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

Percentage Accurate: 99.7% → 99.7%

Time: 14.3s

Alternatives: 17

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Final simplification 99.7%

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

Alternative 2: 90.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+17} \lor \neg \left(y \leq 2.8 \cdot 10^{+91}\right):\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -8e+17) (not (<= y 2.8e+91)))
   (* y (* -0.3333333333333333 (sqrt (/ 1.0 x))))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -8e+17) || !(y <= 2.8e+91)) {
		tmp = y * (-0.3333333333333333 * sqrt((1.0 / x)));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-8d+17)) .or. (.not. (y <= 2.8d+91))) then
        tmp = y * ((-0.3333333333333333d0) * sqrt((1.0d0 / x)))
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -8e+17) || !(y <= 2.8e+91)) {
		tmp = y * (-0.3333333333333333 * Math.sqrt((1.0 / x)));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -8e+17) or not (y <= 2.8e+91):
		tmp = y * (-0.3333333333333333 * math.sqrt((1.0 / x)))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -8e+17) || !(y <= 2.8e+91))
		tmp = Float64(y * Float64(-0.3333333333333333 * sqrt(Float64(1.0 / x))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -8e+17) || ~((y <= 2.8e+91)))
		tmp = y * (-0.3333333333333333 * sqrt((1.0 / x)));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -8e+17], N[Not[LessEqual[y, 2.8e+91]], $MachinePrecision]], N[(y * N[(-0.3333333333333333 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8e17 or 2.7999999999999999e91 < y

   1. Initial program 99.6%

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

      1. frac-2neg 99.6%

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

      2. div-inv 99.5%

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

      3. *-commutative 99.5%

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

      4. distribute-rgt-neg-in 99.5%

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

      5. metadata-eval 99.5%

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

      6. metadata-eval 99.5%

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

      7. div-inv 99.4%

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

      8. clear-num 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in x around 0 95.4%

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

   6. Taylor expanded in y around inf 92.8%

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

      1. associate-*r* 93.5%

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

   8. Simplified 93.5%

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

   if -8e17 < y < 2.7999999999999999e91

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r- 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*r/ 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 98.1%

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

      1. cancel-sign-sub-inv 98.1%

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

      2. metadata-eval 98.1%

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

      3. associate-*r/ 98.2%

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

      4. metadata-eval 98.2%

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

      5. +-commutative 98.2%

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

   7. Simplified 98.2%

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

      1. add-cube-cbrt 97.3%

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

      2. pow3 97.3%

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

   9. Applied egg-rr 97.3%

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

       1. rem-cube-cbrt 98.2%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 45.2%

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

       4. frac-times 45.2%

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

       5. metadata-eval 45.2%

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

       6. metadata-eval 45.2%

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

       7. frac-times 45.2%

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

       8. sqrt-unprod 45.2%

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

       9. add-sqr-sqrt 45.2%

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

       10. metadata-eval 45.2%

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

       11. distribute-neg-frac 45.2%

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

       12. clear-num 45.2%

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

       13. distribute-neg-frac 45.2%

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

       14. metadata-eval 45.2%

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

       15. clear-num 45.2%

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

       16. add-sqr-sqrt 0.0%

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

       17. sqrt-unprod 69.3%

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

       18. frac-times 69.3%

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

       19. metadata-eval 69.3%

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

       20. metadata-eval 69.3%

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

       21. frac-times 69.3%

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

       22. sqrt-unprod 97.9%

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

       23. add-sqr-sqrt 98.1%

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

       24. clear-num 98.1%

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

       25. div-inv 98.3%

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

       26. metadata-eval 98.3%

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

   11. Applied egg-rr 98.3%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+17} \lor \neg \left(y \leq 2.8 \cdot 10^{+91}\right):\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+17} \lor \neg \left(y \leq 2.8 \cdot 10^{+91}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -8e+17) (not (<= y 2.8e+91)))
   (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -8e+17) || !(y <= 2.8e+91)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-8d+17)) .or. (.not. (y <= 2.8d+91))) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -8e+17) || !(y <= 2.8e+91)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -8e+17) or not (y <= 2.8e+91):
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -8e+17) || !(y <= 2.8e+91))
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -8e+17) || ~((y <= 2.8e+91)))
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -8e+17], N[Not[LessEqual[y, 2.8e+91]], $MachinePrecision]], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8e17 or 2.7999999999999999e91 < y

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. distribute-frac-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r- 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r- 99.6%

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

      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-*r/ 99.4%

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

      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* 99.5%

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

      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* 99.4%

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

      15. distribute-neg-frac 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 96.8%

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

      1. associate-*r* 97.6%

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

   7. Simplified 97.6%

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

      1. sqrt-div 97.5%

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

      2. metadata-eval 97.5%

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

      3. div-inv 97.6%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 2.5%

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

      6. sqr-neg 2.5%

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

      7. sqrt-unprod 2.5%

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

      8. add-sqr-sqrt 2.5%

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

      9. neg-sub0 2.5%

         \[\leadsto 1 + \color{blue}{\left(0 - \frac{-0.3333333333333333}{\sqrt{x}}\right)} \cdot y \]

      10. sub-neg 2.5%

          \[\leadsto 1 + \color{blue}{\left(0 + \left(-\frac{-0.3333333333333333}{\sqrt{x}}\right)\right)} \cdot y \]

      11. add-sqr-sqrt 2.5%

          \[\leadsto 1 + \left(0 + \color{blue}{\sqrt{-\frac{-0.3333333333333333}{\sqrt{x}}} \cdot \sqrt{-\frac{-0.3333333333333333}{\sqrt{x}}}}\right) \cdot y \]

      12. sqrt-unprod 2.5%

          \[\leadsto 1 + \left(0 + \color{blue}{\sqrt{\left(-\frac{-0.3333333333333333}{\sqrt{x}}\right) \cdot \left(-\frac{-0.3333333333333333}{\sqrt{x}}\right)}}\right) \cdot y \]

      13. sqr-neg 2.5%

          \[\leadsto 1 + \left(0 + \sqrt{\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}} \cdot \frac{-0.3333333333333333}{\sqrt{x}}}}\right) \cdot y \]

      14. sqrt-unprod 0.0%

          \[\leadsto 1 + \left(0 + \color{blue}{\sqrt{\frac{-0.3333333333333333}{\sqrt{x}}} \cdot \sqrt{\frac{-0.3333333333333333}{\sqrt{x}}}}\right) \cdot y \]

      15. add-sqr-sqrt 97.6%

          \[\leadsto 1 + \left(0 + \color{blue}{\frac{-0.3333333333333333}{\sqrt{x}}}\right) \cdot y \]

   9. Applied egg-rr 97.6%

      \[\leadsto 1 + \color{blue}{\left(0 + \frac{-0.3333333333333333}{\sqrt{x}}\right)} \cdot y \]
   10. Step-by-step derivation

       1. +-lft-identity 97.6%

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

   11. Simplified 97.6%

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

       1. associate-*l/ 97.6%

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

       2. clear-num 97.5%

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

   13. Applied egg-rr 97.5%

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

       1. associate-/r/ 97.5%

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

       2. associate-*l/ 97.6%

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

       3. associate-*r/ 97.6%

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

       4. associate-*r/ 96.8%

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

       5. *-lft-identity 96.8%

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

   15. Simplified 96.8%

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

   if -8e17 < y < 2.7999999999999999e91

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r- 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*r/ 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 98.1%

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

      1. cancel-sign-sub-inv 98.1%

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

      2. metadata-eval 98.1%

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

      3. associate-*r/ 98.2%

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

      4. metadata-eval 98.2%

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

      5. +-commutative 98.2%

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

   7. Simplified 98.2%

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

      1. add-cube-cbrt 97.3%

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

      2. pow3 97.3%

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

   9. Applied egg-rr 97.3%

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

       1. rem-cube-cbrt 98.2%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 45.2%

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

       4. frac-times 45.2%

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

       5. metadata-eval 45.2%

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

       6. metadata-eval 45.2%

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

       7. frac-times 45.2%

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

       8. sqrt-unprod 45.2%

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

       9. add-sqr-sqrt 45.2%

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

       10. metadata-eval 45.2%

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

       11. distribute-neg-frac 45.2%

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

       12. clear-num 45.2%

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

       13. distribute-neg-frac 45.2%

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

       14. metadata-eval 45.2%

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

       15. clear-num 45.2%

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

       16. add-sqr-sqrt 0.0%

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

       17. sqrt-unprod 69.3%

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

       18. frac-times 69.3%

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

       19. metadata-eval 69.3%

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

       20. metadata-eval 69.3%

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

       21. frac-times 69.3%

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

       22. sqrt-unprod 97.9%

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

       23. add-sqr-sqrt 98.1%

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

       24. clear-num 98.1%

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

       25. div-inv 98.3%

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

       26. metadata-eval 98.3%

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

   11. Applied egg-rr 98.3%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+17} \lor \neg \left(y \leq 2.8 \cdot 10^{+91}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+17} \lor \neg \left(y \leq 2.8 \cdot 10^{+91}\right):\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -7.8e+17) (not (<= y 2.8e+91)))
   (+ 1.0 (* y (/ -0.3333333333333333 (sqrt x))))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -7.8e+17) || !(y <= 2.8e+91)) {
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-7.8d+17)) .or. (.not. (y <= 2.8d+91))) then
        tmp = 1.0d0 + (y * ((-0.3333333333333333d0) / sqrt(x)))
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -7.8e+17) || !(y <= 2.8e+91)) {
		tmp = 1.0 + (y * (-0.3333333333333333 / Math.sqrt(x)));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -7.8e+17) or not (y <= 2.8e+91):
		tmp = 1.0 + (y * (-0.3333333333333333 / math.sqrt(x)))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -7.8e+17) || !(y <= 2.8e+91))
		tmp = Float64(1.0 + Float64(y * Float64(-0.3333333333333333 / sqrt(x))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -7.8e+17) || ~((y <= 2.8e+91)))
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -7.8e+17], N[Not[LessEqual[y, 2.8e+91]], $MachinePrecision]], N[(1.0 + N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.8e17 or 2.7999999999999999e91 < y

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. distribute-frac-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r- 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r- 99.6%

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

      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-*r/ 99.4%

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

      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* 99.5%

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

      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* 99.4%

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

      15. distribute-neg-frac 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 96.8%

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

      1. associate-*r* 97.6%

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

   7. Simplified 97.6%

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

      1. sqrt-div 97.5%

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

      2. metadata-eval 97.5%

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

      3. div-inv 97.6%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 2.5%

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

      6. sqr-neg 2.5%

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

      7. sqrt-unprod 2.5%

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

      8. add-sqr-sqrt 2.5%

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

      9. neg-sub0 2.5%

         \[\leadsto 1 + \color{blue}{\left(0 - \frac{-0.3333333333333333}{\sqrt{x}}\right)} \cdot y \]

      10. sub-neg 2.5%

          \[\leadsto 1 + \color{blue}{\left(0 + \left(-\frac{-0.3333333333333333}{\sqrt{x}}\right)\right)} \cdot y \]

      11. add-sqr-sqrt 2.5%

          \[\leadsto 1 + \left(0 + \color{blue}{\sqrt{-\frac{-0.3333333333333333}{\sqrt{x}}} \cdot \sqrt{-\frac{-0.3333333333333333}{\sqrt{x}}}}\right) \cdot y \]

      12. sqrt-unprod 2.5%

          \[\leadsto 1 + \left(0 + \color{blue}{\sqrt{\left(-\frac{-0.3333333333333333}{\sqrt{x}}\right) \cdot \left(-\frac{-0.3333333333333333}{\sqrt{x}}\right)}}\right) \cdot y \]

      13. sqr-neg 2.5%

          \[\leadsto 1 + \left(0 + \sqrt{\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}} \cdot \frac{-0.3333333333333333}{\sqrt{x}}}}\right) \cdot y \]

      14. sqrt-unprod 0.0%

          \[\leadsto 1 + \left(0 + \color{blue}{\sqrt{\frac{-0.3333333333333333}{\sqrt{x}}} \cdot \sqrt{\frac{-0.3333333333333333}{\sqrt{x}}}}\right) \cdot y \]

      15. add-sqr-sqrt 97.6%

          \[\leadsto 1 + \left(0 + \color{blue}{\frac{-0.3333333333333333}{\sqrt{x}}}\right) \cdot y \]

   9. Applied egg-rr 97.6%

      \[\leadsto 1 + \color{blue}{\left(0 + \frac{-0.3333333333333333}{\sqrt{x}}\right)} \cdot y \]
   10. Step-by-step derivation

       1. +-lft-identity 97.6%

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

   11. Simplified 97.6%

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

   if -7.8e17 < y < 2.7999999999999999e91

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r- 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*r/ 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 98.1%

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

      1. cancel-sign-sub-inv 98.1%

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

      2. metadata-eval 98.1%

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

      3. associate-*r/ 98.2%

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

      4. metadata-eval 98.2%

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

      5. +-commutative 98.2%

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

   7. Simplified 98.2%

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

      1. add-cube-cbrt 97.3%

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

      2. pow3 97.3%

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

   9. Applied egg-rr 97.3%

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

       1. rem-cube-cbrt 98.2%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 45.2%

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

       4. frac-times 45.2%

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

       5. metadata-eval 45.2%

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

       6. metadata-eval 45.2%

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

       7. frac-times 45.2%

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

       8. sqrt-unprod 45.2%

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

       9. add-sqr-sqrt 45.2%

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

       10. metadata-eval 45.2%

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

       11. distribute-neg-frac 45.2%

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

       12. clear-num 45.2%

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

       13. distribute-neg-frac 45.2%

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

       14. metadata-eval 45.2%

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

       15. clear-num 45.2%

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

       16. add-sqr-sqrt 0.0%

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

       17. sqrt-unprod 69.3%

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

       18. frac-times 69.3%

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

       19. metadata-eval 69.3%

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

       20. metadata-eval 69.3%

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

       21. frac-times 69.3%

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

       22. sqrt-unprod 97.9%

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

       23. add-sqr-sqrt 98.1%

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

       24. clear-num 98.1%

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

       25. div-inv 98.3%

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

       26. metadata-eval 98.3%

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

   11. Applied egg-rr 98.3%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+17} \lor \neg \left(y \leq 2.8 \cdot 10^{+91}\right):\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+17} \lor \neg \left(y \leq 2.8 \cdot 10^{+91}\right):\\ \;\;\;\;1 + \frac{\frac{y}{-3}}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -8e+17) (not (<= y 2.8e+91)))
   (+ 1.0 (/ (/ y -3.0) (sqrt x)))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -8e+17) || !(y <= 2.8e+91)) {
		tmp = 1.0 + ((y / -3.0) / sqrt(x));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-8d+17)) .or. (.not. (y <= 2.8d+91))) then
        tmp = 1.0d0 + ((y / (-3.0d0)) / sqrt(x))
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -8e+17) || !(y <= 2.8e+91)) {
		tmp = 1.0 + ((y / -3.0) / Math.sqrt(x));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -8e+17) or not (y <= 2.8e+91):
		tmp = 1.0 + ((y / -3.0) / math.sqrt(x))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -8e+17) || !(y <= 2.8e+91))
		tmp = Float64(1.0 + Float64(Float64(y / -3.0) / sqrt(x)));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -8e+17) || ~((y <= 2.8e+91)))
		tmp = 1.0 + ((y / -3.0) / sqrt(x));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -8e+17], N[Not[LessEqual[y, 2.8e+91]], $MachinePrecision]], N[(1.0 + N[(N[(y / -3.0), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8e17 or 2.7999999999999999e91 < y

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. distribute-frac-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r- 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r- 99.6%

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

      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-*r/ 99.4%

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

      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* 99.5%

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

      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* 99.4%

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

      15. distribute-neg-frac 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 96.8%

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

      1. associate-*r* 97.6%

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

   7. Simplified 97.6%

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

      1. sqrt-div 97.5%

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

      2. metadata-eval 97.5%

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

      3. div-inv 97.6%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 2.5%

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

      6. sqr-neg 2.5%

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

      7. sqrt-unprod 2.5%

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

      8. add-sqr-sqrt 2.5%

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

      9. neg-sub0 2.5%

         \[\leadsto 1 + \color{blue}{\left(0 - \frac{-0.3333333333333333}{\sqrt{x}}\right)} \cdot y \]

      10. sub-neg 2.5%

          \[\leadsto 1 + \color{blue}{\left(0 + \left(-\frac{-0.3333333333333333}{\sqrt{x}}\right)\right)} \cdot y \]

      11. add-sqr-sqrt 2.5%

          \[\leadsto 1 + \left(0 + \color{blue}{\sqrt{-\frac{-0.3333333333333333}{\sqrt{x}}} \cdot \sqrt{-\frac{-0.3333333333333333}{\sqrt{x}}}}\right) \cdot y \]

      12. sqrt-unprod 2.5%

          \[\leadsto 1 + \left(0 + \color{blue}{\sqrt{\left(-\frac{-0.3333333333333333}{\sqrt{x}}\right) \cdot \left(-\frac{-0.3333333333333333}{\sqrt{x}}\right)}}\right) \cdot y \]

      13. sqr-neg 2.5%

          \[\leadsto 1 + \left(0 + \sqrt{\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}} \cdot \frac{-0.3333333333333333}{\sqrt{x}}}}\right) \cdot y \]

      14. sqrt-unprod 0.0%

          \[\leadsto 1 + \left(0 + \color{blue}{\sqrt{\frac{-0.3333333333333333}{\sqrt{x}}} \cdot \sqrt{\frac{-0.3333333333333333}{\sqrt{x}}}}\right) \cdot y \]

      15. add-sqr-sqrt 97.6%

          \[\leadsto 1 + \left(0 + \color{blue}{\frac{-0.3333333333333333}{\sqrt{x}}}\right) \cdot y \]

   9. Applied egg-rr 97.6%

      \[\leadsto 1 + \color{blue}{\left(0 + \frac{-0.3333333333333333}{\sqrt{x}}\right)} \cdot y \]
   10. Step-by-step derivation

       1. +-lft-identity 97.6%

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

   11. Simplified 97.6%

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

       1. associate-*l/ 97.6%

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

       2. *-un-lft-identity 97.6%

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

       3. times-frac 96.8%

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

       4. metadata-eval 96.8%

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

       5. metadata-eval 96.8%

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

       6. times-frac 97.7%

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

       7. *-un-lft-identity 97.7%

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

       8. associate-/r* 97.8%

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

   13. Applied egg-rr 97.8%

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

   if -8e17 < y < 2.7999999999999999e91

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r- 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*r/ 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 98.1%

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

      1. cancel-sign-sub-inv 98.1%

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

      2. metadata-eval 98.1%

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

      3. associate-*r/ 98.2%

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

      4. metadata-eval 98.2%

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

      5. +-commutative 98.2%

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

   7. Simplified 98.2%

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

      1. add-cube-cbrt 97.3%

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

      2. pow3 97.3%

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

   9. Applied egg-rr 97.3%

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

       1. rem-cube-cbrt 98.2%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 45.2%

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

       4. frac-times 45.2%

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

       5. metadata-eval 45.2%

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

       6. metadata-eval 45.2%

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

       7. frac-times 45.2%

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

       8. sqrt-unprod 45.2%

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

       9. add-sqr-sqrt 45.2%

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

       10. metadata-eval 45.2%

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

       11. distribute-neg-frac 45.2%

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

       12. clear-num 45.2%

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

       13. distribute-neg-frac 45.2%

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

       14. metadata-eval 45.2%

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

       15. clear-num 45.2%

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

       16. add-sqr-sqrt 0.0%

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

       17. sqrt-unprod 69.3%

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

       18. frac-times 69.3%

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

       19. metadata-eval 69.3%

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

       20. metadata-eval 69.3%

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

       21. frac-times 69.3%

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

       22. sqrt-unprod 97.9%

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

       23. add-sqr-sqrt 98.1%

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

       24. clear-num 98.1%

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

       25. div-inv 98.3%

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

       26. metadata-eval 98.3%

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

   11. Applied egg-rr 98.3%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+17} \lor \neg \left(y \leq 2.8 \cdot 10^{+91}\right):\\ \;\;\;\;1 + \frac{\frac{y}{-3}}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+17}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+91}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.8e+17)
   (+ 1.0 (/ -0.3333333333333333 (/ (sqrt x) y)))
   (if (<= y 2.8e+91)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (+ 1.0 (* y (/ -0.3333333333333333 (sqrt x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.8e+17) {
		tmp = 1.0 + (-0.3333333333333333 / (sqrt(x) / y));
	} else if (y <= 2.8e+91) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-7.8d+17)) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) / (sqrt(x) / y))
    else if (y <= 2.8d+91) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = 1.0d0 + (y * ((-0.3333333333333333d0) / sqrt(x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7.8e+17) {
		tmp = 1.0 + (-0.3333333333333333 / (Math.sqrt(x) / y));
	} else if (y <= 2.8e+91) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 + (y * (-0.3333333333333333 / Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.8e+17:
		tmp = 1.0 + (-0.3333333333333333 / (math.sqrt(x) / y))
	elif y <= 2.8e+91:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = 1.0 + (y * (-0.3333333333333333 / math.sqrt(x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.8e+17)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 / Float64(sqrt(x) / y)));
	elseif (y <= 2.8e+91)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(1.0 + Float64(y * Float64(-0.3333333333333333 / sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.8e+17)
		tmp = 1.0 + (-0.3333333333333333 / (sqrt(x) / y));
	elseif (y <= 2.8e+91)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.8e+17], N[(1.0 + N[(-0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.8e+91], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.8e17

   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. distribute-frac-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r- 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-+r- 99.5%

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

      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-*r/ 99.4%

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

      10. fma-neg 99.4%

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

      11. associate-/r* 99.4%

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

      12. metadata-eval 99.4%

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

      13. *-commutative 99.4%

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

      14. associate-/r* 99.4%

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

      15. distribute-neg-frac 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 97.8%

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

      1. associate-*r* 97.9%

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

   7. Simplified 97.9%

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

      1. sqrt-div 97.9%

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

      2. metadata-eval 97.9%

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

      3. div-inv 97.8%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 1.9%

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

      6. sqr-neg 1.9%

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

      7. sqrt-unprod 1.9%

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

      8. add-sqr-sqrt 1.9%

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

      9. neg-sub0 1.9%

         \[\leadsto 1 + \color{blue}{\left(0 - \frac{-0.3333333333333333}{\sqrt{x}}\right)} \cdot y \]

      10. sub-neg 1.9%

          \[\leadsto 1 + \color{blue}{\left(0 + \left(-\frac{-0.3333333333333333}{\sqrt{x}}\right)\right)} \cdot y \]

      11. add-sqr-sqrt 1.9%

          \[\leadsto 1 + \left(0 + \color{blue}{\sqrt{-\frac{-0.3333333333333333}{\sqrt{x}}} \cdot \sqrt{-\frac{-0.3333333333333333}{\sqrt{x}}}}\right) \cdot y \]

      12. sqrt-unprod 1.9%

          \[\leadsto 1 + \left(0 + \color{blue}{\sqrt{\left(-\frac{-0.3333333333333333}{\sqrt{x}}\right) \cdot \left(-\frac{-0.3333333333333333}{\sqrt{x}}\right)}}\right) \cdot y \]

      13. sqr-neg 1.9%

          \[\leadsto 1 + \left(0 + \sqrt{\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}} \cdot \frac{-0.3333333333333333}{\sqrt{x}}}}\right) \cdot y \]

      14. sqrt-unprod 0.0%

          \[\leadsto 1 + \left(0 + \color{blue}{\sqrt{\frac{-0.3333333333333333}{\sqrt{x}}} \cdot \sqrt{\frac{-0.3333333333333333}{\sqrt{x}}}}\right) \cdot y \]

      15. add-sqr-sqrt 97.8%

          \[\leadsto 1 + \left(0 + \color{blue}{\frac{-0.3333333333333333}{\sqrt{x}}}\right) \cdot y \]

   9. Applied egg-rr 97.8%

      \[\leadsto 1 + \color{blue}{\left(0 + \frac{-0.3333333333333333}{\sqrt{x}}\right)} \cdot y \]
   10. Step-by-step derivation

       1. +-lft-identity 97.8%

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

   11. Simplified 97.8%

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

       1. associate-*l/ 97.9%

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

       2. associate-/l* 97.9%

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

   13. Applied egg-rr 97.9%

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

   if -7.8e17 < y < 2.7999999999999999e91

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r- 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*r/ 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 98.1%

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

      1. cancel-sign-sub-inv 98.1%

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

      2. metadata-eval 98.1%

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

      3. associate-*r/ 98.2%

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

      4. metadata-eval 98.2%

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

      5. +-commutative 98.2%

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

   7. Simplified 98.2%

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

      1. add-cube-cbrt 97.3%

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

      2. pow3 97.3%

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

   9. Applied egg-rr 97.3%

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

       1. rem-cube-cbrt 98.2%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 45.2%

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

       4. frac-times 45.2%

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

       5. metadata-eval 45.2%

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

       6. metadata-eval 45.2%

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

       7. frac-times 45.2%

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

       8. sqrt-unprod 45.2%

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

       9. add-sqr-sqrt 45.2%

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

       10. metadata-eval 45.2%

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

       11. distribute-neg-frac 45.2%

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

       12. clear-num 45.2%

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

       13. distribute-neg-frac 45.2%

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

       14. metadata-eval 45.2%

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

       15. clear-num 45.2%

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

       16. add-sqr-sqrt 0.0%

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

       17. sqrt-unprod 69.3%

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

       18. frac-times 69.3%

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

       19. metadata-eval 69.3%

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

       20. metadata-eval 69.3%

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

       21. frac-times 69.3%

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

       22. sqrt-unprod 97.9%

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

       23. add-sqr-sqrt 98.1%

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

       24. clear-num 98.1%

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

       25. div-inv 98.3%

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

       26. metadata-eval 98.3%

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

   11. Applied egg-rr 98.3%

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

   if 2.7999999999999999e91 < y

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. distribute-frac-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r- 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r- 99.6%

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

      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-*r/ 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* 99.5%

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

      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 inf 96.0%

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

      1. associate-*r* 97.3%

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

   7. Simplified 97.3%

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

      1. sqrt-div 97.2%

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

      2. metadata-eval 97.2%

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

      3. div-inv 97.3%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 3.1%

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

      6. sqr-neg 3.1%

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

      7. sqrt-unprod 3.1%

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

      8. add-sqr-sqrt 3.1%

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

      9. neg-sub0 3.1%

         \[\leadsto 1 + \color{blue}{\left(0 - \frac{-0.3333333333333333}{\sqrt{x}}\right)} \cdot y \]

      10. sub-neg 3.1%

          \[\leadsto 1 + \color{blue}{\left(0 + \left(-\frac{-0.3333333333333333}{\sqrt{x}}\right)\right)} \cdot y \]

      11. add-sqr-sqrt 3.1%

          \[\leadsto 1 + \left(0 + \color{blue}{\sqrt{-\frac{-0.3333333333333333}{\sqrt{x}}} \cdot \sqrt{-\frac{-0.3333333333333333}{\sqrt{x}}}}\right) \cdot y \]

      12. sqrt-unprod 3.1%

          \[\leadsto 1 + \left(0 + \color{blue}{\sqrt{\left(-\frac{-0.3333333333333333}{\sqrt{x}}\right) \cdot \left(-\frac{-0.3333333333333333}{\sqrt{x}}\right)}}\right) \cdot y \]

      13. sqr-neg 3.1%

          \[\leadsto 1 + \left(0 + \sqrt{\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}} \cdot \frac{-0.3333333333333333}{\sqrt{x}}}}\right) \cdot y \]

      14. sqrt-unprod 0.0%

          \[\leadsto 1 + \left(0 + \color{blue}{\sqrt{\frac{-0.3333333333333333}{\sqrt{x}}} \cdot \sqrt{\frac{-0.3333333333333333}{\sqrt{x}}}}\right) \cdot y \]

      15. add-sqr-sqrt 97.3%

          \[\leadsto 1 + \left(0 + \color{blue}{\frac{-0.3333333333333333}{\sqrt{x}}}\right) \cdot y \]

   9. Applied egg-rr 97.3%

      \[\leadsto 1 + \color{blue}{\left(0 + \frac{-0.3333333333333333}{\sqrt{x}}\right)} \cdot y \]
   10. Step-by-step derivation

       1. +-lft-identity 97.3%

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

   11. Simplified 97.3%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+17}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+91}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.106:\\ \;\;\;\;\frac{-0.1111111111111111}{x} - \frac{y}{\sqrt{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{y}{-3}}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.106)
   (- (/ -0.1111111111111111 x) (* (/ y (sqrt x)) 0.3333333333333333))
   (+ 1.0 (/ (/ y -3.0) (sqrt x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.106) {
		tmp = (-0.1111111111111111 / x) - ((y / sqrt(x)) * 0.3333333333333333);
	} else {
		tmp = 1.0 + ((y / -3.0) / sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.106d0) then
        tmp = ((-0.1111111111111111d0) / x) - ((y / sqrt(x)) * 0.3333333333333333d0)
    else
        tmp = 1.0d0 + ((y / (-3.0d0)) / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.106) {
		tmp = (-0.1111111111111111 / x) - ((y / Math.sqrt(x)) * 0.3333333333333333);
	} else {
		tmp = 1.0 + ((y / -3.0) / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.106:
		tmp = (-0.1111111111111111 / x) - ((y / math.sqrt(x)) * 0.3333333333333333)
	else:
		tmp = 1.0 + ((y / -3.0) / math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.106)
		tmp = Float64(Float64(-0.1111111111111111 / x) - Float64(Float64(y / sqrt(x)) * 0.3333333333333333));
	else
		tmp = Float64(1.0 + Float64(Float64(y / -3.0) / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.106)
		tmp = (-0.1111111111111111 / x) - ((y / sqrt(x)) * 0.3333333333333333);
	else
		tmp = 1.0 + ((y / -3.0) / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.106], N[(N[(-0.1111111111111111 / x), $MachinePrecision] - N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y / -3.0), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.105999999999999997

   1. Initial program 99.6%

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

      1. frac-2neg 99.6%

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

      2. div-inv 99.6%

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

      3. *-commutative 99.6%

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

      4. distribute-rgt-neg-in 99.6%

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

      5. metadata-eval 99.6%

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

      6. metadata-eval 99.6%

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

      7. div-inv 99.5%

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

      8. clear-num 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around 0 96.7%

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

      1. distribute-lft-neg-out 96.7%

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

      2. *-commutative 96.7%

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

      3. div-inv 96.7%

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

      4. metadata-eval 96.7%

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

      5. sqrt-div 96.7%

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

      6. neg-sub0 96.7%

         \[\leadsto \frac{-0.1111111111111111}{x} - \color{blue}{\left(0 - \left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right) \cdot y\right)} \]

      7. associate-*l* 96.1%

         \[\leadsto \frac{-0.1111111111111111}{x} - \left(0 - \color{blue}{-0.3333333333333333 \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right)}\right) \]

      8. sqrt-div 96.1%

         \[\leadsto \frac{-0.1111111111111111}{x} - \left(0 - -0.3333333333333333 \cdot \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{x}}} \cdot y\right)\right) \]

      9. metadata-eval 96.1%

         \[\leadsto \frac{-0.1111111111111111}{x} - \left(0 - -0.3333333333333333 \cdot \left(\frac{\color{blue}{1}}{\sqrt{x}} \cdot y\right)\right) \]

      10. associate-*l/ 96.1%

          \[\leadsto \frac{-0.1111111111111111}{x} - \left(0 - -0.3333333333333333 \cdot \color{blue}{\frac{1 \cdot y}{\sqrt{x}}}\right) \]

      11. *-un-lft-identity 96.1%

          \[\leadsto \frac{-0.1111111111111111}{x} - \left(0 - -0.3333333333333333 \cdot \frac{\color{blue}{y}}{\sqrt{x}}\right) \]

   7. Applied egg-rr 96.1%

      \[\leadsto \frac{-0.1111111111111111}{x} - \color{blue}{\left(0 - -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\right)} \]
   8. Step-by-step derivation

      1. neg-sub0 96.1%

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

      2. distribute-lft-neg-in 96.1%

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

      3. metadata-eval 96.1%

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

   9. Simplified 96.1%

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

   if 0.105999999999999997 < x

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r- 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*r/ 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 inf 99.3%

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

      1. associate-*r* 99.3%

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

   7. Simplified 99.3%

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

      1. sqrt-div 99.3%

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

      2. metadata-eval 99.3%

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

      3. div-inv 99.3%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 54.8%

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

      6. sqr-neg 54.8%

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

      7. sqrt-unprod 54.8%

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

      8. add-sqr-sqrt 54.8%

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

      9. neg-sub0 54.8%

         \[\leadsto 1 + \color{blue}{\left(0 - \frac{-0.3333333333333333}{\sqrt{x}}\right)} \cdot y \]

      10. sub-neg 54.8%

          \[\leadsto 1 + \color{blue}{\left(0 + \left(-\frac{-0.3333333333333333}{\sqrt{x}}\right)\right)} \cdot y \]

      11. add-sqr-sqrt 54.8%

          \[\leadsto 1 + \left(0 + \color{blue}{\sqrt{-\frac{-0.3333333333333333}{\sqrt{x}}} \cdot \sqrt{-\frac{-0.3333333333333333}{\sqrt{x}}}}\right) \cdot y \]

      12. sqrt-unprod 54.8%

          \[\leadsto 1 + \left(0 + \color{blue}{\sqrt{\left(-\frac{-0.3333333333333333}{\sqrt{x}}\right) \cdot \left(-\frac{-0.3333333333333333}{\sqrt{x}}\right)}}\right) \cdot y \]

      13. sqr-neg 54.8%

          \[\leadsto 1 + \left(0 + \sqrt{\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}} \cdot \frac{-0.3333333333333333}{\sqrt{x}}}}\right) \cdot y \]

      14. sqrt-unprod 0.0%

          \[\leadsto 1 + \left(0 + \color{blue}{\sqrt{\frac{-0.3333333333333333}{\sqrt{x}}} \cdot \sqrt{\frac{-0.3333333333333333}{\sqrt{x}}}}\right) \cdot y \]

      15. add-sqr-sqrt 99.3%

          \[\leadsto 1 + \left(0 + \color{blue}{\frac{-0.3333333333333333}{\sqrt{x}}}\right) \cdot y \]

   9. Applied egg-rr 99.3%

      \[\leadsto 1 + \color{blue}{\left(0 + \frac{-0.3333333333333333}{\sqrt{x}}\right)} \cdot y \]
   10. Step-by-step derivation

       1. +-lft-identity 99.3%

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

   11. Simplified 99.3%

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

       1. associate-*l/ 99.3%

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

       2. *-un-lft-identity 99.3%

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

       3. times-frac 99.2%

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

       4. metadata-eval 99.2%

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

       5. metadata-eval 99.2%

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

       6. times-frac 99.3%

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

       7. *-un-lft-identity 99.3%

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

       8. associate-/r* 99.3%

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

   13. Applied egg-rr 99.3%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.106:\\ \;\;\;\;\frac{-0.1111111111111111}{x} - \frac{y}{\sqrt{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{y}{-3}}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.11d0) then
        tmp = ((-0.1111111111111111d0) / x) + (y * ((-0.3333333333333333d0) / sqrt(x)))
    else
        tmp = 1.0d0 + ((y / (-3.0d0)) / sqrt(x))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   1. Initial program 99.6%

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

      1. frac-2neg 99.6%

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

      2. div-inv 99.6%

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

      3. *-commutative 99.6%

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

      4. distribute-rgt-neg-in 99.6%

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

      5. metadata-eval 99.6%

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

      6. metadata-eval 99.6%

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

      7. div-inv 99.6%

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

      8. clear-num 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around 0 96.7%

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

   if 0.110000000000000001 < x

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r- 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*r/ 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 inf 99.3%

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

      1. associate-*r* 99.3%

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

   7. Simplified 99.3%

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

      1. sqrt-div 99.3%

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

      2. metadata-eval 99.3%

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

      3. div-inv 99.3%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 55.2%

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

      6. sqr-neg 55.2%

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

      7. sqrt-unprod 55.2%

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

      8. add-sqr-sqrt 55.2%

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

      9. neg-sub0 55.2%

         \[\leadsto 1 + \color{blue}{\left(0 - \frac{-0.3333333333333333}{\sqrt{x}}\right)} \cdot y \]

      10. sub-neg 55.2%

          \[\leadsto 1 + \color{blue}{\left(0 + \left(-\frac{-0.3333333333333333}{\sqrt{x}}\right)\right)} \cdot y \]

      11. add-sqr-sqrt 55.2%

          \[\leadsto 1 + \left(0 + \color{blue}{\sqrt{-\frac{-0.3333333333333333}{\sqrt{x}}} \cdot \sqrt{-\frac{-0.3333333333333333}{\sqrt{x}}}}\right) \cdot y \]

      12. sqrt-unprod 55.2%

          \[\leadsto 1 + \left(0 + \color{blue}{\sqrt{\left(-\frac{-0.3333333333333333}{\sqrt{x}}\right) \cdot \left(-\frac{-0.3333333333333333}{\sqrt{x}}\right)}}\right) \cdot y \]

      13. sqr-neg 55.2%

          \[\leadsto 1 + \left(0 + \sqrt{\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}} \cdot \frac{-0.3333333333333333}{\sqrt{x}}}}\right) \cdot y \]

      14. sqrt-unprod 0.0%

          \[\leadsto 1 + \left(0 + \color{blue}{\sqrt{\frac{-0.3333333333333333}{\sqrt{x}}} \cdot \sqrt{\frac{-0.3333333333333333}{\sqrt{x}}}}\right) \cdot y \]

      15. add-sqr-sqrt 99.3%

          \[\leadsto 1 + \left(0 + \color{blue}{\frac{-0.3333333333333333}{\sqrt{x}}}\right) \cdot y \]

   9. Applied egg-rr 99.3%

      \[\leadsto 1 + \color{blue}{\left(0 + \frac{-0.3333333333333333}{\sqrt{x}}\right)} \cdot y \]
   10. Step-by-step derivation

       1. +-lft-identity 99.3%

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

   11. Simplified 99.3%

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

       1. associate-*l/ 99.3%

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

       2. *-un-lft-identity 99.3%

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

       3. times-frac 99.2%

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

       4. metadata-eval 99.2%

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

       5. metadata-eval 99.2%

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

       6. times-frac 99.3%

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

       7. *-un-lft-identity 99.3%

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

       8. associate-/r* 99.3%

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

   13. Applied egg-rr 99.3%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{-0.1111111111111111}{x} + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{y}{-3}}{\sqrt{x}}\\ \end{array} \]
5. 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* 99.6%

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

   4. metadata-eval 99.6%

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

   5. distribute-frac-neg 99.6%

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

   6. neg-mul-1 99.6%

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

   7. times-frac 99.3%

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

   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. Final simplification 99.3%

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

Alternative 10: 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* 99.6%

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

   4. metadata-eval 99.6%

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

   5. distribute-frac-neg 99.6%

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

   6. neg-mul-1 99.6%

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

   7. times-frac 99.3%

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

   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. associate-*r/ 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 11: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around 0 99.6%

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

4. Final simplification 99.6%

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

Alternative 12: 64.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{0.1111111111111111}{x}\\ t_1 := \frac{1}{t_0}\\ t_2 := \frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}\\ \mathbf{if}\;y \leq -8 \cdot 10^{+17}:\\ \;\;\;\;t_1 + t_2 \cdot \frac{-1}{1 + \frac{-0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+149}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;t_1 - \frac{t_2}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ 0.1111111111111111 x)))
        (t_1 (/ 1.0 t_0))
        (t_2 (* (/ -0.1111111111111111 x) (/ -0.1111111111111111 x))))
   (if (<= y -8e+17)
     (+ t_1 (* t_2 (/ -1.0 (+ 1.0 (/ -0.1111111111111111 x)))))
     (if (<= y 3.4e+149) (+ 1.0 (/ -1.0 (* x 9.0))) (- t_1 (/ t_2 t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (0.1111111111111111 / x);
	double t_1 = 1.0 / t_0;
	double t_2 = (-0.1111111111111111 / x) * (-0.1111111111111111 / x);
	double tmp;
	if (y <= -8e+17) {
		tmp = t_1 + (t_2 * (-1.0 / (1.0 + (-0.1111111111111111 / x))));
	} else if (y <= 3.4e+149) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = t_1 - (t_2 / 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 + (0.1111111111111111d0 / x)
    t_1 = 1.0d0 / t_0
    t_2 = ((-0.1111111111111111d0) / x) * ((-0.1111111111111111d0) / x)
    if (y <= (-8d+17)) then
        tmp = t_1 + (t_2 * ((-1.0d0) / (1.0d0 + ((-0.1111111111111111d0) / x))))
    else if (y <= 3.4d+149) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = t_1 - (t_2 / 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 t_1 = 1.0 / t_0;
	double t_2 = (-0.1111111111111111 / x) * (-0.1111111111111111 / x);
	double tmp;
	if (y <= -8e+17) {
		tmp = t_1 + (t_2 * (-1.0 / (1.0 + (-0.1111111111111111 / x))));
	} else if (y <= 3.4e+149) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = t_1 - (t_2 / t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (0.1111111111111111 / x)
	t_1 = 1.0 / t_0
	t_2 = (-0.1111111111111111 / x) * (-0.1111111111111111 / x)
	tmp = 0
	if y <= -8e+17:
		tmp = t_1 + (t_2 * (-1.0 / (1.0 + (-0.1111111111111111 / x))))
	elif y <= 3.4e+149:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = t_1 - (t_2 / t_0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(0.1111111111111111 / x))
	t_1 = Float64(1.0 / t_0)
	t_2 = Float64(Float64(-0.1111111111111111 / x) * Float64(-0.1111111111111111 / x))
	tmp = 0.0
	if (y <= -8e+17)
		tmp = Float64(t_1 + Float64(t_2 * Float64(-1.0 / Float64(1.0 + Float64(-0.1111111111111111 / x)))));
	elseif (y <= 3.4e+149)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(t_1 - Float64(t_2 / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (0.1111111111111111 / x);
	t_1 = 1.0 / t_0;
	t_2 = (-0.1111111111111111 / x) * (-0.1111111111111111 / x);
	tmp = 0.0;
	if (y <= -8e+17)
		tmp = t_1 + (t_2 * (-1.0 / (1.0 + (-0.1111111111111111 / x))));
	elseif (y <= 3.4e+149)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = t_1 - (t_2 / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-0.1111111111111111 / x), $MachinePrecision] * N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8e+17], N[(t$95$1 + N[(t$95$2 * N[(-1.0 / N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.4e+149], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - N[(t$95$2 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8e17

   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. distribute-frac-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r- 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-+r- 99.5%

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

      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-*r/ 99.4%

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

      10. fma-neg 99.4%

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

      11. associate-/r* 99.4%

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

      12. metadata-eval 99.4%

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

      13. *-commutative 99.4%

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

      14. associate-/r* 99.4%

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

      15. distribute-neg-frac 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around 0 4.8%

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

      1. cancel-sign-sub-inv 4.8%

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

      2. metadata-eval 4.8%

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

      3. associate-*r/ 4.8%

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

      4. metadata-eval 4.8%

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

      5. +-commutative 4.8%

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

   7. Simplified 4.8%

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

      1. add-cube-cbrt 4.8%

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

      2. pow3 4.8%

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

   9. Applied egg-rr 4.8%

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

       1. rem-cube-cbrt 4.8%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 16.6%

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

       4. frac-times 16.6%

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

       5. metadata-eval 16.6%

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

       6. metadata-eval 16.6%

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

       7. frac-times 16.6%

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

       8. sqrt-unprod 7.6%

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

       9. add-sqr-sqrt 7.6%

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

       10. +-commutative 7.6%

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

       11. div-inv 7.6%

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

       12. metadata-eval 7.6%

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

       13. cancel-sign-sub-inv 7.6%

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

       14. div-inv 7.6%

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

       15. flip-- 16.6%

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

       16. metadata-eval 16.6%

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

       17. rem-cube-cbrt 16.6%

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

       18. rem-cube-cbrt 16.6%

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

       19. +-commutative 16.6%

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

       20. div-sub 16.6%

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

   11. Applied egg-rr 4.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{0.1111111111111111}{x} + 1} - \frac{\frac{0.012345679012345678}{{x}^{2}}}{\frac{0.1111111111111111}{x} + 1}} \]
   12. Step-by-step derivation

       1. div-inv 4.8%

          \[\leadsto \frac{1}{\frac{0.1111111111111111}{x} + 1} - \color{blue}{\frac{0.012345679012345678}{{x}^{2}} \cdot \frac{1}{\frac{0.1111111111111111}{x} + 1}} \]

       2. metadata-eval 4.8%

          \[\leadsto \frac{1}{\frac{0.1111111111111111}{x} + 1} - \frac{\color{blue}{0.1111111111111111 \cdot 0.1111111111111111}}{{x}^{2}} \cdot \frac{1}{\frac{0.1111111111111111}{x} + 1} \]

       3. unpow2 4.8%

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

       4. frac-times 4.8%

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

       5. metadata-eval 4.8%

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

       6. associate-*l/ 4.8%

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

       7. metadata-eval 4.8%

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

       8. associate-*l/ 4.8%

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

       9. associate-*l* 4.8%

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

       10. associate-*l/ 4.8%

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

       11. metadata-eval 4.8%

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

       12. metadata-eval 4.8%

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

       13. add-sqr-sqrt 4.8%

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

       14. sqrt-prod 4.8%

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

       15. unpow2 4.8%

           \[\leadsto \frac{1}{\frac{0.1111111111111111}{x} + 1} - \frac{\sqrt{0.012345679012345678}}{\sqrt{\color{blue}{{x}^{2}}}} \cdot \left(\left(\frac{1}{x} \cdot 0.1111111111111111\right) \cdot \frac{1}{\frac{0.1111111111111111}{x} + 1}\right) \]

       16. sqrt-div 4.8%

           \[\leadsto \frac{1}{\frac{0.1111111111111111}{x} + 1} - \color{blue}{\sqrt{\frac{0.012345679012345678}{{x}^{2}}}} \cdot \left(\left(\frac{1}{x} \cdot 0.1111111111111111\right) \cdot \frac{1}{\frac{0.1111111111111111}{x} + 1}\right) \]

       17. metadata-eval 4.8%

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

       18. unpow2 4.8%

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

       19. frac-times 4.8%

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

       20. sqrt-unprod 0.0%

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

       21. add-sqr-sqrt 7.6%

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

   13. Applied egg-rr 7.6%

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

       1. associate-*r* 16.6%

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

       2. +-commutative 16.6%

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

   15. Simplified 16.6%

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

   if -8e17 < y < 3.3999999999999998e149

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r- 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*r/ 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* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 89.9%

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

      1. cancel-sign-sub-inv 89.9%

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

      2. metadata-eval 89.9%

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

      3. associate-*r/ 89.9%

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

      4. metadata-eval 89.9%

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

      5. +-commutative 89.9%

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

   7. Simplified 89.9%

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

      1. add-cube-cbrt 89.2%

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

      2. pow3 89.2%

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

   9. Applied egg-rr 89.2%

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

       1. rem-cube-cbrt 89.9%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 41.5%

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

       4. frac-times 41.5%

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

       5. metadata-eval 41.5%

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

       6. metadata-eval 41.5%

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

       7. frac-times 41.5%

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

       8. sqrt-unprod 41.6%

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

       9. add-sqr-sqrt 41.6%

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

       10. metadata-eval 41.6%

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

       11. distribute-neg-frac 41.6%

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

       12. clear-num 41.6%

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

       13. distribute-neg-frac 41.6%

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

       14. metadata-eval 41.6%

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

       15. clear-num 41.6%

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

       16. add-sqr-sqrt 0.0%

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

       17. sqrt-unprod 63.4%

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

       18. frac-times 63.4%

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

       19. metadata-eval 63.4%

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

       20. metadata-eval 63.4%

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

       21. frac-times 63.4%

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

       22. sqrt-unprod 89.7%

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

       23. add-sqr-sqrt 89.9%

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

       24. clear-num 89.9%

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

       25. div-inv 90.1%

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

       26. metadata-eval 90.1%

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

   11. Applied egg-rr 90.1%

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

   if 3.3999999999999998e149 < y

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. distribute-frac-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r- 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r- 99.7%

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

      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-*r/ 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* 99.4%

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

      12. metadata-eval 99.4%

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

      13. *-commutative 99.4%

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

      14. associate-/r* 99.4%

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

      15. distribute-neg-frac 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around 0 3.6%

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

      1. cancel-sign-sub-inv 3.6%

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

      2. metadata-eval 3.6%

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

      3. associate-*r/ 3.6%

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

      4. metadata-eval 3.6%

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

      5. +-commutative 3.6%

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

   7. Simplified 3.6%

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

      1. add-cube-cbrt 3.6%

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

      2. pow3 3.6%

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

   9. Applied egg-rr 3.6%

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

       1. rem-cube-cbrt 3.6%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 0.6%

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

       4. frac-times 0.6%

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

       5. metadata-eval 0.6%

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

       6. metadata-eval 0.6%

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

       7. frac-times 0.6%

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

       8. sqrt-unprod 0.7%

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

       9. add-sqr-sqrt 0.7%

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

       10. +-commutative 0.7%

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

       11. div-inv 0.7%

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

       12. metadata-eval 0.7%

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

       13. cancel-sign-sub-inv 0.7%

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

       14. div-inv 0.7%

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

       15. flip-- 0.6%

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

       16. metadata-eval 0.6%

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

       17. rem-cube-cbrt 0.6%

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

       18. rem-cube-cbrt 0.6%

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

       19. +-commutative 0.6%

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

       20. div-sub 0.6%

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

   11. Applied egg-rr 19.9%

       \[\leadsto \color{blue}{\frac{1}{\frac{0.1111111111111111}{x} + 1} - \frac{\frac{0.012345679012345678}{{x}^{2}}}{\frac{0.1111111111111111}{x} + 1}} \]
   12. Step-by-step derivation

       1. metadata-eval 19.9%

          \[\leadsto \frac{1}{\frac{0.1111111111111111}{x} + 1} - \frac{\frac{\color{blue}{-0.1111111111111111 \cdot -0.1111111111111111}}{{x}^{2}}}{\frac{0.1111111111111111}{x} + 1} \]

       2. unpow2 19.9%

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

       3. frac-times 19.9%

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

   13. Applied egg-rr 19.9%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+17}:\\ \;\;\;\;\frac{1}{1 + \frac{0.1111111111111111}{x}} + \left(\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}\right) \cdot \frac{-1}{1 + \frac{-0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+149}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{0.1111111111111111}{x}} - \frac{\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}{1 + \frac{0.1111111111111111}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.0% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{0.1111111111111111}{x}\\ \mathbf{if}\;y \leq 3.9 \cdot 10^{+146}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0} - \frac{\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ 0.1111111111111111 x))))
   (if (<= y 3.9e+146)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (-
      (/ 1.0 t_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 <= 3.9e+146) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (1.0 / t_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 <= 3.9d+146) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = (1.0d0 / t_0) - ((((-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 <= 3.9e+146) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (1.0 / t_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 <= 3.9e+146:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = (1.0 / t_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 <= 3.9e+146)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(Float64(1.0 / t_0) - Float64(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 <= 3.9e+146)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = (1.0 / t_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, 3.9e+146], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] - N[(N[(N[(-0.1111111111111111 / x), $MachinePrecision] * N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 3.9e146

   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. distribute-frac-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r- 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r- 99.7%

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

      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-*r/ 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* 99.7%

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

      12. metadata-eval 99.7%

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

      13. *-commutative 99.7%

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

      14. associate-/r* 99.6%

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

      15. distribute-neg-frac 99.6%

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

      16. metadata-eval 99.6%

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

      17. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 70.5%

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

      1. cancel-sign-sub-inv 70.5%

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

      2. metadata-eval 70.5%

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

      3. associate-*r/ 70.5%

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

      4. metadata-eval 70.5%

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

      5. +-commutative 70.5%

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

   7. Simplified 70.5%

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

      1. add-cube-cbrt 69.9%

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

      2. pow3 69.9%

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

   9. Applied egg-rr 69.9%

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

       1. rem-cube-cbrt 70.5%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 35.9%

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

       4. frac-times 35.9%

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

       5. metadata-eval 35.9%

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

       6. metadata-eval 35.9%

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

       7. frac-times 35.9%

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

       8. sqrt-unprod 33.8%

          \[\leadsto \color{blue}{\sqrt{\frac{0.1111111111111111}{x}} \cdot \sqrt{\frac{0.1111111111111111}{x}}} + 1 \]

       9. add-sqr-sqrt 33.8%

          \[\leadsto \color{blue}{\frac{0.1111111111111111}{x}} + 1 \]

       10. metadata-eval 33.8%

           \[\leadsto \frac{\color{blue}{--0.1111111111111111}}{x} + 1 \]

       11. distribute-neg-frac 33.8%

           \[\leadsto \color{blue}{\left(-\frac{-0.1111111111111111}{x}\right)} + 1 \]

       12. clear-num 33.8%

           \[\leadsto \left(-\color{blue}{\frac{1}{\frac{x}{-0.1111111111111111}}}\right) + 1 \]

       13. distribute-neg-frac 33.8%

           \[\leadsto \color{blue}{\frac{-1}{\frac{x}{-0.1111111111111111}}} + 1 \]

       14. metadata-eval 33.8%

           \[\leadsto \frac{\color{blue}{-1}}{\frac{x}{-0.1111111111111111}} + 1 \]

       15. clear-num 33.8%

           \[\leadsto \frac{-1}{\color{blue}{\frac{1}{\frac{-0.1111111111111111}{x}}}} + 1 \]

       16. add-sqr-sqrt 0.0%

           \[\leadsto \frac{-1}{\frac{1}{\color{blue}{\sqrt{\frac{-0.1111111111111111}{x}} \cdot \sqrt{\frac{-0.1111111111111111}{x}}}}} + 1 \]

       17. sqrt-unprod 50.0%

           \[\leadsto \frac{-1}{\frac{1}{\color{blue}{\sqrt{\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}}}} + 1 \]

       18. frac-times 50.0%

           \[\leadsto \frac{-1}{\frac{1}{\sqrt{\color{blue}{\frac{-0.1111111111111111 \cdot -0.1111111111111111}{x \cdot x}}}}} + 1 \]

       19. metadata-eval 50.0%

           \[\leadsto \frac{-1}{\frac{1}{\sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}}}} + 1 \]

       20. metadata-eval 50.0%

           \[\leadsto \frac{-1}{\frac{1}{\sqrt{\frac{\color{blue}{0.1111111111111111 \cdot 0.1111111111111111}}{x \cdot x}}}} + 1 \]

       21. frac-times 50.0%

           \[\leadsto \frac{-1}{\frac{1}{\sqrt{\color{blue}{\frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x}}}}} + 1 \]

       22. sqrt-unprod 70.3%

           \[\leadsto \frac{-1}{\frac{1}{\color{blue}{\sqrt{\frac{0.1111111111111111}{x}} \cdot \sqrt{\frac{0.1111111111111111}{x}}}}} + 1 \]

       23. add-sqr-sqrt 70.4%

           \[\leadsto \frac{-1}{\frac{1}{\color{blue}{\frac{0.1111111111111111}{x}}}} + 1 \]

       24. clear-num 70.4%

           \[\leadsto \frac{-1}{\color{blue}{\frac{x}{0.1111111111111111}}} + 1 \]

       25. div-inv 70.6%

           \[\leadsto \frac{-1}{\color{blue}{x \cdot \frac{1}{0.1111111111111111}}} + 1 \]

       26. metadata-eval 70.6%

           \[\leadsto \frac{-1}{x \cdot \color{blue}{9}} + 1 \]

   11. Applied egg-rr 70.6%

       \[\leadsto \color{blue}{\frac{-1}{x \cdot 9}} + 1 \]

   if 3.9e146 < y

   1. Initial program 99.7%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. sub-neg 99.7%

         \[\leadsto \color{blue}{\left(1 - \frac{1}{x \cdot 9}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. distribute-frac-neg 99.7%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} \]

      3. +-commutative 99.7%

         \[\leadsto \color{blue}{\frac{-y}{3 \cdot \sqrt{x}} + \left(1 - \frac{1}{x \cdot 9}\right)} \]

      4. associate-+r- 99.7%

         \[\leadsto \color{blue}{\left(\frac{-y}{3 \cdot \sqrt{x}} + 1\right) - \frac{1}{x \cdot 9}} \]

      5. +-commutative 99.7%

         \[\leadsto \color{blue}{\left(1 + \frac{-y}{3 \cdot \sqrt{x}}\right)} - \frac{1}{x \cdot 9} \]

      6. associate-+r- 99.7%

         \[\leadsto \color{blue}{1 + \left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

      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-*r/ 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 3.6%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv 3.6%

         \[\leadsto \color{blue}{1 + \left(-0.1111111111111111\right) \cdot \frac{1}{x}} \]

      2. metadata-eval 3.6%

         \[\leadsto 1 + \color{blue}{-0.1111111111111111} \cdot \frac{1}{x} \]

      3. associate-*r/ 3.6%

         \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111 \cdot 1}{x}} \]

      4. metadata-eval 3.6%

         \[\leadsto 1 + \frac{\color{blue}{-0.1111111111111111}}{x} \]

      5. +-commutative 3.6%

         \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]

   7. Simplified 3.6%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]
   8. Step-by-step derivation

      1. add-cube-cbrt 3.6%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{-0.1111111111111111}{x}} \cdot \sqrt[3]{\frac{-0.1111111111111111}{x}}\right) \cdot \sqrt[3]{\frac{-0.1111111111111111}{x}}} + 1 \]

      2. pow3 3.6%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{-0.1111111111111111}{x}}\right)}^{3}} + 1 \]

   9. Applied egg-rr 3.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{-0.1111111111111111}{x}}\right)}^{3}} + 1 \]
   10. Step-by-step derivation

       1. rem-cube-cbrt 3.6%

          \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x}} + 1 \]

       2. add-sqr-sqrt 0.0%

          \[\leadsto \color{blue}{\sqrt{\frac{-0.1111111111111111}{x}} \cdot \sqrt{\frac{-0.1111111111111111}{x}}} + 1 \]

       3. sqrt-unprod 0.7%

          \[\leadsto \color{blue}{\sqrt{\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}} + 1 \]

       4. frac-times 0.7%

          \[\leadsto \sqrt{\color{blue}{\frac{-0.1111111111111111 \cdot -0.1111111111111111}{x \cdot x}}} + 1 \]

       5. metadata-eval 0.7%

          \[\leadsto \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} + 1 \]

       6. metadata-eval 0.7%

          \[\leadsto \sqrt{\frac{\color{blue}{0.1111111111111111 \cdot 0.1111111111111111}}{x \cdot x}} + 1 \]

       7. frac-times 0.7%

          \[\leadsto \sqrt{\color{blue}{\frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x}}} + 1 \]

       8. sqrt-unprod 0.7%

          \[\leadsto \color{blue}{\sqrt{\frac{0.1111111111111111}{x}} \cdot \sqrt{\frac{0.1111111111111111}{x}}} + 1 \]

       9. add-sqr-sqrt 0.7%

          \[\leadsto \color{blue}{\frac{0.1111111111111111}{x}} + 1 \]

       10. +-commutative 0.7%

           \[\leadsto \color{blue}{1 + \frac{0.1111111111111111}{x}} \]

       11. div-inv 0.7%

           \[\leadsto 1 + \color{blue}{0.1111111111111111 \cdot \frac{1}{x}} \]

       12. metadata-eval 0.7%

           \[\leadsto 1 + \color{blue}{\left(--0.1111111111111111\right)} \cdot \frac{1}{x} \]

       13. cancel-sign-sub-inv 0.7%

           \[\leadsto \color{blue}{1 - -0.1111111111111111 \cdot \frac{1}{x}} \]

       14. div-inv 0.7%

           \[\leadsto 1 - \color{blue}{\frac{-0.1111111111111111}{x}} \]

       15. flip-- 0.7%

           \[\leadsto \color{blue}{\frac{1 \cdot 1 - \frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}{1 + \frac{-0.1111111111111111}{x}}} \]

       16. metadata-eval 0.7%

           \[\leadsto \frac{\color{blue}{1} - \frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}{1 + \frac{-0.1111111111111111}{x}} \]

       17. rem-cube-cbrt 0.7%

           \[\leadsto \frac{1 - \color{blue}{{\left(\sqrt[3]{\frac{-0.1111111111111111}{x}}\right)}^{3}} \cdot \frac{-0.1111111111111111}{x}}{1 + \frac{-0.1111111111111111}{x}} \]

       18. rem-cube-cbrt 0.7%

           \[\leadsto \frac{1 - {\left(\sqrt[3]{\frac{-0.1111111111111111}{x}}\right)}^{3} \cdot \color{blue}{{\left(\sqrt[3]{\frac{-0.1111111111111111}{x}}\right)}^{3}}}{1 + \frac{-0.1111111111111111}{x}} \]

       19. +-commutative 0.7%

           \[\leadsto \frac{1 - {\left(\sqrt[3]{\frac{-0.1111111111111111}{x}}\right)}^{3} \cdot {\left(\sqrt[3]{\frac{-0.1111111111111111}{x}}\right)}^{3}}{\color{blue}{\frac{-0.1111111111111111}{x} + 1}} \]

       20. div-sub 0.7%

           \[\leadsto \color{blue}{\frac{1}{\frac{-0.1111111111111111}{x} + 1} - \frac{{\left(\sqrt[3]{\frac{-0.1111111111111111}{x}}\right)}^{3} \cdot {\left(\sqrt[3]{\frac{-0.1111111111111111}{x}}\right)}^{3}}{\frac{-0.1111111111111111}{x} + 1}} \]

   11. Applied egg-rr 19.5%

       \[\leadsto \color{blue}{\frac{1}{\frac{0.1111111111111111}{x} + 1} - \frac{\frac{0.012345679012345678}{{x}^{2}}}{\frac{0.1111111111111111}{x} + 1}} \]
   12. Step-by-step derivation

       1. metadata-eval 19.5%

          \[\leadsto \frac{1}{\frac{0.1111111111111111}{x} + 1} - \frac{\frac{\color{blue}{-0.1111111111111111 \cdot -0.1111111111111111}}{{x}^{2}}}{\frac{0.1111111111111111}{x} + 1} \]

       2. unpow2 19.5%

          \[\leadsto \frac{1}{\frac{0.1111111111111111}{x} + 1} - \frac{\frac{-0.1111111111111111 \cdot -0.1111111111111111}{\color{blue}{x \cdot x}}}{\frac{0.1111111111111111}{x} + 1} \]

       3. frac-times 19.5%

          \[\leadsto \frac{1}{\frac{0.1111111111111111}{x} + 1} - \frac{\color{blue}{\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}}{\frac{0.1111111111111111}{x} + 1} \]

   13. Applied egg-rr 19.5%

       \[\leadsto \frac{1}{\frac{0.1111111111111111}{x} + 1} - \frac{\color{blue}{\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}}{\frac{0.1111111111111111}{x} + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{+146}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{0.1111111111111111}{x}} - \frac{\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}{1 + \frac{0.1111111111111111}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.0% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.106:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.106) (/ -0.1111111111111111 x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.106) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.106d0) then
        tmp = (-0.1111111111111111d0) / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.106) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.106:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.106)
		tmp = Float64(-0.1111111111111111 / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.106)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.106], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.105999999999999997

   1. Initial program 99.6%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. frac-2neg 99.6%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{\frac{-y}{-3 \cdot \sqrt{x}}} \]

      2. div-inv 99.6%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{\left(-y\right) \cdot \frac{1}{-3 \cdot \sqrt{x}}} \]

      3. *-commutative 99.6%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \left(-y\right) \cdot \frac{1}{-\color{blue}{\sqrt{x} \cdot 3}} \]

      4. distribute-rgt-neg-in 99.6%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \left(-y\right) \cdot \frac{1}{\color{blue}{\sqrt{x} \cdot \left(-3\right)}} \]

      5. metadata-eval 99.6%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \left(-y\right) \cdot \frac{1}{\sqrt{x} \cdot \color{blue}{-3}} \]

      6. metadata-eval 99.6%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \left(-y\right) \cdot \frac{1}{\sqrt{x} \cdot \color{blue}{\frac{1}{-0.3333333333333333}}} \]

      7. div-inv 99.5%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \left(-y\right) \cdot \frac{1}{\color{blue}{\frac{\sqrt{x}}{-0.3333333333333333}}} \]

      8. clear-num 99.6%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \left(-y\right) \cdot \color{blue}{\frac{-0.3333333333333333}{\sqrt{x}}} \]

   4. Applied egg-rr 99.6%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{\left(-y\right) \cdot \frac{-0.3333333333333333}{\sqrt{x}}} \]

   5. Taylor expanded in x around 0 96.7%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x}} - \left(-y\right) \cdot \frac{-0.3333333333333333}{\sqrt{x}} \]

   6. Taylor expanded in x around 0 60.0%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x}} \]

   if 0.105999999999999997 < x

   1. Initial program 99.8%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. sub-neg 99.8%

         \[\leadsto \color{blue}{\left(1 - \frac{1}{x \cdot 9}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. distribute-frac-neg 99.8%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} \]

      3. +-commutative 99.8%

         \[\leadsto \color{blue}{\frac{-y}{3 \cdot \sqrt{x}} + \left(1 - \frac{1}{x \cdot 9}\right)} \]

      4. associate-+r- 99.8%

         \[\leadsto \color{blue}{\left(\frac{-y}{3 \cdot \sqrt{x}} + 1\right) - \frac{1}{x \cdot 9}} \]

      5. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(1 + \frac{-y}{3 \cdot \sqrt{x}}\right)} - \frac{1}{x \cdot 9} \]

      6. associate-+r- 99.8%

         \[\leadsto \color{blue}{1 + \left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

      7. neg-mul-1 99.8%

         \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

      8. *-commutative 99.8%

         \[\leadsto 1 + \left(\frac{\color{blue}{y \cdot -1}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

      9. associate-*r/ 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 x around inf 56.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.106:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 62.0% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{-1}{x \cdot 9} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (/ -1.0 (* x 9.0))))

C

double code(double x, double y) {
	return 1.0 + (-1.0 / (x * 9.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
end function

Java

public static double code(double x, double y) {
	return 1.0 + (-1.0 / (x * 9.0));
}

Python

def code(x, y):
	return 1.0 + (-1.0 / (x * 9.0))

Julia

function code(x, y)
	return Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + (-1.0 / (x * 9.0));
end

Wolfram

code[x_, y_] := N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. sub-neg 99.7%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{x \cdot 9}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)} \]

   2. distribute-frac-neg 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} \]

   3. +-commutative 99.7%

      \[\leadsto \color{blue}{\frac{-y}{3 \cdot \sqrt{x}} + \left(1 - \frac{1}{x \cdot 9}\right)} \]

   4. associate-+r- 99.7%

      \[\leadsto \color{blue}{\left(\frac{-y}{3 \cdot \sqrt{x}} + 1\right) - \frac{1}{x \cdot 9}} \]

   5. +-commutative 99.7%

      \[\leadsto \color{blue}{\left(1 + \frac{-y}{3 \cdot \sqrt{x}}\right)} - \frac{1}{x \cdot 9} \]

   6. associate-+r- 99.7%

      \[\leadsto \color{blue}{1 + \left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

   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-*r/ 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* 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, -\frac{1}{x \cdot 9}\right) \]

   12. metadata-eval 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

   13. *-commutative 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

   14. associate-/r* 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

   15. distribute-neg-frac 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

   16. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

   17. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 59.5%

   \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
6. Step-by-step derivation

   1. cancel-sign-sub-inv 59.5%

      \[\leadsto \color{blue}{1 + \left(-0.1111111111111111\right) \cdot \frac{1}{x}} \]

   2. metadata-eval 59.5%

      \[\leadsto 1 + \color{blue}{-0.1111111111111111} \cdot \frac{1}{x} \]

   3. associate-*r/ 59.5%

      \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111 \cdot 1}{x}} \]

   4. metadata-eval 59.5%

      \[\leadsto 1 + \frac{\color{blue}{-0.1111111111111111}}{x} \]

   5. +-commutative 59.5%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]

7. Simplified 59.5%

   \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]
8. Step-by-step derivation

   1. add-cube-cbrt 59.0%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{-0.1111111111111111}{x}} \cdot \sqrt[3]{\frac{-0.1111111111111111}{x}}\right) \cdot \sqrt[3]{\frac{-0.1111111111111111}{x}}} + 1 \]

   2. pow3 59.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{-0.1111111111111111}{x}}\right)}^{3}} + 1 \]

9. Applied egg-rr 59.0%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{-0.1111111111111111}{x}}\right)}^{3}} + 1 \]
10. Step-by-step derivation

    1. rem-cube-cbrt 59.5%

       \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x}} + 1 \]

    2. add-sqr-sqrt 0.0%

       \[\leadsto \color{blue}{\sqrt{\frac{-0.1111111111111111}{x}} \cdot \sqrt{\frac{-0.1111111111111111}{x}}} + 1 \]

    3. sqrt-unprod 30.1%

       \[\leadsto \color{blue}{\sqrt{\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}} + 1 \]

    4. frac-times 30.1%

       \[\leadsto \sqrt{\color{blue}{\frac{-0.1111111111111111 \cdot -0.1111111111111111}{x \cdot x}}} + 1 \]

    5. metadata-eval 30.1%

       \[\leadsto \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} + 1 \]

    6. metadata-eval 30.1%

       \[\leadsto \sqrt{\frac{\color{blue}{0.1111111111111111 \cdot 0.1111111111111111}}{x \cdot x}} + 1 \]

    7. frac-times 30.1%

       \[\leadsto \sqrt{\color{blue}{\frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x}}} + 1 \]

    8. sqrt-unprod 28.4%

       \[\leadsto \color{blue}{\sqrt{\frac{0.1111111111111111}{x}} \cdot \sqrt{\frac{0.1111111111111111}{x}}} + 1 \]

    9. add-sqr-sqrt 28.4%

       \[\leadsto \color{blue}{\frac{0.1111111111111111}{x}} + 1 \]

    10. metadata-eval 28.4%

        \[\leadsto \frac{\color{blue}{--0.1111111111111111}}{x} + 1 \]

    11. distribute-neg-frac 28.4%

        \[\leadsto \color{blue}{\left(-\frac{-0.1111111111111111}{x}\right)} + 1 \]

    12. clear-num 28.4%

        \[\leadsto \left(-\color{blue}{\frac{1}{\frac{x}{-0.1111111111111111}}}\right) + 1 \]

    13. distribute-neg-frac 28.4%

        \[\leadsto \color{blue}{\frac{-1}{\frac{x}{-0.1111111111111111}}} + 1 \]

    14. metadata-eval 28.4%

        \[\leadsto \frac{\color{blue}{-1}}{\frac{x}{-0.1111111111111111}} + 1 \]

    15. clear-num 28.4%

        \[\leadsto \frac{-1}{\color{blue}{\frac{1}{\frac{-0.1111111111111111}{x}}}} + 1 \]

    16. add-sqr-sqrt 0.0%

        \[\leadsto \frac{-1}{\frac{1}{\color{blue}{\sqrt{\frac{-0.1111111111111111}{x}} \cdot \sqrt{\frac{-0.1111111111111111}{x}}}}} + 1 \]

    17. sqrt-unprod 45.0%

        \[\leadsto \frac{-1}{\frac{1}{\color{blue}{\sqrt{\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}}}} + 1 \]

    18. frac-times 45.0%

        \[\leadsto \frac{-1}{\frac{1}{\sqrt{\color{blue}{\frac{-0.1111111111111111 \cdot -0.1111111111111111}{x \cdot x}}}}} + 1 \]

    19. metadata-eval 45.0%

        \[\leadsto \frac{-1}{\frac{1}{\sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}}}} + 1 \]

    20. metadata-eval 45.0%

        \[\leadsto \frac{-1}{\frac{1}{\sqrt{\frac{\color{blue}{0.1111111111111111 \cdot 0.1111111111111111}}{x \cdot x}}}} + 1 \]

    21. frac-times 45.0%

        \[\leadsto \frac{-1}{\frac{1}{\sqrt{\color{blue}{\frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x}}}}} + 1 \]

    22. sqrt-unprod 59.3%

        \[\leadsto \frac{-1}{\frac{1}{\color{blue}{\sqrt{\frac{0.1111111111111111}{x}} \cdot \sqrt{\frac{0.1111111111111111}{x}}}}} + 1 \]

    23. add-sqr-sqrt 59.5%

        \[\leadsto \frac{-1}{\frac{1}{\color{blue}{\frac{0.1111111111111111}{x}}}} + 1 \]

    24. clear-num 59.4%

        \[\leadsto \frac{-1}{\color{blue}{\frac{x}{0.1111111111111111}}} + 1 \]

    25. div-inv 59.6%

        \[\leadsto \frac{-1}{\color{blue}{x \cdot \frac{1}{0.1111111111111111}}} + 1 \]

    26. metadata-eval 59.6%

        \[\leadsto \frac{-1}{x \cdot \color{blue}{9}} + 1 \]

11. Applied egg-rr 59.6%

    \[\leadsto \color{blue}{\frac{-1}{x \cdot 9}} + 1 \]

12. Final simplification 59.6%

    \[\leadsto 1 + \frac{-1}{x \cdot 9} \]
13. Add Preprocessing

Alternative 16: 62.0% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{-0.1111111111111111}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (/ -0.1111111111111111 x)))

C

double code(double x, double y) {
	return 1.0 + (-0.1111111111111111 / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + ((-0.1111111111111111d0) / x)
end function

Java

public static double code(double x, double y) {
	return 1.0 + (-0.1111111111111111 / x);
}

Python

def code(x, y):
	return 1.0 + (-0.1111111111111111 / x)

Julia

function code(x, y)
	return Float64(1.0 + Float64(-0.1111111111111111 / x))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + (-0.1111111111111111 / x);
end

Wolfram

code[x_, y_] := N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. sub-neg 99.7%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{x \cdot 9}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)} \]

   2. distribute-frac-neg 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} \]

   3. +-commutative 99.7%

      \[\leadsto \color{blue}{\frac{-y}{3 \cdot \sqrt{x}} + \left(1 - \frac{1}{x \cdot 9}\right)} \]

   4. associate-+r- 99.7%

      \[\leadsto \color{blue}{\left(\frac{-y}{3 \cdot \sqrt{x}} + 1\right) - \frac{1}{x \cdot 9}} \]

   5. +-commutative 99.7%

      \[\leadsto \color{blue}{\left(1 + \frac{-y}{3 \cdot \sqrt{x}}\right)} - \frac{1}{x \cdot 9} \]

   6. associate-+r- 99.7%

      \[\leadsto \color{blue}{1 + \left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

   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-*r/ 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* 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, -\frac{1}{x \cdot 9}\right) \]

   12. metadata-eval 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

   13. *-commutative 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

   14. associate-/r* 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

   15. distribute-neg-frac 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

   16. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

   17. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 59.5%

   \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
6. Step-by-step derivation

   1. cancel-sign-sub-inv 59.5%

      \[\leadsto \color{blue}{1 + \left(-0.1111111111111111\right) \cdot \frac{1}{x}} \]

   2. metadata-eval 59.5%

      \[\leadsto 1 + \color{blue}{-0.1111111111111111} \cdot \frac{1}{x} \]

   3. associate-*r/ 59.5%

      \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111 \cdot 1}{x}} \]

   4. metadata-eval 59.5%

      \[\leadsto 1 + \frac{\color{blue}{-0.1111111111111111}}{x} \]

   5. +-commutative 59.5%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]

7. Simplified 59.5%

   \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]

8. Final simplification 59.5%

   \[\leadsto 1 + \frac{-0.1111111111111111}{x} \]
9. Add Preprocessing

Alternative 17: 31.8% accurate, 113.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. sub-neg 99.7%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{x \cdot 9}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)} \]

   2. distribute-frac-neg 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} \]

   3. +-commutative 99.7%

      \[\leadsto \color{blue}{\frac{-y}{3 \cdot \sqrt{x}} + \left(1 - \frac{1}{x \cdot 9}\right)} \]

   4. associate-+r- 99.7%

      \[\leadsto \color{blue}{\left(\frac{-y}{3 \cdot \sqrt{x}} + 1\right) - \frac{1}{x \cdot 9}} \]

   5. +-commutative 99.7%

      \[\leadsto \color{blue}{\left(1 + \frac{-y}{3 \cdot \sqrt{x}}\right)} - \frac{1}{x \cdot 9} \]

   6. associate-+r- 99.7%

      \[\leadsto \color{blue}{1 + \left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

   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-*r/ 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* 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, -\frac{1}{x \cdot 9}\right) \]

   12. metadata-eval 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

   13. *-commutative 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

   14. associate-/r* 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

   15. distribute-neg-frac 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

   16. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

   17. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 28.3%

   \[\leadsto \color{blue}{1} \]

6. Final simplification 28.3%

   \[\leadsto 1 \]
7. 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 2024018 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (- (- 1.0 (/ (/ 1.0 x) 9.0)) (/ y (* 3.0 (sqrt x))))

  (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))