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

Percentage Accurate: 99.7% → 99.7%

Time: 10.7s

Alternatives: 16

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: 94.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{+78}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+81}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{y}{3}}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.12e+78)
   (- 1.0 (/ y (* 3.0 (sqrt x))))
   (if (<= y 8.2e+81)
     (+ 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 <= -1.12e+78) {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	} else if (y <= 8.2e+81) {
		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 <= (-1.12d+78)) then
        tmp = 1.0d0 - (y / (3.0d0 * sqrt(x)))
    else if (y <= 8.2d+81) 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 <= -1.12e+78) {
		tmp = 1.0 - (y / (3.0 * Math.sqrt(x)));
	} else if (y <= 8.2e+81) {
		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 <= -1.12e+78:
		tmp = 1.0 - (y / (3.0 * math.sqrt(x)))
	elif y <= 8.2e+81:
		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 <= -1.12e+78)
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	elseif (y <= 8.2e+81)
		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 <= -1.12e+78)
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	elseif (y <= 8.2e+81)
		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, -1.12e+78], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.2e+81], 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 < -1.12e78

   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 93.7%

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

   if -1.12e78 < y < 8.20000000000000024e81

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

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

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

      3. sub-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac2 N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 96.7%

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

   7. Simplified 96.7%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. clear-num N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

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

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

      13. associate-/r/ N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 96.9%

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

   9. Applied egg-rr 96.9%

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

   if 8.20000000000000024e81 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 94.7%

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

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

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

      2. associate-/r* N/A

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

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

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

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

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

      5. sqrt-lowering-sqrt.f64 94.8%

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

   7. Applied egg-rr 94.8%

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

Alternative 3: 94.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{if}\;y \leq -4.6 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+81}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y (* 3.0 (sqrt x))))))
   (if (<= y -4.6e+77)
     t_0
     (if (<= y 8.2e+81) (+ 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 <= -4.6e+77) {
		tmp = t_0;
	} else if (y <= 8.2e+81) {
		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 <= (-4.6d+77)) then
        tmp = t_0
    else if (y <= 8.2d+81) 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 <= -4.6e+77) {
		tmp = t_0;
	} else if (y <= 8.2e+81) {
		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 <= -4.6e+77:
		tmp = t_0
	elif y <= 8.2e+81:
		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 <= -4.6e+77)
		tmp = t_0;
	elseif (y <= 8.2e+81)
		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 <= -4.6e+77)
		tmp = t_0;
	elseif (y <= 8.2e+81)
		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, -4.6e+77], t$95$0, If[LessEqual[y, 8.2e+81], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.5999999999999999e77 or 8.20000000000000024e81 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 94.1%

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

   if -4.5999999999999999e77 < y < 8.20000000000000024e81

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

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

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

      3. sub-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac2 N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 96.7%

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

   7. Simplified 96.7%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. clear-num N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

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

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

      13. associate-/r/ N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 96.9%

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

   9. Applied egg-rr 96.9%

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

Alternative 4: 92.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+81}:\\ \;\;\;\;\left(y \cdot {x}^{-0.5}\right) \cdot -0.3333333333333333\\ \mathbf{elif}\;y \leq 6.9 \cdot 10^{+93}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.8e+81)
   (* (* y (pow x -0.5)) -0.3333333333333333)
   (if (<= y 6.9e+93)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (/ (* y -0.3333333333333333) (sqrt x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.8e+81) {
		tmp = (y * pow(x, -0.5)) * -0.3333333333333333;
	} else if (y <= 6.9e+93) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (y * -0.3333333333333333) / sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.8d+81)) then
        tmp = (y * (x ** (-0.5d0))) * (-0.3333333333333333d0)
    else if (y <= 6.9d+93) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = (y * (-0.3333333333333333d0)) / sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.8e+81) {
		tmp = (y * Math.pow(x, -0.5)) * -0.3333333333333333;
	} else if (y <= 6.9e+93) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (y * -0.3333333333333333) / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.8e+81:
		tmp = (y * math.pow(x, -0.5)) * -0.3333333333333333
	elif y <= 6.9e+93:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = (y * -0.3333333333333333) / math.sqrt(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.8e+81)
		tmp = Float64(Float64(y * (x ^ -0.5)) * -0.3333333333333333);
	elseif (y <= 6.9e+93)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(Float64(y * -0.3333333333333333) / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.8e+81)
		tmp = (y * (x ^ -0.5)) * -0.3333333333333333;
	elseif (y <= 6.9e+93)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = (y * -0.3333333333333333) / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.8e+81], N[(N[(y * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision], If[LessEqual[y, 6.9e+93], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * -0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.8e81

   1. Initial program 99.7%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. 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. *-commutative N/A

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

      8. *-lowering-*.f64 90.8%

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

   7. Simplified 90.8%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. pow1/2 N/A

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

      6. inv-pow N/A

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

      7. pow-pow N/A

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

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

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

      9. metadata-eval 90.9%

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

   9. Applied egg-rr 90.9%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

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

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

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

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

       5. pow-lowering-pow.f64 90.9%

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

   11. Applied egg-rr 90.9%

       \[\leadsto \color{blue}{\left(y \cdot {x}^{-0.5}\right) \cdot -0.3333333333333333} \]

   if -3.8e81 < y < 6.8999999999999995e93

   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 95.7%

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

   7. Simplified 95.7%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. clear-num N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

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

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

      13. associate-/r/ N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 95.8%

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

   9. Applied egg-rr 95.8%

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

   if 6.8999999999999995e93 < y

   1. Initial program 99.6%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in 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. *-commutative N/A

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

      8. *-lowering-*.f64 93.3%

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

   7. Simplified 93.3%

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

      1. *-commutative N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. un-div-inv N/A

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

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

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

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

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

      7. sqrt-lowering-sqrt.f64 93.4%

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

   9. Applied egg-rr 93.4%

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

Alternative 5: 92.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{+82}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+92}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* y -0.3333333333333333) (sqrt x))))
   (if (<= y -5.5e+82)
     t_0
     (if (<= y 2.3e+92) (+ 1.0 (/ -1.0 (* x 9.0))) t_0))))

C

double code(double x, double y) {
	double t_0 = (y * -0.3333333333333333) / sqrt(x);
	double tmp;
	if (y <= -5.5e+82) {
		tmp = t_0;
	} else if (y <= 2.3e+92) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y * (-0.3333333333333333d0)) / sqrt(x)
    if (y <= (-5.5d+82)) then
        tmp = t_0
    else if (y <= 2.3d+92) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * -0.3333333333333333) / Math.sqrt(x);
	double tmp;
	if (y <= -5.5e+82) {
		tmp = t_0;
	} else if (y <= 2.3e+92) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * -0.3333333333333333) / math.sqrt(x)
	tmp = 0
	if y <= -5.5e+82:
		tmp = t_0
	elif y <= 2.3e+92:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * -0.3333333333333333) / sqrt(x))
	tmp = 0.0
	if (y <= -5.5e+82)
		tmp = t_0;
	elseif (y <= 2.3e+92)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * -0.3333333333333333) / sqrt(x);
	tmp = 0.0;
	if (y <= -5.5e+82)
		tmp = t_0;
	elseif (y <= 2.3e+92)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * -0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.5e+82], t$95$0, If[LessEqual[y, 2.3e+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 < -5.49999999999999997e82 or 2.29999999999999998e92 < y

   1. Initial program 99.6%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in 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. *-commutative N/A

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

      8. *-lowering-*.f64 91.8%

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

   7. Simplified 91.8%

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

      1. *-commutative N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. un-div-inv N/A

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

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

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

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

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

      7. sqrt-lowering-sqrt.f64 91.9%

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

   9. Applied egg-rr 91.9%

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

   if -5.49999999999999997e82 < y < 2.29999999999999998e92

   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 95.7%

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

   7. Simplified 95.7%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. clear-num N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

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

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

      13. associate-/r/ N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 95.8%

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

   9. Applied egg-rr 95.8%

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

Alternative 6: 92.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.6 \cdot 10^{+83}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+97}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.6e+83)
   (* -0.3333333333333333 (/ y (sqrt x)))
   (if (<= y 4.5e+97)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (* y (/ -0.3333333333333333 (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.6e+83) {
		tmp = -0.3333333333333333 * (y / sqrt(x));
	} else if (y <= 4.5e+97) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = y * (-0.3333333333333333 / sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-7.6d+83)) then
        tmp = (-0.3333333333333333d0) * (y / sqrt(x))
    else if (y <= 4.5d+97) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = y * ((-0.3333333333333333d0) / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7.6e+83) {
		tmp = -0.3333333333333333 * (y / Math.sqrt(x));
	} else if (y <= 4.5e+97) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = y * (-0.3333333333333333 / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.6e+83:
		tmp = -0.3333333333333333 * (y / math.sqrt(x))
	elif y <= 4.5e+97:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = y * (-0.3333333333333333 / math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.6e+83)
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	elseif (y <= 4.5e+97)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(y * Float64(-0.3333333333333333 / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.6e+83)
		tmp = -0.3333333333333333 * (y / sqrt(x));
	elseif (y <= 4.5e+97)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = y * (-0.3333333333333333 / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.6e+83], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.5e+97], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.6000000000000004e83

   1. Initial program 99.7%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. 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. *-commutative N/A

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

      8. *-lowering-*.f64 90.8%

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

   7. Simplified 90.8%

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

      1. associate-*r* N/A

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

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

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

      3. *-commutative N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. un-div-inv N/A

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

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

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

      8. sqrt-lowering-sqrt.f64 90.8%

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

   9. Applied egg-rr 90.8%

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

   if -7.6000000000000004e83 < y < 4.49999999999999976e97

   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 95.7%

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

   7. Simplified 95.7%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. clear-num N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

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

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

      13. associate-/r/ N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 95.8%

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

   9. Applied egg-rr 95.8%

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

   if 4.49999999999999976e97 < y

   1. Initial program 99.6%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in 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. *-commutative N/A

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

      8. *-lowering-*.f64 93.3%

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

   7. Simplified 93.3%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. un-div-inv N/A

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

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

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

      8. sqrt-lowering-sqrt.f64 93.3%

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

   9. Applied egg-rr 93.3%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.6 \cdot 10^{+83}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+97}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{if}\;y \leq -2.4 \cdot 10^{+78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+92}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (/ -0.3333333333333333 (sqrt x)))))
   (if (<= y -2.4e+78)
     t_0
     (if (<= y 2.8e+92) (+ 1.0 (/ -1.0 (* x 9.0))) t_0))))

C

double code(double x, double y) {
	double t_0 = y * (-0.3333333333333333 / sqrt(x));
	double tmp;
	if (y <= -2.4e+78) {
		tmp = t_0;
	} else if (y <= 2.8e+92) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * ((-0.3333333333333333d0) / sqrt(x))
    if (y <= (-2.4d+78)) then
        tmp = t_0
    else if (y <= 2.8d+92) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (-0.3333333333333333 / Math.sqrt(x));
	double tmp;
	if (y <= -2.4e+78) {
		tmp = t_0;
	} else if (y <= 2.8e+92) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (-0.3333333333333333 / math.sqrt(x))
	tmp = 0
	if y <= -2.4e+78:
		tmp = t_0
	elif y <= 2.8e+92:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(-0.3333333333333333 / sqrt(x)))
	tmp = 0.0
	if (y <= -2.4e+78)
		tmp = t_0;
	elseif (y <= 2.8e+92)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (-0.3333333333333333 / sqrt(x));
	tmp = 0.0;
	if (y <= -2.4e+78)
		tmp = t_0;
	elseif (y <= 2.8e+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[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.4e+78], t$95$0, If[LessEqual[y, 2.8e+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 < -2.3999999999999999e78 or 2.80000000000000001e92 < y

   1. Initial program 99.6%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in 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. *-commutative N/A

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

      8. *-lowering-*.f64 91.8%

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

   7. Simplified 91.8%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. un-div-inv N/A

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

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

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

      8. sqrt-lowering-sqrt.f64 91.8%

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

   9. Applied egg-rr 91.8%

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

   if -2.3999999999999999e78 < y < 2.80000000000000001e92

   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 95.7%

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

   7. Simplified 95.7%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. clear-num N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

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

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

      13. associate-/r/ N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 95.8%

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

   9. Applied egg-rr 95.8%

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

Alternative 8: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = (-0.1111111111111111 / x) - (y / (3.0 * sqrt(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) / x) - (y / (3.0d0 * sqrt(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 / x) - (y / (3.0 * Math.sqrt(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 / x) - (y / (3.0 * math.sqrt(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 / x) - Float64(y / Float64(3.0 * sqrt(x))));
	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 (x <= 0.11)
		tmp = (-0.1111111111111111 / x) - (y / (3.0 * sqrt(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 / x), $MachinePrecision] - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(y / 3.0), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 98.4%

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

   5. Simplified 98.4%

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

   if 0.110000000000000001 < x

   1. Initial program 99.9%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   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 98.3%

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

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

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

      2. associate-/r* N/A

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

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

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

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

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

      5. sqrt-lowering-sqrt.f64 98.4%

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

   7. Applied egg-rr 98.4%

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

Alternative 9: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

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

   2. frac-2neg N/A

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

   3. distribute-frac-neg2 N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. associate-/r* N/A

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

   7. *-commutative N/A

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

   8. sub-neg N/A

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

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

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

   10. *-commutative N/A

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

   11. associate-/r* N/A

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

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

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

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

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

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

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

   15. sub-neg N/A

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

   16. *-commutative N/A

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

   17. associate-/r* N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. distribute-frac-neg2 N/A

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

   21. frac-2neg N/A

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

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

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

   23. /-lowering-/.f64 99.7%

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

6. Applied egg-rr 99.7%

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

Alternative 10: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg N/A

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

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

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

   3. sub-neg N/A

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

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

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. distribute-neg-frac N/A

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

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

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac2 N/A

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

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

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

   13. *-commutative N/A

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

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

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

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

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

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

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

   17. metadata-eval 99.7%

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

3. Simplified 99.7%

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

Alternative 11: 66.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+124}:\\ \;\;\;\;\frac{1}{1 + \frac{\frac{0.012345679012345678}{x} - -0.1111111111111111}{x}} + \frac{\frac{\frac{1.8816764231589208 \cdot 10^{-6}}{x \cdot x} - 0.00015241579027587258}{x} - -0.0013717421124828531}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.3e+124)
   (+
    (/ 1.0 (+ 1.0 (/ (- (/ 0.012345679012345678 x) -0.1111111111111111) x)))
    (/
     (-
      (/ (- (/ 1.8816764231589208e-6 (* x x)) 0.00015241579027587258) x)
      -0.0013717421124828531)
     (* x (* x x))))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.3e+124) {
		tmp = (1.0 / (1.0 + (((0.012345679012345678 / x) - -0.1111111111111111) / x))) + (((((1.8816764231589208e-6 / (x * x)) - 0.00015241579027587258) / x) - -0.0013717421124828531) / (x * (x * x)));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.3d+124)) then
        tmp = (1.0d0 / (1.0d0 + (((0.012345679012345678d0 / x) - (-0.1111111111111111d0)) / x))) + (((((1.8816764231589208d-6 / (x * x)) - 0.00015241579027587258d0) / x) - (-0.0013717421124828531d0)) / (x * (x * x)))
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.3e+124) {
		tmp = (1.0 / (1.0 + (((0.012345679012345678 / x) - -0.1111111111111111) / x))) + (((((1.8816764231589208e-6 / (x * x)) - 0.00015241579027587258) / x) - -0.0013717421124828531) / (x * (x * x)));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.3e+124:
		tmp = (1.0 / (1.0 + (((0.012345679012345678 / x) - -0.1111111111111111) / x))) + (((((1.8816764231589208e-6 / (x * x)) - 0.00015241579027587258) / x) - -0.0013717421124828531) / (x * (x * x)))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.3e+124)
		tmp = Float64(Float64(1.0 / Float64(1.0 + Float64(Float64(Float64(0.012345679012345678 / x) - -0.1111111111111111) / x))) + Float64(Float64(Float64(Float64(Float64(1.8816764231589208e-6 / Float64(x * x)) - 0.00015241579027587258) / x) - -0.0013717421124828531) / Float64(x * Float64(x * x))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.3e+124)
		tmp = (1.0 / (1.0 + (((0.012345679012345678 / x) - -0.1111111111111111) / x))) + (((((1.8816764231589208e-6 / (x * x)) - 0.00015241579027587258) / x) - -0.0013717421124828531) / (x * (x * x)));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.3e+124], N[(N[(1.0 / N[(1.0 + N[(N[(N[(0.012345679012345678 / x), $MachinePrecision] - -0.1111111111111111), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(1.8816764231589208e-6 / N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.00015241579027587258), $MachinePrecision] / x), $MachinePrecision] - -0.0013717421124828531), $MachinePrecision] / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3e124

   1. Initial program 99.6%

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

      1. sub-neg N/A

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

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

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

      3. sub-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac2 N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 2.5%

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

   7. Simplified 2.5%

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

   9. Applied egg-rr 3.7%

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

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{1}{81}, x\right), \frac{-1}{9}\right), x\right)\right)\right), \color{blue}{\left(\frac{\frac{1}{6561} \cdot \frac{1}{x} - \left(\frac{1}{729} + \frac{\frac{1}{531441}}{{x}^{3}}\right)}{{x}^{3}}\right)}\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{1}{81}, x\right), \frac{-1}{9}\right), x\right)\right)\right), \mathsf{/.f64}\left(\left(\frac{1}{6561} \cdot \frac{1}{x} - \left(\frac{1}{729} + \frac{\frac{1}{531441}}{{x}^{3}}\right)\right), \color{blue}{\left({x}^{3}\right)}\right)\right) \]

   12. Simplified 27.8%

       \[\leadsto \frac{1}{1 + \frac{\frac{0.012345679012345678}{x} - -0.1111111111111111}{x}} - \color{blue}{\frac{-0.0013717421124828531 + \frac{0.00015241579027587258 - \frac{1.8816764231589208 \cdot 10^{-6}}{x \cdot x}}{x}}{x \cdot \left(x \cdot x\right)}} \]

   if -1.3e124 < y

   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 79.2%

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

   7. Simplified 79.2%

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

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. clear-num N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

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

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

      13. associate-/r/ N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 79.3%

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

   9. Applied egg-rr 79.3%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+124}:\\ \;\;\;\;\frac{1}{1 + \frac{\frac{0.012345679012345678}{x} - -0.1111111111111111}{x}} + \frac{\frac{\frac{1.8816764231589208 \cdot 10^{-6}}{x \cdot x} - 0.00015241579027587258}{x} - -0.0013717421124828531}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.1% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+124}:\\ \;\;\;\;1 - \frac{0.1111111111111111 + \frac{-0.0027434842249657062}{x \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.3e+124)
   (- 1.0 (/ (+ 0.1111111111111111 (/ -0.0027434842249657062 (* x x))) x))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.3e+124) {
		tmp = 1.0 - ((0.1111111111111111 + (-0.0027434842249657062 / (x * x))) / x);
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.3d+124)) then
        tmp = 1.0d0 - ((0.1111111111111111d0 + ((-0.0027434842249657062d0) / (x * x))) / x)
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.3e+124) {
		tmp = 1.0 - ((0.1111111111111111 + (-0.0027434842249657062 / (x * x))) / x);
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.3e+124:
		tmp = 1.0 - ((0.1111111111111111 + (-0.0027434842249657062 / (x * x))) / x)
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.3e+124)
		tmp = Float64(1.0 - Float64(Float64(0.1111111111111111 + Float64(-0.0027434842249657062 / Float64(x * x))) / x));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.3e+124)
		tmp = 1.0 - ((0.1111111111111111 + (-0.0027434842249657062 / (x * x))) / x);
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.3e+124], N[(1.0 - N[(N[(0.1111111111111111 + N[(-0.0027434842249657062 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3e124

   1. Initial program 99.6%

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

      1. sub-neg N/A

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

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

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

      3. sub-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac2 N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 2.5%

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

   7. Simplified 2.5%

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

   9. Applied egg-rr 3.7%

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

   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. +-commutative N/A

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

       2. associate--l+ N/A

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

       3. +-commutative N/A

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

       4. associate--r- N/A

          \[\leadsto 1 - \color{blue}{\left(\frac{1}{9} \cdot \frac{1}{x} - \frac{\frac{2}{729}}{{x}^{3}}\right)} \]

       5. associate-*r/ N/A

          \[\leadsto 1 - \left(\frac{\frac{1}{9} \cdot 1}{x} - \frac{\color{blue}{\frac{2}{729}}}{{x}^{3}}\right) \]

       6. metadata-eval N/A

          \[\leadsto 1 - \left(\frac{\frac{1}{9}}{x} - \frac{\frac{2}{729}}{{x}^{3}}\right) \]

       7. unpow3 N/A

          \[\leadsto 1 - \left(\frac{\frac{1}{9}}{x} - \frac{\frac{2}{729}}{\left(x \cdot x\right) \cdot \color{blue}{x}}\right) \]

       8. unpow2 N/A

          \[\leadsto 1 - \left(\frac{\frac{1}{9}}{x} - \frac{\frac{2}{729}}{{x}^{2} \cdot x}\right) \]

       9. associate-/r* N/A

          \[\leadsto 1 - \left(\frac{\frac{1}{9}}{x} - \frac{\frac{\frac{2}{729}}{{x}^{2}}}{\color{blue}{x}}\right) \]

       10. metadata-eval N/A

           \[\leadsto 1 - \left(\frac{\frac{1}{9}}{x} - \frac{\frac{\frac{2}{729} \cdot 1}{{x}^{2}}}{x}\right) \]

       11. associate-*r/ N/A

           \[\leadsto 1 - \left(\frac{\frac{1}{9}}{x} - \frac{\frac{2}{729} \cdot \frac{1}{{x}^{2}}}{x}\right) \]

       12. div-sub N/A

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

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

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

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

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

   12. Simplified 25.7%

       \[\leadsto \color{blue}{1 - \frac{0.1111111111111111 + \frac{-0.0027434842249657062}{x \cdot x}}{x}} \]

   if -1.3e124 < y

   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 79.2%

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

   7. Simplified 79.2%

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

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. clear-num N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

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

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

      13. associate-/r/ N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 79.3%

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

   9. Applied egg-rr 79.3%

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

Alternative 13: 62.1% accurate, 14.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   1. Initial program 99.6%

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

      1. sub-neg N/A

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

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

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

      3. sub-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac2 N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 69.1%

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

   7. Simplified 69.1%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 68.0%

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

   10. Simplified 68.0%

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

   if 0.110000000000000001 < x

   1. Initial program 99.9%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   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 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(1 + \frac{-0.1111111111111111}{x}\right) + \frac{y}{\sqrt{x} \cdot -3}} \]
   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 65.8%

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

   7. Simplified 65.8%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 64.3%

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

Alternative 14: 63.2% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg N/A

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

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

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

   3. sub-neg N/A

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

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

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. distribute-neg-frac N/A

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

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

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac2 N/A

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

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

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

   13. *-commutative N/A

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

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

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

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

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

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

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

   17. metadata-eval 99.7%

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

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 67.5%

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

7. Simplified 67.5%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

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

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. div-inv N/A

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

   9. clear-num N/A

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

   10. distribute-neg-frac N/A

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

   11. metadata-eval N/A

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

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

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

   13. associate-/r/ N/A

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

   14. metadata-eval N/A

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

   15. *-commutative N/A

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

   16. *-lowering-*.f64 67.6%

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

9. Applied egg-rr 67.6%

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

Alternative 15: 63.1% 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 67.5%

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

7. Simplified 67.5%

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

Alternative 16: 31.8% 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 67.5%

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

7. Simplified 67.5%

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

8. Taylor expanded in x around inf

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

10. Simplified 31.4%

    \[\leadsto \color{blue}{1} \]
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 2024161 
(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)))))