/r/ N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 98.1%

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

   9. Applied egg-rr 98.1%

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

   if 1.5e16 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 94.6%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. associate-/l* N/A

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

      4. unpow-prod-down N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. inv-pow N/A

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

      8. clear-num N/A

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

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

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

      10. metadata-eval N/A

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

      11. /-lowering-/.f64 N/A

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

      12. sqrt-lowering-sqrt.f64 94.6%

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

   7. Applied egg-rr 94.6%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+42}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+16}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x}} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.4 \cdot 10^{+42}:\\ \;\;\;\;1 + \frac{\frac{y}{-3}}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+16}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x}} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.4e+42)
   (+ 1.0 (/ (/ y -3.0) (sqrt x)))
   (if (<= y 1.5e+16)
     (+ 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 <= -8.4e+42) {
		tmp = 1.0 + ((y / -3.0) / sqrt(x));
	} else if (y <= 1.5e+16) {
		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 <= (-8.4d+42)) then
        tmp = 1.0d0 + ((y / (-3.0d0)) / sqrt(x))
    else if (y <= 1.5d+16) 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 <= -8.4e+42) {
		tmp = 1.0 + ((y / -3.0) / Math.sqrt(x));
	} else if (y <= 1.5e+16) {
		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 <= -8.4e+42:
		tmp = 1.0 + ((y / -3.0) / math.sqrt(x))
	elif y <= 1.5e+16:
		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 <= -8.4e+42)
		tmp = Float64(1.0 + Float64(Float64(y / -3.0) / sqrt(x)));
	elseif (y <= 1.5e+16)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(1.0 - Float64(Float64(y / sqrt(x)) * 0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8.4e+42)
		tmp = 1.0 + ((y / -3.0) / sqrt(x));
	elseif (y <= 1.5e+16)
		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, -8.4e+42], N[(1.0 + N[(N[(y / -3.0), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+16], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.39999999999999982e42

   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 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 92.1%

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

   7. Simplified 92.1%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

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

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

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

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

      10. sqrt-lowering-sqrt.f64 92.1%

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

   9. Applied egg-rr 92.1%

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

   if -8.39999999999999982e42 < y < 1.5e16

   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 98.0%

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

   7. Simplified 98.0%

      \[\leadsto \color{blue}{1 + \frac{-0.1111111111111111}{x}} \]
   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. clear-num N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

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

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

      13. associate-/r/ N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 98.1%

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

   9. Applied egg-rr 98.1%

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

   if 1.5e16 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 94.6%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. associate-/l* N/A

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

      4. unpow-prod-down N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. inv-pow N/A

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

      8. clear-num N/A

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

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

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

      10. metadata-eval N/A

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

      11. /-lowering-/.f64 N/A

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

      12. sqrt-lowering-sqrt.f64 94.6%

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

   7. Applied egg-rr 94.6%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.4 \cdot 10^{+42}:\\ \;\;\;\;1 + \frac{\frac{y}{-3}}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+16}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x}} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{\frac{y}{-3}}{\sqrt{x}}\\ \mathbf{if}\;y \leq -2.9 \cdot 10^{+43}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+16}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (/ y -3.0) (sqrt x)))))
   (if (<= y -2.9e+43)
     t_0
     (if (<= y 1.5e+16) (+ 1.0 (/ -1.0 (* x 9.0))) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((y / -3.0) / sqrt(x));
	double tmp;
	if (y <= -2.9e+43) {
		tmp = t_0;
	} else if (y <= 1.5e+16) {
		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 / (-3.0d0)) / sqrt(x))
    if (y <= (-2.9d+43)) then
        tmp = t_0
    else if (y <= 1.5d+16) 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 / -3.0) / Math.sqrt(x));
	double tmp;
	if (y <= -2.9e+43) {
		tmp = t_0;
	} else if (y <= 1.5e+16) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + ((y / -3.0) / math.sqrt(x))
	tmp = 0
	if y <= -2.9e+43:
		tmp = t_0
	elif y <= 1.5e+16:
		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(Float64(y / -3.0) / sqrt(x)))
	tmp = 0.0
	if (y <= -2.9e+43)
		tmp = t_0;
	elseif (y <= 1.5e+16)
		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 / -3.0) / sqrt(x));
	tmp = 0.0;
	if (y <= -2.9e+43)
		tmp = t_0;
	elseif (y <= 1.5e+16)
		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[(N[(y / -3.0), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.9e+43], t$95$0, If[LessEqual[y, 1.5e+16], 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 < -2.9000000000000002e43 or 1.5e16 < 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 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 93.3%

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

   7. Simplified 93.3%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

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

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

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

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

      10. sqrt-lowering-sqrt.f64 93.4%

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

   9. Applied egg-rr 93.4%

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

   if -2.9000000000000002e43 < y < 1.5e16

   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 98.0%

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

   7. Simplified 98.0%

      \[\leadsto \color{blue}{1 + \frac{-0.1111111111111111}{x}} \]
   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. clear-num N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

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

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

      13. associate-/r/ N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 98.1%

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

   9. Applied egg-rr 98.1%

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

Alternative 7: 94.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{if}\;y \leq -4.3 \cdot 10^{+43}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+16}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* y (/ -0.3333333333333333 (sqrt x))))))
   (if (<= y -4.3e+43)
     t_0
     (if (<= y 1.5e+16) (+ 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 <= -4.3e+43) {
		tmp = t_0;
	} else if (y <= 1.5e+16) {
		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 <= (-4.3d+43)) then
        tmp = t_0
    else if (y <= 1.5d+16) 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 <= -4.3e+43) {
		tmp = t_0;
	} else if (y <= 1.5e+16) {
		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 <= -4.3e+43:
		tmp = t_0
	elif y <= 1.5e+16:
		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 <= -4.3e+43)
		tmp = t_0;
	elseif (y <= 1.5e+16)
		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 <= -4.3e+43)
		tmp = t_0;
	elseif (y <= 1.5e+16)
		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, -4.3e+43], t$95$0, If[LessEqual[y, 1.5e+16], 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 < -4.3e43 or 1.5e16 < 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 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 93.3%

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

   7. Simplified 93.3%

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

      1. associate-*r* N/A

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

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

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. associate-*l/ N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 93.2%

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

   9. Applied egg-rr 93.2%

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

   if -4.3e43 < y < 1.5e16

   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 98.0%

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

   7. Simplified 98.0%

      \[\leadsto \color{blue}{1 + \frac{-0.1111111111111111}{x}} \]
   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. clear-num N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

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

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

      13. associate-/r/ N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 98.1%

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

   9. Applied egg-rr 98.1%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{+43}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+16}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot -0.3333333333333333\right) \cdot \sqrt{\frac{1}{x}}\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{+84}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+78}:\\ \;\;\;\;1 + \frac{\frac{1}{x}}{-9}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y -0.3333333333333333) (sqrt (/ 1.0 x)))))
   (if (<= y -5.8e+84)
     t_0
     (if (<= y 9.5e+78) (+ 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 <= -5.8e+84) {
		tmp = t_0;
	} else if (y <= 9.5e+78) {
		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 <= (-5.8d+84)) then
        tmp = t_0
    else if (y <= 9.5d+78) 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 <= -5.8e+84) {
		tmp = t_0;
	} else if (y <= 9.5e+78) {
		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 <= -5.8e+84:
		tmp = t_0
	elif y <= 9.5e+78:
		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 <= -5.8e+84)
		tmp = t_0;
	elseif (y <= 9.5e+78)
		tmp = Float64(1.0 + Float64(Float64(1.0 / x) / -9.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * -0.3333333333333333) * sqrt((1.0 / x));
	tmp = 0.0;
	if (y <= -5.8e+84)
		tmp = t_0;
	elseif (y <= 9.5e+78)
		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, -5.8e+84], t$95$0, If[LessEqual[y, 9.5e+78], N[(1.0 + N[(N[(1.0 / x), $MachinePrecision] / -9.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.79999999999999977e84 or 9.5000000000000006e78 < y

   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 99.5%

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

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\left(1 + \frac{-0.1111111111111111}{x}\right) + \frac{y}{\sqrt{x} \cdot -3}} \]
   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 93.8%

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

   7. Simplified 93.8%

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

   if -5.79999999999999977e84 < y < 9.5000000000000006e78

   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 91.8%

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

   7. Simplified 91.8%

      \[\leadsto \color{blue}{1 + \frac{-0.1111111111111111}{x}} \]
   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. distribute-neg-frac2 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. metadata-eval 91.9%

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

   9. Applied egg-rr 91.9%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+84}:\\ \;\;\;\;\left(y \cdot -0.3333333333333333\right) \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+78}:\\ \;\;\;\;1 + \frac{\frac{1}{x}}{-9}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -0.3333333333333333\right) \cdot \sqrt{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{+14}:\\ \;\;\;\;1 + \frac{-0.1111111111111111 - y \cdot \left(\sqrt{x} \cdot 0.3333333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3.2e+14)
   (+ 1.0 (/ (- -0.1111111111111111 (* y (* (sqrt x) 0.3333333333333333))) x))
   (- 1.0 (/ y (* 3.0 (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3.2e+14) {
		tmp = 1.0 + ((-0.1111111111111111 - (y * (sqrt(x) * 0.3333333333333333))) / x);
	} else {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 3.2d+14) then
        tmp = 1.0d0 + (((-0.1111111111111111d0) - (y * (sqrt(x) * 0.3333333333333333d0))) / x)
    else
        tmp = 1.0d0 - (y / (3.0d0 * sqrt(x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 3.2e+14) {
		tmp = 1.0 + ((-0.1111111111111111 - (y * (Math.sqrt(x) * 0.3333333333333333))) / x);
	} else {
		tmp = 1.0 - (y / (3.0 * Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 3.2e+14:
		tmp = 1.0 + ((-0.1111111111111111 - (y * (math.sqrt(x) * 0.3333333333333333))) / x)
	else:
		tmp = 1.0 - (y / (3.0 * math.sqrt(x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3.2e+14)
		tmp = Float64(1.0 + Float64(Float64(-0.1111111111111111 - Float64(y * Float64(sqrt(x) * 0.3333333333333333))) / x));
	else
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 3.2e+14)
		tmp = 1.0 + ((-0.1111111111111111 - (y * (sqrt(x) * 0.3333333333333333))) / x);
	else
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3.2e+14], N[(1.0 + N[(N[(-0.1111111111111111 - N[(y * N[(N[Sqrt[x], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.2e14

   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 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. metadata-eval N/A

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

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

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

      4. div-sub N/A

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

      5. associate--r+ N/A

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

      6. +-commutative N/A

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

      7. div-sub N/A

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

      8. sub-neg N/A

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

      9. *-inverses N/A

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

      10. mul-1-neg N/A

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

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

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

      12. mul-1-neg N/A

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

      13. distribute-neg-frac N/A

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

      14. /-lowering-/.f64 N/A

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

   7. Simplified 99.5%

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

   if 3.2e14 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.8%

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

Alternative 10: 98.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.11)
   (/ (+ -0.1111111111111111 (* (sqrt x) (* y -0.3333333333333333))) x)
   (- 1.0 (/ y (* 3.0 (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = (-0.1111111111111111 + (sqrt(x) * (y * -0.3333333333333333))) / x;
	} else {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.11d0) then
        tmp = ((-0.1111111111111111d0) + (sqrt(x) * (y * (-0.3333333333333333d0)))) / x
    else
        tmp = 1.0d0 - (y / (3.0d0 * sqrt(x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = (-0.1111111111111111 + (Math.sqrt(x) * (y * -0.3333333333333333))) / x;
	} else {
		tmp = 1.0 - (y / (3.0 * Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.11:
		tmp = (-0.1111111111111111 + (math.sqrt(x) * (y * -0.3333333333333333))) / x
	else:
		tmp = 1.0 - (y / (3.0 * math.sqrt(x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.11)
		tmp = Float64(Float64(-0.1111111111111111 + Float64(sqrt(x) * Float64(y * -0.3333333333333333))) / x);
	else
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.11)
		tmp = (-0.1111111111111111 + (sqrt(x) * (y * -0.3333333333333333))) / x;
	else
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.11], N[(N[(-0.1111111111111111 + N[(N[Sqrt[x], $MachinePrecision] * N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   1. Initial program 99.6%

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

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

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

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

   7. Simplified 98.6%

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

   if 0.110000000000000001 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 98.9%

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

Alternative 11: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{-0.1111111111111111}{x} + \frac{\frac{y}{\sqrt{x}}}{-3}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.11)
   (+ (/ -0.1111111111111111 x) (/ (/ y (sqrt x)) -3.0))
   (- 1.0 (/ y (* 3.0 (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = (-0.1111111111111111 / x) + ((y / sqrt(x)) / -3.0);
	} 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 (x <= 0.11d0) then
        tmp = ((-0.1111111111111111d0) / x) + ((y / sqrt(x)) / (-3.0d0))
    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 (x <= 0.11) {
		tmp = (-0.1111111111111111 / x) + ((y / Math.sqrt(x)) / -3.0);
	} else {
		tmp = 1.0 - (y / (3.0 * Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.11:
		tmp = (-0.1111111111111111 / x) + ((y / math.sqrt(x)) / -3.0)
	else:
		tmp = 1.0 - (y / (3.0 * math.sqrt(x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.11)
		tmp = Float64(Float64(-0.1111111111111111 / x) + Float64(Float64(y / sqrt(x)) / -3.0));
	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 (x <= 0.11)
		tmp = (-0.1111111111111111 / x) + ((y / sqrt(x)) / -3.0);
	else
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.11], N[(N[(-0.1111111111111111 / x), $MachinePrecision] + N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   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 99.4%

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

   3. Simplified 99.4%

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

      1. associate-/r* N/A

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

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

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

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

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

      4. sqrt-lowering-sqrt.f64 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 98.6%

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

   9. Simplified 98.6%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x}} + \frac{\frac{y}{\sqrt{x}}}{-3} \]

   if 0.110000000000000001 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 98.9%

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

Alternative 12: 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 13: 99.6% accurate, 1.0× speedup

Math

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

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(Float64(y / 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[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. 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. associate-/r* N/A

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

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

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

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

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

   4. sqrt-lowering-sqrt.f64 99.6%

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

6. Applied egg-rr 99.6%

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

Alternative 14: 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 99.6%

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

3. Simplified 99.6%

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

Alternative 15: 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 99.6%

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

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 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 16: 68.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{0.012345679012345678}{x \cdot x}\\ \mathbf{if}\;y \leq -1.6 \cdot 10^{+106}:\\ \;\;\;\;\left(1 + \frac{-0.1111111111111111}{x}\right) \cdot t\_0\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+129}:\\ \;\;\;\;1 + \frac{\frac{1}{x}}{-9}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 - \frac{0.1111111111111111 + \frac{-0.012345679012345678}{x}}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ 0.012345679012345678 (* x x)))))
   (if (<= y -1.6e+106)
     (* (+ 1.0 (/ -0.1111111111111111 x)) t_0)
     (if (<= y 5.8e+129)
       (+ 1.0 (/ (/ 1.0 x) -9.0))
       (*
        t_0
        (- 1.0 (/ (+ 0.1111111111111111 (/ -0.012345679012345678 x)) x)))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (0.012345679012345678 / (x * x));
	double tmp;
	if (y <= -1.6e+106) {
		tmp = (1.0 + (-0.1111111111111111 / x)) * t_0;
	} else if (y <= 5.8e+129) {
		tmp = 1.0 + ((1.0 / x) / -9.0);
	} else {
		tmp = t_0 * (1.0 - ((0.1111111111111111 + (-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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (0.012345679012345678d0 / (x * x))
    if (y <= (-1.6d+106)) then
        tmp = (1.0d0 + ((-0.1111111111111111d0) / x)) * t_0
    else if (y <= 5.8d+129) then
        tmp = 1.0d0 + ((1.0d0 / x) / (-9.0d0))
    else
        tmp = t_0 * (1.0d0 - ((0.1111111111111111d0 + ((-0.012345679012345678d0) / x)) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - (0.012345679012345678 / (x * x));
	double tmp;
	if (y <= -1.6e+106) {
		tmp = (1.0 + (-0.1111111111111111 / x)) * t_0;
	} else if (y <= 5.8e+129) {
		tmp = 1.0 + ((1.0 / x) / -9.0);
	} else {
		tmp = t_0 * (1.0 - ((0.1111111111111111 + (-0.012345679012345678 / x)) / x));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (0.012345679012345678 / (x * x))
	tmp = 0
	if y <= -1.6e+106:
		tmp = (1.0 + (-0.1111111111111111 / x)) * t_0
	elif y <= 5.8e+129:
		tmp = 1.0 + ((1.0 / x) / -9.0)
	else:
		tmp = t_0 * (1.0 - ((0.1111111111111111 + (-0.012345679012345678 / x)) / x))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(0.012345679012345678 / Float64(x * x)))
	tmp = 0.0
	if (y <= -1.6e+106)
		tmp = Float64(Float64(1.0 + Float64(-0.1111111111111111 / x)) * t_0);
	elseif (y <= 5.8e+129)
		tmp = Float64(1.0 + Float64(Float64(1.0 / x) / -9.0));
	else
		tmp = Float64(t_0 * Float64(1.0 - Float64(Float64(0.1111111111111111 + Float64(-0.012345679012345678 / x)) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (0.012345679012345678 / (x * x));
	tmp = 0.0;
	if (y <= -1.6e+106)
		tmp = (1.0 + (-0.1111111111111111 / x)) * t_0;
	elseif (y <= 5.8e+129)
		tmp = 1.0 + ((1.0 / x) / -9.0);
	else
		tmp = t_0 * (1.0 - ((0.1111111111111111 + (-0.012345679012345678 / x)) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(0.012345679012345678 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.6e+106], N[(N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y, 5.8e+129], N[(1.0 + N[(N[(1.0 / x), $MachinePrecision] / -9.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 - N[(N[(0.1111111111111111 + N[(-0.012345679012345678 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.5999999999999999e106

   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 99.5%

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

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\left(1 + \frac{-0.1111111111111111}{x}\right) + \frac{y}{\sqrt{x} \cdot -3}} \]
   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 3.2%

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

   7. Simplified 3.2%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. frac-times N/A

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

      7. /-lowering-/.f64 N/A

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

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{1 - \frac{\frac{-1}{9}}{x}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{1 - \frac{\frac{-1}{9}}{x}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(1 - \frac{\frac{-1}{9}}{x}\right)}\right)\right) \]

      11. div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \left(1 - \frac{-1}{9} \cdot \color{blue}{\frac{1}{x}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{9}\right)\right) \cdot \frac{1}{x}}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \left(1 + \frac{1}{9} \cdot \frac{\color{blue}{1}}{x}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \left(1 + \frac{1}{9} \cdot \frac{\color{blue}{1}}{x}\right)\right)\right) \]

      15. associate-/r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \left(1 + \frac{1}{\color{blue}{\frac{9}{\frac{1}{x}}}}\right)\right)\right) \]

      16. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \left(1 + \frac{\frac{1}{x}}{\color{blue}{9}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{1}{x}}{9}\right)}\right)\right)\right) \]

      18. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{1}{\color{blue}{9 \cdot x}}\right)\right)\right)\right) \]

      19. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{9}}{x}\right)\right)\right)\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{-1}{9}\right)}{x}\right)\right)\right)\right) \]

      22. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{-1}{9}\right)\right), \color{blue}{x}\right)\right)\right)\right) \]

      23. metadata-eval 3.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{9}, x\right)\right)\right)\right) \]

   9. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{\left(1 - \frac{0.012345679012345678}{x \cdot x}\right) \cdot \frac{1}{1 + \frac{0.1111111111111111}{x}}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \color{blue}{\left(1 - \frac{1}{9} \cdot \frac{1}{x}\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right)\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right)\right) \]

       5. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right)\right) \]

       7. /-lowering-/.f64 21.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right)\right) \]

   12. Simplified 21.9%

       \[\leadsto \left(1 - \frac{0.012345679012345678}{x \cdot x}\right) \cdot \color{blue}{\left(1 + \frac{-0.1111111111111111}{x}\right)} \]

   if -1.5999999999999999e106 < y < 5.80000000000000005e129

   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 85.4%

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

   7. Simplified 85.4%

      \[\leadsto \color{blue}{1 + \frac{-0.1111111111111111}{x}} \]
   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. distribute-neg-frac2 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. metadata-eval 85.5%

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

   9. Applied egg-rr 85.5%

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

   if 5.80000000000000005e129 < 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 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 3.3%

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

   7. Simplified 3.3%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. frac-times N/A

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

      7. /-lowering-/.f64 N/A

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

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{1 - \frac{\frac{-1}{9}}{x}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{1 - \frac{\frac{-1}{9}}{x}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(1 - \frac{\frac{-1}{9}}{x}\right)}\right)\right) \]

      11. div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \left(1 - \frac{-1}{9} \cdot \color{blue}{\frac{1}{x}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{9}\right)\right) \cdot \frac{1}{x}}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \left(1 + \frac{1}{9} \cdot \frac{\color{blue}{1}}{x}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \left(1 + \frac{1}{9} \cdot \frac{\color{blue}{1}}{x}\right)\right)\right) \]

      15. associate-/r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \left(1 + \frac{1}{\color{blue}{\frac{9}{\frac{1}{x}}}}\right)\right)\right) \]

      16. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \left(1 + \frac{\frac{1}{x}}{\color{blue}{9}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{1}{x}}{9}\right)}\right)\right)\right) \]

      18. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{1}{\color{blue}{9 \cdot x}}\right)\right)\right)\right) \]

      19. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{9}}{x}\right)\right)\right)\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{-1}{9}\right)}{x}\right)\right)\right)\right) \]

      22. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{-1}{9}\right)\right), \color{blue}{x}\right)\right)\right)\right) \]

      23. metadata-eval 13.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{9}, x\right)\right)\right)\right) \]

   9. Applied egg-rr 13.5%

      \[\leadsto \color{blue}{\left(1 - \frac{0.012345679012345678}{x \cdot x}\right) \cdot \frac{1}{1 + \frac{0.1111111111111111}{x}}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \color{blue}{\left(\left(1 + \frac{\frac{1}{81}}{{x}^{2}}\right) - \frac{1}{9} \cdot \frac{1}{x}\right)}\right) \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\left(\frac{\frac{1}{81}}{{x}^{2}} + 1\right) - \color{blue}{\frac{1}{9}} \cdot \frac{1}{x}\right)\right) \]

       2. associate--l+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{81}}{{x}^{2}} + \color{blue}{\left(1 - \frac{1}{9} \cdot \frac{1}{x}\right)}\right)\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\left(1 - \frac{1}{9} \cdot \frac{1}{x}\right) + \color{blue}{\frac{\frac{1}{81}}{{x}^{2}}}\right)\right) \]

       4. associate--r- N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(1 - \color{blue}{\left(\frac{1}{9} \cdot \frac{1}{x} - \frac{\frac{1}{81}}{{x}^{2}}\right)}\right)\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(1 - \left(\frac{\frac{1}{9} \cdot 1}{x} - \frac{\color{blue}{\frac{1}{81}}}{{x}^{2}}\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(1 - \left(\frac{\frac{1}{9}}{x} - \frac{\frac{1}{81}}{{x}^{2}}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(1 - \left(\frac{\frac{1}{9}}{x} - \frac{\frac{1}{81}}{x \cdot \color{blue}{x}}\right)\right)\right) \]

       8. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(1 - \left(\frac{\frac{1}{9}}{x} - \frac{\frac{\frac{1}{81}}{x}}{\color{blue}{x}}\right)\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(1 - \left(\frac{\frac{1}{9}}{x} - \frac{\frac{\frac{1}{81} \cdot 1}{x}}{x}\right)\right)\right) \]

       10. associate-*r/ N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(1 - \left(\frac{\frac{1}{9}}{x} - \frac{\frac{1}{81} \cdot \frac{1}{x}}{x}\right)\right)\right) \]

       11. div-sub N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(1 - \frac{\frac{1}{9} - \frac{1}{81} \cdot \frac{1}{x}}{\color{blue}{x}}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{\frac{1}{9} - \frac{1}{81} \cdot \frac{1}{x}}{x}\right)}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{9} - \frac{1}{81} \cdot \frac{1}{x}\right), \color{blue}{x}\right)\right)\right) \]

       14. sub-neg N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{9} + \left(\mathsf{neg}\left(\frac{1}{81} \cdot \frac{1}{x}\right)\right)\right), x\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \left(\mathsf{neg}\left(\frac{1}{81} \cdot \frac{1}{x}\right)\right)\right), x\right)\right)\right) \]

       16. associate-*r/ N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \left(\mathsf{neg}\left(\frac{\frac{1}{81} \cdot 1}{x}\right)\right)\right), x\right)\right)\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \left(\mathsf{neg}\left(\frac{\frac{1}{81}}{x}\right)\right)\right), x\right)\right)\right) \]

       18. distribute-neg-frac N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \left(\frac{\mathsf{neg}\left(\frac{1}{81}\right)}{x}\right)\right), x\right)\right)\right) \]

       19. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \left(\frac{\frac{-1}{81}}{x}\right)\right), x\right)\right)\right) \]

       20. /-lowering-/.f64 27.2%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{/.f64}\left(\frac{-1}{81}, x\right)\right), x\right)\right)\right) \]

   12. Simplified 27.2%

       \[\leadsto \left(1 - \frac{0.012345679012345678}{x \cdot x}\right) \cdot \color{blue}{\left(1 - \frac{0.1111111111111111 + \frac{-0.012345679012345678}{x}}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+106}:\\ \;\;\;\;\left(1 + \frac{-0.1111111111111111}{x}\right) \cdot \left(1 - \frac{0.012345679012345678}{x \cdot x}\right)\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+129}:\\ \;\;\;\;1 + \frac{\frac{1}{x}}{-9}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{0.012345679012345678}{x \cdot x}\right) \cdot \left(1 - \frac{0.1111111111111111 + \frac{-0.012345679012345678}{x}}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 68.0% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+108}:\\ \;\;\;\;\left(1 + \frac{-0.1111111111111111}{x}\right) \cdot \left(1 - \frac{0.012345679012345678}{x \cdot x}\right)\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+128}:\\ \;\;\;\;1 + \frac{\frac{1}{x}}{-9}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{0.024691358024691357 + \frac{-0.0054869684499314125}{x}}{x} - 0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.5e+108)
   (*
    (+ 1.0 (/ -0.1111111111111111 x))
    (- 1.0 (/ 0.012345679012345678 (* x x))))
   (if (<= y 3.8e+128)
     (+ 1.0 (/ (/ 1.0 x) -9.0))
     (+
      1.0
      (/
       (-
        (/ (+ 0.024691358024691357 (/ -0.0054869684499314125 x)) x)
        0.1111111111111111)
       x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.5e+108) {
		tmp = (1.0 + (-0.1111111111111111 / x)) * (1.0 - (0.012345679012345678 / (x * x)));
	} else if (y <= 3.8e+128) {
		tmp = 1.0 + ((1.0 / x) / -9.0);
	} else {
		tmp = 1.0 + ((((0.024691358024691357 + (-0.0054869684499314125 / x)) / x) - 0.1111111111111111) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.5d+108)) then
        tmp = (1.0d0 + ((-0.1111111111111111d0) / x)) * (1.0d0 - (0.012345679012345678d0 / (x * x)))
    else if (y <= 3.8d+128) then
        tmp = 1.0d0 + ((1.0d0 / x) / (-9.0d0))
    else
        tmp = 1.0d0 + ((((0.024691358024691357d0 + ((-0.0054869684499314125d0) / x)) / x) - 0.1111111111111111d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.5e+108) {
		tmp = (1.0 + (-0.1111111111111111 / x)) * (1.0 - (0.012345679012345678 / (x * x)));
	} else if (y <= 3.8e+128) {
		tmp = 1.0 + ((1.0 / x) / -9.0);
	} else {
		tmp = 1.0 + ((((0.024691358024691357 + (-0.0054869684499314125 / x)) / x) - 0.1111111111111111) / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.5e+108:
		tmp = (1.0 + (-0.1111111111111111 / x)) * (1.0 - (0.012345679012345678 / (x * x)))
	elif y <= 3.8e+128:
		tmp = 1.0 + ((1.0 / x) / -9.0)
	else:
		tmp = 1.0 + ((((0.024691358024691357 + (-0.0054869684499314125 / x)) / x) - 0.1111111111111111) / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.5e+108)
		tmp = Float64(Float64(1.0 + Float64(-0.1111111111111111 / x)) * Float64(1.0 - Float64(0.012345679012345678 / Float64(x * x))));
	elseif (y <= 3.8e+128)
		tmp = Float64(1.0 + Float64(Float64(1.0 / x) / -9.0));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(0.024691358024691357 + Float64(-0.0054869684499314125 / x)) / x) - 0.1111111111111111) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.5e+108)
		tmp = (1.0 + (-0.1111111111111111 / x)) * (1.0 - (0.012345679012345678 / (x * x)));
	elseif (y <= 3.8e+128)
		tmp = 1.0 + ((1.0 / x) / -9.0);
	else
		tmp = 1.0 + ((((0.024691358024691357 + (-0.0054869684499314125 / x)) / x) - 0.1111111111111111) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.5e+108], N[(N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.012345679012345678 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.8e+128], N[(1.0 + N[(N[(1.0 / x), $MachinePrecision] / -9.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(N[(0.024691358024691357 + N[(-0.0054869684499314125 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.1111111111111111), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.5e108

   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 99.5%

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

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\left(1 + \frac{-0.1111111111111111}{x}\right) + \frac{y}{\sqrt{x} \cdot -3}} \]
   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 3.2%

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

   7. Simplified 3.2%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. frac-times N/A

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

      7. /-lowering-/.f64 N/A

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

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{1 - \frac{\frac{-1}{9}}{x}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{1 - \frac{\frac{-1}{9}}{x}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(1 - \frac{\frac{-1}{9}}{x}\right)}\right)\right) \]

      11. div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \left(1 - \frac{-1}{9} \cdot \color{blue}{\frac{1}{x}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{9}\right)\right) \cdot \frac{1}{x}}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \left(1 + \frac{1}{9} \cdot \frac{\color{blue}{1}}{x}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \left(1 + \frac{1}{9} \cdot \frac{\color{blue}{1}}{x}\right)\right)\right) \]

      15. associate-/r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \left(1 + \frac{1}{\color{blue}{\frac{9}{\frac{1}{x}}}}\right)\right)\right) \]

      16. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \left(1 + \frac{\frac{1}{x}}{\color{blue}{9}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{1}{x}}{9}\right)}\right)\right)\right) \]

      18. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{1}{\color{blue}{9 \cdot x}}\right)\right)\right)\right) \]

      19. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{9}}{x}\right)\right)\right)\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{-1}{9}\right)}{x}\right)\right)\right)\right) \]

      22. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{-1}{9}\right)\right), \color{blue}{x}\right)\right)\right)\right) \]

      23. metadata-eval 3.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{9}, x\right)\right)\right)\right) \]

   9. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{\left(1 - \frac{0.012345679012345678}{x \cdot x}\right) \cdot \frac{1}{1 + \frac{0.1111111111111111}{x}}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \color{blue}{\left(1 - \frac{1}{9} \cdot \frac{1}{x}\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right)\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right)\right) \]

       5. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right)\right) \]

       7. /-lowering-/.f64 21.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right)\right) \]

   12. Simplified 21.9%

       \[\leadsto \left(1 - \frac{0.012345679012345678}{x \cdot x}\right) \cdot \color{blue}{\left(1 + \frac{-0.1111111111111111}{x}\right)} \]

   if -4.5e108 < y < 3.7999999999999999e128

   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 85.4%

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

   7. Simplified 85.4%

      \[\leadsto \color{blue}{1 + \frac{-0.1111111111111111}{x}} \]
   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. distribute-neg-frac2 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. metadata-eval 85.5%

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

   9. Applied egg-rr 85.5%

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

   if 3.7999999999999999e128 < 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 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 3.3%

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

   7. Simplified 3.3%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

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

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

   9. Applied egg-rr 13.0%

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

   11. Applied egg-rr 0.7%

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

   12. Taylor expanded in x around -inf

       \[\leadsto \color{blue}{1 + -1 \cdot \frac{\frac{1}{9} + -1 \cdot \frac{\frac{2}{81} - \frac{4}{729} \cdot \frac{1}{x}}{x}}{x}} \]
   13. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto 1 + \left(\mathsf{neg}\left(\frac{\frac{1}{9} + -1 \cdot \frac{\frac{2}{81} - \frac{4}{729} \cdot \frac{1}{x}}{x}}{x}\right)\right) \]

       2. unsub-neg N/A

          \[\leadsto 1 - \color{blue}{\frac{\frac{1}{9} + -1 \cdot \frac{\frac{2}{81} - \frac{4}{729} \cdot \frac{1}{x}}{x}}{x}} \]

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

          \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{\frac{1}{9} + -1 \cdot \frac{\frac{2}{81} - \frac{4}{729} \cdot \frac{1}{x}}{x}}{x}\right)}\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{9} + -1 \cdot \frac{\frac{2}{81} - \frac{4}{729} \cdot \frac{1}{x}}{x}\right), \color{blue}{x}\right)\right) \]

       5. mul-1-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{9} + \left(\mathsf{neg}\left(\frac{\frac{2}{81} - \frac{4}{729} \cdot \frac{1}{x}}{x}\right)\right)\right), x\right)\right) \]

       6. unsub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{9} - \frac{\frac{2}{81} - \frac{4}{729} \cdot \frac{1}{x}}{x}\right), x\right)\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \left(\frac{\frac{2}{81} - \frac{4}{729} \cdot \frac{1}{x}}{x}\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{/.f64}\left(\left(\frac{2}{81} - \frac{4}{729} \cdot \frac{1}{x}\right), x\right)\right), x\right)\right) \]

       9. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{/.f64}\left(\left(\frac{2}{81} + \left(\mathsf{neg}\left(\frac{4}{729} \cdot \frac{1}{x}\right)\right)\right), x\right)\right), x\right)\right) \]

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

           \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{2}{81}, \left(\mathsf{neg}\left(\frac{4}{729} \cdot \frac{1}{x}\right)\right)\right), x\right)\right), x\right)\right) \]

       11. associate-*r/ N/A

           \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{2}{81}, \left(\mathsf{neg}\left(\frac{\frac{4}{729} \cdot 1}{x}\right)\right)\right), x\right)\right), x\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{2}{81}, \left(\mathsf{neg}\left(\frac{\frac{4}{729}}{x}\right)\right)\right), x\right)\right), x\right)\right) \]

       13. distribute-neg-frac N/A

           \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{2}{81}, \left(\frac{\mathsf{neg}\left(\frac{4}{729}\right)}{x}\right)\right), x\right)\right), x\right)\right) \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{2}{81}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{4}{729}\right)\right), x\right)\right), x\right)\right), x\right)\right) \]

       15. metadata-eval 24.7%

           \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{2}{81}, \mathsf{/.f64}\left(\frac{-4}{729}, x\right)\right), x\right)\right), x\right)\right) \]

   14. Simplified 24.7%

       \[\leadsto \color{blue}{1 - \frac{0.1111111111111111 - \frac{0.024691358024691357 + \frac{-0.0054869684499314125}{x}}{x}}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+108}:\\ \;\;\;\;\left(1 + \frac{-0.1111111111111111}{x}\right) \cdot \left(1 - \frac{0.012345679012345678}{x \cdot x}\right)\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+128}:\\ \;\;\;\;1 + \frac{\frac{1}{x}}{-9}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{0.024691358024691357 + \frac{-0.0054869684499314125}{x}}{x} - 0.1111111111111111}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 65.3% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+110}:\\ \;\;\;\;\left(1 + \frac{-0.1111111111111111}{x}\right) \cdot \left(1 - \frac{0.012345679012345678}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{1}{x}}{-9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.1e+110)
   (*
    (+ 1.0 (/ -0.1111111111111111 x))
    (- 1.0 (/ 0.012345679012345678 (* x x))))
   (+ 1.0 (/ (/ 1.0 x) -9.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.1e+110) {
		tmp = (1.0 + (-0.1111111111111111 / x)) * (1.0 - (0.012345679012345678 / (x * x)));
	} else {
		tmp = 1.0 + ((1.0 / x) / -9.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.1d+110)) then
        tmp = (1.0d0 + ((-0.1111111111111111d0) / x)) * (1.0d0 - (0.012345679012345678d0 / (x * x)))
    else
        tmp = 1.0d0 + ((1.0d0 / x) / (-9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.1e+110) {
		tmp = (1.0 + (-0.1111111111111111 / x)) * (1.0 - (0.012345679012345678 / (x * x)));
	} else {
		tmp = 1.0 + ((1.0 / x) / -9.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.1e+110:
		tmp = (1.0 + (-0.1111111111111111 / x)) * (1.0 - (0.012345679012345678 / (x * x)))
	else:
		tmp = 1.0 + ((1.0 / x) / -9.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.1e+110)
		tmp = Float64(Float64(1.0 + Float64(-0.1111111111111111 / x)) * Float64(1.0 - Float64(0.012345679012345678 / Float64(x * x))));
	else
		tmp = Float64(1.0 + Float64(Float64(1.0 / x) / -9.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.1e+110)
		tmp = (1.0 + (-0.1111111111111111 / x)) * (1.0 - (0.012345679012345678 / (x * x)));
	else
		tmp = 1.0 + ((1.0 / x) / -9.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.1e+110], N[(N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.012345679012345678 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(1.0 / x), $MachinePrecision] / -9.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.10000000000000015e110

   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 99.5%

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

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\left(1 + \frac{-0.1111111111111111}{x}\right) + \frac{y}{\sqrt{x} \cdot -3}} \]
   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 3.2%

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

   7. Simplified 3.2%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. frac-times N/A

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

      7. /-lowering-/.f64 N/A

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

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{1 - \frac{\frac{-1}{9}}{x}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{1 - \frac{\frac{-1}{9}}{x}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(1 - \frac{\frac{-1}{9}}{x}\right)}\right)\right) \]

      11. div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \left(1 - \frac{-1}{9} \cdot \color{blue}{\frac{1}{x}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{9}\right)\right) \cdot \frac{1}{x}}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \left(1 + \frac{1}{9} \cdot \frac{\color{blue}{1}}{x}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \left(1 + \frac{1}{9} \cdot \frac{\color{blue}{1}}{x}\right)\right)\right) \]

      15. associate-/r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \left(1 + \frac{1}{\color{blue}{\frac{9}{\frac{1}{x}}}}\right)\right)\right) \]

      16. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \left(1 + \frac{\frac{1}{x}}{\color{blue}{9}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{1}{x}}{9}\right)}\right)\right)\right) \]

      18. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{1}{\color{blue}{9 \cdot x}}\right)\right)\right)\right) \]

      19. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{9}}{x}\right)\right)\right)\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{-1}{9}\right)}{x}\right)\right)\right)\right) \]

      22. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{-1}{9}\right)\right), \color{blue}{x}\right)\right)\right)\right) \]

      23. metadata-eval 3.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{9}, x\right)\right)\right)\right) \]

   9. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{\left(1 - \frac{0.012345679012345678}{x \cdot x}\right) \cdot \frac{1}{1 + \frac{0.1111111111111111}{x}}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \color{blue}{\left(1 - \frac{1}{9} \cdot \frac{1}{x}\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right)\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right)\right) \]

       5. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right)\right) \]

       7. /-lowering-/.f64 21.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right)\right) \]

   12. Simplified 21.9%

       \[\leadsto \left(1 - \frac{0.012345679012345678}{x \cdot x}\right) \cdot \color{blue}{\left(1 + \frac{-0.1111111111111111}{x}\right)} \]

   if -2.10000000000000015e110 < 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 99.6%

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

   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 72.0%

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

   7. Simplified 72.0%

      \[\leadsto \color{blue}{1 + \frac{-0.1111111111111111}{x}} \]
   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. distribute-neg-frac2 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. metadata-eval 72.1%

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

   9. Applied egg-rr 72.1%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+110}:\\ \;\;\;\;\left(1 + \frac{-0.1111111111111111}{x}\right) \cdot \left(1 - \frac{0.012345679012345678}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{1}{x}}{-9}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 64.9% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+108}:\\ \;\;\;\;1 - \frac{0.1111111111111111 + \frac{-0.024691358024691357}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{1}{x}}{-9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9.5e+108)
   (- 1.0 (/ (+ 0.1111111111111111 (/ -0.024691358024691357 x)) x))
   (+ 1.0 (/ (/ 1.0 x) -9.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9.5e+108) {
		tmp = 1.0 - ((0.1111111111111111 + (-0.024691358024691357 / x)) / x);
	} else {
		tmp = 1.0 + ((1.0 / x) / -9.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-9.5d+108)) then
        tmp = 1.0d0 - ((0.1111111111111111d0 + ((-0.024691358024691357d0) / x)) / x)
    else
        tmp = 1.0d0 + ((1.0d0 / x) / (-9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -9.5e+108) {
		tmp = 1.0 - ((0.1111111111111111 + (-0.024691358024691357 / x)) / x);
	} else {
		tmp = 1.0 + ((1.0 / x) / -9.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9.5e+108:
		tmp = 1.0 - ((0.1111111111111111 + (-0.024691358024691357 / x)) / x)
	else:
		tmp = 1.0 + ((1.0 / x) / -9.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9.5e+108)
		tmp = Float64(1.0 - Float64(Float64(0.1111111111111111 + Float64(-0.024691358024691357 / x)) / x));
	else
		tmp = Float64(1.0 + Float64(Float64(1.0 / x) / -9.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -9.5e+108)
		tmp = 1.0 - ((0.1111111111111111 + (-0.024691358024691357 / x)) / x);
	else
		tmp = 1.0 + ((1.0 / x) / -9.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9.5e+108], N[(1.0 - N[(N[(0.1111111111111111 + N[(-0.024691358024691357 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(1.0 / x), $MachinePrecision] / -9.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.50000000000000097e108

   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 99.5%

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

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\left(1 + \frac{-0.1111111111111111}{x}\right) + \frac{y}{\sqrt{x} \cdot -3}} \]
   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 3.2%

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

   7. Simplified 3.2%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

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

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

   9. Applied egg-rr 3.1%

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

   11. Applied egg-rr 6.8%

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

   12. Taylor expanded in x around -inf

       \[\leadsto \color{blue}{1 + -1 \cdot \frac{\frac{1}{9} - \frac{2}{81} \cdot \frac{1}{x}}{x}} \]
   13. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto 1 + \left(\mathsf{neg}\left(\frac{\frac{1}{9} - \frac{2}{81} \cdot \frac{1}{x}}{x}\right)\right) \]

       2. unsub-neg N/A

          \[\leadsto 1 - \color{blue}{\frac{\frac{1}{9} - \frac{2}{81} \cdot \frac{1}{x}}{x}} \]

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

          \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{\frac{1}{9} - \frac{2}{81} \cdot \frac{1}{x}}{x}\right)}\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{9} - \frac{2}{81} \cdot \frac{1}{x}\right), \color{blue}{x}\right)\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{9} + \left(\mathsf{neg}\left(\frac{2}{81} \cdot \frac{1}{x}\right)\right)\right), x\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \left(\mathsf{neg}\left(\frac{2}{81} \cdot \frac{1}{x}\right)\right)\right), x\right)\right) \]

       7. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \left(\mathsf{neg}\left(\frac{\frac{2}{81} \cdot 1}{x}\right)\right)\right), x\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \left(\mathsf{neg}\left(\frac{\frac{2}{81}}{x}\right)\right)\right), x\right)\right) \]

       9. distribute-neg-frac N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \left(\frac{\mathsf{neg}\left(\frac{2}{81}\right)}{x}\right)\right), x\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{2}{81}\right)\right), x\right)\right), x\right)\right) \]

       11. metadata-eval 19.3%

           \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{/.f64}\left(\frac{-2}{81}, x\right)\right), x\right)\right) \]

   14. Simplified 19.3%

       \[\leadsto \color{blue}{1 - \frac{0.1111111111111111 + \frac{-0.024691358024691357}{x}}{x}} \]

   if -9.50000000000000097e108 < 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 99.6%

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

   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 72.0%

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

   7. Simplified 72.0%

      \[\leadsto \color{blue}{1 + \frac{-0.1111111111111111}{x}} \]
   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. distribute-neg-frac2 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. metadata-eval 72.1%

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

   9. Applied egg-rr 72.1%

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

Alternative 20: 61.2% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{1}{x} \cdot -0.1111111111111111\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = (1.0 / x) * -0.1111111111111111;
	} 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 <= 0.11d0) then
        tmp = (1.0d0 / x) * (-0.1111111111111111d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if x <= 0.11:
		tmp = (1.0 / x) * -0.1111111111111111
	else:
		tmp = 1.0
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.11)
		tmp = (1.0 / x) * -0.1111111111111111;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   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 99.4%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left(1 + \frac{-0.1111111111111111}{x}\right) + \frac{y}{\sqrt{x} \cdot -3}} \]
   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 62.4%

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

   7. Simplified 62.4%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 61.6%

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

   10. Simplified 61.6%

       \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x}} \]
   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 61.6%

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

   12. Applied egg-rr 61.6%

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

   if 0.110000000000000001 < 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 62.2%

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

   7. Simplified 62.2%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 61.8%

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

Alternative 21: 61.3% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		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 <= 0.11d0) 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 <= 0.11) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.11:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.11)
		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 <= 0.11)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   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 99.4%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left(1 + \frac{-0.1111111111111111}{x}\right) + \frac{y}{\sqrt{x} \cdot -3}} \]
   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 62.4%

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

   7. Simplified 62.4%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 61.6%

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

   10. Simplified 61.6%

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

   if 0.110000000000000001 < x

   1. Initial program 99.8%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. 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 62.2%

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

   7. Simplified 62.2%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 61.8%

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

Alternative 22: 62.2% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. 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 62.3%

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

7. Simplified 62.3%

   \[\leadsto \color{blue}{1 + \frac{-0.1111111111111111}{x}} \]
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. distribute-neg-frac2 N/A

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

   10. /-lowering-/.f64 N/A

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

   11. /-lowering-/.f64 N/A

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

   12. metadata-eval 62.4%

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

9. Applied egg-rr 62.4%

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

Alternative 23: 62.2% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. 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 62.3%

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

7. Simplified 62.3%

   \[\leadsto \color{blue}{1 + \frac{-0.1111111111111111}{x}} \]
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. clear-num N/A

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

   10. distribute-neg-frac N/A

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

   11. metadata-eval N/A

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

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

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

   13. associate-/r/ N/A

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

   14. metadata-eval N/A

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

   15. *-commutative N/A

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

   16. *-lowering-*.f64 62.4%

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

9. Applied egg-rr 62.4%

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

Alternative 24: 62.2% 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 99.6%

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

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 62.3%

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

7. Simplified 62.3%

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

Alternative 25: 31.2% 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 99.6%

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

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 62.3%

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

7. Simplified 62.3%

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

8. Taylor expanded in x around inf

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

10. Simplified 31.7%

    \[\leadsto \color{blue}{1} \]
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 2024158 
(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)))))