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

Percentage Accurate: 99.7% → 99.7%

Time: 11.1s

Alternatives: 18

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Final simplification 99.7%

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

Alternative 2: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+72}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+71}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{y}{3}}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.5e+72)
   (- 1.0 (/ y (* 3.0 (sqrt x))))
   (if (<= y 1.75e+71)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (- 1.0 (/ (/ y 3.0) (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6.5e+72) {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	} else if (y <= 1.75e+71) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 - ((y / 3.0) / sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-6.5d+72)) then
        tmp = 1.0d0 - (y / (3.0d0 * sqrt(x)))
    else if (y <= 1.75d+71) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = 1.0d0 - ((y / 3.0d0) / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -6.5e+72) {
		tmp = 1.0 - (y / (3.0 * Math.sqrt(x)));
	} else if (y <= 1.75e+71) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 - ((y / 3.0) / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6.5e+72:
		tmp = 1.0 - (y / (3.0 * math.sqrt(x)))
	elif y <= 1.75e+71:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = 1.0 - ((y / 3.0) / math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6.5e+72)
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	elseif (y <= 1.75e+71)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(1.0 - Float64(Float64(y / 3.0) / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -6.5e+72)
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	elseif (y <= 1.75e+71)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = 1.0 - ((y / 3.0) / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6.5e+72], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.75e+71], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(y / 3.0), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.5000000000000001e72

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 97.4%

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

   if -6.5000000000000001e72 < y < 1.75e71

   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 94.8%

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

   7. Simplified 94.8%

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

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

      6. distribute-neg-frac2 N/A

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

      7. clear-num N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 94.8%

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

   9. Applied egg-rr 94.8%

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

   if 1.75e71 < y

   1. Initial program 99.4%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   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 98.4%

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

      1. associate-/r* N/A

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

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

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

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

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

      4. sqrt-lowering-sqrt.f64 98.5%

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

   7. Applied egg-rr 98.5%

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

Alternative 3: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{+74}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+70}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y (* 3.0 (sqrt x))))))
   (if (<= y -2.3e+74)
     t_0
     (if (<= y 4.8e+70) (+ 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 <= -2.3e+74) {
		tmp = t_0;
	} else if (y <= 4.8e+70) {
		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 <= (-2.3d+74)) then
        tmp = t_0
    else if (y <= 4.8d+70) 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 <= -2.3e+74) {
		tmp = t_0;
	} else if (y <= 4.8e+70) {
		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 <= -2.3e+74:
		tmp = t_0
	elif y <= 4.8e+70:
		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(3.0 * sqrt(x))))
	tmp = 0.0
	if (y <= -2.3e+74)
		tmp = t_0;
	elseif (y <= 4.8e+70)
		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 <= -2.3e+74)
		tmp = t_0;
	elseif (y <= 4.8e+70)
		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[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.3e+74], t$95$0, If[LessEqual[y, 4.8e+70], 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 < -2.2999999999999999e74 or 4.79999999999999974e70 < 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 97.9%

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

   if -2.2999999999999999e74 < y < 4.79999999999999974e70

   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 94.8%

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

   7. Simplified 94.8%

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

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

      6. distribute-neg-frac2 N/A

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

      7. clear-num N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 94.8%

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

   9. Applied egg-rr 94.8%

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

Alternative 4: 92.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+73}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+93}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.2e+73)
   (/ y (* (sqrt x) -3.0))
   (if (<= y 5.3e+93)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (* -0.3333333333333333 (* y (pow x -0.5))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.2e+73) {
		tmp = y / (sqrt(x) * -3.0);
	} else if (y <= 5.3e+93) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = -0.3333333333333333 * (y * pow(x, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.2d+73)) then
        tmp = y / (sqrt(x) * (-3.0d0))
    else if (y <= 5.3d+93) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = (-0.3333333333333333d0) * (y * (x ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.2e+73) {
		tmp = y / (Math.sqrt(x) * -3.0);
	} else if (y <= 5.3e+93) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = -0.3333333333333333 * (y * Math.pow(x, -0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.2e+73:
		tmp = y / (math.sqrt(x) * -3.0)
	elif y <= 5.3e+93:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = -0.3333333333333333 * (y * math.pow(x, -0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.2e+73)
		tmp = Float64(y / Float64(sqrt(x) * -3.0));
	elseif (y <= 5.3e+93)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(-0.3333333333333333 * Float64(y * (x ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.2e+73)
		tmp = y / (sqrt(x) * -3.0);
	elseif (y <= 5.3e+93)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = -0.3333333333333333 * (y * (x ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.2e+73], N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.3e+93], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(y * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.20000000000000001e73

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

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

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

      3. sub-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac2 N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 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 95.5%

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

   7. Simplified 95.5%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. metadata-eval N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. neg-mul-1 N/A

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

      9. distribute-neg-frac N/A

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

      10. distribute-neg-frac2 N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

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

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

      16. metadata-eval 95.8%

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

   9. Applied egg-rr 95.8%

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

   if -1.20000000000000001e73 < y < 5.3000000000000004e93

   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 94.4%

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

   7. Simplified 94.4%

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

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

      6. distribute-neg-frac2 N/A

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

      7. clear-num N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 94.4%

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

   9. Applied egg-rr 94.4%

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

   if 5.3000000000000004e93 < y

   1. Initial program 99.4%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   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 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 98.2%

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

   7. Simplified 98.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. pow1/2 N/A

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

      6. inv-pow N/A

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

      7. pow-pow N/A

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

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

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

      9. metadata-eval 98.3%

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

   9. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{\left({x}^{-0.5} \cdot y\right) \cdot -0.3333333333333333} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+73}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+93}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+84}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+92}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.4e+84)
   (/ y (* (sqrt x) -3.0))
   (if (<= y 1.25e+92)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (* y (* -0.3333333333333333 (pow x -0.5))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.4e+84) {
		tmp = y / (sqrt(x) * -3.0);
	} else if (y <= 1.25e+92) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = y * (-0.3333333333333333 * pow(x, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.4d+84)) then
        tmp = y / (sqrt(x) * (-3.0d0))
    else if (y <= 1.25d+92) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = y * ((-0.3333333333333333d0) * (x ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.4e+84) {
		tmp = y / (Math.sqrt(x) * -3.0);
	} else if (y <= 1.25e+92) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = y * (-0.3333333333333333 * Math.pow(x, -0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.4e+84:
		tmp = y / (math.sqrt(x) * -3.0)
	elif y <= 1.25e+92:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = y * (-0.3333333333333333 * math.pow(x, -0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.4e+84)
		tmp = Float64(y / Float64(sqrt(x) * -3.0));
	elseif (y <= 1.25e+92)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(y * Float64(-0.3333333333333333 * (x ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.4e+84)
		tmp = y / (sqrt(x) * -3.0);
	elseif (y <= 1.25e+92)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = y * (-0.3333333333333333 * (x ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.4e+84], N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.25e+92], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(-0.3333333333333333 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.4e84

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

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

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

      3. sub-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac2 N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 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 95.5%

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

   7. Simplified 95.5%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. metadata-eval N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. neg-mul-1 N/A

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

      9. distribute-neg-frac N/A

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

      10. distribute-neg-frac2 N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

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

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

      16. metadata-eval 95.8%

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

   9. Applied egg-rr 95.8%

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

   if -2.4e84 < y < 1.25000000000000005e92

   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 94.4%

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

   7. Simplified 94.4%

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

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

      6. distribute-neg-frac2 N/A

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

      7. clear-num N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 94.4%

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

   9. Applied egg-rr 94.4%

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

   if 1.25000000000000005e92 < y

   1. Initial program 99.4%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   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 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 98.2%

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

   7. Simplified 98.2%

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

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

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

      2. pow1/2 N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 98.2%

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

   9. Applied egg-rr 98.2%

      \[\leadsto \color{blue}{{x}^{-0.5} \cdot \left(y \cdot -0.3333333333333333\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

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

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

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

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

       5. pow-lowering-pow.f64 98.2%

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

   11. Applied egg-rr 98.2%

       \[\leadsto \color{blue}{\left({x}^{-0.5} \cdot -0.3333333333333333\right) \cdot y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+84}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+92}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+72}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+92}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt{x}} \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.8e+72)
   (/ y (* (sqrt x) -3.0))
   (if (<= y 1.4e+92)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (* (/ y (sqrt x)) -0.3333333333333333))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.8e+72) {
		tmp = y / (sqrt(x) * -3.0);
	} else if (y <= 1.4e+92) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (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 <= (-2.8d+72)) then
        tmp = y / (sqrt(x) * (-3.0d0))
    else if (y <= 1.4d+92) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = (y / sqrt(x)) * (-0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.8e+72) {
		tmp = y / (Math.sqrt(x) * -3.0);
	} else if (y <= 1.4e+92) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (y / Math.sqrt(x)) * -0.3333333333333333;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.8e+72:
		tmp = y / (math.sqrt(x) * -3.0)
	elif y <= 1.4e+92:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = (y / math.sqrt(x)) * -0.3333333333333333
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.8e+72)
		tmp = Float64(y / Float64(sqrt(x) * -3.0));
	elseif (y <= 1.4e+92)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(Float64(y / sqrt(x)) * -0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.8e+72)
		tmp = y / (sqrt(x) * -3.0);
	elseif (y <= 1.4e+92)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = (y / sqrt(x)) * -0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.8e+72], N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.4e+92], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.7999999999999999e72

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

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

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

      3. sub-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac2 N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 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 95.5%

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

   7. Simplified 95.5%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. metadata-eval N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. neg-mul-1 N/A

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

      9. distribute-neg-frac N/A

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

      10. distribute-neg-frac2 N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

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

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

      16. metadata-eval 95.8%

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

   9. Applied egg-rr 95.8%

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

   if -2.7999999999999999e72 < y < 1.4e92

   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 94.4%

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

   7. Simplified 94.4%

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

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

      6. distribute-neg-frac2 N/A

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

      7. clear-num N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 94.4%

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

   9. Applied egg-rr 94.4%

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

   if 1.4e92 < y

   1. Initial program 99.4%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   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 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 98.2%

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

   7. Simplified 98.2%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

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

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

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

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

      8. sqrt-lowering-sqrt.f64 98.2%

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

   9. Applied egg-rr 98.2%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+72}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+92}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt{x}} \cdot -0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+76}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+92}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt{x}} \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.9e+76)
   (* y (/ -0.3333333333333333 (sqrt x)))
   (if (<= y 1.25e+92)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (* (/ y (sqrt x)) -0.3333333333333333))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.9e+76) {
		tmp = y * (-0.3333333333333333 / sqrt(x));
	} else if (y <= 1.25e+92) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (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 <= (-2.9d+76)) then
        tmp = y * ((-0.3333333333333333d0) / sqrt(x))
    else if (y <= 1.25d+92) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = (y / sqrt(x)) * (-0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.9e+76) {
		tmp = y * (-0.3333333333333333 / Math.sqrt(x));
	} else if (y <= 1.25e+92) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (y / Math.sqrt(x)) * -0.3333333333333333;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.9e+76:
		tmp = y * (-0.3333333333333333 / math.sqrt(x))
	elif y <= 1.25e+92:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = (y / math.sqrt(x)) * -0.3333333333333333
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.9e+76)
		tmp = Float64(y * Float64(-0.3333333333333333 / sqrt(x)));
	elseif (y <= 1.25e+92)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(Float64(y / sqrt(x)) * -0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.9e+76)
		tmp = y * (-0.3333333333333333 / sqrt(x));
	elseif (y <= 1.25e+92)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = (y / sqrt(x)) * -0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.9e+76], N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.25e+92], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.9000000000000002e76

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

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

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

      3. sub-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac2 N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 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 95.5%

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

   7. Simplified 95.5%

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

      1. associate-*r* N/A

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

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

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. associate-*l/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. sqrt-lowering-sqrt.f64 95.8%

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

   9. Applied egg-rr 95.8%

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

   if -2.9000000000000002e76 < y < 1.25000000000000005e92

   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 94.4%

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

   7. Simplified 94.4%

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

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

      6. distribute-neg-frac2 N/A

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

      7. clear-num N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 94.4%

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

   9. Applied egg-rr 94.4%

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

   if 1.25000000000000005e92 < y

   1. Initial program 99.4%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   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 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 98.2%

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

   7. Simplified 98.2%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

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

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

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

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

      8. sqrt-lowering-sqrt.f64 98.2%

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

   9. Applied egg-rr 98.2%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+76}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+92}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt{x}} \cdot -0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{\sqrt{x}} \cdot -0.3333333333333333\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+92}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ y (sqrt x)) -0.3333333333333333)))
   (if (<= y -1.7e+77)
     t_0
     (if (<= y 1.6e+92) (+ 1.0 (/ -1.0 (* x 9.0))) t_0))))

C

double code(double x, double y) {
	double t_0 = (y / sqrt(x)) * -0.3333333333333333;
	double tmp;
	if (y <= -1.7e+77) {
		tmp = t_0;
	} else if (y <= 1.6e+92) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y / sqrt(x)) * (-0.3333333333333333d0)
    if (y <= (-1.7d+77)) then
        tmp = t_0
    else if (y <= 1.6d+92) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y / Math.sqrt(x)) * -0.3333333333333333;
	double tmp;
	if (y <= -1.7e+77) {
		tmp = t_0;
	} else if (y <= 1.6e+92) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y / math.sqrt(x)) * -0.3333333333333333
	tmp = 0
	if y <= -1.7e+77:
		tmp = t_0
	elif y <= 1.6e+92:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y / sqrt(x)) * -0.3333333333333333)
	tmp = 0.0
	if (y <= -1.7e+77)
		tmp = t_0;
	elseif (y <= 1.6e+92)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y / sqrt(x)) * -0.3333333333333333;
	tmp = 0.0;
	if (y <= -1.7e+77)
		tmp = t_0;
	elseif (y <= 1.6e+92)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]}, If[LessEqual[y, -1.7e+77], t$95$0, If[LessEqual[y, 1.6e+92], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.69999999999999998e77 or 1.60000000000000013e92 < y

   1. Initial program 99.6%

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

      1. sub-neg N/A

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

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

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

      3. sub-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac2 N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 96.8%

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

   7. Simplified 96.8%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

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

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

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

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

      8. sqrt-lowering-sqrt.f64 96.9%

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

   9. Applied egg-rr 96.9%

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

   if -1.69999999999999998e77 < y < 1.60000000000000013e92

   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 94.4%

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

   7. Simplified 94.4%

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

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

      6. distribute-neg-frac2 N/A

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

      7. clear-num N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 94.4%

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

   9. Applied egg-rr 94.4%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+77}:\\ \;\;\;\;\frac{y}{\sqrt{x}} \cdot -0.3333333333333333\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+92}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt{x}} \cdot -0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.11)
   (/ (- -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 <= 0.11) {
		tmp = (-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 <= 0.11d0) then
        tmp = ((-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 <= 0.11) {
		tmp = (-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 <= 0.11:
		tmp = (-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 <= 0.11)
		tmp = 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 <= 0.11)
		tmp = (-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, 0.11], N[(N[(-0.1111111111111111 - N[(y * N[(N[Sqrt[x], $MachinePrecision] * 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. Step-by-step derivation

      1. sub-neg N/A

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

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

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

      3. sub-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac2 N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

      4. distribute-neg-in N/A

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

      5. +-commutative N/A

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

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

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

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

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

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

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

      14. *-commutative N/A

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

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

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

      16. sqrt-lowering-sqrt.f64 98.4%

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

   7. Simplified 98.4%

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

   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 97.9%

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

Alternative 10: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg N/A

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

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

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

   3. sub-neg N/A

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

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

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. distribute-neg-frac N/A

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

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

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac2 N/A

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

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

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

   13. *-commutative N/A

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

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

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

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

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

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

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

   17. metadata-eval 99.7%

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

3. Simplified 99.7%

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

Alternative 11: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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)) * (-0.3333333333333333d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg N/A

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

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

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

   3. sub-neg N/A

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

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

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. distribute-neg-frac N/A

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

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

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac2 N/A

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

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

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

   13. *-commutative N/A

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

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

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

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

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

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

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

   17. metadata-eval 99.7%

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

3. Simplified 99.7%

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

   1. 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. div-inv N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

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

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

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

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

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

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

   9. metadata-eval N/A

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

   10. metadata-eval 99.7%

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

6. Applied egg-rr 99.7%

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

Alternative 12: 69.1% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;y \leq -3.7 \cdot 10^{+129}:\\ \;\;\;\;\frac{1}{\frac{0.012345679012345678 + x \cdot 0.1111111111111111}{x \cdot x}} - \frac{-0.0013717421124828531}{t\_0}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+98}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.00030483158055174517}{x \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= y -3.7e+129)
     (-
      (/ 1.0 (/ (+ 0.012345679012345678 (* x 0.1111111111111111)) (* x x)))
      (/ -0.0013717421124828531 t_0))
     (if (<= y 2.2e+98)
       (+ 1.0 (/ -1.0 (* x 9.0)))
       (/ -0.00030483158055174517 (* x t_0))))))

C

double code(double x, double y) {
	double t_0 = x * (x * x);
	double tmp;
	if (y <= -3.7e+129) {
		tmp = (1.0 / ((0.012345679012345678 + (x * 0.1111111111111111)) / (x * x))) - (-0.0013717421124828531 / t_0);
	} else if (y <= 2.2e+98) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = -0.00030483158055174517 / (x * t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x * x)
    if (y <= (-3.7d+129)) then
        tmp = (1.0d0 / ((0.012345679012345678d0 + (x * 0.1111111111111111d0)) / (x * x))) - ((-0.0013717421124828531d0) / t_0)
    else if (y <= 2.2d+98) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = (-0.00030483158055174517d0) / (x * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (x * x);
	double tmp;
	if (y <= -3.7e+129) {
		tmp = (1.0 / ((0.012345679012345678 + (x * 0.1111111111111111)) / (x * x))) - (-0.0013717421124828531 / t_0);
	} else if (y <= 2.2e+98) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = -0.00030483158055174517 / (x * t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (x * x)
	tmp = 0
	if y <= -3.7e+129:
		tmp = (1.0 / ((0.012345679012345678 + (x * 0.1111111111111111)) / (x * x))) - (-0.0013717421124828531 / t_0)
	elif y <= 2.2e+98:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = -0.00030483158055174517 / (x * t_0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (y <= -3.7e+129)
		tmp = Float64(Float64(1.0 / Float64(Float64(0.012345679012345678 + Float64(x * 0.1111111111111111)) / Float64(x * x))) - Float64(-0.0013717421124828531 / t_0));
	elseif (y <= 2.2e+98)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(-0.00030483158055174517 / Float64(x * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (x * x);
	tmp = 0.0;
	if (y <= -3.7e+129)
		tmp = (1.0 / ((0.012345679012345678 + (x * 0.1111111111111111)) / (x * x))) - (-0.0013717421124828531 / t_0);
	elseif (y <= 2.2e+98)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = -0.00030483158055174517 / (x * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.7e+129], N[(N[(1.0 / N[(N[(0.012345679012345678 + N[(x * 0.1111111111111111), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-0.0013717421124828531 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e+98], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.00030483158055174517 / N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.69999999999999978e129

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

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

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

      3. sub-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac2 N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 2.4%

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

   7. Simplified 2.4%

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

   9. Applied egg-rr 11.9%

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

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{1}{81} + \frac{1}{9} \cdot x}{{x}^{2}}\right)}\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{729}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-1}{9}, x\right), 1\right), \mathsf{/.f64}\left(x, \frac{-1}{9}\right)\right)\right)\right)\right) \]
   11. Step-by-step derivation

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

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

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

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

       3. *-commutative N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 12.2%

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

   12. Simplified 12.2%

       \[\leadsto \frac{1}{\color{blue}{\frac{0.012345679012345678 + x \cdot 0.1111111111111111}{x \cdot x}}} - \frac{\frac{-0.0013717421124828531}{x \cdot \left(x \cdot x\right)}}{1 + \frac{\frac{-0.1111111111111111}{x} - 1}{\frac{x}{-0.1111111111111111}}} \]

   13. Taylor expanded in x around inf

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, \frac{1}{9}\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \color{blue}{\left(\frac{\frac{-1}{729}}{{x}^{3}}\right)}\right) \]
   14. Step-by-step derivation

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(x, \frac{1}{9}\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{729}, \color{blue}{\left({x}^{3}\right)}\right)\right) \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 32.9%

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

   15. Simplified 32.9%

       \[\leadsto \frac{1}{\frac{0.012345679012345678 + x \cdot 0.1111111111111111}{x \cdot x}} - \color{blue}{\frac{-0.0013717421124828531}{x \cdot \left(x \cdot x\right)}} \]

   if -3.69999999999999978e129 < y < 2.20000000000000009e98

   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 91.1%

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

   7. Simplified 91.1%

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

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

      6. distribute-neg-frac2 N/A

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

      7. clear-num N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 91.2%

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

   9. Applied egg-rr 91.2%

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

   if 2.20000000000000009e98 < y

   1. Initial program 99.4%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   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 3.0%

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

   7. Simplified 3.0%

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

   9. Applied egg-rr 0.8%

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

   10. Taylor expanded in x around -inf

       \[\leadsto \color{blue}{1 + -1 \cdot \frac{\left(\frac{1}{9} + \frac{\frac{2}{6561}}{{x}^{3}}\right) - \frac{2}{729} \cdot \frac{1}{{x}^{2}}}{x}} \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(-1 \cdot \frac{\left(\frac{1}{9} + \frac{\frac{2}{6561}}{{x}^{3}}\right) - \frac{2}{729} \cdot \frac{1}{{x}^{2}}}{x}\right)}\right) \]

       2. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{-1 \cdot \left(\left(\frac{1}{9} + \frac{\frac{2}{6561}}{{x}^{3}}\right) - \frac{2}{729} \cdot \frac{1}{{x}^{2}}\right)}{\color{blue}{x}}\right)\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\left(\left(\frac{1}{9} + \frac{\frac{2}{6561}}{{x}^{3}}\right) - \frac{2}{729} \cdot \frac{1}{{x}^{2}}\right)\right)}{x}\right)\right) \]

       4. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\left(\left(\frac{1}{9} + \frac{\frac{2}{6561}}{{x}^{3}}\right) + \left(\mathsf{neg}\left(\frac{2}{729} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)}{x}\right)\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{2}{729} \cdot \frac{1}{{x}^{2}}\right)\right) + \left(\frac{1}{9} + \frac{\frac{2}{6561}}{{x}^{3}}\right)\right)\right)}{x}\right)\right) \]

       6. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{2}{729} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right) + \left(\mathsf{neg}\left(\left(\frac{1}{9} + \frac{\frac{2}{6561}}{{x}^{3}}\right)\right)\right)}{x}\right)\right) \]

       7. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{2}{729} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\left(\frac{1}{9} + \frac{\frac{2}{6561}}{{x}^{3}}\right)\right)\right)}{x}\right)\right) \]

       8. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{2}{729} \cdot \frac{1}{{x}^{2}} - \left(\frac{1}{9} + \frac{\frac{2}{6561}}{{x}^{3}}\right)}{x}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{2}{729} \cdot \frac{1}{{x}^{2}} - \left(\frac{1}{9} + \frac{\frac{2}{6561}}{{x}^{3}}\right)\right), \color{blue}{x}\right)\right) \]

   12. Simplified 6.9%

       \[\leadsto \color{blue}{1 + \frac{-0.1111111111111111 - \left(\frac{0.00030483158055174517}{x \cdot \left(x \cdot x\right)} + \frac{-0.0027434842249657062}{x \cdot x}\right)}{x}} \]

   13. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{\frac{-2}{6561}}{{x}^{4}}} \]
   14. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(\frac{-2}{6561}, \color{blue}{\left({x}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-2}{6561}, \left({x}^{\left(3 + \color{blue}{1}\right)}\right)\right) \]

       3. pow-plus N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-2}{6561}, \left({x}^{3} \cdot \color{blue}{x}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-2}{6561}, \left(x \cdot \color{blue}{{x}^{3}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\frac{-2}{6561}, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3}\right)}\right)\right) \]

       6. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-2}{6561}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-2}{6561}, \mathsf{*.f64}\left(x, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\frac{-2}{6561}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-2}{6561}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 16.4%

           \[\leadsto \mathsf{/.f64}\left(\frac{-2}{6561}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   15. Simplified 16.4%

       \[\leadsto \color{blue}{\frac{-0.00030483158055174517}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 69.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+133}:\\ \;\;\;\;1 + \frac{-0.1111111111111111 + \frac{0.0027434842249657062}{x \cdot x}}{x}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+98}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.00030483158055174517}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.6e+133)
   (+ 1.0 (/ (+ -0.1111111111111111 (/ 0.0027434842249657062 (* x x))) x))
   (if (<= y 2.2e+98)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (/ -0.00030483158055174517 (* x (* x (* x x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.6e+133) {
		tmp = 1.0 + ((-0.1111111111111111 + (0.0027434842249657062 / (x * x))) / x);
	} else if (y <= 2.2e+98) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = -0.00030483158055174517 / (x * (x * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.6d+133)) then
        tmp = 1.0d0 + (((-0.1111111111111111d0) + (0.0027434842249657062d0 / (x * x))) / x)
    else if (y <= 2.2d+98) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = (-0.00030483158055174517d0) / (x * (x * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.6e+133) {
		tmp = 1.0 + ((-0.1111111111111111 + (0.0027434842249657062 / (x * x))) / x);
	} else if (y <= 2.2e+98) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = -0.00030483158055174517 / (x * (x * (x * x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.6e+133:
		tmp = 1.0 + ((-0.1111111111111111 + (0.0027434842249657062 / (x * x))) / x)
	elif y <= 2.2e+98:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = -0.00030483158055174517 / (x * (x * (x * x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.6e+133)
		tmp = Float64(1.0 + Float64(Float64(-0.1111111111111111 + Float64(0.0027434842249657062 / Float64(x * x))) / x));
	elseif (y <= 2.2e+98)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(-0.00030483158055174517 / Float64(x * Float64(x * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.6e+133)
		tmp = 1.0 + ((-0.1111111111111111 + (0.0027434842249657062 / (x * x))) / x);
	elseif (y <= 2.2e+98)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = -0.00030483158055174517 / (x * (x * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.6e+133], N[(1.0 + N[(N[(-0.1111111111111111 + N[(0.0027434842249657062 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e+98], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.00030483158055174517 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.59999999999999978e133

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

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

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

      3. sub-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac2 N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 2.2%

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

   7. Simplified 2.2%

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

   9. Applied egg-rr 12.2%

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

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\left(1 + \frac{\frac{2}{729}}{{x}^{3}}\right) - \frac{1}{9} \cdot \frac{1}{x}} \]
   11. Step-by-step derivation

       1. associate--l+ N/A

          \[\leadsto 1 + \color{blue}{\left(\frac{\frac{2}{729}}{{x}^{3}} - \frac{1}{9} \cdot \frac{1}{x}\right)} \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{2}{729}}{{x}^{3}} - \frac{1}{9} \cdot \frac{1}{x}\right)}\right) \]

       3. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{2}{729}}{{x}^{3}} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right) + \color{blue}{\frac{\frac{2}{729}}{{x}^{3}}}\right)\right) \]

       5. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{2}{729}}{{x}^{3}}\right)\right)\right)\right)\right)\right) \]

       6. unpow3 N/A

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

       7. unpow2 N/A

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

       8. associate-/r* N/A

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

       9. metadata-eval N/A

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

       10. associate-*r/ N/A

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

       11. distribute-neg-in N/A

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

       12. sub-neg N/A

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

       13. associate-*r/ N/A

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

       14. metadata-eval N/A

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

       15. div-sub N/A

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

       16. distribute-neg-frac N/A

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

   12. Simplified 34.1%

       \[\leadsto \color{blue}{1 + \frac{-0.1111111111111111 + \frac{0.0027434842249657062}{x \cdot x}}{x}} \]

   if -3.59999999999999978e133 < y < 2.20000000000000009e98

   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 90.2%

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

   7. Simplified 90.2%

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

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

      6. distribute-neg-frac2 N/A

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

      7. clear-num N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 90.2%

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

   9. Applied egg-rr 90.2%

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

   if 2.20000000000000009e98 < y

   1. Initial program 99.4%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   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 3.0%

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

   7. Simplified 3.0%

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

   9. Applied egg-rr 0.8%

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

   10. Taylor expanded in x around -inf

       \[\leadsto \color{blue}{1 + -1 \cdot \frac{\left(\frac{1}{9} + \frac{\frac{2}{6561}}{{x}^{3}}\right) - \frac{2}{729} \cdot \frac{1}{{x}^{2}}}{x}} \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(-1 \cdot \frac{\left(\frac{1}{9} + \frac{\frac{2}{6561}}{{x}^{3}}\right) - \frac{2}{729} \cdot \frac{1}{{x}^{2}}}{x}\right)}\right) \]

       2. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{-1 \cdot \left(\left(\frac{1}{9} + \frac{\frac{2}{6561}}{{x}^{3}}\right) - \frac{2}{729} \cdot \frac{1}{{x}^{2}}\right)}{\color{blue}{x}}\right)\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\left(\left(\frac{1}{9} + \frac{\frac{2}{6561}}{{x}^{3}}\right) - \frac{2}{729} \cdot \frac{1}{{x}^{2}}\right)\right)}{x}\right)\right) \]

       4. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\left(\left(\frac{1}{9} + \frac{\frac{2}{6561}}{{x}^{3}}\right) + \left(\mathsf{neg}\left(\frac{2}{729} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)}{x}\right)\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{2}{729} \cdot \frac{1}{{x}^{2}}\right)\right) + \left(\frac{1}{9} + \frac{\frac{2}{6561}}{{x}^{3}}\right)\right)\right)}{x}\right)\right) \]

       6. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{2}{729} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right) + \left(\mathsf{neg}\left(\left(\frac{1}{9} + \frac{\frac{2}{6561}}{{x}^{3}}\right)\right)\right)}{x}\right)\right) \]

       7. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{2}{729} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\left(\frac{1}{9} + \frac{\frac{2}{6561}}{{x}^{3}}\right)\right)\right)}{x}\right)\right) \]

       8. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{2}{729} \cdot \frac{1}{{x}^{2}} - \left(\frac{1}{9} + \frac{\frac{2}{6561}}{{x}^{3}}\right)}{x}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{2}{729} \cdot \frac{1}{{x}^{2}} - \left(\frac{1}{9} + \frac{\frac{2}{6561}}{{x}^{3}}\right)\right), \color{blue}{x}\right)\right) \]

   12. Simplified 6.9%

       \[\leadsto \color{blue}{1 + \frac{-0.1111111111111111 - \left(\frac{0.00030483158055174517}{x \cdot \left(x \cdot x\right)} + \frac{-0.0027434842249657062}{x \cdot x}\right)}{x}} \]

   13. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{\frac{-2}{6561}}{{x}^{4}}} \]
   14. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(\frac{-2}{6561}, \color{blue}{\left({x}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-2}{6561}, \left({x}^{\left(3 + \color{blue}{1}\right)}\right)\right) \]

       3. pow-plus N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-2}{6561}, \left({x}^{3} \cdot \color{blue}{x}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-2}{6561}, \left(x \cdot \color{blue}{{x}^{3}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\frac{-2}{6561}, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3}\right)}\right)\right) \]

       6. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-2}{6561}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-2}{6561}, \mathsf{*.f64}\left(x, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\frac{-2}{6561}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-2}{6561}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 16.4%

           \[\leadsto \mathsf{/.f64}\left(\frac{-2}{6561}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   15. Simplified 16.4%

       \[\leadsto \color{blue}{\frac{-0.00030483158055174517}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 66.1% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{+98}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.00030483158055174517}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.2e+98)
   (+ 1.0 (/ -1.0 (* x 9.0)))
   (/ -0.00030483158055174517 (* x (* x (* x x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.2e+98) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = -0.00030483158055174517 / (x * (x * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.2d+98) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = (-0.00030483158055174517d0) / (x * (x * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.2e+98) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = -0.00030483158055174517 / (x * (x * (x * x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.2e+98:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = -0.00030483158055174517 / (x * (x * (x * x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.2e+98)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(-0.00030483158055174517 / Float64(x * Float64(x * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.2e+98)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = -0.00030483158055174517 / (x * (x * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.2e+98], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.00030483158055174517 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.20000000000000009e98

   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 76.9%

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

   7. Simplified 76.9%

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

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

      6. distribute-neg-frac2 N/A

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

      7. clear-num N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 76.9%

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

   9. Applied egg-rr 76.9%

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

   if 2.20000000000000009e98 < y

   1. Initial program 99.4%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   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 3.0%

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

   7. Simplified 3.0%

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

   9. Applied egg-rr 0.8%

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

   10. Taylor expanded in x around -inf

       \[\leadsto \color{blue}{1 + -1 \cdot \frac{\left(\frac{1}{9} + \frac{\frac{2}{6561}}{{x}^{3}}\right) - \frac{2}{729} \cdot \frac{1}{{x}^{2}}}{x}} \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(-1 \cdot \frac{\left(\frac{1}{9} + \frac{\frac{2}{6561}}{{x}^{3}}\right) - \frac{2}{729} \cdot \frac{1}{{x}^{2}}}{x}\right)}\right) \]

       2. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{-1 \cdot \left(\left(\frac{1}{9} + \frac{\frac{2}{6561}}{{x}^{3}}\right) - \frac{2}{729} \cdot \frac{1}{{x}^{2}}\right)}{\color{blue}{x}}\right)\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\left(\left(\frac{1}{9} + \frac{\frac{2}{6561}}{{x}^{3}}\right) - \frac{2}{729} \cdot \frac{1}{{x}^{2}}\right)\right)}{x}\right)\right) \]

       4. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\left(\left(\frac{1}{9} + \frac{\frac{2}{6561}}{{x}^{3}}\right) + \left(\mathsf{neg}\left(\frac{2}{729} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)}{x}\right)\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{2}{729} \cdot \frac{1}{{x}^{2}}\right)\right) + \left(\frac{1}{9} + \frac{\frac{2}{6561}}{{x}^{3}}\right)\right)\right)}{x}\right)\right) \]

       6. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{2}{729} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right) + \left(\mathsf{neg}\left(\left(\frac{1}{9} + \frac{\frac{2}{6561}}{{x}^{3}}\right)\right)\right)}{x}\right)\right) \]

       7. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{2}{729} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\left(\frac{1}{9} + \frac{\frac{2}{6561}}{{x}^{3}}\right)\right)\right)}{x}\right)\right) \]

       8. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{2}{729} \cdot \frac{1}{{x}^{2}} - \left(\frac{1}{9} + \frac{\frac{2}{6561}}{{x}^{3}}\right)}{x}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{2}{729} \cdot \frac{1}{{x}^{2}} - \left(\frac{1}{9} + \frac{\frac{2}{6561}}{{x}^{3}}\right)\right), \color{blue}{x}\right)\right) \]

   12. Simplified 6.9%

       \[\leadsto \color{blue}{1 + \frac{-0.1111111111111111 - \left(\frac{0.00030483158055174517}{x \cdot \left(x \cdot x\right)} + \frac{-0.0027434842249657062}{x \cdot x}\right)}{x}} \]

   13. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{\frac{-2}{6561}}{{x}^{4}}} \]
   14. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(\frac{-2}{6561}, \color{blue}{\left({x}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-2}{6561}, \left({x}^{\left(3 + \color{blue}{1}\right)}\right)\right) \]

       3. pow-plus N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-2}{6561}, \left({x}^{3} \cdot \color{blue}{x}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-2}{6561}, \left(x \cdot \color{blue}{{x}^{3}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\frac{-2}{6561}, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3}\right)}\right)\right) \]

       6. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-2}{6561}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-2}{6561}, \mathsf{*.f64}\left(x, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\frac{-2}{6561}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-2}{6561}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 16.4%

           \[\leadsto \mathsf{/.f64}\left(\frac{-2}{6561}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   15. Simplified 16.4%

       \[\leadsto \color{blue}{\frac{-0.00030483158055174517}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 62.4% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.155:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.155) (/ -0.1111111111111111 x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.155) {
		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 <= 0.155d0) 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 <= 0.155) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.155:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.155)
		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 <= 0.155)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.155], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.154999999999999999

   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 68.6%

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

   7. Simplified 68.6%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 67.5%

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

   10. Simplified 67.5%

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

   if 0.154999999999999999 < 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 63.1%

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

   7. Simplified 63.1%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 61.2%

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

Alternative 16: 63.6% 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 99.7%

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

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 65.9%

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

7. Simplified 65.9%

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

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

   4. metadata-eval N/A

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

   5. div-inv N/A

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

   6. distribute-neg-frac2 N/A

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

   7. clear-num N/A

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

   8. distribute-neg-frac N/A

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

   9. metadata-eval N/A

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

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

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

   11. associate-/r/ N/A

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. *-lowering-*.f64 66.0%

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

9. Applied egg-rr 66.0%

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

Alternative 17: 63.5% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{-0.1111111111111111}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return 1.0 + (-0.1111111111111111 / x);
}

Python

def code(x, y):
	return 1.0 + (-0.1111111111111111 / x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg N/A

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

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

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

   3. sub-neg N/A

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

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

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. distribute-neg-frac N/A

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

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

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

   14. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

   16. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

   17. metadata-eval 99.7%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{\left(1 + \frac{-0.1111111111111111}{x}\right) + \frac{y}{\sqrt{x} \cdot -3}} \]
4. Add Preprocessing

5. Taylor expanded in y around 0

   \[\leadsto \color{blue}{1 - \frac{1}{9} \cdot \frac{1}{x}} \]
6. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto 1 + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)} \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

   5. distribute-neg-frac N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

   7. /-lowering-/.f64 65.9%

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

7. Simplified 65.9%

   \[\leadsto \color{blue}{1 + \frac{-0.1111111111111111}{x}} \]
8. Add Preprocessing

Alternative 18: 32.4% accurate, 113.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)} \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(1 - \frac{1}{x \cdot 9}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{x \cdot 9}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{y}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{9 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

   6. associate-/r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

   7. distribute-neg-frac N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{x}\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3 \cdot \sqrt{x}}}\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), x\right)\right), \left(\mathsf{neg}\left(\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\mathsf{neg}\left(\frac{y}{\color{blue}{3} \cdot \sqrt{x}}\right)\right)\right) \]

   11. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \left(\frac{y}{\color{blue}{\mathsf{neg}\left(3 \cdot \sqrt{x}\right)}}\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{x}\right)\right)}\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\sqrt{x} \cdot 3\right)\right)\right)\right) \]

   14. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

   16. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right)\right) \]

   17. metadata-eval 99.7%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, x\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right)\right)\right) \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{\left(1 + \frac{-0.1111111111111111}{x}\right) + \frac{y}{\sqrt{x} \cdot -3}} \]
4. Add Preprocessing

5. Taylor expanded in y around 0

   \[\leadsto \color{blue}{1 - \frac{1}{9} \cdot \frac{1}{x}} \]
6. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto 1 + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)} \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{x}\right)\right)}\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9} \cdot 1}{x}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{x}\right)\right)\right) \]

   5. distribute-neg-frac N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{\color{blue}{x}}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{9}}{x}\right)\right) \]

   7. /-lowering-/.f64 65.9%

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{9}, \color{blue}{x}\right)\right) \]

7. Simplified 65.9%

   \[\leadsto \color{blue}{1 + \frac{-0.1111111111111111}{x}} \]

8. Taylor expanded in x around inf

   \[\leadsto \color{blue}{1} \]
9. Step-by-step derivation

10. Simplified 30.4%

    \[\leadsto \color{blue}{1} \]
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 2024160 
(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)))))