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

Percentage Accurate: 99.7% → 99.7%

Time: 10.3s

Alternatives: 15

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: 95.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+56} \lor \neg \left(y \leq 3 \cdot 10^{+63}\right):\\ \;\;\;\;1 - 0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.5e+56) (not (<= y 3e+63)))
   (- 1.0 (* 0.3333333333333333 (* y (sqrt (/ 1.0 x)))))
   (- 1.0 (pow (* x 9.0) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.5e+56) || !(y <= 3e+63)) {
		tmp = 1.0 - (0.3333333333333333 * (y * sqrt((1.0 / x))));
	} else {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-4.5d+56)) .or. (.not. (y <= 3d+63))) then
        tmp = 1.0d0 - (0.3333333333333333d0 * (y * sqrt((1.0d0 / x))))
    else
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -4.5e+56) || !(y <= 3e+63)) {
		tmp = 1.0 - (0.3333333333333333 * (y * Math.sqrt((1.0 / x))));
	} else {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.5e+56) or not (y <= 3e+63):
		tmp = 1.0 - (0.3333333333333333 * (y * math.sqrt((1.0 / x))))
	else:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.5e+56) || !(y <= 3e+63))
		tmp = Float64(1.0 - Float64(0.3333333333333333 * Float64(y * sqrt(Float64(1.0 / x)))));
	else
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4.5e+56) || ~((y <= 3e+63)))
		tmp = 1.0 - (0.3333333333333333 * (y * sqrt((1.0 / x))));
	else
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.5e+56], N[Not[LessEqual[y, 3e+63]], $MachinePrecision]], N[(1.0 - N[(0.3333333333333333 * N[(y * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.5000000000000003e56 or 2.99999999999999999e63 < 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 92.5%

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

   if -4.5000000000000003e56 < y < 2.99999999999999999e63

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fmm-def 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 97.3%

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

      1. associate-*r/ 97.4%

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

      2. metadata-eval 97.4%

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

   7. Simplified 97.4%

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

      1. metadata-eval 97.4%

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

      2. associate-/r* 97.4%

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

      3. *-commutative 97.4%

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

      4. inv-pow 97.4%

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

   9. Applied egg-rr 97.4%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+56} \lor \neg \left(y \leq 3 \cdot 10^{+63}\right):\\ \;\;\;\;1 - 0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+58} \lor \neg \left(y \leq 6.2 \cdot 10^{+62}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.45e+58) (not (<= y 6.2e+62)))
   (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))
   (- 1.0 (pow (* x 9.0) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.45e+58) || !(y <= 6.2e+62)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	} else {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.45d+58)) .or. (.not. (y <= 6.2d+62))) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    else
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.45e+58) || !(y <= 6.2e+62)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.45e+58) or not (y <= 6.2e+62):
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	else:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.45e+58) || !(y <= 6.2e+62))
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	else
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.45e+58) || ~((y <= 6.2e+62)))
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	else
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.45e+58], N[Not[LessEqual[y, 6.2e+62]], $MachinePrecision]], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.45000000000000001e58 or 6.20000000000000029e62 < y

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. *-commutative 99.6%

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

      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 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf 92.5%

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

   if -1.45000000000000001e58 < y < 6.20000000000000029e62

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fmm-def 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 97.3%

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

      1. associate-*r/ 97.4%

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

      2. metadata-eval 97.4%

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

   7. Simplified 97.4%

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

      1. metadata-eval 97.4%

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

      2. associate-/r* 97.4%

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

      3. *-commutative 97.4%

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

      4. inv-pow 97.4%

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

   9. Applied egg-rr 97.4%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+58} \lor \neg \left(y \leq 6.2 \cdot 10^{+62}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+59} \lor \neg \left(y \leq 4.9 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.1e+59) (not (<= y 4.9e+67)))
   (/ y (* (sqrt x) -3.0))
   (- 1.0 (pow (* x 9.0) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.1e+59) || !(y <= 4.9e+67)) {
		tmp = y / (sqrt(x) * -3.0);
	} else {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-3.1d+59)) .or. (.not. (y <= 4.9d+67))) then
        tmp = y / (sqrt(x) * (-3.0d0))
    else
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -3.1e+59) || !(y <= 4.9e+67)) {
		tmp = y / (Math.sqrt(x) * -3.0);
	} else {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.1e+59) or not (y <= 4.9e+67):
		tmp = y / (math.sqrt(x) * -3.0)
	else:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.1e+59) || !(y <= 4.9e+67))
		tmp = Float64(y / Float64(sqrt(x) * -3.0));
	else
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -3.1e+59) || ~((y <= 4.9e+67)))
		tmp = y / (sqrt(x) * -3.0);
	else
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.1e+59], N[Not[LessEqual[y, 4.9e+67]], $MachinePrecision]], N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.10000000000000015e59 or 4.8999999999999999e67 < 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* 99.6%

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

      10. fmm-def 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* 99.7%

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

      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* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 87.1%

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

      1. *-commutative 87.1%

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

      2. associate-*l* 87.0%

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

      3. *-commutative 87.0%

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

   7. Simplified 87.0%

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

      1. *-commutative 87.0%

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

      2. associate-*r* 87.1%

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

      3. metadata-eval 87.1%

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

      4. distribute-rgt-neg-in 87.1%

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

      5. associate-*l* 87.0%

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

      6. sqrt-div 87.0%

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

      7. metadata-eval 87.0%

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

      8. metadata-eval 87.0%

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

      9. div-inv 87.0%

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

      10. times-frac 87.2%

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

      11. *-un-lft-identity 87.2%

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

      12. *-commutative 87.2%

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

      13. distribute-neg-frac2 87.2%

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

      14. *-commutative 87.2%

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

      15. distribute-rgt-neg-in 87.2%

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

      16. metadata-eval 87.2%

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

   9. Applied egg-rr 87.2%

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

   if -3.10000000000000015e59 < y < 4.8999999999999999e67

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fmm-def 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 97.3%

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

      1. associate-*r/ 97.4%

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

      2. metadata-eval 97.4%

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

   7. Simplified 97.4%

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

      1. metadata-eval 97.4%

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

      2. associate-/r* 97.4%

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

      3. *-commutative 97.4%

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

      4. inv-pow 97.4%

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

   9. Applied egg-rr 97.4%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+59} \lor \neg \left(y \leq 4.9 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+61} \lor \neg \left(y \leq 3.6 \cdot 10^{+80}\right):\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.45e+61) (not (<= y 3.6e+80)))
   (/ y (* (sqrt x) -3.0))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.45e+61) || !(y <= 3.6e+80)) {
		tmp = y / (sqrt(x) * -3.0);
	} 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 <= (-1.45d+61)) .or. (.not. (y <= 3.6d+80))) then
        tmp = y / (sqrt(x) * (-3.0d0))
    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 <= -1.45e+61) || !(y <= 3.6e+80)) {
		tmp = y / (Math.sqrt(x) * -3.0);
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.45e+61) or not (y <= 3.6e+80):
		tmp = y / (math.sqrt(x) * -3.0)
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.45e+61) || !(y <= 3.6e+80))
		tmp = Float64(y / Float64(sqrt(x) * -3.0));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.45e+61) || ~((y <= 3.6e+80)))
		tmp = y / (sqrt(x) * -3.0);
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.45e+61], N[Not[LessEqual[y, 3.6e+80]], $MachinePrecision]], N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.45e61 or 3.59999999999999995e80 < 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* 99.6%

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

      10. fmm-def 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* 99.7%

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

      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* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 87.1%

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

      1. *-commutative 87.1%

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

      2. associate-*l* 87.0%

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

      3. *-commutative 87.0%

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

   7. Simplified 87.0%

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

      1. *-commutative 87.0%

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

      2. associate-*r* 87.1%

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

      3. metadata-eval 87.1%

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

      4. distribute-rgt-neg-in 87.1%

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

      5. associate-*l* 87.0%

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

      6. sqrt-div 87.0%

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

      7. metadata-eval 87.0%

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

      8. metadata-eval 87.0%

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

      9. div-inv 87.0%

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

      10. times-frac 87.2%

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

      11. *-un-lft-identity 87.2%

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

      12. *-commutative 87.2%

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

      13. distribute-neg-frac2 87.2%

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

      14. *-commutative 87.2%

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

      15. distribute-rgt-neg-in 87.2%

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

      16. metadata-eval 87.2%

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

   9. Applied egg-rr 87.2%

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

   if -1.45e61 < y < 3.59999999999999995e80

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fmm-def 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 97.3%

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

      1. associate-*r/ 97.4%

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

      2. metadata-eval 97.4%

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

   7. Simplified 97.4%

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

      1. expm1-log1p-u 93.1%

         \[\leadsto 1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.1111111111111111}{x}\right)\right)} \]

      2. expm1-undefine 93.0%

         \[\leadsto 1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.1111111111111111}{x}\right)} - 1\right)} \]

      3. log1p-undefine 93.0%

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

      4. add-exp-log 97.3%

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

      5. add-sqr-sqrt 97.1%

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

      6. sqrt-unprod 75.0%

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

      7. frac-times 75.1%

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

      8. metadata-eval 75.1%

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

      9. metadata-eval 75.1%

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

      10. frac-times 75.0%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 43.3%

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

   9. Applied egg-rr 43.3%

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

       1. +-commutative 43.3%

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

       2. associate--l+ 43.3%

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

       3. metadata-eval 43.3%

          \[\leadsto 1 - \left(\frac{-0.1111111111111111}{x} + \color{blue}{0}\right) \]

   11. Simplified 43.3%

       \[\leadsto 1 - \color{blue}{\left(\frac{-0.1111111111111111}{x} + 0\right)} \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto 1 - \color{blue}{\sqrt{\frac{-0.1111111111111111}{x} + 0} \cdot \sqrt{\frac{-0.1111111111111111}{x} + 0}} \]

       2. sqrt-unprod 75.1%

          \[\leadsto 1 - \color{blue}{\sqrt{\left(\frac{-0.1111111111111111}{x} + 0\right) \cdot \left(\frac{-0.1111111111111111}{x} + 0\right)}} \]

       3. +-rgt-identity 75.1%

          \[\leadsto 1 - \sqrt{\left(\frac{-0.1111111111111111}{x} + 0\right) \cdot \color{blue}{\frac{-0.1111111111111111}{x}}} \]

       4. clear-num 75.1%

          \[\leadsto 1 - \sqrt{\left(\frac{-0.1111111111111111}{x} + 0\right) \cdot \color{blue}{\frac{1}{\frac{x}{-0.1111111111111111}}}} \]

       5. un-div-inv 75.1%

          \[\leadsto 1 - \sqrt{\color{blue}{\frac{\frac{-0.1111111111111111}{x} + 0}{\frac{x}{-0.1111111111111111}}}} \]

       6. +-rgt-identity 75.1%

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

       7. clear-num 75.0%

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

       8. div-inv 75.1%

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

       9. metadata-eval 75.1%

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

       10. metadata-eval 75.1%

           \[\leadsto 1 - \sqrt{\frac{\frac{1}{x \cdot \color{blue}{\left(-9\right)}}}{\frac{x}{-0.1111111111111111}}} \]

       11. distribute-rgt-neg-in 75.1%

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

       12. add-sqr-sqrt 75.0%

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

       13. pow2 75.0%

           \[\leadsto 1 - \sqrt{\frac{\frac{1}{-\color{blue}{{\left(\sqrt{x \cdot 9}\right)}^{2}}}}{\frac{x}{-0.1111111111111111}}} \]

       14. sqrt-prod 75.0%

           \[\leadsto 1 - \sqrt{\frac{\frac{1}{-{\color{blue}{\left(\sqrt{x} \cdot \sqrt{9}\right)}}^{2}}}{\frac{x}{-0.1111111111111111}}} \]

       15. metadata-eval 75.0%

           \[\leadsto 1 - \sqrt{\frac{\frac{1}{-{\left(\sqrt{x} \cdot \color{blue}{3}\right)}^{2}}}{\frac{x}{-0.1111111111111111}}} \]

       16. distribute-neg-frac2 75.0%

           \[\leadsto 1 - \sqrt{\frac{\color{blue}{-\frac{1}{{\left(\sqrt{x} \cdot 3\right)}^{2}}}}{\frac{x}{-0.1111111111111111}}} \]

       17. neg-mul-1 75.0%

           \[\leadsto 1 - \sqrt{\frac{\color{blue}{-1 \cdot \frac{1}{{\left(\sqrt{x} \cdot 3\right)}^{2}}}}{\frac{x}{-0.1111111111111111}}} \]

       18. div-inv 75.0%

           \[\leadsto 1 - \sqrt{\frac{-1 \cdot \frac{1}{{\left(\sqrt{x} \cdot 3\right)}^{2}}}{\color{blue}{x \cdot \frac{1}{-0.1111111111111111}}}} \]

       19. metadata-eval 75.0%

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

       20. metadata-eval 75.0%

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

       21. distribute-rgt-neg-in 75.0%

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

       22. add-sqr-sqrt 75.0%

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

       23. pow2 75.0%

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

       24. sqrt-prod 74.9%

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

       25. metadata-eval 74.9%

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

   13. Applied egg-rr 97.4%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+61} \lor \neg \left(y \leq 3.6 \cdot 10^{+80}\right):\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+61} \lor \neg \left(y \leq 1.75 \cdot 10^{+77}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.85e+61) (not (<= y 1.75e+77)))
   (* -0.3333333333333333 (/ y (sqrt x)))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.85e+61) || !(y <= 1.75e+77)) {
		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 <= (-1.85d+61)) .or. (.not. (y <= 1.75d+77))) 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 <= -1.85e+61) || !(y <= 1.75e+77)) {
		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 <= -1.85e+61) or not (y <= 1.75e+77):
		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 <= -1.85e+61) || !(y <= 1.75e+77))
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.85e+61) || ~((y <= 1.75e+77)))
		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, -1.85e+61], N[Not[LessEqual[y, 1.75e+77]], $MachinePrecision]], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.85000000000000001e61 or 1.7500000000000001e77 < 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* 99.6%

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

      10. fmm-def 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* 99.7%

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

      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* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 87.1%

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

      1. *-commutative 87.1%

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

      2. associate-*l* 87.0%

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

      3. *-commutative 87.0%

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

   7. Simplified 87.0%

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

      1. *-commutative 87.0%

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

      2. associate-*r* 87.1%

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

      3. metadata-eval 87.1%

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

      4. distribute-rgt-neg-in 87.1%

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

      5. associate-*l* 87.0%

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

      6. sqrt-div 87.0%

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

      7. metadata-eval 87.0%

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

      8. metadata-eval 87.0%

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

      9. div-inv 87.0%

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

      10. times-frac 87.2%

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

      11. *-un-lft-identity 87.2%

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

      12. *-commutative 87.2%

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

      13. neg-mul-1 87.2%

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

      14. clear-num 87.1%

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

      15. *-commutative 87.1%

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

      16. associate-*r/ 87.1%

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

      17. div-inv 87.1%

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

      18. associate-/r* 87.1%

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

      19. clear-num 87.0%

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

   9. Applied egg-rr 87.0%

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

       1. associate-/r/ 87.0%

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

       2. associate-*l/ 87.1%

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

       3. *-lft-identity 87.1%

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

       4. associate-/l/ 87.1%

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

       5. associate-/r* 87.1%

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

       6. associate-/r/ 87.1%

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

       7. metadata-eval 87.1%

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

   11. Simplified 87.1%

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

       1. associate-/l* 87.1%

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

       2. *-commutative 87.1%

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

   13. Applied egg-rr 87.1%

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

   if -1.85000000000000001e61 < y < 1.7500000000000001e77

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fmm-def 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 97.3%

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

      1. associate-*r/ 97.4%

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

      2. metadata-eval 97.4%

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

   7. Simplified 97.4%

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

      1. expm1-log1p-u 93.1%

         \[\leadsto 1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.1111111111111111}{x}\right)\right)} \]

      2. expm1-undefine 93.0%

         \[\leadsto 1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.1111111111111111}{x}\right)} - 1\right)} \]

      3. log1p-undefine 93.0%

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

      4. add-exp-log 97.3%

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

      5. add-sqr-sqrt 97.1%

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

      6. sqrt-unprod 75.0%

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

      7. frac-times 75.1%

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

      8. metadata-eval 75.1%

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

      9. metadata-eval 75.1%

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

      10. frac-times 75.0%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 43.3%

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

   9. Applied egg-rr 43.3%

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

       1. +-commutative 43.3%

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

       2. associate--l+ 43.3%

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

       3. metadata-eval 43.3%

          \[\leadsto 1 - \left(\frac{-0.1111111111111111}{x} + \color{blue}{0}\right) \]

   11. Simplified 43.3%

       \[\leadsto 1 - \color{blue}{\left(\frac{-0.1111111111111111}{x} + 0\right)} \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto 1 - \color{blue}{\sqrt{\frac{-0.1111111111111111}{x} + 0} \cdot \sqrt{\frac{-0.1111111111111111}{x} + 0}} \]

       2. sqrt-unprod 75.1%

          \[\leadsto 1 - \color{blue}{\sqrt{\left(\frac{-0.1111111111111111}{x} + 0\right) \cdot \left(\frac{-0.1111111111111111}{x} + 0\right)}} \]

       3. +-rgt-identity 75.1%

          \[\leadsto 1 - \sqrt{\left(\frac{-0.1111111111111111}{x} + 0\right) \cdot \color{blue}{\frac{-0.1111111111111111}{x}}} \]

       4. clear-num 75.1%

          \[\leadsto 1 - \sqrt{\left(\frac{-0.1111111111111111}{x} + 0\right) \cdot \color{blue}{\frac{1}{\frac{x}{-0.1111111111111111}}}} \]

       5. un-div-inv 75.1%

          \[\leadsto 1 - \sqrt{\color{blue}{\frac{\frac{-0.1111111111111111}{x} + 0}{\frac{x}{-0.1111111111111111}}}} \]

       6. +-rgt-identity 75.1%

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

       7. clear-num 75.0%

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

       8. div-inv 75.1%

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

       9. metadata-eval 75.1%

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

       10. metadata-eval 75.1%

           \[\leadsto 1 - \sqrt{\frac{\frac{1}{x \cdot \color{blue}{\left(-9\right)}}}{\frac{x}{-0.1111111111111111}}} \]

       11. distribute-rgt-neg-in 75.1%

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

       12. add-sqr-sqrt 75.0%

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

       13. pow2 75.0%

           \[\leadsto 1 - \sqrt{\frac{\frac{1}{-\color{blue}{{\left(\sqrt{x \cdot 9}\right)}^{2}}}}{\frac{x}{-0.1111111111111111}}} \]

       14. sqrt-prod 75.0%

           \[\leadsto 1 - \sqrt{\frac{\frac{1}{-{\color{blue}{\left(\sqrt{x} \cdot \sqrt{9}\right)}}^{2}}}{\frac{x}{-0.1111111111111111}}} \]

       15. metadata-eval 75.0%

           \[\leadsto 1 - \sqrt{\frac{\frac{1}{-{\left(\sqrt{x} \cdot \color{blue}{3}\right)}^{2}}}{\frac{x}{-0.1111111111111111}}} \]

       16. distribute-neg-frac2 75.0%

           \[\leadsto 1 - \sqrt{\frac{\color{blue}{-\frac{1}{{\left(\sqrt{x} \cdot 3\right)}^{2}}}}{\frac{x}{-0.1111111111111111}}} \]

       17. neg-mul-1 75.0%

           \[\leadsto 1 - \sqrt{\frac{\color{blue}{-1 \cdot \frac{1}{{\left(\sqrt{x} \cdot 3\right)}^{2}}}}{\frac{x}{-0.1111111111111111}}} \]

       18. div-inv 75.0%

           \[\leadsto 1 - \sqrt{\frac{-1 \cdot \frac{1}{{\left(\sqrt{x} \cdot 3\right)}^{2}}}{\color{blue}{x \cdot \frac{1}{-0.1111111111111111}}}} \]

       19. metadata-eval 75.0%

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

       20. metadata-eval 75.0%

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

       21. distribute-rgt-neg-in 75.0%

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

       22. add-sqr-sqrt 75.0%

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

       23. pow2 75.0%

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

       24. sqrt-prod 74.9%

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

       25. metadata-eval 74.9%

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

   13. Applied egg-rr 97.4%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+61} \lor \neg \left(y \leq 1.75 \cdot 10^{+77}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+61}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.36 \cdot 10^{+70}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.85e+61)
   (* -0.3333333333333333 (/ y (sqrt x)))
   (if (<= y 1.36e+70)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (/ -0.3333333333333333 (/ (sqrt x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.85e+61) {
		tmp = -0.3333333333333333 * (y / sqrt(x));
	} else if (y <= 1.36e+70) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = -0.3333333333333333 / (sqrt(x) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.85d+61)) then
        tmp = (-0.3333333333333333d0) * (y / sqrt(x))
    else if (y <= 1.36d+70) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = (-0.3333333333333333d0) / (sqrt(x) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.85e+61) {
		tmp = -0.3333333333333333 * (y / Math.sqrt(x));
	} else if (y <= 1.36e+70) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = -0.3333333333333333 / (Math.sqrt(x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.85e+61:
		tmp = -0.3333333333333333 * (y / math.sqrt(x))
	elif y <= 1.36e+70:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = -0.3333333333333333 / (math.sqrt(x) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.85e+61)
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	elseif (y <= 1.36e+70)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(-0.3333333333333333 / Float64(sqrt(x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.85e+61)
		tmp = -0.3333333333333333 * (y / sqrt(x));
	elseif (y <= 1.36e+70)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = -0.3333333333333333 / (sqrt(x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.85e+61], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.36e+70], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.85000000000000001e61

   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. fmm-def 99.4%

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

      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* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 84.6%

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

      1. *-commutative 84.6%

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

      2. associate-*l* 84.5%

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

      3. *-commutative 84.5%

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

   7. Simplified 84.5%

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

      1. *-commutative 84.5%

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

      2. associate-*r* 84.6%

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

      3. metadata-eval 84.6%

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

      4. distribute-rgt-neg-in 84.6%

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

      5. associate-*l* 84.5%

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

      6. sqrt-div 84.5%

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

      7. metadata-eval 84.5%

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

      8. metadata-eval 84.5%

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

      9. div-inv 84.5%

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

      10. times-frac 84.7%

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

      11. *-un-lft-identity 84.7%

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

      12. *-commutative 84.7%

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

      13. neg-mul-1 84.7%

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

      14. clear-num 84.7%

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

      15. *-commutative 84.7%

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

      16. associate-*r/ 84.6%

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

      17. div-inv 84.6%

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

      18. associate-/r* 84.7%

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

      19. clear-num 84.6%

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

   9. Applied egg-rr 84.6%

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

       1. associate-/r/ 84.6%

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

       2. associate-*l/ 84.7%

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

       3. *-lft-identity 84.7%

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

       4. associate-/l/ 84.6%

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

       5. associate-/r* 84.6%

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

       6. associate-/r/ 84.6%

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

       7. metadata-eval 84.6%

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

   11. Simplified 84.6%

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

       1. associate-/l* 84.6%

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

       2. *-commutative 84.6%

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

   13. Applied egg-rr 84.6%

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

   if -1.85000000000000001e61 < y < 1.35999999999999995e70

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fmm-def 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 97.3%

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

      1. associate-*r/ 97.4%

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

      2. metadata-eval 97.4%

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

   7. Simplified 97.4%

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

      1. expm1-log1p-u 93.1%

         \[\leadsto 1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.1111111111111111}{x}\right)\right)} \]

      2. expm1-undefine 93.0%

         \[\leadsto 1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.1111111111111111}{x}\right)} - 1\right)} \]

      3. log1p-undefine 93.0%

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

      4. add-exp-log 97.3%

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

      5. add-sqr-sqrt 97.1%

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

      6. sqrt-unprod 75.0%

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

      7. frac-times 75.1%

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

      8. metadata-eval 75.1%

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

      9. metadata-eval 75.1%

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

      10. frac-times 75.0%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 43.3%

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

   9. Applied egg-rr 43.3%

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

       1. +-commutative 43.3%

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

       2. associate--l+ 43.3%

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

       3. metadata-eval 43.3%

          \[\leadsto 1 - \left(\frac{-0.1111111111111111}{x} + \color{blue}{0}\right) \]

   11. Simplified 43.3%

       \[\leadsto 1 - \color{blue}{\left(\frac{-0.1111111111111111}{x} + 0\right)} \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto 1 - \color{blue}{\sqrt{\frac{-0.1111111111111111}{x} + 0} \cdot \sqrt{\frac{-0.1111111111111111}{x} + 0}} \]

       2. sqrt-unprod 75.1%

          \[\leadsto 1 - \color{blue}{\sqrt{\left(\frac{-0.1111111111111111}{x} + 0\right) \cdot \left(\frac{-0.1111111111111111}{x} + 0\right)}} \]

       3. +-rgt-identity 75.1%

          \[\leadsto 1 - \sqrt{\left(\frac{-0.1111111111111111}{x} + 0\right) \cdot \color{blue}{\frac{-0.1111111111111111}{x}}} \]

       4. clear-num 75.1%

          \[\leadsto 1 - \sqrt{\left(\frac{-0.1111111111111111}{x} + 0\right) \cdot \color{blue}{\frac{1}{\frac{x}{-0.1111111111111111}}}} \]

       5. un-div-inv 75.1%

          \[\leadsto 1 - \sqrt{\color{blue}{\frac{\frac{-0.1111111111111111}{x} + 0}{\frac{x}{-0.1111111111111111}}}} \]

       6. +-rgt-identity 75.1%

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

       7. clear-num 75.0%

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

       8. div-inv 75.1%

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

       9. metadata-eval 75.1%

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

       10. metadata-eval 75.1%

           \[\leadsto 1 - \sqrt{\frac{\frac{1}{x \cdot \color{blue}{\left(-9\right)}}}{\frac{x}{-0.1111111111111111}}} \]

       11. distribute-rgt-neg-in 75.1%

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

       12. add-sqr-sqrt 75.0%

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

       13. pow2 75.0%

           \[\leadsto 1 - \sqrt{\frac{\frac{1}{-\color{blue}{{\left(\sqrt{x \cdot 9}\right)}^{2}}}}{\frac{x}{-0.1111111111111111}}} \]

       14. sqrt-prod 75.0%

           \[\leadsto 1 - \sqrt{\frac{\frac{1}{-{\color{blue}{\left(\sqrt{x} \cdot \sqrt{9}\right)}}^{2}}}{\frac{x}{-0.1111111111111111}}} \]

       15. metadata-eval 75.0%

           \[\leadsto 1 - \sqrt{\frac{\frac{1}{-{\left(\sqrt{x} \cdot \color{blue}{3}\right)}^{2}}}{\frac{x}{-0.1111111111111111}}} \]

       16. distribute-neg-frac2 75.0%

           \[\leadsto 1 - \sqrt{\frac{\color{blue}{-\frac{1}{{\left(\sqrt{x} \cdot 3\right)}^{2}}}}{\frac{x}{-0.1111111111111111}}} \]

       17. neg-mul-1 75.0%

           \[\leadsto 1 - \sqrt{\frac{\color{blue}{-1 \cdot \frac{1}{{\left(\sqrt{x} \cdot 3\right)}^{2}}}}{\frac{x}{-0.1111111111111111}}} \]

       18. div-inv 75.0%

           \[\leadsto 1 - \sqrt{\frac{-1 \cdot \frac{1}{{\left(\sqrt{x} \cdot 3\right)}^{2}}}{\color{blue}{x \cdot \frac{1}{-0.1111111111111111}}}} \]

       19. metadata-eval 75.0%

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

       20. metadata-eval 75.0%

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

       21. distribute-rgt-neg-in 75.0%

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

       22. add-sqr-sqrt 75.0%

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

       23. pow2 75.0%

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

       24. sqrt-prod 74.9%

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

       25. metadata-eval 74.9%

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

   13. Applied egg-rr 97.4%

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

   if 1.35999999999999995e70 < 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* 99.7%

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

      10. fmm-def 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* 99.6%

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

      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* 99.6%

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

      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 inf 89.7%

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

      1. *-commutative 89.7%

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

      2. associate-*l* 89.6%

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

      3. *-commutative 89.6%

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

   7. Simplified 89.6%

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

      1. *-commutative 89.6%

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

      2. associate-*r* 89.7%

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

      3. metadata-eval 89.7%

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

      4. distribute-rgt-neg-in 89.7%

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

      5. associate-*l* 89.6%

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

      6. sqrt-div 89.6%

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

      7. metadata-eval 89.6%

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

      8. metadata-eval 89.6%

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

      9. div-inv 89.6%

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

      10. times-frac 89.8%

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

      11. *-un-lft-identity 89.8%

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

      12. *-commutative 89.8%

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

      13. neg-mul-1 89.8%

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

      14. clear-num 89.6%

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

      15. *-commutative 89.6%

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

      16. associate-*r/ 89.6%

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

      17. div-inv 89.6%

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

      18. associate-/r* 89.6%

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

      19. clear-num 89.5%

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

   9. Applied egg-rr 89.5%

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

       1. associate-/r/ 89.5%

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

       2. associate-*l/ 89.6%

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

       3. *-lft-identity 89.6%

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

       4. associate-/l/ 89.6%

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

       5. associate-/r* 89.7%

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

       6. associate-/r/ 89.7%

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

       7. metadata-eval 89.7%

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

   11. Simplified 89.7%

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

       1. clear-num 89.6%

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

       2. inv-pow 89.6%

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

       3. *-un-lft-identity 89.6%

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

       4. times-frac 89.6%

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

       5. metadata-eval 89.6%

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

   13. Applied egg-rr 89.6%

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

       1. unpow-1 89.6%

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

       2. associate-/r* 89.7%

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

       3. metadata-eval 89.7%

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

   15. Simplified 89.7%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+61}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.36 \cdot 10^{+70}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = (-0.1111111111111111 / x) - (y / (3.0 * sqrt(x)));
	} else {
		tmp = 1.0 - (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 (x <= 0.11d0) then
        tmp = ((-0.1111111111111111d0) / x) - (y / (3.0d0 * sqrt(x)))
    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 (x <= 0.11) {
		tmp = (-0.1111111111111111 / x) - (y / (3.0 * Math.sqrt(x)));
	} else {
		tmp = 1.0 - (0.3333333333333333 * (y * Math.sqrt((1.0 / x))));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   1. Initial program 99.6%

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

      1. add-sqr-sqrt 99.2%

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

      2. pow2 99.2%

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

      3. sqrt-prod 99.1%

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

      4. metadata-eval 99.1%

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in x around 0 98.0%

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

   if 0.110000000000000001 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 96.8%

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

4. Final simplification 97.4%

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

Alternative 9: 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 99.7%

   \[\leadsto \left(1 - \color{blue}{\frac{0.1111111111111111}{x}}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
4. 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* 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. neg-mul-1 99.7%

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

   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. Add Preprocessing

Alternative 11: 65.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{+113}:\\ \;\;\;\;\frac{1 - \frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x}}{1 - \frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.9e+113)
   (/
    (- 1.0 (* (/ 0.1111111111111111 x) (/ 0.1111111111111111 x)))
    (- 1.0 (/ 0.1111111111111111 x)))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.9e+113) {
		tmp = (1.0 - ((0.1111111111111111 / x) * (0.1111111111111111 / x))) / (1.0 - (0.1111111111111111 / 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 <= (-3.9d+113)) then
        tmp = (1.0d0 - ((0.1111111111111111d0 / x) * (0.1111111111111111d0 / x))) / (1.0d0 - (0.1111111111111111d0 / 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 <= -3.9e+113) {
		tmp = (1.0 - ((0.1111111111111111 / x) * (0.1111111111111111 / x))) / (1.0 - (0.1111111111111111 / x));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.9e+113:
		tmp = (1.0 - ((0.1111111111111111 / x) * (0.1111111111111111 / x))) / (1.0 - (0.1111111111111111 / x))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.9e+113)
		tmp = Float64(Float64(1.0 - Float64(Float64(0.1111111111111111 / x) * Float64(0.1111111111111111 / x))) / Float64(1.0 - Float64(0.1111111111111111 / x)));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.9e+113)
		tmp = (1.0 - ((0.1111111111111111 / x) * (0.1111111111111111 / x))) / (1.0 - (0.1111111111111111 / x));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.9e+113], N[(N[(1.0 - N[(N[(0.1111111111111111 / x), $MachinePrecision] * N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.9000000000000002e113

   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* 99.5%

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

      10. fmm-def 99.4%

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

      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* 99.6%

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

      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 3.2%

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

      1. associate-*r/ 3.2%

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

      2. metadata-eval 3.2%

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

   7. Simplified 3.2%

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

      1. expm1-log1p-u 3.2%

         \[\leadsto 1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.1111111111111111}{x}\right)\right)} \]

      2. expm1-undefine 3.2%

         \[\leadsto 1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.1111111111111111}{x}\right)} - 1\right)} \]

      3. log1p-undefine 3.2%

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

      4. add-exp-log 3.2%

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

      5. add-sqr-sqrt 3.2%

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

      6. sqrt-unprod 3.2%

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

      7. frac-times 3.2%

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

      8. metadata-eval 3.2%

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

      9. metadata-eval 3.2%

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

      10. frac-times 3.2%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 6.5%

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

   9. Applied egg-rr 6.5%

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

       1. +-commutative 6.5%

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

       2. associate--l+ 6.5%

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

       3. metadata-eval 6.5%

          \[\leadsto 1 - \left(\frac{-0.1111111111111111}{x} + \color{blue}{0}\right) \]

   11. Simplified 6.5%

       \[\leadsto 1 - \color{blue}{\left(\frac{-0.1111111111111111}{x} + 0\right)} \]
   12. Step-by-step derivation

       1. +-rgt-identity 6.5%

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

       2. clear-num 6.5%

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

       3. div-inv 6.5%

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

       4. metadata-eval 6.5%

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

       5. metadata-eval 6.5%

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

       6. distribute-rgt-neg-in 6.5%

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

       7. add-sqr-sqrt 6.5%

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

       8. pow2 6.5%

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

       9. sqrt-prod 6.5%

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

       10. metadata-eval 6.5%

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

       11. distribute-neg-frac2 6.5%

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

       12. distribute-neg-frac 6.5%

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

       13. metadata-eval 6.5%

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

       14. unpow2 6.5%

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

       15. swap-sqr 6.5%

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

       16. add-sqr-sqrt 6.5%

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

       17. metadata-eval 6.5%

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

   13. Applied egg-rr 6.5%

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

       1. sub-neg 6.5%

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

       2. flip-+ 24.1%

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

       3. metadata-eval 24.1%

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

       4. distribute-neg-frac 24.1%

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

       5. metadata-eval 24.1%

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

       6. *-commutative 24.1%

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

       7. associate-/r* 24.1%

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

       8. metadata-eval 24.1%

          \[\leadsto \frac{1 - \frac{\color{blue}{0.1111111111111111}}{x} \cdot \left(-\frac{-1}{x \cdot 9}\right)}{1 - \left(-\frac{-1}{x \cdot 9}\right)} \]

       9. distribute-neg-frac 24.1%

          \[\leadsto \frac{1 - \frac{0.1111111111111111}{x} \cdot \color{blue}{\frac{--1}{x \cdot 9}}}{1 - \left(-\frac{-1}{x \cdot 9}\right)} \]

       10. metadata-eval 24.1%

           \[\leadsto \frac{1 - \frac{0.1111111111111111}{x} \cdot \frac{\color{blue}{1}}{x \cdot 9}}{1 - \left(-\frac{-1}{x \cdot 9}\right)} \]

       11. *-commutative 24.1%

           \[\leadsto \frac{1 - \frac{0.1111111111111111}{x} \cdot \frac{1}{\color{blue}{9 \cdot x}}}{1 - \left(-\frac{-1}{x \cdot 9}\right)} \]

       12. associate-/r* 24.1%

           \[\leadsto \frac{1 - \frac{0.1111111111111111}{x} \cdot \color{blue}{\frac{\frac{1}{9}}{x}}}{1 - \left(-\frac{-1}{x \cdot 9}\right)} \]

       13. metadata-eval 24.1%

           \[\leadsto \frac{1 - \frac{0.1111111111111111}{x} \cdot \frac{\color{blue}{0.1111111111111111}}{x}}{1 - \left(-\frac{-1}{x \cdot 9}\right)} \]

       14. distribute-neg-frac 24.1%

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

       15. metadata-eval 24.1%

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

       16. *-commutative 24.1%

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

       17. associate-/r* 24.1%

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

       18. metadata-eval 24.1%

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

   15. Applied egg-rr 24.1%

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

   if -3.9000000000000002e113 < y

   1. Initial program 99.7%

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

      1. 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. fmm-def 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 72.1%

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

      1. associate-*r/ 72.1%

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

      2. metadata-eval 72.1%

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

   7. Simplified 72.1%

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

      1. expm1-log1p-u 69.0%

         \[\leadsto 1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.1111111111111111}{x}\right)\right)} \]

      2. expm1-undefine 69.0%

         \[\leadsto 1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.1111111111111111}{x}\right)} - 1\right)} \]

      3. log1p-undefine 69.0%

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

      4. add-exp-log 72.1%

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

      5. add-sqr-sqrt 71.9%

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

      6. sqrt-unprod 57.7%

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

      7. frac-times 57.7%

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

      8. metadata-eval 57.7%

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

      9. metadata-eval 57.7%

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

      10. frac-times 57.7%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 32.6%

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

   9. Applied egg-rr 32.6%

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

       1. +-commutative 32.6%

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

       2. associate--l+ 32.6%

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

       3. metadata-eval 32.6%

          \[\leadsto 1 - \left(\frac{-0.1111111111111111}{x} + \color{blue}{0}\right) \]

   11. Simplified 32.6%

       \[\leadsto 1 - \color{blue}{\left(\frac{-0.1111111111111111}{x} + 0\right)} \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto 1 - \color{blue}{\sqrt{\frac{-0.1111111111111111}{x} + 0} \cdot \sqrt{\frac{-0.1111111111111111}{x} + 0}} \]

       2. sqrt-unprod 57.7%

          \[\leadsto 1 - \color{blue}{\sqrt{\left(\frac{-0.1111111111111111}{x} + 0\right) \cdot \left(\frac{-0.1111111111111111}{x} + 0\right)}} \]

       3. +-rgt-identity 57.7%

          \[\leadsto 1 - \sqrt{\left(\frac{-0.1111111111111111}{x} + 0\right) \cdot \color{blue}{\frac{-0.1111111111111111}{x}}} \]

       4. clear-num 57.7%

          \[\leadsto 1 - \sqrt{\left(\frac{-0.1111111111111111}{x} + 0\right) \cdot \color{blue}{\frac{1}{\frac{x}{-0.1111111111111111}}}} \]

       5. un-div-inv 57.7%

          \[\leadsto 1 - \sqrt{\color{blue}{\frac{\frac{-0.1111111111111111}{x} + 0}{\frac{x}{-0.1111111111111111}}}} \]

       6. +-rgt-identity 57.7%

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

       7. clear-num 57.7%

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

       8. div-inv 57.7%

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

       9. metadata-eval 57.7%

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

       10. metadata-eval 57.7%

           \[\leadsto 1 - \sqrt{\frac{\frac{1}{x \cdot \color{blue}{\left(-9\right)}}}{\frac{x}{-0.1111111111111111}}} \]

       11. distribute-rgt-neg-in 57.7%

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

       12. add-sqr-sqrt 57.7%

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

       13. pow2 57.7%

           \[\leadsto 1 - \sqrt{\frac{\frac{1}{-\color{blue}{{\left(\sqrt{x \cdot 9}\right)}^{2}}}}{\frac{x}{-0.1111111111111111}}} \]

       14. sqrt-prod 57.7%

           \[\leadsto 1 - \sqrt{\frac{\frac{1}{-{\color{blue}{\left(\sqrt{x} \cdot \sqrt{9}\right)}}^{2}}}{\frac{x}{-0.1111111111111111}}} \]

       15. metadata-eval 57.7%

           \[\leadsto 1 - \sqrt{\frac{\frac{1}{-{\left(\sqrt{x} \cdot \color{blue}{3}\right)}^{2}}}{\frac{x}{-0.1111111111111111}}} \]

       16. distribute-neg-frac2 57.7%

           \[\leadsto 1 - \sqrt{\frac{\color{blue}{-\frac{1}{{\left(\sqrt{x} \cdot 3\right)}^{2}}}}{\frac{x}{-0.1111111111111111}}} \]

       17. neg-mul-1 57.7%

           \[\leadsto 1 - \sqrt{\frac{\color{blue}{-1 \cdot \frac{1}{{\left(\sqrt{x} \cdot 3\right)}^{2}}}}{\frac{x}{-0.1111111111111111}}} \]

       18. div-inv 57.6%

           \[\leadsto 1 - \sqrt{\frac{-1 \cdot \frac{1}{{\left(\sqrt{x} \cdot 3\right)}^{2}}}{\color{blue}{x \cdot \frac{1}{-0.1111111111111111}}}} \]

       19. metadata-eval 57.6%

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

       20. metadata-eval 57.6%

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

       21. distribute-rgt-neg-in 57.6%

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

       22. add-sqr-sqrt 57.6%

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

       23. pow2 57.6%

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

       24. sqrt-prod 57.6%

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

       25. metadata-eval 57.6%

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

   13. Applied egg-rr 72.2%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{+113}:\\ \;\;\;\;\frac{1 - \frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x}}{1 - \frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.7% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00095:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.00095) (/ -0.1111111111111111 x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.00095) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.00095d0) then
        tmp = (-0.1111111111111111d0) / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.00095) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.00095:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.00095)
		tmp = Float64(-0.1111111111111111 / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.00095)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.00095], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 9.49999999999999998e-4

   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* 99.6%

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

      10. fmm-def 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* 99.7%

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

      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 x around 0 98.3%

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

   6. Taylor expanded in y around 0 63.5%

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

   if 9.49999999999999998e-4 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 96.4%

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

   4. Taylor expanded in y around 0 57.3%

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

Alternative 13: 62.9% 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* 99.7%

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

   10. fmm-def 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* 99.7%

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

   15. distribute-neg-frac 99.7%

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

   16. metadata-eval 99.7%

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

   17. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in y around 0 62.4%

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

   1. associate-*r/ 62.5%

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

   2. metadata-eval 62.5%

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

7. Simplified 62.5%

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

   1. expm1-log1p-u 59.8%

      \[\leadsto 1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.1111111111111111}{x}\right)\right)} \]

   2. expm1-undefine 59.7%

      \[\leadsto 1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.1111111111111111}{x}\right)} - 1\right)} \]

   3. log1p-undefine 59.7%

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

   4. add-exp-log 62.4%

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

   5. add-sqr-sqrt 62.3%

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

   6. sqrt-unprod 50.0%

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

   7. frac-times 50.0%

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

   8. metadata-eval 50.0%

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

   9. metadata-eval 50.0%

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

   10. frac-times 50.0%

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

   11. sqrt-unprod 0.0%

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

   12. add-sqr-sqrt 28.9%

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

9. Applied egg-rr 28.9%

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

    1. +-commutative 28.9%

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

    2. associate--l+ 28.9%

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

    3. metadata-eval 28.9%

       \[\leadsto 1 - \left(\frac{-0.1111111111111111}{x} + \color{blue}{0}\right) \]

11. Simplified 28.9%

    \[\leadsto 1 - \color{blue}{\left(\frac{-0.1111111111111111}{x} + 0\right)} \]
12. Step-by-step derivation

    1. add-sqr-sqrt 0.0%

       \[\leadsto 1 - \color{blue}{\sqrt{\frac{-0.1111111111111111}{x} + 0} \cdot \sqrt{\frac{-0.1111111111111111}{x} + 0}} \]

    2. sqrt-unprod 50.0%

       \[\leadsto 1 - \color{blue}{\sqrt{\left(\frac{-0.1111111111111111}{x} + 0\right) \cdot \left(\frac{-0.1111111111111111}{x} + 0\right)}} \]

    3. +-rgt-identity 50.0%

       \[\leadsto 1 - \sqrt{\left(\frac{-0.1111111111111111}{x} + 0\right) \cdot \color{blue}{\frac{-0.1111111111111111}{x}}} \]

    4. clear-num 50.0%

       \[\leadsto 1 - \sqrt{\left(\frac{-0.1111111111111111}{x} + 0\right) \cdot \color{blue}{\frac{1}{\frac{x}{-0.1111111111111111}}}} \]

    5. un-div-inv 50.0%

       \[\leadsto 1 - \sqrt{\color{blue}{\frac{\frac{-0.1111111111111111}{x} + 0}{\frac{x}{-0.1111111111111111}}}} \]

    6. +-rgt-identity 50.0%

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

    7. clear-num 50.0%

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

    8. div-inv 50.0%

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

    9. metadata-eval 50.0%

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

    10. metadata-eval 50.0%

        \[\leadsto 1 - \sqrt{\frac{\frac{1}{x \cdot \color{blue}{\left(-9\right)}}}{\frac{x}{-0.1111111111111111}}} \]

    11. distribute-rgt-neg-in 50.0%

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

    12. add-sqr-sqrt 50.0%

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

    13. pow2 50.0%

        \[\leadsto 1 - \sqrt{\frac{\frac{1}{-\color{blue}{{\left(\sqrt{x \cdot 9}\right)}^{2}}}}{\frac{x}{-0.1111111111111111}}} \]

    14. sqrt-prod 50.0%

        \[\leadsto 1 - \sqrt{\frac{\frac{1}{-{\color{blue}{\left(\sqrt{x} \cdot \sqrt{9}\right)}}^{2}}}{\frac{x}{-0.1111111111111111}}} \]

    15. metadata-eval 50.0%

        \[\leadsto 1 - \sqrt{\frac{\frac{1}{-{\left(\sqrt{x} \cdot \color{blue}{3}\right)}^{2}}}{\frac{x}{-0.1111111111111111}}} \]

    16. distribute-neg-frac2 50.0%

        \[\leadsto 1 - \sqrt{\frac{\color{blue}{-\frac{1}{{\left(\sqrt{x} \cdot 3\right)}^{2}}}}{\frac{x}{-0.1111111111111111}}} \]

    17. neg-mul-1 50.0%

        \[\leadsto 1 - \sqrt{\frac{\color{blue}{-1 \cdot \frac{1}{{\left(\sqrt{x} \cdot 3\right)}^{2}}}}{\frac{x}{-0.1111111111111111}}} \]

    18. div-inv 50.0%

        \[\leadsto 1 - \sqrt{\frac{-1 \cdot \frac{1}{{\left(\sqrt{x} \cdot 3\right)}^{2}}}{\color{blue}{x \cdot \frac{1}{-0.1111111111111111}}}} \]

    19. metadata-eval 50.0%

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

    20. metadata-eval 50.0%

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

    21. distribute-rgt-neg-in 50.0%

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

    22. add-sqr-sqrt 50.0%

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

    23. pow2 50.0%

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

    24. sqrt-prod 49.9%

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

    25. metadata-eval 49.9%

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

13. Applied egg-rr 62.5%

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

14. Final simplification 62.5%

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

Alternative 14: 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* 99.7%

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

   10. fmm-def 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* 99.7%

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

   15. distribute-neg-frac 99.7%

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

   16. metadata-eval 99.7%

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

   17. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in y around 0 62.4%

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

   1. associate-*r/ 62.5%

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

   2. metadata-eval 62.5%

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

7. Simplified 62.5%

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

Alternative 15: 32.1% accurate, 113.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

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

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around inf 64.1%

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

4. Taylor expanded in y around 0 28.8%

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

Developer Target 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024181 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :alt
  (! :herbie-platform default (- (- 1 (/ (/ 1 x) 9)) (/ y (* 3 (sqrt x)))))

  (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))