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

Percentage Accurate: 99.7% → 99.7%

Time: 10.7s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Final simplification 99.7%

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

Alternative 2: 92.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+87}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 4600000000000:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;1 + \sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.4e+87)
   (* -0.3333333333333333 (/ y (sqrt x)))
   (if (<= y 4600000000000.0)
     (+ 1.0 (/ (/ -1.0 x) 9.0))
     (+ 1.0 (* (sqrt (/ 1.0 x)) (* y -0.3333333333333333))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.4e+87) {
		tmp = -0.3333333333333333 * (y / sqrt(x));
	} else if (y <= 4600000000000.0) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = 1.0 + (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 <= (-1.4d+87)) then
        tmp = (-0.3333333333333333d0) * (y / sqrt(x))
    else if (y <= 4600000000000.0d0) then
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    else
        tmp = 1.0d0 + (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 <= -1.4e+87) {
		tmp = -0.3333333333333333 * (y / Math.sqrt(x));
	} else if (y <= 4600000000000.0) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = 1.0 + (Math.sqrt((1.0 / x)) * (y * -0.3333333333333333));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.4e+87:
		tmp = -0.3333333333333333 * (y / math.sqrt(x))
	elif y <= 4600000000000.0:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	else:
		tmp = 1.0 + (math.sqrt((1.0 / x)) * (y * -0.3333333333333333))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.4e+87)
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	elseif (y <= 4600000000000.0)
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	else
		tmp = Float64(1.0 + 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 <= -1.4e+87)
		tmp = -0.3333333333333333 * (y / sqrt(x));
	elseif (y <= 4600000000000.0)
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	else
		tmp = 1.0 + (sqrt((1.0 / x)) * (y * -0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.4e+87], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4600000000000.0], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.40000000000000008e87

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 99.6%

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

   4. Taylor expanded in y around inf 93.3%

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

      1. *-commutative 93.3%

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

      2. *-commutative 93.3%

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

   6. Simplified 93.3%

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

      1. sqrt-div 93.2%

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

      2. metadata-eval 93.2%

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

      3. un-div-inv 93.4%

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

   8. Applied egg-rr 93.4%

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

   if -1.40000000000000008e87 < y < 4.6e12

   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. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.7%

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

      10. fma-neg 99.7%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      13. *-commutative 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      14. associate-/r* 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      15. distribute-neg-frac 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      16. metadata-eval 99.6%

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

      17. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 96.8%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 43.0%

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

      3. frac-times 43.0%

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

      4. metadata-eval 43.0%

         \[\leadsto 1 + \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} \]

      5. metadata-eval 43.0%

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

      6. frac-times 43.0%

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

      7. sqrt-unprod 43.1%

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

      8. add-sqr-sqrt 43.1%

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

      9. frac-2neg 43.1%

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

      10. metadata-eval 43.1%

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

      11. distribute-frac-neg2 43.1%

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

   7. Applied egg-rr 43.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 74.5%

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

      3. frac-times 74.6%

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

      4. metadata-eval 74.6%

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

      5. metadata-eval 74.6%

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

      6. frac-times 74.5%

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

      7. sqrt-unprod 96.6%

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

      8. add-sqr-sqrt 96.8%

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

      9. clear-num 96.8%

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

      10. div-inv 96.9%

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

      11. metadata-eval 96.9%

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

      12. inv-pow 96.9%

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

   9. Applied egg-rr 96.9%

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

       1. unpow-1 96.9%

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

       2. associate-/r* 96.9%

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

   11. Applied egg-rr 96.9%

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

   if 4.6e12 < 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. associate--l- 99.6%

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

      2. sub-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. distribute-neg-in 99.6%

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

      5. distribute-frac-neg 99.6%

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

      6. sub-neg 99.6%

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

      7. neg-mul-1 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-/l* 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.4%

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

      12. metadata-eval 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      13. *-commutative 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      14. associate-/r* 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      15. distribute-neg-frac 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 94.2%

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

      1. associate-*r* 94.0%

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

      2. *-commutative 94.0%

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

      3. associate-*l* 94.2%

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

   7. Simplified 94.2%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+87}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 4600000000000:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;1 + \sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+86}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 4600000000000:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;1 - 0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.45e+86)
   (* -0.3333333333333333 (/ y (sqrt x)))
   (if (<= y 4600000000000.0)
     (+ 1.0 (/ (/ -1.0 x) 9.0))
     (- 1.0 (* 0.3333333333333333 (* y (sqrt (/ 1.0 x))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.45e+86) {
		tmp = -0.3333333333333333 * (y / sqrt(x));
	} else if (y <= 4600000000000.0) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = 1.0 - (0.3333333333333333 * (y * sqrt((1.0 / x))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.45d+86)) then
        tmp = (-0.3333333333333333d0) * (y / sqrt(x))
    else if (y <= 4600000000000.0d0) then
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    else
        tmp = 1.0d0 - (0.3333333333333333d0 * (y * sqrt((1.0d0 / x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.45e+86) {
		tmp = -0.3333333333333333 * (y / Math.sqrt(x));
	} else if (y <= 4600000000000.0) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = 1.0 - (0.3333333333333333 * (y * Math.sqrt((1.0 / x))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.45e+86:
		tmp = -0.3333333333333333 * (y / math.sqrt(x))
	elif y <= 4600000000000.0:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	else:
		tmp = 1.0 - (0.3333333333333333 * (y * math.sqrt((1.0 / x))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.45e+86)
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	elseif (y <= 4600000000000.0)
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	else
		tmp = Float64(1.0 - Float64(0.3333333333333333 * Float64(y * sqrt(Float64(1.0 / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.45e+86)
		tmp = -0.3333333333333333 * (y / sqrt(x));
	elseif (y <= 4600000000000.0)
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	else
		tmp = 1.0 - (0.3333333333333333 * (y * sqrt((1.0 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.45e+86], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4600000000000.0], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.3333333333333333 * N[(y * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.44999999999999995e86

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 99.6%

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

   4. Taylor expanded in y around inf 93.3%

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

      1. *-commutative 93.3%

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

      2. *-commutative 93.3%

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

   6. Simplified 93.3%

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

      1. sqrt-div 93.2%

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

      2. metadata-eval 93.2%

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

      3. un-div-inv 93.4%

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

   8. Applied egg-rr 93.4%

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

   if -1.44999999999999995e86 < y < 4.6e12

   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. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.7%

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

      10. fma-neg 99.7%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      13. *-commutative 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      14. associate-/r* 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      15. distribute-neg-frac 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      16. metadata-eval 99.6%

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

      17. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 96.8%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 43.0%

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

      3. frac-times 43.0%

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

      4. metadata-eval 43.0%

         \[\leadsto 1 + \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} \]

      5. metadata-eval 43.0%

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

      6. frac-times 43.0%

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

      7. sqrt-unprod 43.1%

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

      8. add-sqr-sqrt 43.1%

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

      9. frac-2neg 43.1%

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

      10. metadata-eval 43.1%

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

      11. distribute-frac-neg2 43.1%

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

   7. Applied egg-rr 43.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 74.5%

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

      3. frac-times 74.6%

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

      4. metadata-eval 74.6%

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

      5. metadata-eval 74.6%

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

      6. frac-times 74.5%

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

      7. sqrt-unprod 96.6%

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

      8. add-sqr-sqrt 96.8%

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

      9. clear-num 96.8%

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

      10. div-inv 96.9%

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

      11. metadata-eval 96.9%

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

      12. inv-pow 96.9%

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

   9. Applied egg-rr 96.9%

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

       1. unpow-1 96.9%

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

       2. associate-/r* 96.9%

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

   11. Applied egg-rr 96.9%

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

   if 4.6e12 < 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 94.2%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+86}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 4600000000000:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;1 - 0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{\sqrt{x}}\\ \mathbf{if}\;y \leq -1.5 \cdot 10^{+86}:\\ \;\;\;\;-0.3333333333333333 \cdot t\_0\\ \mathbf{elif}\;y \leq 4600000000000:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;1 - t\_0 \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (sqrt x))))
   (if (<= y -1.5e+86)
     (* -0.3333333333333333 t_0)
     (if (<= y 4600000000000.0)
       (+ 1.0 (/ (/ -1.0 x) 9.0))
       (- 1.0 (* t_0 0.3333333333333333))))))

C

double code(double x, double y) {
	double t_0 = y / sqrt(x);
	double tmp;
	if (y <= -1.5e+86) {
		tmp = -0.3333333333333333 * t_0;
	} else if (y <= 4600000000000.0) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = 1.0 - (t_0 * 0.3333333333333333);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / sqrt(x)
    if (y <= (-1.5d+86)) then
        tmp = (-0.3333333333333333d0) * t_0
    else if (y <= 4600000000000.0d0) then
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    else
        tmp = 1.0d0 - (t_0 * 0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y / Math.sqrt(x);
	double tmp;
	if (y <= -1.5e+86) {
		tmp = -0.3333333333333333 * t_0;
	} else if (y <= 4600000000000.0) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = 1.0 - (t_0 * 0.3333333333333333);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / math.sqrt(x)
	tmp = 0
	if y <= -1.5e+86:
		tmp = -0.3333333333333333 * t_0
	elif y <= 4600000000000.0:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	else:
		tmp = 1.0 - (t_0 * 0.3333333333333333)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / sqrt(x))
	tmp = 0.0
	if (y <= -1.5e+86)
		tmp = Float64(-0.3333333333333333 * t_0);
	elseif (y <= 4600000000000.0)
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	else
		tmp = Float64(1.0 - Float64(t_0 * 0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / sqrt(x);
	tmp = 0.0;
	if (y <= -1.5e+86)
		tmp = -0.3333333333333333 * t_0;
	elseif (y <= 4600000000000.0)
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	else
		tmp = 1.0 - (t_0 * 0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.5e+86], N[(-0.3333333333333333 * t$95$0), $MachinePrecision], If[LessEqual[y, 4600000000000.0], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(t$95$0 * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.49999999999999988e86

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 99.6%

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

   4. Taylor expanded in y around inf 93.3%

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

      1. *-commutative 93.3%

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

      2. *-commutative 93.3%

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

   6. Simplified 93.3%

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

      1. sqrt-div 93.2%

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

      2. metadata-eval 93.2%

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

      3. un-div-inv 93.4%

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

   8. Applied egg-rr 93.4%

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

   if -1.49999999999999988e86 < y < 4.6e12

   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. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.7%

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

      10. fma-neg 99.7%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      13. *-commutative 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      14. associate-/r* 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      15. distribute-neg-frac 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      16. metadata-eval 99.6%

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

      17. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 96.8%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 43.0%

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

      3. frac-times 43.0%

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

      4. metadata-eval 43.0%

         \[\leadsto 1 + \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} \]

      5. metadata-eval 43.0%

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

      6. frac-times 43.0%

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

      7. sqrt-unprod 43.1%

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

      8. add-sqr-sqrt 43.1%

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

      9. frac-2neg 43.1%

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

      10. metadata-eval 43.1%

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

      11. distribute-frac-neg2 43.1%

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

   7. Applied egg-rr 43.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 74.5%

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

      3. frac-times 74.6%

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

      4. metadata-eval 74.6%

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

      5. metadata-eval 74.6%

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

      6. frac-times 74.5%

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

      7. sqrt-unprod 96.6%

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

      8. add-sqr-sqrt 96.8%

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

      9. clear-num 96.8%

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

      10. div-inv 96.9%

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

      11. metadata-eval 96.9%

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

      12. inv-pow 96.9%

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

   9. Applied egg-rr 96.9%

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

       1. unpow-1 96.9%

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

       2. associate-/r* 96.9%

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

   11. Applied egg-rr 96.9%

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

   if 4.6e12 < 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 94.2%

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

      1. pow1 94.2%

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

      2. associate-*r* 94.0%

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

      3. metadata-eval 94.0%

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

      4. sqrt-prod 94.1%

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

      5. div-inv 94.1%

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

      6. sqrt-div 93.9%

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

      7. metadata-eval 93.9%

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

   5. Applied egg-rr 93.9%

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

      1. unpow1 93.9%

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

      2. associate-*l/ 94.1%

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

      3. associate-*r/ 94.2%

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

   7. Simplified 94.2%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+86}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 4600000000000:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x}} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+86}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+114}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.7e+86)
   (* -0.3333333333333333 (/ y (sqrt x)))
   (if (<= y 3.1e+114)
     (+ 1.0 (/ (/ -1.0 x) 9.0))
     (* -0.3333333333333333 (* y (pow x -0.5))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.7e+86) {
		tmp = -0.3333333333333333 * (y / sqrt(x));
	} else if (y <= 3.1e+114) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = -0.3333333333333333 * (y * pow(x, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.7d+86)) then
        tmp = (-0.3333333333333333d0) * (y / sqrt(x))
    else if (y <= 3.1d+114) then
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    else
        tmp = (-0.3333333333333333d0) * (y * (x ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.7e+86) {
		tmp = -0.3333333333333333 * (y / Math.sqrt(x));
	} else if (y <= 3.1e+114) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = -0.3333333333333333 * (y * Math.pow(x, -0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.7e+86:
		tmp = -0.3333333333333333 * (y / math.sqrt(x))
	elif y <= 3.1e+114:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	else:
		tmp = -0.3333333333333333 * (y * math.pow(x, -0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.7e+86)
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	elseif (y <= 3.1e+114)
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	else
		tmp = Float64(-0.3333333333333333 * Float64(y * (x ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.7e+86)
		tmp = -0.3333333333333333 * (y / sqrt(x));
	elseif (y <= 3.1e+114)
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	else
		tmp = -0.3333333333333333 * (y * (x ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.7e+86], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.1e+114], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(y * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6999999999999999e86

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 99.6%

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

   4. Taylor expanded in y around inf 93.3%

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

      1. *-commutative 93.3%

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

      2. *-commutative 93.3%

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

   6. Simplified 93.3%

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

      1. sqrt-div 93.2%

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

      2. metadata-eval 93.2%

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

      3. un-div-inv 93.4%

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

   8. Applied egg-rr 93.4%

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

   if -1.6999999999999999e86 < y < 3.1e114

   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. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.7%

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

      10. fma-neg 99.7%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      13. *-commutative 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      14. associate-/r* 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      15. distribute-neg-frac 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      16. metadata-eval 99.6%

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

      17. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 92.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 43.3%

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

      3. frac-times 43.3%

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

      4. metadata-eval 43.3%

         \[\leadsto 1 + \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} \]

      5. metadata-eval 43.3%

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

      6. frac-times 43.3%

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

      7. sqrt-unprod 43.4%

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

      8. add-sqr-sqrt 43.4%

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

      9. frac-2neg 43.4%

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

      10. metadata-eval 43.4%

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

      11. distribute-frac-neg2 43.4%

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

   7. Applied egg-rr 43.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 71.3%

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

      3. frac-times 71.4%

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

      4. metadata-eval 71.4%

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

      5. metadata-eval 71.4%

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

      6. frac-times 71.3%

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

      7. sqrt-unprod 91.9%

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

      8. add-sqr-sqrt 92.1%

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

      9. clear-num 92.1%

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

      10. div-inv 92.2%

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

      11. metadata-eval 92.2%

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

      12. inv-pow 92.2%

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

   9. Applied egg-rr 92.2%

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

       1. unpow-1 92.2%

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

       2. associate-/r* 92.2%

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

   11. Applied egg-rr 92.2%

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

   if 3.1e114 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 99.5%

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

   4. Taylor expanded in y around inf 98.2%

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

      1. *-commutative 98.2%

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

      2. *-commutative 98.2%

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

   6. Simplified 98.2%

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

      1. *-un-lft-identity 98.2%

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

      2. inv-pow 98.2%

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

      3. sqrt-pow1 98.2%

         \[\leadsto \left(y \cdot \left(1 \cdot \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)\right) \cdot -0.3333333333333333 \]

      4. metadata-eval 98.2%

         \[\leadsto \left(y \cdot \left(1 \cdot {x}^{\color{blue}{-0.5}}\right)\right) \cdot -0.3333333333333333 \]

   8. Applied egg-rr 98.2%

      \[\leadsto \left(y \cdot \color{blue}{\left(1 \cdot {x}^{-0.5}\right)}\right) \cdot -0.3333333333333333 \]
   9. Step-by-step derivation

      1. *-lft-identity 98.2%

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

   10. Simplified 98.2%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+86}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+114}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+89}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+114}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.4e+89)
   (* -0.3333333333333333 (/ y (sqrt x)))
   (if (<= y 3.1e+114)
     (+ 1.0 (/ (/ -1.0 x) 9.0))
     (* (pow x -0.5) (* y -0.3333333333333333)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.4e+89) {
		tmp = -0.3333333333333333 * (y / sqrt(x));
	} else if (y <= 3.1e+114) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = pow(x, -0.5) * (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 <= (-1.4d+89)) then
        tmp = (-0.3333333333333333d0) * (y / sqrt(x))
    else if (y <= 3.1d+114) then
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    else
        tmp = (x ** (-0.5d0)) * (y * (-0.3333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.4e+89) {
		tmp = -0.3333333333333333 * (y / Math.sqrt(x));
	} else if (y <= 3.1e+114) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = Math.pow(x, -0.5) * (y * -0.3333333333333333);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.4e+89:
		tmp = -0.3333333333333333 * (y / math.sqrt(x))
	elif y <= 3.1e+114:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	else:
		tmp = math.pow(x, -0.5) * (y * -0.3333333333333333)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.4e+89)
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	elseif (y <= 3.1e+114)
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	else
		tmp = Float64((x ^ -0.5) * Float64(y * -0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.4e+89)
		tmp = -0.3333333333333333 * (y / sqrt(x));
	elseif (y <= 3.1e+114)
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	else
		tmp = (x ^ -0.5) * (y * -0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.4e+89], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.1e+114], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.3999999999999999e89

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 99.6%

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

   4. Taylor expanded in y around inf 93.3%

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

      1. *-commutative 93.3%

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

      2. *-commutative 93.3%

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

   6. Simplified 93.3%

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

      1. sqrt-div 93.2%

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

      2. metadata-eval 93.2%

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

      3. un-div-inv 93.4%

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

   8. Applied egg-rr 93.4%

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

   if -1.3999999999999999e89 < y < 3.1e114

   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. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.7%

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

      10. fma-neg 99.7%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      13. *-commutative 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      14. associate-/r* 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      15. distribute-neg-frac 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      16. metadata-eval 99.6%

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

      17. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 92.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 43.3%

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

      3. frac-times 43.3%

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

      4. metadata-eval 43.3%

         \[\leadsto 1 + \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} \]

      5. metadata-eval 43.3%

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

      6. frac-times 43.3%

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

      7. sqrt-unprod 43.4%

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

      8. add-sqr-sqrt 43.4%

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

      9. frac-2neg 43.4%

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

      10. metadata-eval 43.4%

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

      11. distribute-frac-neg2 43.4%

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

   7. Applied egg-rr 43.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 71.3%

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

      3. frac-times 71.4%

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

      4. metadata-eval 71.4%

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

      5. metadata-eval 71.4%

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

      6. frac-times 71.3%

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

      7. sqrt-unprod 91.9%

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

      8. add-sqr-sqrt 92.1%

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

      9. clear-num 92.1%

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

      10. div-inv 92.2%

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

      11. metadata-eval 92.2%

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

      12. inv-pow 92.2%

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

   9. Applied egg-rr 92.2%

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

       1. unpow-1 92.2%

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

       2. associate-/r* 92.2%

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

   11. Applied egg-rr 92.2%

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

   if 3.1e114 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 99.5%

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

   4. Taylor expanded in y around inf 98.2%

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

      1. *-commutative 98.2%

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

      2. *-commutative 98.2%

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

   6. Simplified 98.2%

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

      1. pow1 98.2%

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

      2. *-commutative 98.2%

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

      3. sqrt-div 98.2%

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

      4. metadata-eval 98.2%

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

      5. un-div-inv 98.2%

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

   8. Applied egg-rr 98.2%

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

      1. unpow1 98.2%

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

      2. associate-*r/ 98.1%

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

   10. Simplified 98.1%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot y}{\sqrt{x}}} \]
   11. Step-by-step derivation

       1. clear-num 98.1%

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

       2. associate-/r/ 98.1%

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

       3. pow1/2 98.1%

          \[\leadsto \frac{1}{\color{blue}{{x}^{0.5}}} \cdot \left(-0.3333333333333333 \cdot y\right) \]

       4. pow-flip 98.2%

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

       5. metadata-eval 98.2%

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

       6. *-commutative 98.2%

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

   12. Applied egg-rr 98.2%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+89}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+114}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{+88} \lor \neg \left(y \leq 3.1 \cdot 10^{+114}\right):\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.32e+88) (not (<= y 3.1e+114)))
   (* y (/ -0.3333333333333333 (sqrt x)))
   (+ 1.0 (/ (/ -1.0 x) 9.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.32e+88) || !(y <= 3.1e+114)) {
		tmp = y * (-0.3333333333333333 / sqrt(x));
	} else {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.32d+88)) .or. (.not. (y <= 3.1d+114))) then
        tmp = y * ((-0.3333333333333333d0) / sqrt(x))
    else
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.32e+88) || !(y <= 3.1e+114)) {
		tmp = y * (-0.3333333333333333 / Math.sqrt(x));
	} else {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.32e+88) or not (y <= 3.1e+114):
		tmp = y * (-0.3333333333333333 / math.sqrt(x))
	else:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.32e+88) || !(y <= 3.1e+114))
		tmp = Float64(y * Float64(-0.3333333333333333 / sqrt(x)));
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.32e+88) || ~((y <= 3.1e+114)))
		tmp = y * (-0.3333333333333333 / sqrt(x));
	else
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.32e+88], N[Not[LessEqual[y, 3.1e+114]], $MachinePrecision]], N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3200000000000001e88 or 3.1e114 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 99.5%

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

   4. Taylor expanded in y around inf 95.3%

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

      1. *-commutative 95.3%

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

      2. *-commutative 95.3%

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

   6. Simplified 95.3%

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

      1. pow1 95.3%

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

      2. *-commutative 95.3%

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

      3. sqrt-div 95.3%

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

      4. metadata-eval 95.3%

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

      5. un-div-inv 95.4%

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

   8. Applied egg-rr 95.4%

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

      1. unpow1 95.4%

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

      2. associate-*r/ 95.3%

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

   10. Simplified 95.3%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot y}{\sqrt{x}}} \]
   11. Step-by-step derivation

       1. *-commutative 95.3%

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

       2. associate-/l* 95.2%

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

   12. Applied egg-rr 95.2%

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

   if -1.3200000000000001e88 < y < 3.1e114

   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. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.7%

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

      10. fma-neg 99.7%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      13. *-commutative 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      14. associate-/r* 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      15. distribute-neg-frac 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      16. metadata-eval 99.6%

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

      17. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 92.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 43.3%

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

      3. frac-times 43.3%

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

      4. metadata-eval 43.3%

         \[\leadsto 1 + \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} \]

      5. metadata-eval 43.3%

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

      6. frac-times 43.3%

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

      7. sqrt-unprod 43.4%

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

      8. add-sqr-sqrt 43.4%

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

      9. frac-2neg 43.4%

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

      10. metadata-eval 43.4%

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

      11. distribute-frac-neg2 43.4%

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

   7. Applied egg-rr 43.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 71.3%

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

      3. frac-times 71.4%

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

      4. metadata-eval 71.4%

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

      5. metadata-eval 71.4%

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

      6. frac-times 71.3%

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

      7. sqrt-unprod 91.9%

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

      8. add-sqr-sqrt 92.1%

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

      9. clear-num 92.1%

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

      10. div-inv 92.2%

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

      11. metadata-eval 92.2%

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

      12. inv-pow 92.2%

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

   9. Applied egg-rr 92.2%

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

       1. unpow-1 92.2%

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

       2. associate-/r* 92.2%

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

   11. Applied egg-rr 92.2%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{+88} \lor \neg \left(y \leq 3.1 \cdot 10^{+114}\right):\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+86} \lor \neg \left(y \leq 3.1 \cdot 10^{+114}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.5e+86) (not (<= y 3.1e+114)))
   (* -0.3333333333333333 (/ y (sqrt x)))
   (+ 1.0 (/ (/ -1.0 x) 9.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.5e+86) || !(y <= 3.1e+114)) {
		tmp = -0.3333333333333333 * (y / sqrt(x));
	} else {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.5d+86)) .or. (.not. (y <= 3.1d+114))) then
        tmp = (-0.3333333333333333d0) * (y / sqrt(x))
    else
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.5e+86) || !(y <= 3.1e+114)) {
		tmp = -0.3333333333333333 * (y / Math.sqrt(x));
	} else {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.5e+86) or not (y <= 3.1e+114):
		tmp = -0.3333333333333333 * (y / math.sqrt(x))
	else:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.5e+86) || !(y <= 3.1e+114))
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.5e+86) || ~((y <= 3.1e+114)))
		tmp = -0.3333333333333333 * (y / sqrt(x));
	else
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.5e+86], N[Not[LessEqual[y, 3.1e+114]], $MachinePrecision]], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.49999999999999988e86 or 3.1e114 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 99.5%

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

   4. Taylor expanded in y around inf 95.3%

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

      1. *-commutative 95.3%

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

      2. *-commutative 95.3%

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

   6. Simplified 95.3%

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

      1. sqrt-div 95.3%

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

      2. metadata-eval 95.3%

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

      3. un-div-inv 95.4%

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

   8. Applied egg-rr 95.4%

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

   if -1.49999999999999988e86 < y < 3.1e114

   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. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.7%

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

      10. fma-neg 99.7%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      13. *-commutative 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      14. associate-/r* 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      15. distribute-neg-frac 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      16. metadata-eval 99.6%

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

      17. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 92.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 43.3%

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

      3. frac-times 43.3%

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

      4. metadata-eval 43.3%

         \[\leadsto 1 + \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} \]

      5. metadata-eval 43.3%

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

      6. frac-times 43.3%

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

      7. sqrt-unprod 43.4%

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

      8. add-sqr-sqrt 43.4%

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

      9. frac-2neg 43.4%

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

      10. metadata-eval 43.4%

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

      11. distribute-frac-neg2 43.4%

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

   7. Applied egg-rr 43.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 71.3%

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

      3. frac-times 71.4%

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

      4. metadata-eval 71.4%

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

      5. metadata-eval 71.4%

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

      6. frac-times 71.3%

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

      7. sqrt-unprod 91.9%

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

      8. add-sqr-sqrt 92.1%

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

      9. clear-num 92.1%

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

      10. div-inv 92.2%

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

      11. metadata-eval 92.2%

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

      12. inv-pow 92.2%

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

   9. Applied egg-rr 92.2%

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

       1. unpow-1 92.2%

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

       2. associate-/r* 92.2%

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

   11. Applied egg-rr 92.2%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+86} \lor \neg \left(y \leq 3.1 \cdot 10^{+114}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00072:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(y \cdot \sqrt{x}\right) - 0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x}} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.00072)
   (/ (- (* -0.3333333333333333 (* y (sqrt x))) 0.1111111111111111) x)
   (- 1.0 (* (/ y (sqrt x)) 0.3333333333333333))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.00072) {
		tmp = ((-0.3333333333333333 * (y * sqrt(x))) - 0.1111111111111111) / x;
	} 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 (x <= 0.00072d0) then
        tmp = (((-0.3333333333333333d0) * (y * sqrt(x))) - 0.1111111111111111d0) / x
    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 (x <= 0.00072) {
		tmp = ((-0.3333333333333333 * (y * Math.sqrt(x))) - 0.1111111111111111) / x;
	} else {
		tmp = 1.0 - ((y / Math.sqrt(x)) * 0.3333333333333333);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.00072:
		tmp = ((-0.3333333333333333 * (y * math.sqrt(x))) - 0.1111111111111111) / x
	else:
		tmp = 1.0 - ((y / math.sqrt(x)) * 0.3333333333333333)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.00072)
		tmp = Float64(Float64(Float64(-0.3333333333333333 * Float64(y * sqrt(x))) - 0.1111111111111111) / x);
	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 (x <= 0.00072)
		tmp = ((-0.3333333333333333 * (y * sqrt(x))) - 0.1111111111111111) / x;
	else
		tmp = 1.0 - ((y / sqrt(x)) * 0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.00072], N[(N[(N[(-0.3333333333333333 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.1111111111111111), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.20000000000000045e-4

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-/r* 99.4%

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

      4. metadata-eval 99.4%

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

      5. distribute-frac-neg 99.4%

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

      6. neg-mul-1 99.4%

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

      7. times-frac 99.4%

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

      8. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 99.4%

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

   6. Taylor expanded in x around 0 99.0%

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

   if 7.20000000000000045e-4 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 98.5%

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

      1. pow1 98.5%

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

      2. associate-*r* 98.4%

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

      3. metadata-eval 98.4%

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

      4. sqrt-prod 98.5%

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

      5. div-inv 98.5%

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

      6. sqrt-div 98.4%

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

      7. metadata-eval 98.4%

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

   5. Applied egg-rr 98.4%

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

      1. unpow1 98.4%

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

      2. associate-*l/ 98.4%

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

      3. associate-*r/ 98.5%

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

   7. Simplified 98.5%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00072:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(y \cdot \sqrt{x}\right) - 0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x}} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-/r* 99.6%

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

   4. metadata-eval 99.6%

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

   5. distribute-frac-neg 99.6%

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

   6. neg-mul-1 99.6%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 11: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) - (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 - (0.1111111111111111d0 / x)) - (y / (3.0d0 * sqrt(x)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(0.1111111111111111 / x), $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. Taylor expanded in x around 0 99.6%

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

4. Final simplification 99.6%

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

Alternative 12: 65.9% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{y} \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.5e+154)
   (+ 1.0 (/ (/ -1.0 x) 9.0))
   (- 1.0 (* (/ 0.1111111111111111 y) (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.5e+154) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = 1.0 - ((0.1111111111111111 / y) * (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.5d+154) then
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    else
        tmp = 1.0d0 - ((0.1111111111111111d0 / y) * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.5e+154) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = 1.0 - ((0.1111111111111111 / y) * (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.5e+154:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	else:
		tmp = 1.0 - ((0.1111111111111111 / y) * (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.5e+154)
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	else
		tmp = Float64(1.0 - Float64(Float64(0.1111111111111111 / y) * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.5e+154)
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	else
		tmp = 1.0 - ((0.1111111111111111 / y) * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.5e+154], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(0.1111111111111111 / y), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.50000000000000013e154

   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. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.6%

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

      10. fma-neg 99.6%

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

      11. associate-/r* 99.6%

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

      12. metadata-eval 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      13. *-commutative 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      14. associate-/r* 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      15. distribute-neg-frac 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      16. metadata-eval 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 74.2%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 36.0%

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

      3. frac-times 36.0%

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

      4. metadata-eval 36.0%

         \[\leadsto 1 + \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} \]

      5. metadata-eval 36.0%

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

      6. frac-times 36.0%

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

      7. sqrt-unprod 35.2%

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

      8. add-sqr-sqrt 35.2%

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

      9. frac-2neg 35.2%

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

      10. metadata-eval 35.2%

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

      11. distribute-frac-neg2 35.2%

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

   7. Applied egg-rr 35.2%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 57.1%

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

      3. frac-times 57.1%

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

      4. metadata-eval 57.1%

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

      5. metadata-eval 57.1%

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

      6. frac-times 57.1%

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

      7. sqrt-unprod 74.1%

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

      8. add-sqr-sqrt 74.2%

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

      9. clear-num 74.2%

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

      10. div-inv 74.3%

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

      11. metadata-eval 74.3%

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

      12. inv-pow 74.3%

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

   9. Applied egg-rr 74.3%

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

       1. unpow-1 74.3%

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

       2. associate-/r* 74.3%

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

   11. Applied egg-rr 74.3%

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

   if 1.50000000000000013e154 < 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 y around inf 99.3%

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

   4. Taylor expanded in x around 0 87.1%

      \[\leadsto \color{blue}{\frac{x + -1 \cdot \left(y \cdot \left(0.3333333333333333 \cdot \sqrt{x} + 0.1111111111111111 \cdot \frac{1}{y}\right)\right)}{x}} \]
   5. Step-by-step derivation

      1. mul-1-neg 87.1%

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

      2. unsub-neg 87.1%

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

      3. div-sub 87.1%

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

      4. *-inverses 87.1%

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

      5. *-commutative 87.1%

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

      6. associate-/l* 91.8%

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

      7. fma-define 91.8%

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

      8. associate-*r/ 91.8%

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

      9. metadata-eval 91.8%

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

   6. Simplified 91.8%

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

   7. Taylor expanded in y around 0 37.0%

      \[\leadsto 1 - \color{blue}{\frac{0.1111111111111111}{y}} \cdot \frac{y}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{y} \cdot \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.0% 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. associate--l- 99.7%

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

   2. sub-neg 99.7%

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

   3. +-commutative 99.7%

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

   4. distribute-neg-in 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. sub-neg 99.7%

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

   7. neg-mul-1 99.7%

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

   8. *-commutative 99.7%

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

   9. associate-/l* 99.6%

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

   10. fma-neg 99.6%

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

   11. associate-/r* 99.6%

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

   12. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

   13. *-commutative 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

   14. associate-/r* 99.5%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

   15. distribute-neg-frac 99.5%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

   16. metadata-eval 99.5%

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

   17. metadata-eval 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in y around 0 67.9%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 32.8%

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

   3. frac-times 32.8%

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

   4. metadata-eval 32.8%

      \[\leadsto 1 + \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} \]

   5. metadata-eval 32.8%

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

   6. frac-times 32.8%

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

   7. sqrt-unprod 32.1%

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

   8. add-sqr-sqrt 32.1%

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

   9. frac-2neg 32.1%

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

   10. metadata-eval 32.1%

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

   11. distribute-frac-neg2 32.1%

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

7. Applied egg-rr 32.1%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 53.8%

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

   3. frac-times 53.8%

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

   4. metadata-eval 53.8%

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

   5. metadata-eval 53.8%

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

   6. frac-times 53.8%

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

   7. sqrt-unprod 67.8%

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

   8. add-sqr-sqrt 67.9%

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

   9. clear-num 67.9%

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

   10. div-inv 67.9%

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

   11. metadata-eval 67.9%

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

   12. inv-pow 67.9%

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

9. Applied egg-rr 67.9%

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

    1. unpow-1 67.9%

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

11. Applied egg-rr 67.9%

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

12. Final simplification 67.9%

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

Alternative 14: 63.0% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate--l- 99.7%

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

   2. sub-neg 99.7%

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

   3. +-commutative 99.7%

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

   4. distribute-neg-in 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. sub-neg 99.7%

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

   7. neg-mul-1 99.7%

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

   8. *-commutative 99.7%

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

   9. associate-/l* 99.6%

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

   10. fma-neg 99.6%

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

   11. associate-/r* 99.6%

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

   12. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

   13. *-commutative 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

   14. associate-/r* 99.5%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

   15. distribute-neg-frac 99.5%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

   16. metadata-eval 99.5%

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

   17. metadata-eval 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in y around 0 67.9%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 32.8%

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

   3. frac-times 32.8%

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

   4. metadata-eval 32.8%

      \[\leadsto 1 + \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} \]

   5. metadata-eval 32.8%

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

   6. frac-times 32.8%

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

   7. sqrt-unprod 32.1%

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

   8. add-sqr-sqrt 32.1%

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

   9. frac-2neg 32.1%

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

   10. metadata-eval 32.1%

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

   11. distribute-frac-neg2 32.1%

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

7. Applied egg-rr 32.1%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 53.8%

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

   3. frac-times 53.8%

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

   4. metadata-eval 53.8%

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

   5. metadata-eval 53.8%

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

   6. frac-times 53.8%

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

   7. sqrt-unprod 67.8%

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

   8. add-sqr-sqrt 67.9%

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

   9. clear-num 67.9%

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

   10. div-inv 67.9%

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

   11. metadata-eval 67.9%

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

   12. inv-pow 67.9%

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

9. Applied egg-rr 67.9%

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

    1. unpow-1 67.9%

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

    2. associate-/r* 67.9%

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

11. Applied egg-rr 67.9%

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

12. Final simplification 67.9%

    \[\leadsto 1 + \frac{\frac{-1}{x}}{9} \]
13. Add Preprocessing

Alternative 15: 62.9% 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. associate--l- 99.7%

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

   2. sub-neg 99.7%

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

   3. +-commutative 99.7%

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

   4. distribute-neg-in 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. sub-neg 99.7%

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

   7. neg-mul-1 99.7%

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

   8. *-commutative 99.7%

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

   9. associate-/l* 99.6%

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

   10. fma-neg 99.6%

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

   11. associate-/r* 99.6%

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

   12. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

   13. *-commutative 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

   14. associate-/r* 99.5%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

   15. distribute-neg-frac 99.5%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

   16. metadata-eval 99.5%

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

   17. metadata-eval 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in y around 0 67.9%

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

6. Final simplification 67.9%

   \[\leadsto 1 + \frac{-0.1111111111111111}{x} \]
7. Add Preprocessing

Alternative 16: 32.0% accurate, 28.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.1111111111111111}{-x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 0.1111111111111111 (- x)))

C

double code(double x, double y) {
	return 0.1111111111111111 / -x;
}

Fortran

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

Java

public static double code(double x, double y) {
	return 0.1111111111111111 / -x;
}

Python

def code(x, y):
	return 0.1111111111111111 / -x

Julia

function code(x, y)
	return Float64(0.1111111111111111 / Float64(-x))
end

MATLAB

function tmp = code(x, y)
	tmp = 0.1111111111111111 / -x;
end

Wolfram

code[x_, y_] := N[(0.1111111111111111 / (-x)), $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. Taylor expanded in x around 0 63.3%

   \[\leadsto \color{blue}{-1 \cdot \frac{0.1111111111111111 + 0.3333333333333333 \cdot \left(\sqrt{x} \cdot y\right)}{x}} \]
4. Step-by-step derivation

   1. mul-1-neg 63.3%

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

   2. *-commutative 63.3%

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

5. Simplified 63.3%

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

6. Taylor expanded in y around 0 36.9%

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

7. Final simplification 36.9%

   \[\leadsto \frac{0.1111111111111111}{-x} \]
8. Add Preprocessing

Developer target: 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 2024066 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :alt
  (- (- 1.0 (/ (/ 1.0 x) 9.0)) (/ y (* 3.0 (sqrt x))))

  (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))