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

Percentage Accurate: 99.7% → 99.7%

Time: 9.9s

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: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+45}:\\ \;\;\;\;1 - \frac{\frac{y}{3}}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+99}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3} \cdot {x}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.8e+45)
   (- 1.0 (/ (/ y 3.0) (sqrt x)))
   (if (<= y 7.8e+99)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (- 1.0 (* (/ y 3.0) (pow x -0.5))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.8e+45) {
		tmp = 1.0 - ((y / 3.0) / sqrt(x));
	} else if (y <= 7.8e+99) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 - ((y / 3.0) * pow(x, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.8d+45)) then
        tmp = 1.0d0 - ((y / 3.0d0) / sqrt(x))
    else if (y <= 7.8d+99) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = 1.0d0 - ((y / 3.0d0) * (x ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.8e+45) {
		tmp = 1.0 - ((y / 3.0) / Math.sqrt(x));
	} else if (y <= 7.8e+99) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 - ((y / 3.0) * Math.pow(x, -0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.8e+45:
		tmp = 1.0 - ((y / 3.0) / math.sqrt(x))
	elif y <= 7.8e+99:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = 1.0 - ((y / 3.0) * math.pow(x, -0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.8e+45)
		tmp = Float64(1.0 - Float64(Float64(y / 3.0) / sqrt(x)));
	elseif (y <= 7.8e+99)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(1.0 - Float64(Float64(y / 3.0) * (x ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.8e+45)
		tmp = 1.0 - ((y / 3.0) / sqrt(x));
	elseif (y <= 7.8e+99)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = 1.0 - ((y / 3.0) * (x ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.8e+45], N[(1.0 - N[(N[(y / 3.0), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.8e+99], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(y / 3.0), $MachinePrecision] * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.8e45

   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 94.9%

      \[\leadsto \color{blue}{1} - \frac{y}{3 \cdot \sqrt{x}} \]
   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 94.9%

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

   7. Applied egg-rr 94.9%

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

   if -1.8e45 < y < 7.79999999999999989e99

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

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

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

      3. sub-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac2 N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 94.1%

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

   7. Simplified 94.1%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. metadata-eval N/A

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

      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 94.1%

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

   9. Applied egg-rr 94.1%

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

   if 7.79999999999999989e99 < 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 96.4%

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

      1. *-lft-identity N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

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

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

      6. pow1/2 N/A

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

      7. pow-flip N/A

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

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

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

      9. metadata-eval N/A

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

      10. /-lowering-/.f64 96.5%

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

   7. Applied egg-rr 96.5%

      \[\leadsto 1 - \color{blue}{{x}^{-0.5} \cdot \frac{y}{3}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+45}:\\ \;\;\;\;1 - \frac{\frac{y}{3}}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+99}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3} \cdot {x}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{+45}:\\ \;\;\;\;1 - \frac{\frac{y}{3}}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+99}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.4e+45)
   (- 1.0 (/ (/ y 3.0) (sqrt x)))
   (if (<= y 7.8e+99)
     (+ 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 <= -4.4e+45) {
		tmp = 1.0 - ((y / 3.0) / sqrt(x));
	} else if (y <= 7.8e+99) {
		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 <= (-4.4d+45)) then
        tmp = 1.0d0 - ((y / 3.0d0) / sqrt(x))
    else if (y <= 7.8d+99) 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 <= -4.4e+45) {
		tmp = 1.0 - ((y / 3.0) / Math.sqrt(x));
	} else if (y <= 7.8e+99) {
		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 <= -4.4e+45:
		tmp = 1.0 - ((y / 3.0) / math.sqrt(x))
	elif y <= 7.8e+99:
		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 <= -4.4e+45)
		tmp = Float64(1.0 - Float64(Float64(y / 3.0) / sqrt(x)));
	elseif (y <= 7.8e+99)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.4e+45)
		tmp = 1.0 - ((y / 3.0) / sqrt(x));
	elseif (y <= 7.8e+99)
		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, -4.4e+45], N[(1.0 - N[(N[(y / 3.0), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.8e+99], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.4000000000000001e45

   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 94.9%

      \[\leadsto \color{blue}{1} - \frac{y}{3 \cdot \sqrt{x}} \]
   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 94.9%

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

   7. Applied egg-rr 94.9%

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

   if -4.4000000000000001e45 < y < 7.79999999999999989e99

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

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

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

      3. sub-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac2 N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 94.1%

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

   7. Simplified 94.1%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. metadata-eval N/A

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

      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 94.1%

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

   9. Applied egg-rr 94.1%

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

   if 7.79999999999999989e99 < 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 96.4%

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

Alternative 4: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{if}\;y \leq -1 \cdot 10^{+46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+99}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y (* 3.0 (sqrt x))))))
   (if (<= y -1e+46) t_0 (if (<= y 7.8e+99) (+ 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 <= -1e+46) {
		tmp = t_0;
	} else if (y <= 7.8e+99) {
		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 <= (-1d+46)) then
        tmp = t_0
    else if (y <= 7.8d+99) 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 <= -1e+46) {
		tmp = t_0;
	} else if (y <= 7.8e+99) {
		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 <= -1e+46:
		tmp = t_0
	elif y <= 7.8e+99:
		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 <= -1e+46)
		tmp = t_0;
	elseif (y <= 7.8e+99)
		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 <= -1e+46)
		tmp = t_0;
	elseif (y <= 7.8e+99)
		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, -1e+46], t$95$0, If[LessEqual[y, 7.8e+99], 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 < -9.9999999999999999e45 or 7.79999999999999989e99 < 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 95.5%

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

   if -9.9999999999999999e45 < y < 7.79999999999999989e99

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

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

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

      3. sub-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac2 N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 94.1%

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

   7. Simplified 94.1%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. metadata-eval N/A

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

      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 94.1%

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

   9. Applied egg-rr 94.1%

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

Alternative 5: 93.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+45}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+99}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x}} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.2e+45)
   (+ 1.0 (/ -0.3333333333333333 (/ (sqrt x) y)))
   (if (<= y 7.8e+99)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (- 1.0 (* (/ y (sqrt x)) 0.3333333333333333)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.2e+45) {
		tmp = 1.0 + (-0.3333333333333333 / (sqrt(x) / y));
	} else if (y <= 7.8e+99) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 - ((y / sqrt(x)) * 0.3333333333333333);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.2d+45)) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) / (sqrt(x) / y))
    else if (y <= 7.8d+99) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = 1.0d0 - ((y / sqrt(x)) * 0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.2e+45) {
		tmp = 1.0 + (-0.3333333333333333 / (Math.sqrt(x) / y));
	} else if (y <= 7.8e+99) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 - ((y / Math.sqrt(x)) * 0.3333333333333333);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.2e+45:
		tmp = 1.0 + (-0.3333333333333333 / (math.sqrt(x) / y))
	elif y <= 7.8e+99:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = 1.0 - ((y / math.sqrt(x)) * 0.3333333333333333)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.2e+45)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 / Float64(sqrt(x) / y)));
	elseif (y <= 7.8e+99)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(1.0 - Float64(Float64(y / sqrt(x)) * 0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.2e+45)
		tmp = 1.0 + (-0.3333333333333333 / (sqrt(x) / y));
	elseif (y <= 7.8e+99)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = 1.0 - ((y / sqrt(x)) * 0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.2e+45], N[(1.0 + N[(-0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.8e+99], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.1999999999999999e45

   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 94.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. distribute-neg-frac N/A

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

      5. neg-mul-1 N/A

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

      6. times-frac N/A

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

      7. metadata-eval N/A

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

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

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

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

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

      12. sqrt-lowering-sqrt.f64 94.8%

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

   7. Applied egg-rr 94.8%

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

   if -4.1999999999999999e45 < y < 7.79999999999999989e99

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

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

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

      3. sub-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac2 N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 94.1%

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

   7. Simplified 94.1%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. metadata-eval N/A

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

      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 94.1%

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

   9. Applied egg-rr 94.1%

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

   if 7.79999999999999989e99 < 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 96.4%

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

      1. div-inv N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

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

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

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

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

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

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

      8. metadata-eval 96.3%

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

   7. Applied egg-rr 96.3%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+45}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+99}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x}} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{+45}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+99}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.32e+45)
   (+ 1.0 (/ -0.3333333333333333 (/ (sqrt x) y)))
   (if (<= y 8.2e+99)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (* (sqrt (/ 1.0 x)) (* y -0.3333333333333333)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.32e+45) {
		tmp = 1.0 + (-0.3333333333333333 / (sqrt(x) / y));
	} else if (y <= 8.2e+99) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = sqrt((1.0 / x)) * (y * -0.3333333333333333);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.32d+45)) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) / (sqrt(x) / y))
    else if (y <= 8.2d+99) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = sqrt((1.0d0 / x)) * (y * (-0.3333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.32e+45) {
		tmp = 1.0 + (-0.3333333333333333 / (Math.sqrt(x) / y));
	} else if (y <= 8.2e+99) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = Math.sqrt((1.0 / x)) * (y * -0.3333333333333333);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.32e+45:
		tmp = 1.0 + (-0.3333333333333333 / (math.sqrt(x) / y))
	elif y <= 8.2e+99:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = math.sqrt((1.0 / x)) * (y * -0.3333333333333333)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.32e+45)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 / Float64(sqrt(x) / y)));
	elseif (y <= 8.2e+99)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(sqrt(Float64(1.0 / x)) * Float64(y * -0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.32e+45)
		tmp = 1.0 + (-0.3333333333333333 / (sqrt(x) / y));
	elseif (y <= 8.2e+99)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = sqrt((1.0 / x)) * (y * -0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.32e+45], N[(1.0 + N[(-0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.2e+99], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.32000000000000005e45

   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 94.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. distribute-neg-frac N/A

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

      5. neg-mul-1 N/A

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

      6. times-frac N/A

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

      7. metadata-eval N/A

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

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

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

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

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

      12. sqrt-lowering-sqrt.f64 94.8%

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

   7. Applied egg-rr 94.8%

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

   if -1.32000000000000005e45 < y < 8.19999999999999959e99

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

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

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

      3. sub-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac2 N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 94.1%

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

   7. Simplified 94.1%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. metadata-eval N/A

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

      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 94.1%

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

   9. Applied egg-rr 94.1%

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

   if 8.19999999999999959e99 < y

   1. Initial program 99.6%

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

      1. sub-neg N/A

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

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

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

      3. sub-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac2 N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 96.1%

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

   7. Simplified 96.1%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{+45}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+99}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \mathbf{if}\;y \leq -1.02 \cdot 10^{+46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+99}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt (/ 1.0 x)) (* y -0.3333333333333333))))
   (if (<= y -1.02e+46)
     t_0
     (if (<= y 7.8e+99) (+ 1.0 (/ -1.0 (* x 9.0))) t_0))))

C

double code(double x, double y) {
	double t_0 = sqrt((1.0 / x)) * (y * -0.3333333333333333);
	double tmp;
	if (y <= -1.02e+46) {
		tmp = t_0;
	} else if (y <= 7.8e+99) {
		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 = sqrt((1.0d0 / x)) * (y * (-0.3333333333333333d0))
    if (y <= (-1.02d+46)) then
        tmp = t_0
    else if (y <= 7.8d+99) 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 = Math.sqrt((1.0 / x)) * (y * -0.3333333333333333);
	double tmp;
	if (y <= -1.02e+46) {
		tmp = t_0;
	} else if (y <= 7.8e+99) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt((1.0 / x)) * (y * -0.3333333333333333)
	tmp = 0
	if y <= -1.02e+46:
		tmp = t_0
	elif y <= 7.8e+99:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(Float64(1.0 / x)) * Float64(y * -0.3333333333333333))
	tmp = 0.0
	if (y <= -1.02e+46)
		tmp = t_0;
	elseif (y <= 7.8e+99)
		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 = sqrt((1.0 / x)) * (y * -0.3333333333333333);
	tmp = 0.0;
	if (y <= -1.02e+46)
		tmp = t_0;
	elseif (y <= 7.8e+99)
		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[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.02e+46], t$95$0, If[LessEqual[y, 7.8e+99], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.0199999999999999e46 or 7.79999999999999989e99 < y

   1. Initial program 99.6%

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

      1. sub-neg N/A

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

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

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

      3. sub-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac2 N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 93.5%

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

   7. Simplified 93.5%

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

   if -1.0199999999999999e46 < y < 7.79999999999999989e99

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

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

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

      3. sub-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac2 N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 94.1%

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

   7. Simplified 94.1%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. metadata-eval N/A

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

      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 94.1%

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

   9. Applied egg-rr 94.1%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+46}:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+99}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{-14}:\\ \;\;\;\;\frac{-0.1111111111111111}{x} - \frac{\frac{y}{\sqrt{x}}}{3}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{y}{3}}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.12e-14)
   (- (/ -0.1111111111111111 x) (/ (/ y (sqrt x)) 3.0))
   (- 1.0 (/ (/ y 3.0) (sqrt x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.12e-14) {
		tmp = (-0.1111111111111111 / x) - ((y / sqrt(x)) / 3.0);
	} else {
		tmp = 1.0 - ((y / 3.0) / sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.12d-14) then
        tmp = ((-0.1111111111111111d0) / x) - ((y / sqrt(x)) / 3.0d0)
    else
        tmp = 1.0d0 - ((y / 3.0d0) / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.12e-14) {
		tmp = (-0.1111111111111111 / x) - ((y / Math.sqrt(x)) / 3.0);
	} else {
		tmp = 1.0 - ((y / 3.0) / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.12e-14:
		tmp = (-0.1111111111111111 / x) - ((y / math.sqrt(x)) / 3.0)
	else:
		tmp = 1.0 - ((y / 3.0) / math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.12e-14)
		tmp = Float64(Float64(-0.1111111111111111 / x) - Float64(Float64(y / sqrt(x)) / 3.0));
	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 (x <= 1.12e-14)
		tmp = (-0.1111111111111111 / x) - ((y / sqrt(x)) / 3.0);
	else
		tmp = 1.0 - ((y / 3.0) / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.12e-14], N[(N[(-0.1111111111111111 / x), $MachinePrecision] - N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / 3.0), $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 < 1.12000000000000006e-14

   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 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 99.5%

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

   5. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x}} - \frac{y}{3 \cdot \sqrt{x}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. sqrt-lowering-sqrt.f64 99.6%

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

   7. Applied egg-rr 99.6%

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

   if 1.12000000000000006e-14 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.3%

      \[\leadsto \color{blue}{1} - \frac{y}{3 \cdot \sqrt{x}} \]
   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 99.4%

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

   7. Applied egg-rr 99.4%

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

Alternative 9: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{-14}:\\ \;\;\;\;\frac{-0.1111111111111111}{x} - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{y}{3}}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.12e-14)
   (- (/ -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 <= 1.12e-14) {
		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 <= 1.12d-14) 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 <= 1.12e-14) {
		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 <= 1.12e-14:
		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 <= 1.12e-14)
		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 <= 1.12e-14)
		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, 1.12e-14], 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 < 1.12000000000000006e-14

   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 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 99.5%

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

   5. Simplified 99.5%

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

   if 1.12000000000000006e-14 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.3%

      \[\leadsto \color{blue}{1} - \frac{y}{3 \cdot \sqrt{x}} \]
   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 99.4%

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

   7. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{1 - \frac{\frac{y}{3}}{\sqrt{x}}} \]
3. Recombined 2 regimes into one program.
4. 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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg N/A

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

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

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

   3. sub-neg N/A

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

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

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. distribute-neg-frac N/A

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

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

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac2 N/A

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

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

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

   13. *-commutative N/A

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

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

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

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

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

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

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

   17. metadata-eval 99.7%

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

3. Simplified 99.7%

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

   1. associate-/r* N/A

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

   2. div-inv N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

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

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

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

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

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

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

   9. metadata-eval N/A

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

   10. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

Alternative 12: 64.5% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{+104}:\\ \;\;\;\;1 + \frac{-0.1111111111111111 + \frac{0.024691358024691357}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.6e+104)
   (+ 1.0 (/ (+ -0.1111111111111111 (/ 0.024691358024691357 x)) x))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6.6e+104) {
		tmp = 1.0 + ((-0.1111111111111111 + (0.024691358024691357 / x)) / x);
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-6.6d+104)) then
        tmp = 1.0d0 + (((-0.1111111111111111d0) + (0.024691358024691357d0 / x)) / x)
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -6.6e+104) {
		tmp = 1.0 + ((-0.1111111111111111 + (0.024691358024691357 / x)) / x);
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6.6e+104:
		tmp = 1.0 + ((-0.1111111111111111 + (0.024691358024691357 / x)) / x)
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6.6e+104)
		tmp = Float64(1.0 + Float64(Float64(-0.1111111111111111 + Float64(0.024691358024691357 / 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 <= -6.6e+104)
		tmp = 1.0 + ((-0.1111111111111111 + (0.024691358024691357 / x)) / x);
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6.6e+104], N[(1.0 + N[(N[(-0.1111111111111111 + N[(0.024691358024691357 / x), $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 < -6.59999999999999969e104

   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 2.7%

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

   7. Simplified 2.7%

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

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

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

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

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

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

      12. metadata-eval 2.7%

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

   9. Applied egg-rr 2.7%

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

   11. Applied egg-rr 3.5%

       \[\leadsto \color{blue}{\frac{\frac{\frac{0.0013717421124828531}{x}}{x \cdot x} + -1}{\frac{\frac{\frac{0.1111111111111111}{x} + -1}{x}}{9} + -1}} \]

   12. Taylor expanded in x around inf

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

       1. associate--l+ N/A

          \[\leadsto 1 + \color{blue}{\left(\frac{\frac{2}{81}}{{x}^{2}} - \frac{1}{9} \cdot \frac{1}{x}\right)} \]

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

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

       3. sub-neg N/A

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

       4. +-commutative N/A

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

       5. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(0 - \frac{1}{9} \cdot \frac{1}{x}\right) + \frac{\color{blue}{\frac{2}{81}}}{{x}^{2}}\right)\right) \]

       6. associate--r- N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(0 - \color{blue}{\left(\frac{1}{9} \cdot \frac{1}{x} - \frac{\frac{2}{81}}{{x}^{2}}\right)}\right)\right) \]

       7. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(0 - \left(\frac{\frac{1}{9} \cdot 1}{x} - \frac{\color{blue}{\frac{2}{81}}}{{x}^{2}}\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(0 - \left(\frac{\frac{1}{9}}{x} - \frac{\frac{2}{81}}{{x}^{2}}\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(0 - \left(\frac{\frac{1}{9}}{x} - \frac{\frac{2}{81}}{x \cdot \color{blue}{x}}\right)\right)\right) \]

       10. associate-/r* N/A

           \[\leadsto \mathsf{+.f64}\left(1, \left(0 - \left(\frac{\frac{1}{9}}{x} - \frac{\frac{\frac{2}{81}}{x}}{\color{blue}{x}}\right)\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{+.f64}\left(1, \left(0 - \left(\frac{\frac{1}{9}}{x} - \frac{\frac{\frac{2}{81} \cdot 1}{x}}{x}\right)\right)\right) \]

       12. associate-*r/ N/A

           \[\leadsto \mathsf{+.f64}\left(1, \left(0 - \left(\frac{\frac{1}{9}}{x} - \frac{\frac{2}{81} \cdot \frac{1}{x}}{x}\right)\right)\right) \]

       13. div-sub N/A

           \[\leadsto \mathsf{+.f64}\left(1, \left(0 - \frac{\frac{1}{9} - \frac{2}{81} \cdot \frac{1}{x}}{\color{blue}{x}}\right)\right) \]

       14. neg-sub0 N/A

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

       15. mul-1-neg N/A

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

       16. associate-*r/ N/A

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

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

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

   14. Simplified 31.2%

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

   if -6.59999999999999969e104 < y

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

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

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

      3. sub-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac2 N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 73.2%

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

   7. Simplified 73.2%

      \[\leadsto \color{blue}{1 + \frac{-0.1111111111111111}{x}} \]
   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 73.3%

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

   9. Applied egg-rr 73.3%

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

Alternative 13: 60.9% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 600:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 600.0) (/ -0.1111111111111111 x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 600.0) {
		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 <= 600.0d0) 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 <= 600.0) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 600.0:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 600.0)
		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 <= 600.0)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 600.0], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 600

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

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

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

      3. sub-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac2 N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 61.7%

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

   7. Simplified 61.7%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 61.7%

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

   10. Simplified 61.7%

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

   if 600 < x

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

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

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

      3. sub-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac2 N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 59.8%

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

   7. Simplified 59.8%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 59.6%

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

Alternative 14: 62.0% 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 60.8%

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

7. Simplified 60.8%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

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

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. div-inv N/A

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

   9. 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 60.9%

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

9. Applied egg-rr 60.9%

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

Alternative 15: 62.0% 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 60.8%

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

7. Simplified 60.8%

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

Alternative 16: 30.8% 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 60.8%

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

7. Simplified 60.8%

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

8. Taylor expanded in x around inf

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

10. Simplified 28.7%

    \[\leadsto \color{blue}{1} \]
11. Add Preprocessing

Developer Target 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024152 
(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)))))