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

Percentage Accurate: 99.7% → 99.7%

Time: 10.4s

Alternatives: 18

Speedup: N/A×

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

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

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 2: 94.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+36}:\\ \;\;\;\;1 - y \cdot \sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+26}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{{x}^{-0.5}}{\frac{3}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.65e+36)
   (- 1.0 (* y (sqrt (/ 0.1111111111111111 x))))
   (if (<= y 3.5e+26)
     (+ 1.0 (/ -0.1111111111111111 x))
     (- 1.0 (/ (pow x -0.5) (/ 3.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.65e+36) {
		tmp = 1.0 - (y * sqrt((0.1111111111111111 / x)));
	} else if (y <= 3.5e+26) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 - (pow(x, -0.5) / (3.0 / 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 <= (-1.65d+36)) then
        tmp = 1.0d0 - (y * sqrt((0.1111111111111111d0 / x)))
    else if (y <= 3.5d+26) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = 1.0d0 - ((x ** (-0.5d0)) / (3.0d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.65e+36) {
		tmp = 1.0 - (y * Math.sqrt((0.1111111111111111 / x)));
	} else if (y <= 3.5e+26) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 - (Math.pow(x, -0.5) / (3.0 / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.65e+36:
		tmp = 1.0 - (y * math.sqrt((0.1111111111111111 / x)))
	elif y <= 3.5e+26:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = 1.0 - (math.pow(x, -0.5) / (3.0 / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.65e+36)
		tmp = Float64(1.0 - Float64(y * sqrt(Float64(0.1111111111111111 / x))));
	elseif (y <= 3.5e+26)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(1.0 - Float64((x ^ -0.5) / Float64(3.0 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.65e+36)
		tmp = 1.0 - (y * sqrt((0.1111111111111111 / x)));
	elseif (y <= 3.5e+26)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = 1.0 - ((x ^ -0.5) / (3.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.65e+36], N[(1.0 - N[(y * N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.5e+26], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[Power[x, -0.5], $MachinePrecision] / N[(3.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6499999999999999e36

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf 95.1%

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

      1. metadata-eval 95.1%

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

      2. *-commutative 95.1%

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

      3. sqrt-div 95.1%

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

      4. metadata-eval 95.1%

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

      5. un-div-inv 95.0%

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

      6. times-frac 95.2%

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

      7. *-un-lft-identity 95.2%

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

      8. associate-/r* 95.1%

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

      9. div-inv 95.0%

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

      10. metadata-eval 95.0%

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

   5. Applied egg-rr 95.0%

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

      1. associate-/l* 95.1%

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

      2. div-inv 95.2%

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

      3. metadata-eval 95.2%

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

      4. sqrt-div 95.1%

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

      5. *-commutative 95.1%

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

      6. metadata-eval 95.1%

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

      7. sqrt-prod 95.3%

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

      8. un-div-inv 95.3%

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

   7. Applied egg-rr 95.3%

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

   if -1.6499999999999999e36 < y < 3.4999999999999999e26

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.7%

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

      10. fma-neg 99.7%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

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

      13. *-commutative 99.7%

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

      14. associate-/r* 99.7%

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

      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 98.7%

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

   if 3.4999999999999999e26 < 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 89.4%

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

      1. metadata-eval 89.4%

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

      2. *-commutative 89.4%

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

      3. sqrt-div 89.4%

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

      4. metadata-eval 89.4%

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

      5. un-div-inv 89.4%

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

      6. times-frac 90.8%

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

      7. *-commutative 90.8%

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

      8. times-frac 90.8%

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

      9. metadata-eval 90.8%

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

      10. sqrt-div 90.8%

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

      11. metadata-eval 90.8%

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

      12. add-sqr-sqrt 90.7%

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

      13. frac-times 90.8%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 7.5%

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

      16. clear-num 7.5%

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

      17. div-inv 7.5%

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

      18. add-sqr-sqrt 0.0%

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

      19. sqrt-unprod 90.8%

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

      20. frac-times 90.7%

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

      21. metadata-eval 90.7%

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

      22. add-sqr-sqrt 90.8%

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

   5. Applied egg-rr 90.8%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+36}:\\ \;\;\;\;1 - y \cdot \sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+26}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{{x}^{-0.5}}{\frac{3}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+37} \lor \neg \left(y \leq 6.2 \cdot 10^{+25}\right):\\ \;\;\;\;1 - y \cdot \sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -8.5e+37) (not (<= y 6.2e+25)))
   (- 1.0 (* y (sqrt (/ 0.1111111111111111 x))))
   (+ 1.0 (/ -0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -8.5e+37) || !(y <= 6.2e+25)) {
		tmp = 1.0 - (y * sqrt((0.1111111111111111 / x)));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-8.5d+37)) .or. (.not. (y <= 6.2d+25))) then
        tmp = 1.0d0 - (y * sqrt((0.1111111111111111d0 / x)))
    else
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -8.5e+37) || !(y <= 6.2e+25)) {
		tmp = 1.0 - (y * Math.sqrt((0.1111111111111111 / x)));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -8.5e+37) or not (y <= 6.2e+25):
		tmp = 1.0 - (y * math.sqrt((0.1111111111111111 / x)))
	else:
		tmp = 1.0 + (-0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -8.5e+37) || !(y <= 6.2e+25))
		tmp = Float64(1.0 - Float64(y * sqrt(Float64(0.1111111111111111 / x))));
	else
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -8.5e+37) || ~((y <= 6.2e+25)))
		tmp = 1.0 - (y * sqrt((0.1111111111111111 / x)));
	else
		tmp = 1.0 + (-0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -8.5e+37], N[Not[LessEqual[y, 6.2e+25]], $MachinePrecision]], N[(1.0 - N[(y * N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.4999999999999999e37 or 6.1999999999999996e25 < 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 91.7%

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

      1. metadata-eval 91.7%

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

      2. *-commutative 91.7%

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

      3. sqrt-div 91.7%

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

      4. metadata-eval 91.7%

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

      5. un-div-inv 91.6%

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

      6. times-frac 92.6%

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

      7. *-un-lft-identity 92.6%

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

      8. associate-/r* 92.5%

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

      9. div-inv 92.3%

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

      10. metadata-eval 92.3%

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

   5. Applied egg-rr 92.3%

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

      1. associate-/l* 92.3%

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

      2. div-inv 92.4%

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

      3. metadata-eval 92.4%

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

      4. sqrt-div 92.4%

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

      5. *-commutative 92.4%

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

      6. metadata-eval 92.4%

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

      7. sqrt-prod 92.5%

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

      8. un-div-inv 92.5%

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

   7. Applied egg-rr 92.5%

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

   if -8.4999999999999999e37 < y < 6.1999999999999996e25

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.7%

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

      10. fma-neg 99.7%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

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

      13. *-commutative 99.7%

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

      14. associate-/r* 99.7%

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

      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 98.7%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+37} \lor \neg \left(y \leq 6.2 \cdot 10^{+25}\right):\\ \;\;\;\;1 - y \cdot \sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+48} \lor \neg \left(y \leq 6 \cdot 10^{+56}\right):\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.9e+48) (not (<= y 6e+56)))
   (* y (* -0.3333333333333333 (sqrt (/ 1.0 x))))
   (+ 1.0 (/ -0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.9e+48) || !(y <= 6e+56)) {
		tmp = y * (-0.3333333333333333 * sqrt((1.0 / x)));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-2.9d+48)) .or. (.not. (y <= 6d+56))) then
        tmp = y * ((-0.3333333333333333d0) * sqrt((1.0d0 / x)))
    else
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.9e+48) || !(y <= 6e+56)) {
		tmp = y * (-0.3333333333333333 * Math.sqrt((1.0 / x)));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.9e+48) or not (y <= 6e+56):
		tmp = y * (-0.3333333333333333 * math.sqrt((1.0 / x)))
	else:
		tmp = 1.0 + (-0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.9e+48) || !(y <= 6e+56))
		tmp = Float64(y * Float64(-0.3333333333333333 * sqrt(Float64(1.0 / x))));
	else
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.9e+48) || ~((y <= 6e+56)))
		tmp = y * (-0.3333333333333333 * sqrt((1.0 / x)));
	else
		tmp = 1.0 + (-0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.9e+48], N[Not[LessEqual[y, 6e+56]], $MachinePrecision]], N[(y * N[(-0.3333333333333333 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.8999999999999999e48 or 6.00000000000000012e56 < 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 0 99.6%

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

   4. Taylor expanded in y around inf 86.2%

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

      1. *-commutative 86.2%

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

      2. *-commutative 86.2%

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

      3. associate-*r* 86.9%

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

   6. Simplified 86.9%

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

   if -2.8999999999999999e48 < y < 6.00000000000000012e56

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.7%

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

      10. fma-neg 99.7%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

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

      13. *-commutative 99.7%

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

      14. associate-/r* 99.7%

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

      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 97.9%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+48} \lor \neg \left(y \leq 6 \cdot 10^{+56}\right):\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.3 \cdot 10^{+35}:\\ \;\;\;\;1 - y \cdot \sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+26}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5.3e+35)
   (- 1.0 (* y (sqrt (/ 0.1111111111111111 x))))
   (if (<= y 3.1e+26)
     (+ 1.0 (/ -0.1111111111111111 x))
     (- 1.0 (/ y (* 3.0 (sqrt x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5.3e+35) {
		tmp = 1.0 - (y * sqrt((0.1111111111111111 / x)));
	} else if (y <= 3.1e+26) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-5.3d+35)) then
        tmp = 1.0d0 - (y * sqrt((0.1111111111111111d0 / x)))
    else if (y <= 3.1d+26) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = 1.0d0 - (y / (3.0d0 * sqrt(x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -5.3e+35) {
		tmp = 1.0 - (y * Math.sqrt((0.1111111111111111 / x)));
	} else if (y <= 3.1e+26) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 - (y / (3.0 * Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5.3e+35:
		tmp = 1.0 - (y * math.sqrt((0.1111111111111111 / x)))
	elif y <= 3.1e+26:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = 1.0 - (y / (3.0 * math.sqrt(x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5.3e+35)
		tmp = Float64(1.0 - Float64(y * sqrt(Float64(0.1111111111111111 / x))));
	elseif (y <= 3.1e+26)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -5.3e+35)
		tmp = 1.0 - (y * sqrt((0.1111111111111111 / x)));
	elseif (y <= 3.1e+26)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5.3e+35], N[(1.0 - N[(y * N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.1e+26], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.30000000000000009e35

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf 95.1%

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

      1. metadata-eval 95.1%

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

      2. *-commutative 95.1%

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

      3. sqrt-div 95.1%

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

      4. metadata-eval 95.1%

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

      5. un-div-inv 95.0%

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

      6. times-frac 95.2%

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

      7. *-un-lft-identity 95.2%

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

      8. associate-/r* 95.1%

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

      9. div-inv 95.0%

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

      10. metadata-eval 95.0%

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

   5. Applied egg-rr 95.0%

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

      1. associate-/l* 95.1%

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

      2. div-inv 95.2%

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

      3. metadata-eval 95.2%

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

      4. sqrt-div 95.1%

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

      5. *-commutative 95.1%

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

      6. metadata-eval 95.1%

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

      7. sqrt-prod 95.3%

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

      8. un-div-inv 95.3%

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

   7. Applied egg-rr 95.3%

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

   if -5.30000000000000009e35 < y < 3.1e26

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.7%

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

      10. fma-neg 99.7%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

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

      13. *-commutative 99.7%

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

      14. associate-/r* 99.7%

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

      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 98.7%

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

   if 3.1e26 < 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 89.4%

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

      1. metadata-eval 89.4%

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

      2. *-commutative 89.4%

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

      3. sqrt-div 89.4%

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

      4. metadata-eval 89.4%

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

      5. un-div-inv 89.4%

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

      6. times-frac 90.8%

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

      7. *-un-lft-identity 90.8%

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

   5. Applied egg-rr 90.8%

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

      1. *-commutative 90.8%

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

   7. Simplified 90.8%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.3 \cdot 10^{+35}:\\ \;\;\;\;1 - y \cdot \sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+26}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{x}}\\ \mathbf{if}\;y \leq -2.9 \cdot 10^{+48}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot t\_0\right)\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+57}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 x))))
   (if (<= y -2.9e+48)
     (* -0.3333333333333333 (* y t_0))
     (if (<= y 1.15e+57)
       (+ 1.0 (/ -0.1111111111111111 x))
       (* y (* -0.3333333333333333 t_0))))))

C

double code(double x, double y) {
	double t_0 = sqrt((1.0 / x));
	double tmp;
	if (y <= -2.9e+48) {
		tmp = -0.3333333333333333 * (y * t_0);
	} else if (y <= 1.15e+57) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = y * (-0.3333333333333333 * t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / x))
    if (y <= (-2.9d+48)) then
        tmp = (-0.3333333333333333d0) * (y * t_0)
    else if (y <= 1.15d+57) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = y * ((-0.3333333333333333d0) * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt((1.0 / x));
	double tmp;
	if (y <= -2.9e+48) {
		tmp = -0.3333333333333333 * (y * t_0);
	} else if (y <= 1.15e+57) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = y * (-0.3333333333333333 * t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt((1.0 / x))
	tmp = 0
	if y <= -2.9e+48:
		tmp = -0.3333333333333333 * (y * t_0)
	elif y <= 1.15e+57:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = y * (-0.3333333333333333 * t_0)
	return tmp

Julia

function code(x, y)
	t_0 = sqrt(Float64(1.0 / x))
	tmp = 0.0
	if (y <= -2.9e+48)
		tmp = Float64(-0.3333333333333333 * Float64(y * t_0));
	elseif (y <= 1.15e+57)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(y * Float64(-0.3333333333333333 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt((1.0 / x));
	tmp = 0.0;
	if (y <= -2.9e+48)
		tmp = -0.3333333333333333 * (y * t_0);
	elseif (y <= 1.15e+57)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = y * (-0.3333333333333333 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, -2.9e+48], N[(-0.3333333333333333 * N[(y * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.15e+57], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(y * N[(-0.3333333333333333 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.8999999999999999e48

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 99.5%

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

   4. Taylor expanded in y around inf 87.6%

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

      1. *-commutative 87.6%

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

      2. *-commutative 87.6%

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

   6. Simplified 87.6%

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

   if -2.8999999999999999e48 < y < 1.1499999999999999e57

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.7%

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

      10. fma-neg 99.7%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

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

      13. *-commutative 99.7%

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

      14. associate-/r* 99.7%

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

      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 97.9%

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

   if 1.1499999999999999e57 < 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 0 99.6%

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

   4. Taylor expanded in y around inf 85.2%

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

      1. *-commutative 85.2%

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

      2. *-commutative 85.2%

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

      3. associate-*r* 86.4%

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

   6. Simplified 86.4%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+48}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+57}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(y \cdot \sqrt{x}\right) - 0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{{x}^{-0.5}}{\frac{3}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.112)
   (/ (- (* -0.3333333333333333 (* y (sqrt x))) 0.1111111111111111) x)
   (- 1.0 (/ (pow x -0.5) (/ 3.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = ((-0.3333333333333333 * (y * sqrt(x))) - 0.1111111111111111) / x;
	} else {
		tmp = 1.0 - (pow(x, -0.5) / (3.0 / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.112d0) then
        tmp = (((-0.3333333333333333d0) * (y * sqrt(x))) - 0.1111111111111111d0) / x
    else
        tmp = 1.0d0 - ((x ** (-0.5d0)) / (3.0d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = ((-0.3333333333333333 * (y * Math.sqrt(x))) - 0.1111111111111111) / x;
	} else {
		tmp = 1.0 - (Math.pow(x, -0.5) / (3.0 / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.112:
		tmp = ((-0.3333333333333333 * (y * math.sqrt(x))) - 0.1111111111111111) / x
	else:
		tmp = 1.0 - (math.pow(x, -0.5) / (3.0 / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.112)
		tmp = Float64(Float64(Float64(-0.3333333333333333 * Float64(y * sqrt(x))) - 0.1111111111111111) / x);
	else
		tmp = Float64(1.0 - Float64((x ^ -0.5) / Float64(3.0 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.112)
		tmp = ((-0.3333333333333333 * (y * sqrt(x))) - 0.1111111111111111) / x;
	else
		tmp = 1.0 - ((x ^ -0.5) / (3.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.112], N[(N[(N[(-0.3333333333333333 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.1111111111111111), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - N[(N[Power[x, -0.5], $MachinePrecision] / N[(3.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.112000000000000002

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-/r* 99.5%

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

      4. metadata-eval 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. neg-mul-1 99.5%

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

      7. times-frac 98.8%

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

      8. metadata-eval 98.8%

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

   3. Simplified 98.8%

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

      1. *-commutative 98.8%

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

      2. associate-*l/ 99.4%

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

      3. associate-*r/ 99.5%

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

      4. frac-2neg 99.5%

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

      5. associate-*r/ 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

      8. div-inv 99.5%

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

      9. distribute-neg-frac2 99.5%

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

      10. associate-/r* 99.5%

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

      11. clear-num 99.4%

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

      12. distribute-neg-frac2 99.4%

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

      13. *-commutative 99.4%

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

      14. associate-/l* 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. distribute-frac-neg2 99.5%

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

      2. distribute-neg-frac 99.5%

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

      3. metadata-eval 99.5%

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

      4. associate-/r* 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in x around 0 99.0%

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

   if 0.112000000000000002 < 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 99.7%

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

      1. metadata-eval 99.7%

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

      2. *-commutative 99.7%

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

      3. sqrt-div 99.7%

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

      4. metadata-eval 99.7%

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

      5. un-div-inv 99.7%

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

      6. times-frac 99.8%

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

      7. *-commutative 99.8%

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

      8. times-frac 99.8%

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

      9. metadata-eval 99.8%

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

      10. sqrt-div 99.8%

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

      11. metadata-eval 99.8%

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

      12. add-sqr-sqrt 99.8%

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

      13. frac-times 99.8%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 63.3%

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

      16. clear-num 63.3%

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

      17. div-inv 63.3%

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

      18. add-sqr-sqrt 0.0%

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

      19. sqrt-unprod 99.8%

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

      20. frac-times 99.8%

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

      21. metadata-eval 99.8%

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

      22. add-sqr-sqrt 99.8%

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

   5. Applied egg-rr 99.8%

      \[\leadsto 1 - \color{blue}{\frac{{x}^{-0.5}}{\frac{3}{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(y \cdot \sqrt{x}\right) - 0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{{x}^{-0.5}}{\frac{3}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) + ((y * -0.3333333333333333) / 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 * (-0.3333333333333333d0)) / sqrt(x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg 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 99.3%

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

   8. metadata-eval 99.3%

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

3. Simplified 99.3%

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

   1. clear-num 99.3%

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

   2. un-div-inv 99.6%

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

6. Applied egg-rr 99.6%

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

   1. associate-/r/ 99.6%

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

   2. associate-*l/ 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 9: 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 99.3%

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

   8. metadata-eval 99.3%

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

3. Simplified 99.3%

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

Alternative 10: 67.0% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.08 \cdot 10^{+133}:\\ \;\;\;\;\frac{1 - \frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}{1 - 0.1111111111111111 \cdot \frac{1}{x}}\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{+174}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-1}{\left(x \cdot -9\right) \cdot \left(x \cdot -9\right)}}{1 - \frac{-0.1111111111111111}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.08e+133)
   (/
    (- 1.0 (* (/ -0.1111111111111111 x) (/ -0.1111111111111111 x)))
    (- 1.0 (* 0.1111111111111111 (/ 1.0 x))))
   (if (<= y 7.6e+174)
     (+ 1.0 (/ -0.1111111111111111 x))
     (/
      (+ 1.0 (/ -1.0 (* (* x -9.0) (* x -9.0))))
      (- 1.0 (/ -0.1111111111111111 x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.08e+133) {
		tmp = (1.0 - ((-0.1111111111111111 / x) * (-0.1111111111111111 / x))) / (1.0 - (0.1111111111111111 * (1.0 / x)));
	} else if (y <= 7.6e+174) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (1.0 + (-1.0 / ((x * -9.0) * (x * -9.0)))) / (1.0 - (-0.1111111111111111 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.08d+133)) then
        tmp = (1.0d0 - (((-0.1111111111111111d0) / x) * ((-0.1111111111111111d0) / x))) / (1.0d0 - (0.1111111111111111d0 * (1.0d0 / x)))
    else if (y <= 7.6d+174) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = (1.0d0 + ((-1.0d0) / ((x * (-9.0d0)) * (x * (-9.0d0))))) / (1.0d0 - ((-0.1111111111111111d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.08e+133) {
		tmp = (1.0 - ((-0.1111111111111111 / x) * (-0.1111111111111111 / x))) / (1.0 - (0.1111111111111111 * (1.0 / x)));
	} else if (y <= 7.6e+174) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (1.0 + (-1.0 / ((x * -9.0) * (x * -9.0)))) / (1.0 - (-0.1111111111111111 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.08e+133:
		tmp = (1.0 - ((-0.1111111111111111 / x) * (-0.1111111111111111 / x))) / (1.0 - (0.1111111111111111 * (1.0 / x)))
	elif y <= 7.6e+174:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = (1.0 + (-1.0 / ((x * -9.0) * (x * -9.0)))) / (1.0 - (-0.1111111111111111 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.08e+133)
		tmp = Float64(Float64(1.0 - Float64(Float64(-0.1111111111111111 / x) * Float64(-0.1111111111111111 / x))) / Float64(1.0 - Float64(0.1111111111111111 * Float64(1.0 / x))));
	elseif (y <= 7.6e+174)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(Float64(1.0 + Float64(-1.0 / Float64(Float64(x * -9.0) * Float64(x * -9.0)))) / Float64(1.0 - Float64(-0.1111111111111111 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.08e+133)
		tmp = (1.0 - ((-0.1111111111111111 / x) * (-0.1111111111111111 / x))) / (1.0 - (0.1111111111111111 * (1.0 / x)));
	elseif (y <= 7.6e+174)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = (1.0 + (-1.0 / ((x * -9.0) * (x * -9.0)))) / (1.0 - (-0.1111111111111111 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.08e+133], N[(N[(1.0 - N[(N[(-0.1111111111111111 / x), $MachinePrecision] * N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(0.1111111111111111 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.6e+174], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(-1.0 / N[(N[(x * -9.0), $MachinePrecision] * N[(x * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.08e133

   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. fma-neg 99.6%

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

      11. associate-/r* 99.5%

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

      12. metadata-eval 99.5%

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

      13. *-commutative 99.5%

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

      14. associate-/r* 99.5%

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

      15. distribute-neg-frac 99.5%

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

      16. metadata-eval 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 4.3%

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

      1. sub-neg 4.3%

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

      2. flip-+ 4.2%

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

      3. metadata-eval 4.2%

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

      4. distribute-lft-neg-in 4.2%

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

      5. un-div-inv 4.2%

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

      6. metadata-eval 4.2%

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

      7. distribute-lft-neg-in 4.2%

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

      8. un-div-inv 4.2%

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

      9. metadata-eval 4.2%

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

      10. distribute-lft-neg-in 4.2%

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

      11. un-div-inv 4.2%

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

      12. metadata-eval 4.2%

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

   7. Applied egg-rr 4.2%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 5.2%

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

      3. frac-times 5.2%

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

      4. metadata-eval 5.2%

         \[\leadsto \frac{1 - \frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}{1 - \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}}} \]

      5. metadata-eval 5.2%

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

      6. frac-times 5.2%

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

      7. sqrt-unprod 25.8%

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

      8. add-sqr-sqrt 25.8%

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

      9. clear-num 25.8%

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

      10. associate-/r/ 25.8%

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

   9. Applied egg-rr 25.8%

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

   if -1.08e133 < y < 7.6000000000000004e174

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.6%

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

      10. fma-neg 99.7%

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

      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 0 80.5%

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

   if 7.6000000000000004e174 < 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. fma-neg 99.7%

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

      11. associate-/r* 99.6%

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

      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 4.5%

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

      1. sub-neg 4.5%

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

      2. flip-+ 39.0%

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

      3. metadata-eval 39.0%

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

      4. distribute-lft-neg-in 39.0%

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

      5. un-div-inv 39.0%

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

      6. metadata-eval 39.0%

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

      7. distribute-lft-neg-in 39.0%

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

      8. un-div-inv 39.0%

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

      9. metadata-eval 39.0%

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

      10. distribute-lft-neg-in 39.0%

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

      11. un-div-inv 39.0%

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

      12. metadata-eval 39.0%

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

   7. Applied egg-rr 39.0%

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

      1. clear-num 39.0%

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

      2. clear-num 39.0%

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

      3. frac-times 39.0%

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

      4. metadata-eval 39.0%

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

      5. div-inv 39.0%

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

      6. metadata-eval 39.0%

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

      7. div-inv 39.0%

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

      8. metadata-eval 39.0%

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

   9. Applied egg-rr 39.0%

      \[\leadsto \frac{1 - \color{blue}{\frac{1}{\left(x \cdot -9\right) \cdot \left(x \cdot -9\right)}}}{1 - \frac{-0.1111111111111111}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.08 \cdot 10^{+133}:\\ \;\;\;\;\frac{1 - \frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}{1 - 0.1111111111111111 \cdot \frac{1}{x}}\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{+174}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-1}{\left(x \cdot -9\right) \cdot \left(x \cdot -9\right)}}{1 - \frac{-0.1111111111111111}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+130}:\\ \;\;\;\;\frac{1 - \frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}{1 - 0.1111111111111111 \cdot \frac{1}{x}}\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{+174}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{-0.1111111111111111}{x \cdot \left(x \cdot -9\right)}}{1 - \frac{-0.1111111111111111}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5.5e+130)
   (/
    (- 1.0 (* (/ -0.1111111111111111 x) (/ -0.1111111111111111 x)))
    (- 1.0 (* 0.1111111111111111 (/ 1.0 x))))
   (if (<= y 7.6e+174)
     (+ 1.0 (/ -0.1111111111111111 x))
     (/
      (- 1.0 (/ -0.1111111111111111 (* x (* x -9.0))))
      (- 1.0 (/ -0.1111111111111111 x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5.5e+130) {
		tmp = (1.0 - ((-0.1111111111111111 / x) * (-0.1111111111111111 / x))) / (1.0 - (0.1111111111111111 * (1.0 / x)));
	} else if (y <= 7.6e+174) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (1.0 - (-0.1111111111111111 / (x * (x * -9.0)))) / (1.0 - (-0.1111111111111111 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-5.5d+130)) then
        tmp = (1.0d0 - (((-0.1111111111111111d0) / x) * ((-0.1111111111111111d0) / x))) / (1.0d0 - (0.1111111111111111d0 * (1.0d0 / x)))
    else if (y <= 7.6d+174) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = (1.0d0 - ((-0.1111111111111111d0) / (x * (x * (-9.0d0))))) / (1.0d0 - ((-0.1111111111111111d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -5.5e+130) {
		tmp = (1.0 - ((-0.1111111111111111 / x) * (-0.1111111111111111 / x))) / (1.0 - (0.1111111111111111 * (1.0 / x)));
	} else if (y <= 7.6e+174) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (1.0 - (-0.1111111111111111 / (x * (x * -9.0)))) / (1.0 - (-0.1111111111111111 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5.5e+130:
		tmp = (1.0 - ((-0.1111111111111111 / x) * (-0.1111111111111111 / x))) / (1.0 - (0.1111111111111111 * (1.0 / x)))
	elif y <= 7.6e+174:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = (1.0 - (-0.1111111111111111 / (x * (x * -9.0)))) / (1.0 - (-0.1111111111111111 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5.5e+130)
		tmp = Float64(Float64(1.0 - Float64(Float64(-0.1111111111111111 / x) * Float64(-0.1111111111111111 / x))) / Float64(1.0 - Float64(0.1111111111111111 * Float64(1.0 / x))));
	elseif (y <= 7.6e+174)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(Float64(1.0 - Float64(-0.1111111111111111 / Float64(x * Float64(x * -9.0)))) / Float64(1.0 - Float64(-0.1111111111111111 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -5.5e+130)
		tmp = (1.0 - ((-0.1111111111111111 / x) * (-0.1111111111111111 / x))) / (1.0 - (0.1111111111111111 * (1.0 / x)));
	elseif (y <= 7.6e+174)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = (1.0 - (-0.1111111111111111 / (x * (x * -9.0)))) / (1.0 - (-0.1111111111111111 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5.5e+130], N[(N[(1.0 - N[(N[(-0.1111111111111111 / x), $MachinePrecision] * N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(0.1111111111111111 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.6e+174], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(-0.1111111111111111 / N[(x * N[(x * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.4999999999999997e130

   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. fma-neg 99.6%

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

      11. associate-/r* 99.5%

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

      12. metadata-eval 99.5%

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

      13. *-commutative 99.5%

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

      14. associate-/r* 99.5%

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

      15. distribute-neg-frac 99.5%

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

      16. metadata-eval 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 4.3%

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

      1. sub-neg 4.3%

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

      2. flip-+ 4.2%

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

      3. metadata-eval 4.2%

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

      4. distribute-lft-neg-in 4.2%

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

      5. un-div-inv 4.2%

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

      6. metadata-eval 4.2%

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

      7. distribute-lft-neg-in 4.2%

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

      8. un-div-inv 4.2%

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

      9. metadata-eval 4.2%

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

      10. distribute-lft-neg-in 4.2%

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

      11. un-div-inv 4.2%

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

      12. metadata-eval 4.2%

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

   7. Applied egg-rr 4.2%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 5.2%

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

      3. frac-times 5.2%

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

      4. metadata-eval 5.2%

         \[\leadsto \frac{1 - \frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}{1 - \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}}} \]

      5. metadata-eval 5.2%

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

      6. frac-times 5.2%

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

      7. sqrt-unprod 25.8%

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

      8. add-sqr-sqrt 25.8%

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

      9. clear-num 25.8%

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

      10. associate-/r/ 25.8%

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

   9. Applied egg-rr 25.8%

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

   if -5.4999999999999997e130 < y < 7.6000000000000004e174

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.6%

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

      10. fma-neg 99.7%

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

      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 0 80.5%

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

   if 7.6000000000000004e174 < 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. fma-neg 99.7%

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

      11. associate-/r* 99.6%

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

      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 4.5%

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

      1. sub-neg 4.5%

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

      2. flip-+ 39.0%

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

      3. metadata-eval 39.0%

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

      4. distribute-lft-neg-in 39.0%

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

      5. un-div-inv 39.0%

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

      6. metadata-eval 39.0%

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

      7. distribute-lft-neg-in 39.0%

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

      8. un-div-inv 39.0%

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

      9. metadata-eval 39.0%

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

      10. distribute-lft-neg-in 39.0%

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

      11. un-div-inv 39.0%

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

      12. metadata-eval 39.0%

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

   7. Applied egg-rr 39.0%

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

      1. clear-num 39.0%

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

      2. frac-times 39.0%

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

      3. metadata-eval 39.0%

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

      4. div-inv 39.0%

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

      5. metadata-eval 39.0%

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

   9. Applied egg-rr 39.0%

      \[\leadsto \frac{1 - \color{blue}{\frac{-0.1111111111111111}{\left(x \cdot -9\right) \cdot x}}}{1 - \frac{-0.1111111111111111}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+130}:\\ \;\;\;\;\frac{1 - \frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}{1 - 0.1111111111111111 \cdot \frac{1}{x}}\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{+174}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{-0.1111111111111111}{x \cdot \left(x \cdot -9\right)}}{1 - \frac{-0.1111111111111111}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{-0.1111111111111111}{x}\\ \mathbf{if}\;y \leq -2.4 \cdot 10^{+133}:\\ \;\;\;\;\frac{1 + \frac{\frac{0.012345679012345678}{x}}{x}}{t\_0}\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{+174}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{-0.1111111111111111}{x \cdot \left(x \cdot -9\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ -0.1111111111111111 x))))
   (if (<= y -2.4e+133)
     (/ (+ 1.0 (/ (/ 0.012345679012345678 x) x)) t_0)
     (if (<= y 7.6e+174)
       (+ 1.0 (/ -0.1111111111111111 x))
       (/ (- 1.0 (/ -0.1111111111111111 (* x (* x -9.0)))) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (-0.1111111111111111 / x);
	double tmp;
	if (y <= -2.4e+133) {
		tmp = (1.0 + ((0.012345679012345678 / x) / x)) / t_0;
	} else if (y <= 7.6e+174) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (1.0 - (-0.1111111111111111 / (x * (x * -9.0)))) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - ((-0.1111111111111111d0) / x)
    if (y <= (-2.4d+133)) then
        tmp = (1.0d0 + ((0.012345679012345678d0 / x) / x)) / t_0
    else if (y <= 7.6d+174) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = (1.0d0 - ((-0.1111111111111111d0) / (x * (x * (-9.0d0))))) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - (-0.1111111111111111 / x);
	double tmp;
	if (y <= -2.4e+133) {
		tmp = (1.0 + ((0.012345679012345678 / x) / x)) / t_0;
	} else if (y <= 7.6e+174) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (1.0 - (-0.1111111111111111 / (x * (x * -9.0)))) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (-0.1111111111111111 / x)
	tmp = 0
	if y <= -2.4e+133:
		tmp = (1.0 + ((0.012345679012345678 / x) / x)) / t_0
	elif y <= 7.6e+174:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = (1.0 - (-0.1111111111111111 / (x * (x * -9.0)))) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(-0.1111111111111111 / x))
	tmp = 0.0
	if (y <= -2.4e+133)
		tmp = Float64(Float64(1.0 + Float64(Float64(0.012345679012345678 / x) / x)) / t_0);
	elseif (y <= 7.6e+174)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(Float64(1.0 - Float64(-0.1111111111111111 / Float64(x * Float64(x * -9.0)))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (-0.1111111111111111 / x);
	tmp = 0.0;
	if (y <= -2.4e+133)
		tmp = (1.0 + ((0.012345679012345678 / x) / x)) / t_0;
	elseif (y <= 7.6e+174)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = (1.0 - (-0.1111111111111111 / (x * (x * -9.0)))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.4e+133], N[(N[(1.0 + N[(N[(0.012345679012345678 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 7.6e+174], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(-0.1111111111111111 / N[(x * N[(x * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.3999999999999999e133

   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. fma-neg 99.6%

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

      11. associate-/r* 99.5%

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

      12. metadata-eval 99.5%

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

      13. *-commutative 99.5%

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

      14. associate-/r* 99.5%

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

      15. distribute-neg-frac 99.5%

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

      16. metadata-eval 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 4.3%

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

      1. sub-neg 4.3%

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

      2. flip-+ 4.2%

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

      3. metadata-eval 4.2%

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

      4. distribute-lft-neg-in 4.2%

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

      5. un-div-inv 4.2%

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

      6. metadata-eval 4.2%

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

      7. distribute-lft-neg-in 4.2%

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

      8. un-div-inv 4.2%

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

      9. metadata-eval 4.2%

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

      10. distribute-lft-neg-in 4.2%

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

      11. un-div-inv 4.2%

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

      12. metadata-eval 4.2%

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

   7. Applied egg-rr 4.2%

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

      1. pow1 4.2%

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

      2. metadata-eval 4.2%

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

      3. sqrt-pow1 25.8%

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

      4. pow2 25.8%

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

      5. frac-times 25.8%

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

      6. sqrt-div 25.8%

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

      7. metadata-eval 25.8%

         \[\leadsto \frac{1 - \frac{\sqrt{\color{blue}{0.012345679012345678}}}{\sqrt{x \cdot x}} \cdot \frac{-0.1111111111111111}{x}}{1 - \frac{-0.1111111111111111}{x}} \]

      8. metadata-eval 25.8%

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

      9. sqrt-unprod 25.8%

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

      10. associate-/l/ 25.8%

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

      11. associate-/l/ 25.8%

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

      12. metadata-eval 25.8%

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

      13. add-sqr-sqrt 25.8%

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

      14. distribute-neg-frac 25.8%

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

      15. distribute-lft-neg-in 25.8%

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

      16. associate-*l/ 25.8%

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

      17. distribute-neg-frac2 25.8%

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

      18. associate-*r/ 25.8%

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

      19. metadata-eval 25.8%

          \[\leadsto \frac{1 - \frac{\frac{\color{blue}{0.012345679012345678}}{x}}{-x}}{1 - \frac{-0.1111111111111111}{x}} \]

   9. Applied egg-rr 25.8%

      \[\leadsto \frac{1 - \color{blue}{\frac{\frac{0.012345679012345678}{x}}{-x}}}{1 - \frac{-0.1111111111111111}{x}} \]

   if -2.3999999999999999e133 < y < 7.6000000000000004e174

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.6%

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

      10. fma-neg 99.7%

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

      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 0 80.5%

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

   if 7.6000000000000004e174 < 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. fma-neg 99.7%

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

      11. associate-/r* 99.6%

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

      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 4.5%

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

      1. sub-neg 4.5%

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

      2. flip-+ 39.0%

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

      3. metadata-eval 39.0%

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

      4. distribute-lft-neg-in 39.0%

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

      5. un-div-inv 39.0%

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

      6. metadata-eval 39.0%

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

      7. distribute-lft-neg-in 39.0%

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

      8. un-div-inv 39.0%

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

      9. metadata-eval 39.0%

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

      10. distribute-lft-neg-in 39.0%

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

      11. un-div-inv 39.0%

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

      12. metadata-eval 39.0%

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

   7. Applied egg-rr 39.0%

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

      1. clear-num 39.0%

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

      2. frac-times 39.0%

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

      3. metadata-eval 39.0%

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

      4. div-inv 39.0%

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

      5. metadata-eval 39.0%

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

   9. Applied egg-rr 39.0%

      \[\leadsto \frac{1 - \color{blue}{\frac{-0.1111111111111111}{\left(x \cdot -9\right) \cdot x}}}{1 - \frac{-0.1111111111111111}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+133}:\\ \;\;\;\;\frac{1 + \frac{\frac{0.012345679012345678}{x}}{x}}{1 - \frac{-0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{+174}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{-0.1111111111111111}{x \cdot \left(x \cdot -9\right)}}{1 - \frac{-0.1111111111111111}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 67.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{-0.1111111111111111}{x}\\ \mathbf{if}\;y \leq -4.8 \cdot 10^{+130}:\\ \;\;\;\;\frac{1 + \frac{\frac{0.012345679012345678}{x}}{x}}{t\_0}\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{+174}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ -0.1111111111111111 x))))
   (if (<= y -4.8e+130)
     (/ (+ 1.0 (/ (/ 0.012345679012345678 x) x)) t_0)
     (if (<= y 7.6e+174)
       (+ 1.0 (/ -0.1111111111111111 x))
       (/
        (- 1.0 (* (/ -0.1111111111111111 x) (/ -0.1111111111111111 x)))
        t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (-0.1111111111111111 / x);
	double tmp;
	if (y <= -4.8e+130) {
		tmp = (1.0 + ((0.012345679012345678 / x) / x)) / t_0;
	} else if (y <= 7.6e+174) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (1.0 - ((-0.1111111111111111 / x) * (-0.1111111111111111 / x))) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - ((-0.1111111111111111d0) / x)
    if (y <= (-4.8d+130)) then
        tmp = (1.0d0 + ((0.012345679012345678d0 / x) / x)) / t_0
    else if (y <= 7.6d+174) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = (1.0d0 - (((-0.1111111111111111d0) / x) * ((-0.1111111111111111d0) / x))) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - (-0.1111111111111111 / x);
	double tmp;
	if (y <= -4.8e+130) {
		tmp = (1.0 + ((0.012345679012345678 / x) / x)) / t_0;
	} else if (y <= 7.6e+174) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (1.0 - ((-0.1111111111111111 / x) * (-0.1111111111111111 / x))) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (-0.1111111111111111 / x)
	tmp = 0
	if y <= -4.8e+130:
		tmp = (1.0 + ((0.012345679012345678 / x) / x)) / t_0
	elif y <= 7.6e+174:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = (1.0 - ((-0.1111111111111111 / x) * (-0.1111111111111111 / x))) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(-0.1111111111111111 / x))
	tmp = 0.0
	if (y <= -4.8e+130)
		tmp = Float64(Float64(1.0 + Float64(Float64(0.012345679012345678 / x) / x)) / t_0);
	elseif (y <= 7.6e+174)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(-0.1111111111111111 / x) * Float64(-0.1111111111111111 / x))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (-0.1111111111111111 / x);
	tmp = 0.0;
	if (y <= -4.8e+130)
		tmp = (1.0 + ((0.012345679012345678 / x) / x)) / t_0;
	elseif (y <= 7.6e+174)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = (1.0 - ((-0.1111111111111111 / x) * (-0.1111111111111111 / x))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.8e+130], N[(N[(1.0 + N[(N[(0.012345679012345678 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 7.6e+174], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(-0.1111111111111111 / x), $MachinePrecision] * N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.80000000000000048e130

   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. fma-neg 99.6%

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

      11. associate-/r* 99.5%

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

      12. metadata-eval 99.5%

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

      13. *-commutative 99.5%

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

      14. associate-/r* 99.5%

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

      15. distribute-neg-frac 99.5%

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

      16. metadata-eval 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 4.3%

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

      1. sub-neg 4.3%

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

      2. flip-+ 4.2%

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

      3. metadata-eval 4.2%

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

      4. distribute-lft-neg-in 4.2%

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

      5. un-div-inv 4.2%

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

      6. metadata-eval 4.2%

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

      7. distribute-lft-neg-in 4.2%

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

      8. un-div-inv 4.2%

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

      9. metadata-eval 4.2%

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

      10. distribute-lft-neg-in 4.2%

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

      11. un-div-inv 4.2%

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

      12. metadata-eval 4.2%

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

   7. Applied egg-rr 4.2%

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

      1. pow1 4.2%

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

      2. metadata-eval 4.2%

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

      3. sqrt-pow1 25.8%

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

      4. pow2 25.8%

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

      5. frac-times 25.8%

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

      6. sqrt-div 25.8%

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

      7. metadata-eval 25.8%

         \[\leadsto \frac{1 - \frac{\sqrt{\color{blue}{0.012345679012345678}}}{\sqrt{x \cdot x}} \cdot \frac{-0.1111111111111111}{x}}{1 - \frac{-0.1111111111111111}{x}} \]

      8. metadata-eval 25.8%

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

      9. sqrt-unprod 25.8%

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

      10. associate-/l/ 25.8%

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

      11. associate-/l/ 25.8%

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

      12. metadata-eval 25.8%

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

      13. add-sqr-sqrt 25.8%

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

      14. distribute-neg-frac 25.8%

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

      15. distribute-lft-neg-in 25.8%

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

      16. associate-*l/ 25.8%

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

      17. distribute-neg-frac2 25.8%

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

      18. associate-*r/ 25.8%

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

      19. metadata-eval 25.8%

          \[\leadsto \frac{1 - \frac{\frac{\color{blue}{0.012345679012345678}}{x}}{-x}}{1 - \frac{-0.1111111111111111}{x}} \]

   9. Applied egg-rr 25.8%

      \[\leadsto \frac{1 - \color{blue}{\frac{\frac{0.012345679012345678}{x}}{-x}}}{1 - \frac{-0.1111111111111111}{x}} \]

   if -4.80000000000000048e130 < y < 7.6000000000000004e174

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.6%

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

      10. fma-neg 99.7%

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

      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 0 80.5%

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

   if 7.6000000000000004e174 < 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. fma-neg 99.7%

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

      11. associate-/r* 99.6%

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

      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 4.5%

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

      1. sub-neg 4.5%

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

      2. flip-+ 39.0%

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

      3. metadata-eval 39.0%

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

      4. distribute-lft-neg-in 39.0%

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

      5. un-div-inv 39.0%

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

      6. metadata-eval 39.0%

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

      7. distribute-lft-neg-in 39.0%

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

      8. un-div-inv 39.0%

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

      9. metadata-eval 39.0%

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

      10. distribute-lft-neg-in 39.0%

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

      11. un-div-inv 39.0%

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

      12. metadata-eval 39.0%

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

   7. Applied egg-rr 39.0%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+130}:\\ \;\;\;\;\frac{1 + \frac{\frac{0.012345679012345678}{x}}{x}}{1 - \frac{-0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{+174}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}{1 - \frac{-0.1111111111111111}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 67.0% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{0.012345679012345678}{x}}{x}\\ t_1 := 1 - \frac{-0.1111111111111111}{x}\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{+132}:\\ \;\;\;\;\frac{1 + t\_0}{t\_1}\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{+174}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ 0.012345679012345678 x) x))
        (t_1 (- 1.0 (/ -0.1111111111111111 x))))
   (if (<= y -5.8e+132)
     (/ (+ 1.0 t_0) t_1)
     (if (<= y 7.6e+174)
       (+ 1.0 (/ -0.1111111111111111 x))
       (/ (- 1.0 t_0) t_1)))))

C

double code(double x, double y) {
	double t_0 = (0.012345679012345678 / x) / x;
	double t_1 = 1.0 - (-0.1111111111111111 / x);
	double tmp;
	if (y <= -5.8e+132) {
		tmp = (1.0 + t_0) / t_1;
	} else if (y <= 7.6e+174) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (1.0 - t_0) / t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (0.012345679012345678d0 / x) / x
    t_1 = 1.0d0 - ((-0.1111111111111111d0) / x)
    if (y <= (-5.8d+132)) then
        tmp = (1.0d0 + t_0) / t_1
    else if (y <= 7.6d+174) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = (1.0d0 - t_0) / t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (0.012345679012345678 / x) / x;
	double t_1 = 1.0 - (-0.1111111111111111 / x);
	double tmp;
	if (y <= -5.8e+132) {
		tmp = (1.0 + t_0) / t_1;
	} else if (y <= 7.6e+174) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (1.0 - t_0) / t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (0.012345679012345678 / x) / x
	t_1 = 1.0 - (-0.1111111111111111 / x)
	tmp = 0
	if y <= -5.8e+132:
		tmp = (1.0 + t_0) / t_1
	elif y <= 7.6e+174:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = (1.0 - t_0) / t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(0.012345679012345678 / x) / x)
	t_1 = Float64(1.0 - Float64(-0.1111111111111111 / x))
	tmp = 0.0
	if (y <= -5.8e+132)
		tmp = Float64(Float64(1.0 + t_0) / t_1);
	elseif (y <= 7.6e+174)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(Float64(1.0 - t_0) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (0.012345679012345678 / x) / x;
	t_1 = 1.0 - (-0.1111111111111111 / x);
	tmp = 0.0;
	if (y <= -5.8e+132)
		tmp = (1.0 + t_0) / t_1;
	elseif (y <= 7.6e+174)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = (1.0 - t_0) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(0.012345679012345678 / x), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.8e+132], N[(N[(1.0 + t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[y, 7.6e+174], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.7999999999999997e132

   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. fma-neg 99.6%

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

      11. associate-/r* 99.5%

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

      12. metadata-eval 99.5%

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

      13. *-commutative 99.5%

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

      14. associate-/r* 99.5%

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

      15. distribute-neg-frac 99.5%

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

      16. metadata-eval 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 4.3%

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

      1. sub-neg 4.3%

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

      2. flip-+ 4.2%

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

      3. metadata-eval 4.2%

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

      4. distribute-lft-neg-in 4.2%

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

      5. un-div-inv 4.2%

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

      6. metadata-eval 4.2%

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

      7. distribute-lft-neg-in 4.2%

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

      8. un-div-inv 4.2%

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

      9. metadata-eval 4.2%

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

      10. distribute-lft-neg-in 4.2%

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

      11. un-div-inv 4.2%

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

      12. metadata-eval 4.2%

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

   7. Applied egg-rr 4.2%

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

      1. pow1 4.2%

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

      2. metadata-eval 4.2%

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

      3. sqrt-pow1 25.8%

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

      4. pow2 25.8%

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

      5. frac-times 25.8%

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

      6. sqrt-div 25.8%

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

      7. metadata-eval 25.8%

         \[\leadsto \frac{1 - \frac{\sqrt{\color{blue}{0.012345679012345678}}}{\sqrt{x \cdot x}} \cdot \frac{-0.1111111111111111}{x}}{1 - \frac{-0.1111111111111111}{x}} \]

      8. metadata-eval 25.8%

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

      9. sqrt-unprod 25.8%

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

      10. associate-/l/ 25.8%

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

      11. associate-/l/ 25.8%

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

      12. metadata-eval 25.8%

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

      13. add-sqr-sqrt 25.8%

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

      14. distribute-neg-frac 25.8%

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

      15. distribute-lft-neg-in 25.8%

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

      16. associate-*l/ 25.8%

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

      17. distribute-neg-frac2 25.8%

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

      18. associate-*r/ 25.8%

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

      19. metadata-eval 25.8%

          \[\leadsto \frac{1 - \frac{\frac{\color{blue}{0.012345679012345678}}{x}}{-x}}{1 - \frac{-0.1111111111111111}{x}} \]

   9. Applied egg-rr 25.8%

      \[\leadsto \frac{1 - \color{blue}{\frac{\frac{0.012345679012345678}{x}}{-x}}}{1 - \frac{-0.1111111111111111}{x}} \]

   if -5.7999999999999997e132 < y < 7.6000000000000004e174

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.6%

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

      10. fma-neg 99.7%

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

      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 0 80.5%

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

   if 7.6000000000000004e174 < 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. fma-neg 99.7%

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

      11. associate-/r* 99.6%

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

      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 4.5%

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

      1. sub-neg 4.5%

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

      2. flip-+ 39.0%

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

      3. metadata-eval 39.0%

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

      4. distribute-lft-neg-in 39.0%

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

      5. un-div-inv 39.0%

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

      6. metadata-eval 39.0%

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

      7. distribute-lft-neg-in 39.0%

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

      8. un-div-inv 39.0%

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

      9. metadata-eval 39.0%

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

      10. distribute-lft-neg-in 39.0%

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

      11. un-div-inv 39.0%

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

      12. metadata-eval 39.0%

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

   7. Applied egg-rr 39.0%

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

      1. frac-times 39.0%

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

      2. associate-/r* 39.0%

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

      3. metadata-eval 39.0%

         \[\leadsto \frac{1 - \frac{\frac{\color{blue}{0.012345679012345678}}{x}}{x}}{1 - \frac{-0.1111111111111111}{x}} \]

   9. Applied egg-rr 39.0%

      \[\leadsto \frac{1 - \color{blue}{\frac{\frac{0.012345679012345678}{x}}{x}}}{1 - \frac{-0.1111111111111111}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+132}:\\ \;\;\;\;\frac{1 + \frac{\frac{0.012345679012345678}{x}}{x}}{1 - \frac{-0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{+174}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\frac{0.012345679012345678}{x}}{x}}{1 - \frac{-0.1111111111111111}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 64.6% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.6 \cdot 10^{+174}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\frac{0.012345679012345678}{x}}{x}}{1 - \frac{-0.1111111111111111}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.6e+174)
   (+ 1.0 (/ -0.1111111111111111 x))
   (/
    (- 1.0 (/ (/ 0.012345679012345678 x) x))
    (- 1.0 (/ -0.1111111111111111 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.6e+174) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (1.0 - ((0.012345679012345678 / x) / x)) / (1.0 - (-0.1111111111111111 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 7.6d+174) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = (1.0d0 - ((0.012345679012345678d0 / x) / x)) / (1.0d0 - ((-0.1111111111111111d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 7.6e+174) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (1.0 - ((0.012345679012345678 / x) / x)) / (1.0 - (-0.1111111111111111 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7.6e+174:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = (1.0 - ((0.012345679012345678 / x) / x)) / (1.0 - (-0.1111111111111111 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.6e+174)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(0.012345679012345678 / x) / x)) / Float64(1.0 - Float64(-0.1111111111111111 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7.6e+174)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = (1.0 - ((0.012345679012345678 / x) / x)) / (1.0 - (-0.1111111111111111 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.6e+174], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(0.012345679012345678 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.6000000000000004e174

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.6%

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

      10. fma-neg 99.7%

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

      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 0 70.4%

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

   if 7.6000000000000004e174 < 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. fma-neg 99.7%

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

      11. associate-/r* 99.6%

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

      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 4.5%

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

      1. sub-neg 4.5%

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

      2. flip-+ 39.0%

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

      3. metadata-eval 39.0%

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

      4. distribute-lft-neg-in 39.0%

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

      5. un-div-inv 39.0%

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

      6. metadata-eval 39.0%

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

      7. distribute-lft-neg-in 39.0%

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

      8. un-div-inv 39.0%

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

      9. metadata-eval 39.0%

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

      10. distribute-lft-neg-in 39.0%

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

      11. un-div-inv 39.0%

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

      12. metadata-eval 39.0%

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

   7. Applied egg-rr 39.0%

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

      1. frac-times 39.0%

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

      2. associate-/r* 39.0%

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

      3. metadata-eval 39.0%

         \[\leadsto \frac{1 - \frac{\frac{\color{blue}{0.012345679012345678}}{x}}{x}}{1 - \frac{-0.1111111111111111}{x}} \]

   9. Applied egg-rr 39.0%

      \[\leadsto \frac{1 - \color{blue}{\frac{\frac{0.012345679012345678}{x}}{x}}}{1 - \frac{-0.1111111111111111}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 61.3% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 48000:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 48000.0) (/ -0.1111111111111111 x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 48000.0) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 48000.0d0) then
        tmp = (-0.1111111111111111d0) / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 48000.0) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 48000.0:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 48000.0)
		tmp = Float64(-0.1111111111111111 / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 48000.0)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 48000.0], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 48000

   1. Initial program 99.5%

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

      1. associate--l- 99.5%

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

      2. sub-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. distribute-neg-in 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. sub-neg 99.5%

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

      7. neg-mul-1 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-/l* 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.4%

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

      12. metadata-eval 99.4%

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

      13. *-commutative 99.4%

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

      14. associate-/r* 99.5%

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

      15. distribute-neg-frac 99.5%

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

      16. metadata-eval 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 60.6%

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

   6. Taylor expanded in x around 0 60.2%

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

   if 48000 < 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 99.7%

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

   4. Taylor expanded in y around 0 64.7%

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

Alternative 17: 62.4% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{-0.1111111111111111}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0 + (-0.1111111111111111 / x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate--l- 99.7%

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

   2. sub-neg 99.7%

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

   3. +-commutative 99.7%

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

   4. distribute-neg-in 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. sub-neg 99.7%

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

   7. neg-mul-1 99.7%

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

   8. *-commutative 99.7%

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

   9. associate-/l* 99.6%

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

   10. fma-neg 99.7%

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

   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 0 62.7%

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

Alternative 18: 32.0% 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 69.1%

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

4. Taylor expanded in y around 0 33.0%

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

Developer target: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024103 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

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

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