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

Percentage Accurate: 99.7% → 99.7%

Time: 10.7s

Alternatives: 14

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 + \frac{-0.1111111111111111}{x}\right) - \frac{y}{\sqrt{x \cdot 9}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (- (+ 1.0 (/ -0.1111111111111111 x)) (/ y (sqrt (* x 9.0)))))

C

double code(double x, double y) {
	return (1.0 + (-0.1111111111111111 / x)) - (y / sqrt((x * 9.0)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 + ((-0.1111111111111111d0) / x)) - (y / sqrt((x * 9.0d0)))
end function

Java

public static double code(double x, double y) {
	return (1.0 + (-0.1111111111111111 / x)) - (y / Math.sqrt((x * 9.0)));
}

Python

def code(x, y):
	return (1.0 + (-0.1111111111111111 / x)) - (y / math.sqrt((x * 9.0)))

Julia

function code(x, y)
	return Float64(Float64(1.0 + Float64(-0.1111111111111111 / x)) - Float64(y / sqrt(Float64(x * 9.0))))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 + (-0.1111111111111111 / x)) - (y / sqrt((x * 9.0)));
end

Wolfram

code[x_, y_] := N[(N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. associate--l- 99.6%

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

   2. sub-neg 99.6%

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

   3. +-commutative 99.6%

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

   4. distribute-neg-in 99.6%

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

   5. distribute-frac-neg 99.6%

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

   6. sub-neg 99.6%

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

   7. neg-mul-1 99.6%

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

   8. *-commutative 99.6%

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

   9. associate-/l* 99.6%

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

   10. fma-neg 99.6%

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

   11. associate-/r* 99.7%

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

   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. Step-by-step derivation

   1. +-commutative 99.6%

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

   2. fma-undefine 99.6%

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

   3. associate-+l+ 99.6%

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

   4. associate-*r/ 99.6%

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

   5. associate-*l/ 99.3%

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

   6. *-commutative 99.3%

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

   7. +-commutative 99.3%

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

   8. div-inv 99.3%

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

   9. metadata-eval 99.3%

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

   10. cancel-sign-sub-inv 99.3%

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

   11. div-inv 99.3%

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

   12. +-commutative 99.3%

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

   13. associate-+l- 99.3%

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

   14. cancel-sign-sub-inv 99.3%

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

   15. metadata-eval 99.3%

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

   16. metadata-eval 99.3%

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

   17. times-frac 99.6%

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

   18. *-un-lft-identity 99.6%

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

   19. associate--r+ 99.6%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

   \[\leadsto \left(1 + \frac{-0.1111111111111111}{x}\right) - \frac{y}{\sqrt{x \cdot 9}} \]
8. Add Preprocessing

Alternative 2: 94.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+34}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x}} \cdot 0.3333333333333333\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+72}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.55e+34)
   (- 1.0 (* (/ y (sqrt x)) 0.3333333333333333))
   (if (<= y 1.4e+72)
     (+ 1.0 (/ -0.1111111111111111 x))
     (+ 1.0 (* (sqrt (/ 1.0 x)) (* y -0.3333333333333333))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.55e+34) {
		tmp = 1.0 - ((y / sqrt(x)) * 0.3333333333333333);
	} else if (y <= 1.4e+72) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 + (sqrt((1.0 / x)) * (y * -0.3333333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.55d+34)) then
        tmp = 1.0d0 - ((y / sqrt(x)) * 0.3333333333333333d0)
    else if (y <= 1.4d+72) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = 1.0d0 + (sqrt((1.0d0 / x)) * (y * (-0.3333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.55e+34) {
		tmp = 1.0 - ((y / Math.sqrt(x)) * 0.3333333333333333);
	} else if (y <= 1.4e+72) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 + (Math.sqrt((1.0 / x)) * (y * -0.3333333333333333));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.55e+34:
		tmp = 1.0 - ((y / math.sqrt(x)) * 0.3333333333333333)
	elif y <= 1.4e+72:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = 1.0 + (math.sqrt((1.0 / x)) * (y * -0.3333333333333333))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.55e+34)
		tmp = Float64(1.0 - Float64(Float64(y / sqrt(x)) * 0.3333333333333333));
	elseif (y <= 1.4e+72)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 / x)) * Float64(y * -0.3333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.55e+34)
		tmp = 1.0 - ((y / sqrt(x)) * 0.3333333333333333);
	elseif (y <= 1.4e+72)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = 1.0 + (sqrt((1.0 / x)) * (y * -0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.55e+34], N[(1.0 - N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.4e+72], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.54999999999999989e34

   1. Initial program 99.4%

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

   3. Taylor expanded in x around inf 86.4%

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

      1. *-commutative 86.4%

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

      2. sqrt-div 86.4%

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

      3. metadata-eval 86.4%

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

      4. un-div-inv 86.4%

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

   5. Applied egg-rr 86.4%

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

   if -1.54999999999999989e34 < y < 1.4e72

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.7%

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

      10. fma-neg 99.7%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

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

      13. *-commutative 99.7%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 98.0%

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

   if 1.4e72 < y

   1. Initial program 99.5%

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

      1. associate--l- 99.5%

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

      2. sub-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. distribute-neg-in 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. sub-neg 99.5%

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

      7. neg-mul-1 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-/l* 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.5%

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

      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 94.9%

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

      1. associate-*r* 96.8%

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

      2. *-commutative 96.8%

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

      3. associate-*l* 96.6%

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

   7. Simplified 96.6%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+34}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x}} \cdot 0.3333333333333333\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+72}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+33} \lor \neg \left(y \leq 1.75 \cdot 10^{+73}\right):\\ \;\;\;\;1 - \frac{y}{\sqrt{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -6.8e+33) (not (<= y 1.75e+73)))
   (- 1.0 (* (/ y (sqrt x)) 0.3333333333333333))
   (+ 1.0 (/ -0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -6.8e+33) || !(y <= 1.75e+73)) {
		tmp = 1.0 - ((y / sqrt(x)) * 0.3333333333333333);
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-6.8d+33)) .or. (.not. (y <= 1.75d+73))) then
        tmp = 1.0d0 - ((y / sqrt(x)) * 0.3333333333333333d0)
    else
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -6.8e+33) || !(y <= 1.75e+73)) {
		tmp = 1.0 - ((y / Math.sqrt(x)) * 0.3333333333333333);
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -6.8e+33) or not (y <= 1.75e+73):
		tmp = 1.0 - ((y / math.sqrt(x)) * 0.3333333333333333)
	else:
		tmp = 1.0 + (-0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -6.8e+33) || !(y <= 1.75e+73))
		tmp = Float64(1.0 - Float64(Float64(y / sqrt(x)) * 0.3333333333333333));
	else
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -6.8e+33) || ~((y <= 1.75e+73)))
		tmp = 1.0 - ((y / sqrt(x)) * 0.3333333333333333);
	else
		tmp = 1.0 + (-0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -6.8e+33], N[Not[LessEqual[y, 1.75e+73]], $MachinePrecision]], N[(1.0 - N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.7999999999999999e33 or 1.75000000000000001e73 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf 90.0%

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

      1. *-commutative 90.0%

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

      2. sqrt-div 90.0%

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

      3. metadata-eval 90.0%

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

      4. un-div-inv 90.0%

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

   5. Applied egg-rr 90.0%

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

   if -6.7999999999999999e33 < y < 1.75000000000000001e73

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.7%

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

      10. fma-neg 99.7%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

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

      13. *-commutative 99.7%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 98.0%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+33} \lor \neg \left(y \leq 1.75 \cdot 10^{+73}\right):\\ \;\;\;\;1 - \frac{y}{\sqrt{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+34} \lor \neg \left(y \leq 2.3 \cdot 10^{+95}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.55e+34) (not (<= y 2.3e+95)))
   (* -0.3333333333333333 (/ y (sqrt x)))
   (+ 1.0 (/ -0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.55e+34) || !(y <= 2.3e+95)) {
		tmp = -0.3333333333333333 * (y / sqrt(x));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.55d+34)) .or. (.not. (y <= 2.3d+95))) then
        tmp = (-0.3333333333333333d0) * (y / sqrt(x))
    else
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.55e+34) || !(y <= 2.3e+95)) {
		tmp = -0.3333333333333333 * (y / Math.sqrt(x));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.55e+34) or not (y <= 2.3e+95):
		tmp = -0.3333333333333333 * (y / math.sqrt(x))
	else:
		tmp = 1.0 + (-0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.55e+34) || !(y <= 2.3e+95))
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	else
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.55e+34) || ~((y <= 2.3e+95)))
		tmp = -0.3333333333333333 * (y / sqrt(x));
	else
		tmp = 1.0 + (-0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.55e+34], N[Not[LessEqual[y, 2.3e+95]], $MachinePrecision]], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.54999999999999989e34 or 2.29999999999999997e95 < y

   1. Initial program 99.5%

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

      1. associate--l- 99.5%

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

      2. sub-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. distribute-neg-in 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. sub-neg 99.5%

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

      7. neg-mul-1 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-/l* 99.4%

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

      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* 99.5%

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

      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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. fma-undefine 99.5%

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

      3. associate-+l+ 99.5%

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

      4. associate-*r/ 99.5%

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

      5. associate-*l/ 98.8%

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

      6. *-commutative 98.8%

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

      7. +-commutative 98.8%

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

      8. div-inv 98.7%

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

      9. metadata-eval 98.7%

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

      10. cancel-sign-sub-inv 98.7%

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

      11. div-inv 98.8%

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

      12. +-commutative 98.8%

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

      13. associate-+l- 98.8%

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

      14. cancel-sign-sub-inv 98.8%

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

      15. metadata-eval 98.8%

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

      16. metadata-eval 98.8%

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

      17. times-frac 99.5%

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

      18. *-un-lft-identity 99.5%

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

      19. associate--r+ 99.5%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around inf 87.0%

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

      1. associate-*r* 87.9%

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

      2. *-commutative 87.9%

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

   9. Simplified 87.9%

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

       1. *-commutative 87.9%

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

       2. *-commutative 87.9%

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

       3. sqrt-div 87.8%

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

       4. metadata-eval 87.8%

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

       5. div-inv 87.9%

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

       6. associate-*r/ 87.9%

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

   11. Applied egg-rr 87.9%

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

       1. *-commutative 87.9%

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

       2. associate-/l* 87.1%

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

   13. Simplified 87.1%

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

   if -1.54999999999999989e34 < y < 2.29999999999999997e95

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.7%

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

      10. fma-neg 99.7%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

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

      13. *-commutative 99.7%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 97.4%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+34} \lor \neg \left(y \leq 2.3 \cdot 10^{+95}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+34}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+95}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.55e+34)
   (* -0.3333333333333333 (/ y (sqrt x)))
   (if (<= y 2.05e+95)
     (+ 1.0 (/ -0.1111111111111111 x))
     (/ y (* (sqrt x) -3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.55e+34) {
		tmp = -0.3333333333333333 * (y / sqrt(x));
	} else if (y <= 2.05e+95) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = y / (sqrt(x) * -3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.55d+34)) then
        tmp = (-0.3333333333333333d0) * (y / sqrt(x))
    else if (y <= 2.05d+95) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = y / (sqrt(x) * (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.55e+34) {
		tmp = -0.3333333333333333 * (y / Math.sqrt(x));
	} else if (y <= 2.05e+95) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = y / (Math.sqrt(x) * -3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.55e+34:
		tmp = -0.3333333333333333 * (y / math.sqrt(x))
	elif y <= 2.05e+95:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = y / (math.sqrt(x) * -3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.55e+34)
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	elseif (y <= 2.05e+95)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(y / Float64(sqrt(x) * -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.55e+34)
		tmp = -0.3333333333333333 * (y / sqrt(x));
	elseif (y <= 2.05e+95)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = y / (sqrt(x) * -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.55e+34], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.05e+95], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.54999999999999989e34

   1. Initial program 99.4%

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

      1. associate--l- 99.4%

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

      2. sub-neg 99.4%

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

      3. +-commutative 99.4%

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

      4. distribute-neg-in 99.4%

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

      5. distribute-frac-neg 99.4%

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

      6. sub-neg 99.4%

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

      7. neg-mul-1 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-/l* 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* 99.6%

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

      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. Step-by-step derivation

      1. +-commutative 99.6%

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

      2. fma-undefine 99.5%

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

      3. associate-+l+ 99.5%

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

      4. associate-*r/ 99.6%

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

      5. associate-*l/ 99.5%

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

      6. *-commutative 99.5%

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

      7. +-commutative 99.5%

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

      8. div-inv 99.5%

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

      9. metadata-eval 99.5%

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

      10. cancel-sign-sub-inv 99.5%

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

      11. div-inv 99.5%

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

      12. +-commutative 99.5%

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

      13. associate-+l- 99.5%

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

      14. cancel-sign-sub-inv 99.5%

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

      15. metadata-eval 99.5%

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

      16. metadata-eval 99.5%

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

      17. times-frac 99.5%

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

      18. *-un-lft-identity 99.5%

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

      19. associate--r+ 99.5%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around inf 83.5%

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

      1. associate-*r* 83.6%

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

      2. *-commutative 83.6%

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

   9. Simplified 83.6%

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

       1. *-commutative 83.6%

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

       2. *-commutative 83.6%

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

       3. sqrt-div 83.6%

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

       4. metadata-eval 83.6%

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

       5. div-inv 83.6%

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

       6. associate-*r/ 83.6%

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

   11. Applied egg-rr 83.6%

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

       1. *-commutative 83.6%

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

       2. associate-/l* 83.6%

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

   13. Simplified 83.6%

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

   if -1.54999999999999989e34 < y < 2.04999999999999993e95

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.7%

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

      10. fma-neg 99.7%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

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

      13. *-commutative 99.7%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 97.4%

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

   if 2.04999999999999993e95 < y

   1. Initial program 99.5%

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

      1. associate--l- 99.5%

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

      2. sub-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. distribute-neg-in 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. sub-neg 99.5%

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

      7. neg-mul-1 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-/l* 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.5%

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

      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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. fma-undefine 99.5%

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

      3. associate-+l+ 99.5%

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

      4. associate-*r/ 99.4%

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

      5. associate-*l/ 97.7%

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

      6. *-commutative 97.7%

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

      7. +-commutative 97.7%

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

      8. div-inv 97.7%

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

      9. metadata-eval 97.7%

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

      10. cancel-sign-sub-inv 97.7%

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

      11. div-inv 97.7%

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

      12. +-commutative 97.7%

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

      13. associate-+l- 97.7%

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

      14. cancel-sign-sub-inv 97.7%

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

      15. metadata-eval 97.7%

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

      16. metadata-eval 97.7%

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

      17. times-frac 99.6%

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

      18. *-un-lft-identity 99.6%

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

      19. associate--r+ 99.6%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around inf 92.2%

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

      1. associate-*r* 94.2%

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

      2. *-commutative 94.2%

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

   9. Simplified 94.2%

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

       1. *-commutative 94.2%

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

       2. sqrt-div 94.1%

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

       3. metadata-eval 94.1%

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

       4. associate-/r/ 94.0%

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

       5. un-div-inv 94.1%

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

       6. div-inv 94.2%

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

       7. metadata-eval 94.2%

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

   11. Applied egg-rr 94.2%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+34}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+95}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+34}:\\ \;\;\;\;\frac{\frac{y}{-3}}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+98}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.55e+34)
   (/ (/ y -3.0) (sqrt x))
   (if (<= y 5e+98)
     (+ 1.0 (/ -0.1111111111111111 x))
     (/ y (* (sqrt x) -3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.55e+34) {
		tmp = (y / -3.0) / sqrt(x);
	} else if (y <= 5e+98) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = y / (sqrt(x) * -3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.55d+34)) then
        tmp = (y / (-3.0d0)) / sqrt(x)
    else if (y <= 5d+98) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = y / (sqrt(x) * (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.55e+34) {
		tmp = (y / -3.0) / Math.sqrt(x);
	} else if (y <= 5e+98) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = y / (Math.sqrt(x) * -3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.55e+34:
		tmp = (y / -3.0) / math.sqrt(x)
	elif y <= 5e+98:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = y / (math.sqrt(x) * -3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.55e+34)
		tmp = Float64(Float64(y / -3.0) / sqrt(x));
	elseif (y <= 5e+98)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(y / Float64(sqrt(x) * -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.55e+34)
		tmp = (y / -3.0) / sqrt(x);
	elseif (y <= 5e+98)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = y / (sqrt(x) * -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.55e+34], N[(N[(y / -3.0), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+98], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.54999999999999989e34

   1. Initial program 99.4%

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

      1. associate--l- 99.4%

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

      2. sub-neg 99.4%

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

      3. +-commutative 99.4%

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

      4. distribute-neg-in 99.4%

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

      5. distribute-frac-neg 99.4%

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

      6. sub-neg 99.4%

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

      7. neg-mul-1 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-/l* 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* 99.6%

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

      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. Step-by-step derivation

      1. +-commutative 99.6%

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

      2. fma-undefine 99.5%

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

      3. associate-+l+ 99.5%

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

      4. associate-*r/ 99.6%

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

      5. associate-*l/ 99.5%

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

      6. *-commutative 99.5%

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

      7. +-commutative 99.5%

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

      8. div-inv 99.5%

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

      9. metadata-eval 99.5%

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

      10. cancel-sign-sub-inv 99.5%

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

      11. div-inv 99.5%

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

      12. +-commutative 99.5%

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

      13. associate-+l- 99.5%

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

      14. cancel-sign-sub-inv 99.5%

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

      15. metadata-eval 99.5%

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

      16. metadata-eval 99.5%

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

      17. times-frac 99.5%

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

      18. *-un-lft-identity 99.5%

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

      19. associate--r+ 99.5%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around inf 83.5%

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

      1. associate-*r* 83.6%

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

      2. *-commutative 83.6%

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

   9. Simplified 83.6%

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

       1. *-commutative 83.6%

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

       2. sqrt-div 83.6%

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

       3. metadata-eval 83.6%

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

       4. associate-/r/ 83.5%

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

       5. un-div-inv 83.5%

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

       6. div-inv 83.6%

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

       7. metadata-eval 83.6%

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

   11. Applied egg-rr 83.6%

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

       1. *-commutative 83.6%

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

       2. associate-/r* 83.6%

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

   13. Simplified 83.6%

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

   if -1.54999999999999989e34 < y < 4.9999999999999998e98

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.7%

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

      10. fma-neg 99.7%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

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

      13. *-commutative 99.7%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 97.4%

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

   if 4.9999999999999998e98 < y

   1. Initial program 99.5%

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

      1. associate--l- 99.5%

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

      2. sub-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. distribute-neg-in 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. sub-neg 99.5%

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

      7. neg-mul-1 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-/l* 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.5%

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

      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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. fma-undefine 99.5%

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

      3. associate-+l+ 99.5%

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

      4. associate-*r/ 99.4%

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

      5. associate-*l/ 97.7%

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

      6. *-commutative 97.7%

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

      7. +-commutative 97.7%

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

      8. div-inv 97.7%

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

      9. metadata-eval 97.7%

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

      10. cancel-sign-sub-inv 97.7%

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

      11. div-inv 97.7%

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

      12. +-commutative 97.7%

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

      13. associate-+l- 97.7%

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

      14. cancel-sign-sub-inv 97.7%

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

      15. metadata-eval 97.7%

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

      16. metadata-eval 97.7%

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

      17. times-frac 99.6%

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

      18. *-un-lft-identity 99.6%

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

      19. associate--r+ 99.6%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around inf 92.2%

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

      1. associate-*r* 94.2%

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

      2. *-commutative 94.2%

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

   9. Simplified 94.2%

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

       1. *-commutative 94.2%

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

       2. sqrt-div 94.1%

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

       3. metadata-eval 94.1%

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

       4. associate-/r/ 94.0%

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

       5. un-div-inv 94.1%

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

       6. div-inv 94.2%

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

       7. metadata-eval 94.2%

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

   11. Applied egg-rr 94.2%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+34}:\\ \;\;\;\;\frac{\frac{y}{-3}}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+98}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+34}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+95}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.55e+34)
   (* y (* -0.3333333333333333 (pow x -0.5)))
   (if (<= y 2.1e+95)
     (+ 1.0 (/ -0.1111111111111111 x))
     (/ y (* (sqrt x) -3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.55e+34) {
		tmp = y * (-0.3333333333333333 * pow(x, -0.5));
	} else if (y <= 2.1e+95) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = y / (sqrt(x) * -3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.55d+34)) then
        tmp = y * ((-0.3333333333333333d0) * (x ** (-0.5d0)))
    else if (y <= 2.1d+95) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = y / (sqrt(x) * (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.55e+34) {
		tmp = y * (-0.3333333333333333 * Math.pow(x, -0.5));
	} else if (y <= 2.1e+95) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = y / (Math.sqrt(x) * -3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.55e+34:
		tmp = y * (-0.3333333333333333 * math.pow(x, -0.5))
	elif y <= 2.1e+95:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = y / (math.sqrt(x) * -3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.55e+34)
		tmp = Float64(y * Float64(-0.3333333333333333 * (x ^ -0.5)));
	elseif (y <= 2.1e+95)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(y / Float64(sqrt(x) * -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.55e+34)
		tmp = y * (-0.3333333333333333 * (x ^ -0.5));
	elseif (y <= 2.1e+95)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = y / (sqrt(x) * -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.55e+34], N[(y * N[(-0.3333333333333333 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.1e+95], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.54999999999999989e34

   1. Initial program 99.4%

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

      1. associate--l- 99.4%

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

      2. sub-neg 99.4%

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

      3. +-commutative 99.4%

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

      4. distribute-neg-in 99.4%

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

      5. distribute-frac-neg 99.4%

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

      6. sub-neg 99.4%

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

      7. neg-mul-1 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-/l* 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* 99.6%

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

      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. Step-by-step derivation

      1. +-commutative 99.6%

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

      2. fma-undefine 99.5%

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

      3. associate-+l+ 99.5%

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

      4. associate-*r/ 99.6%

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

      5. associate-*l/ 99.5%

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

      6. *-commutative 99.5%

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

      7. +-commutative 99.5%

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

      8. div-inv 99.5%

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

      9. metadata-eval 99.5%

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

      10. cancel-sign-sub-inv 99.5%

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

      11. div-inv 99.5%

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

      12. +-commutative 99.5%

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

      13. associate-+l- 99.5%

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

      14. cancel-sign-sub-inv 99.5%

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

      15. metadata-eval 99.5%

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

      16. metadata-eval 99.5%

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

      17. times-frac 99.5%

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

      18. *-un-lft-identity 99.5%

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

      19. associate--r+ 99.5%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around inf 83.5%

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

      1. associate-*r* 83.6%

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

      2. *-commutative 83.6%

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

   9. Simplified 83.6%

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

       1. *-un-lft-identity 83.6%

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

       2. inv-pow 83.6%

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

       3. sqrt-pow1 83.6%

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

       4. metadata-eval 83.6%

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

   11. Applied egg-rr 83.6%

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

       1. *-lft-identity 83.6%

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

   13. Simplified 83.6%

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

   if -1.54999999999999989e34 < y < 2.1e95

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.7%

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

      10. fma-neg 99.7%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

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

      13. *-commutative 99.7%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 97.4%

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

   if 2.1e95 < y

   1. Initial program 99.5%

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

      1. associate--l- 99.5%

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

      2. sub-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. distribute-neg-in 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. sub-neg 99.5%

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

      7. neg-mul-1 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-/l* 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.5%

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

      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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. fma-undefine 99.5%

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

      3. associate-+l+ 99.5%

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

      4. associate-*r/ 99.4%

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

      5. associate-*l/ 97.7%

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

      6. *-commutative 97.7%

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

      7. +-commutative 97.7%

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

      8. div-inv 97.7%

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

      9. metadata-eval 97.7%

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

      10. cancel-sign-sub-inv 97.7%

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

      11. div-inv 97.7%

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

      12. +-commutative 97.7%

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

      13. associate-+l- 97.7%

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

      14. cancel-sign-sub-inv 97.7%

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

      15. metadata-eval 97.7%

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

      16. metadata-eval 97.7%

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

      17. times-frac 99.6%

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

      18. *-un-lft-identity 99.6%

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

      19. associate--r+ 99.6%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around inf 92.2%

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

      1. associate-*r* 94.2%

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

      2. *-commutative 94.2%

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

   9. Simplified 94.2%

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

       1. *-commutative 94.2%

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

       2. sqrt-div 94.1%

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

       3. metadata-eval 94.1%

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

       4. associate-/r/ 94.0%

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

       5. un-div-inv 94.1%

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

       6. div-inv 94.2%

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

       7. metadata-eval 94.2%

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

   11. Applied egg-rr 94.2%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+34}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+95}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(y \cdot \left(-\sqrt{x}\right)\right) - 0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x}} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 4.5)
   (/ (- (* 0.3333333333333333 (* y (- (sqrt x)))) 0.1111111111111111) x)
   (- 1.0 (* (/ y (sqrt x)) 0.3333333333333333))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4.5) {
		tmp = ((0.3333333333333333 * (y * -sqrt(x))) - 0.1111111111111111) / x;
	} else {
		tmp = 1.0 - ((y / sqrt(x)) * 0.3333333333333333);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 4.5d0) then
        tmp = ((0.3333333333333333d0 * (y * -sqrt(x))) - 0.1111111111111111d0) / x
    else
        tmp = 1.0d0 - ((y / sqrt(x)) * 0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 4.5) {
		tmp = ((0.3333333333333333 * (y * -Math.sqrt(x))) - 0.1111111111111111) / x;
	} else {
		tmp = 1.0 - ((y / Math.sqrt(x)) * 0.3333333333333333);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 4.5:
		tmp = ((0.3333333333333333 * (y * -math.sqrt(x))) - 0.1111111111111111) / x
	else:
		tmp = 1.0 - ((y / math.sqrt(x)) * 0.3333333333333333)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4.5)
		tmp = Float64(Float64(Float64(0.3333333333333333 * Float64(y * Float64(-sqrt(x)))) - 0.1111111111111111) / x);
	else
		tmp = Float64(1.0 - Float64(Float64(y / sqrt(x)) * 0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 4.5)
		tmp = ((0.3333333333333333 * (y * -sqrt(x))) - 0.1111111111111111) / x;
	else
		tmp = 1.0 - ((y / sqrt(x)) * 0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4.5], N[(N[(N[(0.3333333333333333 * N[(y * (-N[Sqrt[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] - 0.1111111111111111), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.5

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 98.5%

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

      1. mul-1-neg 98.5%

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

      2. *-commutative 98.5%

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

   5. Simplified 98.5%

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

   if 4.5 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 99.1%

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

      1. *-commutative 99.1%

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

      2. sqrt-div 99.2%

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

      3. metadata-eval 99.2%

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

      4. un-div-inv 99.2%

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

   5. Applied egg-rr 99.2%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(y \cdot \left(-\sqrt{x}\right)\right) - 0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x}} \cdot 0.3333333333333333\\ \end{array} \]
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 99.6%

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

   1. sub-neg 99.6%

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

   2. *-commutative 99.6%

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

   3. associate-/r* 99.6%

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

   4. metadata-eval 99.6%

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

   5. distribute-frac-neg 99.6%

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

   6. neg-mul-1 99.6%

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

   7. times-frac 99.3%

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

   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: 68.8% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+134}:\\ \;\;\;\;1 + \frac{y \cdot \frac{0.1111111111111111}{x}}{y}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-0.1111111111111111 \cdot y}{x}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.25e+134)
   (+ 1.0 (/ (* y (/ 0.1111111111111111 x)) y))
   (if (<= y 1.4e+154)
     (+ 1.0 (/ -0.1111111111111111 x))
     (+ 1.0 (/ (/ (* -0.1111111111111111 y) x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.25e+134) {
		tmp = 1.0 + ((y * (0.1111111111111111 / x)) / y);
	} else if (y <= 1.4e+154) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 + (((-0.1111111111111111 * y) / x) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.25d+134)) then
        tmp = 1.0d0 + ((y * (0.1111111111111111d0 / x)) / y)
    else if (y <= 1.4d+154) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = 1.0d0 + ((((-0.1111111111111111d0) * y) / x) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.25e+134) {
		tmp = 1.0 + ((y * (0.1111111111111111 / x)) / y);
	} else if (y <= 1.4e+154) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 + (((-0.1111111111111111 * y) / x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.25e+134:
		tmp = 1.0 + ((y * (0.1111111111111111 / x)) / y)
	elif y <= 1.4e+154:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = 1.0 + (((-0.1111111111111111 * y) / x) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.25e+134)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(0.1111111111111111 / x)) / y));
	elseif (y <= 1.4e+154)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(-0.1111111111111111 * y) / x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.25e+134)
		tmp = 1.0 + ((y * (0.1111111111111111 / x)) / y);
	elseif (y <= 1.4e+154)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = 1.0 + (((-0.1111111111111111 * y) / x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.25e+134], N[(1.0 + N[(N[(y * N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.4e+154], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(-0.1111111111111111 * y), $MachinePrecision] / x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.24999999999999995e134

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.5%

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

      10. 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* 99.6%

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

      12. metadata-eval 99.6%

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

      13. *-commutative 99.6%

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

      14. associate-/r* 99.6%

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

      15. distribute-neg-frac 99.6%

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

      16. metadata-eval 99.6%

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

      17. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 99.6%

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

      1. associate-*r/ 99.6%

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

      2. metadata-eval 99.6%

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

      3. associate-/r* 99.6%

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

   7. Simplified 99.6%

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

      1. sub-neg 99.6%

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

      2. distribute-lft-in 99.6%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 0.4%

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

      5. swap-sqr 0.4%

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

      6. metadata-eval 0.4%

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

      7. add-sqr-sqrt 0.4%

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

      8. div-inv 0.4%

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

      9. sqrt-div 0.4%

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

      10. metadata-eval 0.4%

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

      11. associate-/l/ 0.4%

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

      12. distribute-neg-frac 0.4%

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

      13. metadata-eval 0.4%

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

   9. Applied egg-rr 0.4%

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

       1. distribute-lft-out 0.4%

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

       2. *-commutative 0.4%

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

   11. Simplified 0.4%

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

   12. Taylor expanded in y around 0 2.9%

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

       1. *-commutative 2.9%

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

       2. metadata-eval 2.9%

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

       3. distribute-neg-frac 2.9%

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

       4. associate-/r* 2.9%

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

       5. distribute-neg-frac 2.9%

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

       6. associate-*l/ 2.8%

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

       7. add-sqr-sqrt 2.8%

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

       8. distribute-rgt-neg-in 2.8%

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

       9. sqrt-div 2.8%

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

       10. metadata-eval 2.8%

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

       11. distribute-frac-neg2 2.8%

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

       12. metadata-eval 2.8%

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

       13. frac-2neg 2.8%

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

       14. add-sqr-sqrt 0.0%

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

       15. sqrt-unprod 27.2%

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

       16. frac-times 27.2%

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

       17. metadata-eval 27.2%

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

       18. add-sqr-sqrt 27.2%

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

       19. add-sqr-sqrt 27.2%

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

   14. Applied egg-rr 27.2%

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

   if -1.24999999999999995e134 < y < 1.4e154

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. sub-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. distribute-neg-in 99.6%

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

      5. distribute-frac-neg 99.6%

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

      6. sub-neg 99.6%

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

      7. neg-mul-1 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-/l* 99.6%

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

      10. fma-neg 99.7%

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

      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 86.4%

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

   if 1.4e154 < y

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. sub-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. distribute-neg-in 99.6%

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

      5. distribute-frac-neg 99.6%

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

      6. sub-neg 99.6%

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

      7. neg-mul-1 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-/l* 99.6%

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

      10. fma-neg 99.6%

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

      11. associate-/r* 99.6%

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

      12. metadata-eval 99.6%

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

      13. *-commutative 99.6%

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

      14. associate-/r* 99.6%

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

      15. distribute-neg-frac 99.6%

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

      16. metadata-eval 99.6%

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

      17. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 99.7%

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

      1. associate-*r/ 99.7%

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

      2. metadata-eval 99.7%

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

      3. associate-/r* 99.7%

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

   7. Simplified 99.7%

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

      1. sub-neg 99.7%

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

      2. distribute-lft-in 99.7%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 0.3%

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

      5. swap-sqr 0.3%

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

      6. metadata-eval 0.3%

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

      7. add-sqr-sqrt 0.3%

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

      8. div-inv 0.3%

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

      9. sqrt-div 0.3%

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

      10. metadata-eval 0.3%

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

      11. associate-/l/ 0.3%

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

      12. distribute-neg-frac 0.3%

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

      13. metadata-eval 0.3%

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

   9. Applied egg-rr 0.3%

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

       1. distribute-lft-out 0.3%

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

       2. *-commutative 0.3%

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

   11. Simplified 0.3%

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

   12. Taylor expanded in y around 0 3.5%

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

       1. associate-*r/ 3.5%

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

       2. associate-/r* 32.7%

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

   14. Applied egg-rr 32.7%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+134}:\\ \;\;\;\;1 + \frac{y \cdot \frac{0.1111111111111111}{x}}{y}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-0.1111111111111111 \cdot y}{x}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.1% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+133}:\\ \;\;\;\;1 + \frac{y \cdot \frac{0.1111111111111111}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.1e+133)
   (+ 1.0 (/ (* y (/ 0.1111111111111111 x)) y))
   (+ 1.0 (/ -0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.1e+133) {
		tmp = 1.0 + ((y * (0.1111111111111111 / x)) / y);
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.1d+133)) then
        tmp = 1.0d0 + ((y * (0.1111111111111111d0 / x)) / y)
    else
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.1e+133) {
		tmp = 1.0 + ((y * (0.1111111111111111 / x)) / y);
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.1e+133:
		tmp = 1.0 + ((y * (0.1111111111111111 / x)) / y)
	else:
		tmp = 1.0 + (-0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.1e+133)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(0.1111111111111111 / x)) / y));
	else
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.1e+133)
		tmp = 1.0 + ((y * (0.1111111111111111 / x)) / y);
	else
		tmp = 1.0 + (-0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.1e+133], N[(1.0 + N[(N[(y * N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.10000000000000004e133

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.5%

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

      10. 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* 99.6%

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

      12. metadata-eval 99.6%

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

      13. *-commutative 99.6%

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

      14. associate-/r* 99.6%

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

      15. distribute-neg-frac 99.6%

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

      16. metadata-eval 99.6%

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

      17. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 99.6%

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

      1. associate-*r/ 99.6%

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

      2. metadata-eval 99.6%

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

      3. associate-/r* 99.6%

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

   7. Simplified 99.6%

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

      1. sub-neg 99.6%

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

      2. distribute-lft-in 99.6%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 0.4%

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

      5. swap-sqr 0.4%

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

      6. metadata-eval 0.4%

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

      7. add-sqr-sqrt 0.4%

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

      8. div-inv 0.4%

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

      9. sqrt-div 0.4%

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

      10. metadata-eval 0.4%

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

      11. associate-/l/ 0.4%

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

      12. distribute-neg-frac 0.4%

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

      13. metadata-eval 0.4%

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

   9. Applied egg-rr 0.4%

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

       1. distribute-lft-out 0.4%

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

       2. *-commutative 0.4%

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

   11. Simplified 0.4%

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

   12. Taylor expanded in y around 0 2.9%

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

       1. *-commutative 2.9%

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

       2. metadata-eval 2.9%

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

       3. distribute-neg-frac 2.9%

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

       4. associate-/r* 2.9%

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

       5. distribute-neg-frac 2.9%

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

       6. associate-*l/ 2.8%

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

       7. add-sqr-sqrt 2.8%

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

       8. distribute-rgt-neg-in 2.8%

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

       9. sqrt-div 2.8%

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

       10. metadata-eval 2.8%

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

       11. distribute-frac-neg2 2.8%

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

       12. metadata-eval 2.8%

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

       13. frac-2neg 2.8%

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

       14. add-sqr-sqrt 0.0%

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

       15. sqrt-unprod 27.2%

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

       16. frac-times 27.2%

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

       17. metadata-eval 27.2%

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

       18. add-sqr-sqrt 27.2%

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

       19. add-sqr-sqrt 27.2%

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

   14. Applied egg-rr 27.2%

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

   if -4.10000000000000004e133 < y

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. sub-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. distribute-neg-in 99.6%

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

      5. distribute-frac-neg 99.6%

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

      6. sub-neg 99.6%

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

      7. neg-mul-1 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-/l* 99.6%

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

      10. 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 73.4%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+133}:\\ \;\;\;\;1 + \frac{y \cdot \frac{0.1111111111111111}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.9% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		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 <= 0.11d0) 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 <= 0.11) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.11:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 98.5%

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

      1. mul-1-neg 98.5%

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

      2. *-commutative 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in y around 0 60.4%

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

   if 0.110000000000000001 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 99.1%

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

   4. Taylor expanded in y around 0 63.6%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.8% 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 99.6%

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

   1. associate--l- 99.6%

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

   2. sub-neg 99.6%

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

   3. +-commutative 99.6%

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

   4. distribute-neg-in 99.6%

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

   5. distribute-frac-neg 99.6%

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

   6. sub-neg 99.6%

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

   7. neg-mul-1 99.6%

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

   8. *-commutative 99.6%

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

   9. associate-/l* 99.6%

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

   10. fma-neg 99.6%

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

   11. associate-/r* 99.7%

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

   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 62.6%

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

6. Final simplification 62.6%

   \[\leadsto 1 + \frac{-0.1111111111111111}{x} \]
7. Add Preprocessing

Alternative 14: 31.9% 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 99.6%

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

3. Taylor expanded in x around inf 64.3%

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

4. Taylor expanded in y around 0 29.2%

   \[\leadsto \color{blue}{1} \]

5. Final simplification 29.2%

   \[\leadsto 1 \]
6. 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 2024084 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

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

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