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

Percentage Accurate: 99.7% → 99.7%

Time: 12.1s

Alternatives: 19

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return (1.0 + (-1.0 / (x * 9.0))) - (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 + ((-1.0d0) / (x * 9.0d0))) - (y / sqrt((x * 9.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. metadata-eval 99.7%

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

   3. sqrt-prod 99.7%

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

   4. pow1/2 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

3. Applied egg-rr 99.7%

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

   1. unpow1/2 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+55}:\\ \;\;\;\;1 + \frac{\frac{y}{\sqrt{x}}}{-3}\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{+57}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{{x}^{-0.5}}{-3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.8e+55)
   (+ 1.0 (/ (/ y (sqrt x)) -3.0))
   (if (<= y 2.95e+57)
     (+ 1.0 (/ (/ -1.0 x) 9.0))
     (+ 1.0 (* y (/ (pow x -0.5) -3.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.8e+55) {
		tmp = 1.0 + ((y / sqrt(x)) / -3.0);
	} else if (y <= 2.95e+57) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = 1.0 + (y * (pow(x, -0.5) / -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 <= (-3.8d+55)) then
        tmp = 1.0d0 + ((y / sqrt(x)) / (-3.0d0))
    else if (y <= 2.95d+57) then
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    else
        tmp = 1.0d0 + (y * ((x ** (-0.5d0)) / (-3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.8e+55) {
		tmp = 1.0 + ((y / Math.sqrt(x)) / -3.0);
	} else if (y <= 2.95e+57) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = 1.0 + (y * (Math.pow(x, -0.5) / -3.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.8e+55:
		tmp = 1.0 + ((y / math.sqrt(x)) / -3.0)
	elif y <= 2.95e+57:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	else:
		tmp = 1.0 + (y * (math.pow(x, -0.5) / -3.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.8e+55)
		tmp = Float64(1.0 + Float64(Float64(y / sqrt(x)) / -3.0));
	elseif (y <= 2.95e+57)
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	else
		tmp = Float64(1.0 + Float64(y * Float64((x ^ -0.5) / -3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.8e+55)
		tmp = 1.0 + ((y / sqrt(x)) / -3.0);
	elseif (y <= 2.95e+57)
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	else
		tmp = 1.0 + (y * ((x ^ -0.5) / -3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.8e+55], N[(1.0 + N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.95e+57], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(N[Power[x, -0.5], $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.8e55

   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. +-commutative 99.7%

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

      3. associate--r+ 99.7%

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

      4. sub-neg 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. associate-+r- 99.7%

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

      7. neg-mul-1 99.7%

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

      8. associate-*l/ 99.6%

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

      9. fma-neg 99.6%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

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

      12. distribute-neg-frac 99.4%

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

      13. metadata-eval 99.4%

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

      14. *-commutative 99.4%

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

      15. associate-/r* 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 94.4%

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

      1. *-commutative 94.4%

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

      2. associate-*l* 94.4%

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

      3. *-commutative 94.4%

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

   6. Simplified 94.4%

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

      1. associate-*r* 94.2%

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

      2. metadata-eval 94.2%

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

      3. div-inv 94.5%

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

      4. associate-*l/ 94.5%

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

      5. inv-pow 94.5%

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

      6. sqrt-pow1 94.6%

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

      7. metadata-eval 94.6%

         \[\leadsto 1 + \frac{{x}^{\color{blue}{-0.5}} \cdot y}{-3} \]

   8. Applied egg-rr 94.6%

      \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5} \cdot y}{-3}} \]
   9. Step-by-step derivation

      1. associate-/l* 94.5%

         \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5}}{\frac{-3}{y}}} \]

      2. associate-/r/ 94.5%

         \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5}}{-3} \cdot y} \]

   10. Simplified 94.5%

       \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5}}{-3} \cdot y} \]
   11. Step-by-step derivation

       1. *-commutative 94.5%

          \[\leadsto 1 + \color{blue}{y \cdot \frac{{x}^{-0.5}}{-3}} \]

       2. associate-*r/ 94.6%

          \[\leadsto 1 + \color{blue}{\frac{y \cdot {x}^{-0.5}}{-3}} \]

       3. metadata-eval 94.6%

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

       4. sqrt-pow1 94.5%

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

       5. inv-pow 94.5%

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

       6. sqrt-div 94.5%

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

       7. metadata-eval 94.5%

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

       8. div-inv 94.5%

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

   12. Applied egg-rr 94.5%

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

   if -3.8e55 < y < 2.95000000000000006e57

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--r+ 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. associate-*l/ 99.8%

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

      9. fma-neg 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-/r* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 97.4%

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

      1. *-commutative 97.4%

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

      2. inv-pow 97.4%

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

      3. metadata-eval 97.4%

         \[\leadsto 1 - {x}^{\color{blue}{\left(-0.5 + -0.5\right)}} \cdot 0.1111111111111111 \]

      4. pow-prod-up 97.3%

         \[\leadsto 1 - \color{blue}{\left({x}^{-0.5} \cdot {x}^{-0.5}\right)} \cdot 0.1111111111111111 \]

      5. metadata-eval 97.3%

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

      6. swap-sqr 97.2%

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

      7. metadata-eval 97.2%

         \[\leadsto 1 - \left({x}^{-0.5} \cdot \color{blue}{\frac{1}{-3}}\right) \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right) \]

      8. div-inv 97.3%

         \[\leadsto 1 - \color{blue}{\frac{{x}^{-0.5}}{-3}} \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right) \]

      9. metadata-eval 97.3%

         \[\leadsto 1 - \frac{{x}^{-0.5}}{-3} \cdot \left({x}^{-0.5} \cdot \color{blue}{\frac{1}{-3}}\right) \]

      10. div-inv 97.2%

          \[\leadsto 1 - \frac{{x}^{-0.5}}{-3} \cdot \color{blue}{\frac{{x}^{-0.5}}{-3}} \]

      11. frac-times 97.2%

          \[\leadsto 1 - \color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5}}{-3 \cdot -3}} \]

      12. pow-prod-up 97.5%

          \[\leadsto 1 - \frac{\color{blue}{{x}^{\left(-0.5 + -0.5\right)}}}{-3 \cdot -3} \]

      13. metadata-eval 97.5%

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

      14. inv-pow 97.5%

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

      15. metadata-eval 97.5%

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

   6. Applied egg-rr 97.5%

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

   if 2.95000000000000006e57 < y

   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. +-commutative 99.4%

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

      3. associate--r+ 99.4%

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

      4. sub-neg 99.4%

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

      5. distribute-frac-neg 99.4%

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

      6. associate-+r- 99.4%

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

      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. associate-*l/ 99.3%

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

      9. fma-neg 99.3%

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

      10. associate-/r* 99.5%

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

      11. metadata-eval 99.5%

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

      12. distribute-neg-frac 99.5%

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

      13. metadata-eval 99.5%

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

      14. *-commutative 99.5%

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

      15. associate-/r* 99.5%

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

      16. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 93.7%

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

      1. *-commutative 93.7%

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

      2. associate-*l* 93.8%

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

      3. *-commutative 93.8%

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

   6. Simplified 93.8%

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

      1. associate-*r* 93.8%

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

      2. metadata-eval 93.8%

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

      3. div-inv 94.0%

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

      4. associate-*l/ 93.9%

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

      5. inv-pow 93.9%

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

      6. sqrt-pow1 94.0%

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

      7. metadata-eval 94.0%

         \[\leadsto 1 + \frac{{x}^{\color{blue}{-0.5}} \cdot y}{-3} \]

   8. Applied egg-rr 94.0%

      \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5} \cdot y}{-3}} \]
   9. Step-by-step derivation

      1. associate-/l* 93.7%

         \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5}}{\frac{-3}{y}}} \]

      2. associate-/r/ 94.0%

         \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5}}{-3} \cdot y} \]

   10. Simplified 94.0%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+55}:\\ \;\;\;\;1 + \frac{\frac{y}{\sqrt{x}}}{-3}\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{+57}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{{x}^{-0.5}}{-3}\\ \end{array} \]

Alternative 3: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+55}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x} \cdot 3}\\ \mathbf{elif}\;y \leq 3.75 \cdot 10^{+58}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{{x}^{-0.5}}{-3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8e+55)
   (- 1.0 (/ y (* (sqrt x) 3.0)))
   (if (<= y 3.75e+58)
     (+ 1.0 (/ (/ -1.0 x) 9.0))
     (+ 1.0 (* y (/ (pow x -0.5) -3.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8e+55) {
		tmp = 1.0 - (y / (sqrt(x) * 3.0));
	} else if (y <= 3.75e+58) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = 1.0 + (y * (pow(x, -0.5) / -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 <= (-8d+55)) then
        tmp = 1.0d0 - (y / (sqrt(x) * 3.0d0))
    else if (y <= 3.75d+58) then
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    else
        tmp = 1.0d0 + (y * ((x ** (-0.5d0)) / (-3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8e+55) {
		tmp = 1.0 - (y / (Math.sqrt(x) * 3.0));
	} else if (y <= 3.75e+58) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = 1.0 + (y * (Math.pow(x, -0.5) / -3.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8e+55:
		tmp = 1.0 - (y / (math.sqrt(x) * 3.0))
	elif y <= 3.75e+58:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	else:
		tmp = 1.0 + (y * (math.pow(x, -0.5) / -3.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8e+55)
		tmp = Float64(1.0 - Float64(y / Float64(sqrt(x) * 3.0)));
	elseif (y <= 3.75e+58)
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	else
		tmp = Float64(1.0 + Float64(y * Float64((x ^ -0.5) / -3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8e+55)
		tmp = 1.0 - (y / (sqrt(x) * 3.0));
	elseif (y <= 3.75e+58)
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	else
		tmp = 1.0 + (y * ((x ^ -0.5) / -3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8e+55], N[(1.0 - N[(y / N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.75e+58], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(N[Power[x, -0.5], $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.00000000000000008e55

   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. +-commutative 99.7%

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

      3. associate--r+ 99.7%

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

      4. sub-neg 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. associate-+r- 99.7%

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

      7. neg-mul-1 99.7%

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

      8. associate-*l/ 99.6%

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

      9. fma-neg 99.6%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

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

      12. distribute-neg-frac 99.4%

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

      13. metadata-eval 99.4%

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

      14. *-commutative 99.4%

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

      15. associate-/r* 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 94.4%

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

      1. *-commutative 94.4%

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

      2. associate-*l* 94.4%

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

      3. *-commutative 94.4%

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

   6. Simplified 94.4%

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

      1. associate-*r* 94.2%

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

      2. metadata-eval 94.2%

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

      3. div-inv 94.5%

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

      4. sqrt-div 94.5%

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

      5. metadata-eval 94.5%

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

      6. associate-/r* 94.6%

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

      7. frac-2neg 94.6%

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

      8. metadata-eval 94.6%

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

      9. *-commutative 94.6%

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

      10. distribute-lft-neg-in 94.6%

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

      11. metadata-eval 94.6%

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

      12. associate-*l/ 94.7%

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

      13. neg-mul-1 94.7%

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

   8. Applied egg-rr 94.7%

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

   if -8.00000000000000008e55 < y < 3.75e58

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--r+ 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. associate-*l/ 99.8%

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

      9. fma-neg 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-/r* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 97.4%

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

      1. *-commutative 97.4%

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

      2. inv-pow 97.4%

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

      3. metadata-eval 97.4%

         \[\leadsto 1 - {x}^{\color{blue}{\left(-0.5 + -0.5\right)}} \cdot 0.1111111111111111 \]

      4. pow-prod-up 97.3%

         \[\leadsto 1 - \color{blue}{\left({x}^{-0.5} \cdot {x}^{-0.5}\right)} \cdot 0.1111111111111111 \]

      5. metadata-eval 97.3%

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

      6. swap-sqr 97.2%

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

      7. metadata-eval 97.2%

         \[\leadsto 1 - \left({x}^{-0.5} \cdot \color{blue}{\frac{1}{-3}}\right) \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right) \]

      8. div-inv 97.3%

         \[\leadsto 1 - \color{blue}{\frac{{x}^{-0.5}}{-3}} \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right) \]

      9. metadata-eval 97.3%

         \[\leadsto 1 - \frac{{x}^{-0.5}}{-3} \cdot \left({x}^{-0.5} \cdot \color{blue}{\frac{1}{-3}}\right) \]

      10. div-inv 97.2%

          \[\leadsto 1 - \frac{{x}^{-0.5}}{-3} \cdot \color{blue}{\frac{{x}^{-0.5}}{-3}} \]

      11. frac-times 97.2%

          \[\leadsto 1 - \color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5}}{-3 \cdot -3}} \]

      12. pow-prod-up 97.5%

          \[\leadsto 1 - \frac{\color{blue}{{x}^{\left(-0.5 + -0.5\right)}}}{-3 \cdot -3} \]

      13. metadata-eval 97.5%

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

      14. inv-pow 97.5%

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

      15. metadata-eval 97.5%

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

   6. Applied egg-rr 97.5%

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

   if 3.75e58 < y

   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. +-commutative 99.4%

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

      3. associate--r+ 99.4%

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

      4. sub-neg 99.4%

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

      5. distribute-frac-neg 99.4%

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

      6. associate-+r- 99.4%

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

      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. associate-*l/ 99.3%

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

      9. fma-neg 99.3%

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

      10. associate-/r* 99.5%

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

      11. metadata-eval 99.5%

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

      12. distribute-neg-frac 99.5%

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

      13. metadata-eval 99.5%

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

      14. *-commutative 99.5%

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

      15. associate-/r* 99.5%

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

      16. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 93.7%

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

      1. *-commutative 93.7%

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

      2. associate-*l* 93.8%

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

      3. *-commutative 93.8%

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

   6. Simplified 93.8%

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

      1. associate-*r* 93.8%

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

      2. metadata-eval 93.8%

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

      3. div-inv 94.0%

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

      4. associate-*l/ 93.9%

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

      5. inv-pow 93.9%

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

      6. sqrt-pow1 94.0%

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

      7. metadata-eval 94.0%

         \[\leadsto 1 + \frac{{x}^{\color{blue}{-0.5}} \cdot y}{-3} \]

   8. Applied egg-rr 94.0%

      \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5} \cdot y}{-3}} \]
   9. Step-by-step derivation

      1. associate-/l* 93.7%

         \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5}}{\frac{-3}{y}}} \]

      2. associate-/r/ 94.0%

         \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5}}{-3} \cdot y} \]

   10. Simplified 94.0%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+55}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x} \cdot 3}\\ \mathbf{elif}\;y \leq 3.75 \cdot 10^{+58}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{{x}^{-0.5}}{-3}\\ \end{array} \]

Alternative 4: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+55} \lor \neg \left(y \leq 4 \cdot 10^{+60}\right):\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.1e+55) (not (<= y 4e+60)))
   (+ 1.0 (* y (/ -0.3333333333333333 (sqrt x))))
   (+ 1.0 (/ (/ -1.0 x) 9.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.1e+55) || !(y <= 4e+60)) {
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	} else {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-4.1d+55)) .or. (.not. (y <= 4d+60))) then
        tmp = 1.0d0 + (y * ((-0.3333333333333333d0) / sqrt(x)))
    else
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -4.1e+55) || !(y <= 4e+60)) {
		tmp = 1.0 + (y * (-0.3333333333333333 / Math.sqrt(x)));
	} else {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.1e+55) or not (y <= 4e+60):
		tmp = 1.0 + (y * (-0.3333333333333333 / math.sqrt(x)))
	else:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.1e+55) || !(y <= 4e+60))
		tmp = Float64(1.0 + Float64(y * Float64(-0.3333333333333333 / sqrt(x))));
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4.1e+55) || ~((y <= 4e+60)))
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	else
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.1e+55], N[Not[LessEqual[y, 4e+60]], $MachinePrecision]], N[(1.0 + N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.09999999999999981e55 or 3.9999999999999998e60 < 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. +-commutative 99.6%

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

      3. associate--r+ 99.6%

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

      4. sub-neg 99.6%

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

      5. distribute-frac-neg 99.6%

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

      6. associate-+r- 99.6%

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

      7. neg-mul-1 99.6%

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

      8. associate-*l/ 99.4%

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

      9. fma-neg 99.4%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

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

      12. distribute-neg-frac 99.4%

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

      13. metadata-eval 99.4%

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

      14. *-commutative 99.4%

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

      15. associate-/r* 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 94.0%

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

      1. *-commutative 94.0%

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

      2. associate-*l* 94.1%

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

      3. *-commutative 94.1%

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

   6. Simplified 94.1%

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

      1. associate-*r* 94.0%

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

      2. metadata-eval 94.0%

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

      3. div-inv 94.2%

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

      4. associate-*l/ 94.2%

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

      5. inv-pow 94.2%

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

      6. sqrt-pow1 94.2%

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

      7. metadata-eval 94.2%

         \[\leadsto 1 + \frac{{x}^{\color{blue}{-0.5}} \cdot y}{-3} \]

   8. Applied egg-rr 94.2%

      \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5} \cdot y}{-3}} \]
   9. Step-by-step derivation

      1. associate-/l* 94.1%

         \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5}}{\frac{-3}{y}}} \]

      2. associate-/r/ 94.2%

         \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5}}{-3} \cdot y} \]

   10. Simplified 94.2%

       \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5}}{-3} \cdot y} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 49.3%

          \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{x}^{-0.5}}{-3}\right)\right)} \cdot y \]

       2. expm1-udef 9.5%

          \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{x}^{-0.5}}{-3}\right)} - 1\right)} \cdot y \]

       3. clear-num 9.5%

          \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{-3}{{x}^{-0.5}}}}\right)} - 1\right) \cdot y \]

       4. associate-/r/ 9.5%

          \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{-3} \cdot {x}^{-0.5}}\right)} - 1\right) \cdot y \]

       5. metadata-eval 9.5%

          \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{-0.3333333333333333} \cdot {x}^{-0.5}\right)} - 1\right) \cdot y \]

       6. metadata-eval 9.5%

          \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot {x}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \cdot y \]

       7. sqrt-pow1 9.5%

          \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot \color{blue}{\sqrt{{x}^{-1}}}\right)} - 1\right) \cdot y \]

       8. inv-pow 9.5%

          \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot \sqrt{\color{blue}{\frac{1}{x}}}\right)} - 1\right) \cdot y \]

       9. sqrt-div 9.5%

          \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}\right)} - 1\right) \cdot y \]

       10. metadata-eval 9.5%

           \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot \frac{\color{blue}{1}}{\sqrt{x}}\right)} - 1\right) \cdot y \]

       11. div-inv 9.5%

           \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}}}\right)} - 1\right) \cdot y \]

   12. Applied egg-rr 9.5%

       \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-0.3333333333333333}{\sqrt{x}}\right)} - 1\right)} \cdot y \]
   13. Step-by-step derivation

       1. expm1-def 49.2%

          \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.3333333333333333}{\sqrt{x}}\right)\right)} \cdot y \]

       2. expm1-log1p 94.1%

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

   14. Simplified 94.1%

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

   if -4.09999999999999981e55 < y < 3.9999999999999998e60

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--r+ 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. associate-*l/ 99.8%

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

      9. fma-neg 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-/r* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 97.4%

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

      1. *-commutative 97.4%

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

      2. inv-pow 97.4%

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

      3. metadata-eval 97.4%

         \[\leadsto 1 - {x}^{\color{blue}{\left(-0.5 + -0.5\right)}} \cdot 0.1111111111111111 \]

      4. pow-prod-up 97.3%

         \[\leadsto 1 - \color{blue}{\left({x}^{-0.5} \cdot {x}^{-0.5}\right)} \cdot 0.1111111111111111 \]

      5. metadata-eval 97.3%

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

      6. swap-sqr 97.2%

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

      7. metadata-eval 97.2%

         \[\leadsto 1 - \left({x}^{-0.5} \cdot \color{blue}{\frac{1}{-3}}\right) \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right) \]

      8. div-inv 97.3%

         \[\leadsto 1 - \color{blue}{\frac{{x}^{-0.5}}{-3}} \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right) \]

      9. metadata-eval 97.3%

         \[\leadsto 1 - \frac{{x}^{-0.5}}{-3} \cdot \left({x}^{-0.5} \cdot \color{blue}{\frac{1}{-3}}\right) \]

      10. div-inv 97.2%

          \[\leadsto 1 - \frac{{x}^{-0.5}}{-3} \cdot \color{blue}{\frac{{x}^{-0.5}}{-3}} \]

      11. frac-times 97.2%

          \[\leadsto 1 - \color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5}}{-3 \cdot -3}} \]

      12. pow-prod-up 97.5%

          \[\leadsto 1 - \frac{\color{blue}{{x}^{\left(-0.5 + -0.5\right)}}}{-3 \cdot -3} \]

      13. metadata-eval 97.5%

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

      14. inv-pow 97.5%

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

      15. metadata-eval 97.5%

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

   6. Applied egg-rr 97.5%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+55} \lor \neg \left(y \leq 4 \cdot 10^{+60}\right):\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \end{array} \]

Alternative 5: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{+55} \lor \neg \left(y \leq 5.2 \cdot 10^{+63}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -6.4e+55) (not (<= y 5.2e+63)))
   (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))
   (+ 1.0 (/ (/ -1.0 x) 9.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -6.4e+55) || !(y <= 5.2e+63)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	} else {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-6.4d+55)) .or. (.not. (y <= 5.2d+63))) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    else
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -6.4e+55) || !(y <= 5.2e+63)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -6.4e+55) or not (y <= 5.2e+63):
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	else:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -6.4e+55) || !(y <= 5.2e+63))
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -6.4e+55) || ~((y <= 5.2e+63)))
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	else
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -6.4e+55], N[Not[LessEqual[y, 5.2e+63]], $MachinePrecision]], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.4000000000000005e55 or 5.2000000000000002e63 < 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. +-commutative 99.6%

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

      3. associate--r+ 99.6%

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

      4. sub-neg 99.6%

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

      5. distribute-frac-neg 99.6%

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

      6. associate-+r- 99.6%

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

      7. neg-mul-1 99.6%

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

      8. associate-*l/ 99.4%

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

      9. fma-neg 99.4%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

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

      12. distribute-neg-frac 99.4%

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

      13. metadata-eval 99.4%

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

      14. *-commutative 99.4%

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

      15. associate-/r* 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 94.0%

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

      1. *-commutative 94.0%

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

      2. associate-*l* 94.1%

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

      3. *-commutative 94.1%

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

   6. Simplified 94.1%

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

      1. associate-*r* 94.0%

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

      2. metadata-eval 94.0%

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

      3. div-inv 94.2%

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

      4. associate-*l/ 94.2%

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

      5. inv-pow 94.2%

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

      6. sqrt-pow1 94.2%

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

      7. metadata-eval 94.2%

         \[\leadsto 1 + \frac{{x}^{\color{blue}{-0.5}} \cdot y}{-3} \]

   8. Applied egg-rr 94.2%

      \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5} \cdot y}{-3}} \]
   9. Step-by-step derivation

      1. associate-/l* 94.1%

         \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5}}{\frac{-3}{y}}} \]

      2. associate-/r/ 94.2%

         \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5}}{-3} \cdot y} \]

   10. Simplified 94.2%

       \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5}}{-3} \cdot y} \]
   11. Step-by-step derivation

       1. *-commutative 94.2%

          \[\leadsto 1 + \color{blue}{y \cdot \frac{{x}^{-0.5}}{-3}} \]

       2. associate-*r/ 94.2%

          \[\leadsto 1 + \color{blue}{\frac{y \cdot {x}^{-0.5}}{-3}} \]

       3. metadata-eval 94.2%

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

       4. sqrt-pow1 94.2%

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

       5. inv-pow 94.2%

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

       6. sqrt-div 94.2%

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

       7. metadata-eval 94.2%

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

       8. div-inv 94.2%

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

   12. Applied egg-rr 94.2%

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

       1. div-inv 94.1%

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

       2. metadata-eval 94.1%

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

   14. Applied egg-rr 94.1%

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

   if -6.4000000000000005e55 < y < 5.2000000000000002e63

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--r+ 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. associate-*l/ 99.8%

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

      9. fma-neg 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-/r* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 97.4%

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

      1. *-commutative 97.4%

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

      2. inv-pow 97.4%

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

      3. metadata-eval 97.4%

         \[\leadsto 1 - {x}^{\color{blue}{\left(-0.5 + -0.5\right)}} \cdot 0.1111111111111111 \]

      4. pow-prod-up 97.3%

         \[\leadsto 1 - \color{blue}{\left({x}^{-0.5} \cdot {x}^{-0.5}\right)} \cdot 0.1111111111111111 \]

      5. metadata-eval 97.3%

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

      6. swap-sqr 97.2%

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

      7. metadata-eval 97.2%

         \[\leadsto 1 - \left({x}^{-0.5} \cdot \color{blue}{\frac{1}{-3}}\right) \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right) \]

      8. div-inv 97.3%

         \[\leadsto 1 - \color{blue}{\frac{{x}^{-0.5}}{-3}} \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right) \]

      9. metadata-eval 97.3%

         \[\leadsto 1 - \frac{{x}^{-0.5}}{-3} \cdot \left({x}^{-0.5} \cdot \color{blue}{\frac{1}{-3}}\right) \]

      10. div-inv 97.2%

          \[\leadsto 1 - \frac{{x}^{-0.5}}{-3} \cdot \color{blue}{\frac{{x}^{-0.5}}{-3}} \]

      11. frac-times 97.2%

          \[\leadsto 1 - \color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5}}{-3 \cdot -3}} \]

      12. pow-prod-up 97.5%

          \[\leadsto 1 - \frac{\color{blue}{{x}^{\left(-0.5 + -0.5\right)}}}{-3 \cdot -3} \]

      13. metadata-eval 97.5%

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

      14. inv-pow 97.5%

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

      15. metadata-eval 97.5%

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

   6. Applied egg-rr 97.5%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{+55} \lor \neg \left(y \leq 5.2 \cdot 10^{+63}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \end{array} \]

Alternative 6: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+55} \lor \neg \left(y \leq 3 \cdot 10^{+57}\right):\\ \;\;\;\;1 + \frac{\frac{y}{\sqrt{x}}}{-3}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5.8e+55) (not (<= y 3e+57)))
   (+ 1.0 (/ (/ y (sqrt x)) -3.0))
   (+ 1.0 (/ (/ -1.0 x) 9.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5.8e+55) || !(y <= 3e+57)) {
		tmp = 1.0 + ((y / sqrt(x)) / -3.0);
	} else {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-5.8d+55)) .or. (.not. (y <= 3d+57))) then
        tmp = 1.0d0 + ((y / sqrt(x)) / (-3.0d0))
    else
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -5.8e+55) || !(y <= 3e+57)) {
		tmp = 1.0 + ((y / Math.sqrt(x)) / -3.0);
	} else {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -5.8e+55) or not (y <= 3e+57):
		tmp = 1.0 + ((y / math.sqrt(x)) / -3.0)
	else:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5.8e+55) || !(y <= 3e+57))
		tmp = Float64(1.0 + Float64(Float64(y / sqrt(x)) / -3.0));
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -5.8e+55) || ~((y <= 3e+57)))
		tmp = 1.0 + ((y / sqrt(x)) / -3.0);
	else
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5.8e+55], N[Not[LessEqual[y, 3e+57]], $MachinePrecision]], N[(1.0 + N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.7999999999999997e55 or 3e57 < 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. +-commutative 99.6%

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

      3. associate--r+ 99.6%

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

      4. sub-neg 99.6%

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

      5. distribute-frac-neg 99.6%

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

      6. associate-+r- 99.6%

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

      7. neg-mul-1 99.6%

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

      8. associate-*l/ 99.4%

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

      9. fma-neg 99.4%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

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

      12. distribute-neg-frac 99.4%

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

      13. metadata-eval 99.4%

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

      14. *-commutative 99.4%

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

      15. associate-/r* 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 94.0%

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

      1. *-commutative 94.0%

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

      2. associate-*l* 94.1%

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

      3. *-commutative 94.1%

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

   6. Simplified 94.1%

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

      1. associate-*r* 94.0%

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

      2. metadata-eval 94.0%

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

      3. div-inv 94.2%

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

      4. associate-*l/ 94.2%

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

      5. inv-pow 94.2%

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

      6. sqrt-pow1 94.2%

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

      7. metadata-eval 94.2%

         \[\leadsto 1 + \frac{{x}^{\color{blue}{-0.5}} \cdot y}{-3} \]

   8. Applied egg-rr 94.2%

      \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5} \cdot y}{-3}} \]
   9. Step-by-step derivation

      1. associate-/l* 94.1%

         \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5}}{\frac{-3}{y}}} \]

      2. associate-/r/ 94.2%

         \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5}}{-3} \cdot y} \]

   10. Simplified 94.2%

       \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5}}{-3} \cdot y} \]
   11. Step-by-step derivation

       1. *-commutative 94.2%

          \[\leadsto 1 + \color{blue}{y \cdot \frac{{x}^{-0.5}}{-3}} \]

       2. associate-*r/ 94.2%

          \[\leadsto 1 + \color{blue}{\frac{y \cdot {x}^{-0.5}}{-3}} \]

       3. metadata-eval 94.2%

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

       4. sqrt-pow1 94.2%

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

       5. inv-pow 94.2%

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

       6. sqrt-div 94.2%

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

       7. metadata-eval 94.2%

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

       8. div-inv 94.2%

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

   12. Applied egg-rr 94.2%

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

   if -5.7999999999999997e55 < y < 3e57

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--r+ 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. associate-*l/ 99.8%

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

      9. fma-neg 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-/r* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 97.4%

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

      1. *-commutative 97.4%

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

      2. inv-pow 97.4%

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

      3. metadata-eval 97.4%

         \[\leadsto 1 - {x}^{\color{blue}{\left(-0.5 + -0.5\right)}} \cdot 0.1111111111111111 \]

      4. pow-prod-up 97.3%

         \[\leadsto 1 - \color{blue}{\left({x}^{-0.5} \cdot {x}^{-0.5}\right)} \cdot 0.1111111111111111 \]

      5. metadata-eval 97.3%

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

      6. swap-sqr 97.2%

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

      7. metadata-eval 97.2%

         \[\leadsto 1 - \left({x}^{-0.5} \cdot \color{blue}{\frac{1}{-3}}\right) \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right) \]

      8. div-inv 97.3%

         \[\leadsto 1 - \color{blue}{\frac{{x}^{-0.5}}{-3}} \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right) \]

      9. metadata-eval 97.3%

         \[\leadsto 1 - \frac{{x}^{-0.5}}{-3} \cdot \left({x}^{-0.5} \cdot \color{blue}{\frac{1}{-3}}\right) \]

      10. div-inv 97.2%

          \[\leadsto 1 - \frac{{x}^{-0.5}}{-3} \cdot \color{blue}{\frac{{x}^{-0.5}}{-3}} \]

      11. frac-times 97.2%

          \[\leadsto 1 - \color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5}}{-3 \cdot -3}} \]

      12. pow-prod-up 97.5%

          \[\leadsto 1 - \frac{\color{blue}{{x}^{\left(-0.5 + -0.5\right)}}}{-3 \cdot -3} \]

      13. metadata-eval 97.5%

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

      14. inv-pow 97.5%

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

      15. metadata-eval 97.5%

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

   6. Applied egg-rr 97.5%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+55} \lor \neg \left(y \leq 3 \cdot 10^{+57}\right):\\ \;\;\;\;1 + \frac{\frac{y}{\sqrt{x}}}{-3}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \end{array} \]

Alternative 7: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+55}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+61}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.6e+55)
   (+ 1.0 (/ -0.3333333333333333 (/ (sqrt x) y)))
   (if (<= y 3.3e+61)
     (+ 1.0 (/ (/ -1.0 x) 9.0))
     (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.6e+55) {
		tmp = 1.0 + (-0.3333333333333333 / (sqrt(x) / y));
	} else if (y <= 3.3e+61) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-8.6d+55)) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) / (sqrt(x) / y))
    else if (y <= 3.3d+61) then
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    else
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8.6e+55) {
		tmp = 1.0 + (-0.3333333333333333 / (Math.sqrt(x) / y));
	} else if (y <= 3.3e+61) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.6e+55:
		tmp = 1.0 + (-0.3333333333333333 / (math.sqrt(x) / y))
	elif y <= 3.3e+61:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	else:
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.6e+55)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 / Float64(sqrt(x) / y)));
	elseif (y <= 3.3e+61)
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	else
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8.6e+55)
		tmp = 1.0 + (-0.3333333333333333 / (sqrt(x) / y));
	elseif (y <= 3.3e+61)
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	else
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.6e+55], N[(1.0 + N[(-0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.3e+61], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.5999999999999998e55

   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. +-commutative 99.7%

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

      3. associate--r+ 99.7%

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

      4. sub-neg 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. associate-+r- 99.7%

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

      7. neg-mul-1 99.7%

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

      8. associate-*l/ 99.6%

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

      9. fma-neg 99.6%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

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

      12. distribute-neg-frac 99.4%

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

      13. metadata-eval 99.4%

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

      14. *-commutative 99.4%

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

      15. associate-/r* 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 94.4%

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

      1. *-commutative 94.4%

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

      2. associate-*l* 94.4%

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

      3. *-commutative 94.4%

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

   6. Simplified 94.4%

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

      1. associate-*r* 94.2%

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

      2. metadata-eval 94.2%

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

      3. div-inv 94.5%

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

      4. associate-*l/ 94.5%

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

      5. inv-pow 94.5%

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

      6. sqrt-pow1 94.6%

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

      7. metadata-eval 94.6%

         \[\leadsto 1 + \frac{{x}^{\color{blue}{-0.5}} \cdot y}{-3} \]

   8. Applied egg-rr 94.6%

      \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5} \cdot y}{-3}} \]
   9. Step-by-step derivation

      1. associate-/l* 94.5%

         \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5}}{\frac{-3}{y}}} \]

      2. associate-/r/ 94.5%

         \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5}}{-3} \cdot y} \]

   10. Simplified 94.5%

       \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5}}{-3} \cdot y} \]
   11. Step-by-step derivation

       1. associate-*l/ 94.6%

          \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5} \cdot y}{-3}} \]

       2. clear-num 94.5%

          \[\leadsto 1 + \color{blue}{\frac{1}{\frac{-3}{{x}^{-0.5} \cdot y}}} \]

       3. metadata-eval 94.5%

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

       4. sqrt-pow1 94.5%

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

       5. inv-pow 94.5%

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

       6. sqrt-div 94.5%

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

       7. metadata-eval 94.5%

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

       8. associate-/r/ 94.4%

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

       9. clear-num 94.4%

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

   12. Applied egg-rr 94.4%

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

       1. associate-/r/ 94.4%

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

       2. metadata-eval 94.4%

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

       3. associate-*r/ 94.4%

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

       4. associate-/l* 94.5%

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

   14. Simplified 94.5%

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

   if -8.5999999999999998e55 < y < 3.2999999999999998e61

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--r+ 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. associate-*l/ 99.8%

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

      9. fma-neg 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-/r* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 97.4%

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

      1. *-commutative 97.4%

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

      2. inv-pow 97.4%

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

      3. metadata-eval 97.4%

         \[\leadsto 1 - {x}^{\color{blue}{\left(-0.5 + -0.5\right)}} \cdot 0.1111111111111111 \]

      4. pow-prod-up 97.3%

         \[\leadsto 1 - \color{blue}{\left({x}^{-0.5} \cdot {x}^{-0.5}\right)} \cdot 0.1111111111111111 \]

      5. metadata-eval 97.3%

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

      6. swap-sqr 97.2%

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

      7. metadata-eval 97.2%

         \[\leadsto 1 - \left({x}^{-0.5} \cdot \color{blue}{\frac{1}{-3}}\right) \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right) \]

      8. div-inv 97.3%

         \[\leadsto 1 - \color{blue}{\frac{{x}^{-0.5}}{-3}} \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right) \]

      9. metadata-eval 97.3%

         \[\leadsto 1 - \frac{{x}^{-0.5}}{-3} \cdot \left({x}^{-0.5} \cdot \color{blue}{\frac{1}{-3}}\right) \]

      10. div-inv 97.2%

          \[\leadsto 1 - \frac{{x}^{-0.5}}{-3} \cdot \color{blue}{\frac{{x}^{-0.5}}{-3}} \]

      11. frac-times 97.2%

          \[\leadsto 1 - \color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5}}{-3 \cdot -3}} \]

      12. pow-prod-up 97.5%

          \[\leadsto 1 - \frac{\color{blue}{{x}^{\left(-0.5 + -0.5\right)}}}{-3 \cdot -3} \]

      13. metadata-eval 97.5%

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

      14. inv-pow 97.5%

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

      15. metadata-eval 97.5%

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

   6. Applied egg-rr 97.5%

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

   if 3.2999999999999998e61 < y

   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. +-commutative 99.4%

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

      3. associate--r+ 99.4%

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

      4. sub-neg 99.4%

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

      5. distribute-frac-neg 99.4%

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

      6. associate-+r- 99.4%

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

      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. associate-*l/ 99.3%

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

      9. fma-neg 99.3%

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

      10. associate-/r* 99.5%

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

      11. metadata-eval 99.5%

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

      12. distribute-neg-frac 99.5%

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

      13. metadata-eval 99.5%

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

      14. *-commutative 99.5%

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

      15. associate-/r* 99.5%

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

      16. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 93.7%

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

      1. *-commutative 93.7%

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

      2. associate-*l* 93.8%

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

      3. *-commutative 93.8%

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

   6. Simplified 93.8%

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

      1. associate-*r* 93.8%

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

      2. metadata-eval 93.8%

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

      3. div-inv 94.0%

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

      4. associate-*l/ 93.9%

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

      5. inv-pow 93.9%

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

      6. sqrt-pow1 94.0%

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

      7. metadata-eval 94.0%

         \[\leadsto 1 + \frac{{x}^{\color{blue}{-0.5}} \cdot y}{-3} \]

   8. Applied egg-rr 94.0%

      \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5} \cdot y}{-3}} \]
   9. Step-by-step derivation

      1. associate-/l* 93.7%

         \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5}}{\frac{-3}{y}}} \]

      2. associate-/r/ 94.0%

         \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5}}{-3} \cdot y} \]

   10. Simplified 94.0%

       \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5}}{-3} \cdot y} \]
   11. Step-by-step derivation

       1. *-commutative 94.0%

          \[\leadsto 1 + \color{blue}{y \cdot \frac{{x}^{-0.5}}{-3}} \]

       2. associate-*r/ 94.0%

          \[\leadsto 1 + \color{blue}{\frac{y \cdot {x}^{-0.5}}{-3}} \]

       3. metadata-eval 94.0%

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

       4. sqrt-pow1 93.9%

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

       5. inv-pow 93.9%

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

       6. sqrt-div 93.9%

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

       7. metadata-eval 93.9%

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

       8. div-inv 93.9%

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

   12. Applied egg-rr 93.9%

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

       1. div-inv 93.9%

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

       2. metadata-eval 93.9%

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

   14. Applied egg-rr 93.9%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+55}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+61}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \]

Alternative 8: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+55}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 2.85 \cdot 10^{+63}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.5e+55)
   (+ 1.0 (/ -0.3333333333333333 (/ (sqrt x) y)))
   (if (<= y 2.85e+63)
     (+ 1.0 (/ (/ -1.0 x) 9.0))
     (+ 1.0 (/ (* y -0.3333333333333333) (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.5e+55) {
		tmp = 1.0 + (-0.3333333333333333 / (sqrt(x) / y));
	} else if (y <= 2.85e+63) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = 1.0 + ((y * -0.3333333333333333) / sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-8.5d+55)) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) / (sqrt(x) / y))
    else if (y <= 2.85d+63) then
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    else
        tmp = 1.0d0 + ((y * (-0.3333333333333333d0)) / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8.5e+55) {
		tmp = 1.0 + (-0.3333333333333333 / (Math.sqrt(x) / y));
	} else if (y <= 2.85e+63) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = 1.0 + ((y * -0.3333333333333333) / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.5e+55:
		tmp = 1.0 + (-0.3333333333333333 / (math.sqrt(x) / y))
	elif y <= 2.85e+63:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	else:
		tmp = 1.0 + ((y * -0.3333333333333333) / math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.5e+55)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 / Float64(sqrt(x) / y)));
	elseif (y <= 2.85e+63)
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	else
		tmp = Float64(1.0 + Float64(Float64(y * -0.3333333333333333) / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8.5e+55)
		tmp = 1.0 + (-0.3333333333333333 / (sqrt(x) / y));
	elseif (y <= 2.85e+63)
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	else
		tmp = 1.0 + ((y * -0.3333333333333333) / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.5e+55], N[(1.0 + N[(-0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.85e+63], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * -0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.50000000000000002e55

   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. +-commutative 99.7%

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

      3. associate--r+ 99.7%

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

      4. sub-neg 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. associate-+r- 99.7%

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

      7. neg-mul-1 99.7%

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

      8. associate-*l/ 99.6%

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

      9. fma-neg 99.6%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

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

      12. distribute-neg-frac 99.4%

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

      13. metadata-eval 99.4%

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

      14. *-commutative 99.4%

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

      15. associate-/r* 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 94.4%

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

      1. *-commutative 94.4%

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

      2. associate-*l* 94.4%

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

      3. *-commutative 94.4%

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

   6. Simplified 94.4%

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

      1. associate-*r* 94.2%

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

      2. metadata-eval 94.2%

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

      3. div-inv 94.5%

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

      4. associate-*l/ 94.5%

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

      5. inv-pow 94.5%

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

      6. sqrt-pow1 94.6%

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

      7. metadata-eval 94.6%

         \[\leadsto 1 + \frac{{x}^{\color{blue}{-0.5}} \cdot y}{-3} \]

   8. Applied egg-rr 94.6%

      \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5} \cdot y}{-3}} \]
   9. Step-by-step derivation

      1. associate-/l* 94.5%

         \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5}}{\frac{-3}{y}}} \]

      2. associate-/r/ 94.5%

         \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5}}{-3} \cdot y} \]

   10. Simplified 94.5%

       \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5}}{-3} \cdot y} \]
   11. Step-by-step derivation

       1. associate-*l/ 94.6%

          \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5} \cdot y}{-3}} \]

       2. clear-num 94.5%

          \[\leadsto 1 + \color{blue}{\frac{1}{\frac{-3}{{x}^{-0.5} \cdot y}}} \]

       3. metadata-eval 94.5%

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

       4. sqrt-pow1 94.5%

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

       5. inv-pow 94.5%

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

       6. sqrt-div 94.5%

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

       7. metadata-eval 94.5%

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

       8. associate-/r/ 94.4%

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

       9. clear-num 94.4%

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

   12. Applied egg-rr 94.4%

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

       1. associate-/r/ 94.4%

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

       2. metadata-eval 94.4%

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

       3. associate-*r/ 94.4%

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

       4. associate-/l* 94.5%

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

   14. Simplified 94.5%

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

   if -8.50000000000000002e55 < y < 2.8500000000000001e63

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--r+ 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. associate-*l/ 99.8%

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

      9. fma-neg 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-/r* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 97.4%

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

      1. *-commutative 97.4%

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

      2. inv-pow 97.4%

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

      3. metadata-eval 97.4%

         \[\leadsto 1 - {x}^{\color{blue}{\left(-0.5 + -0.5\right)}} \cdot 0.1111111111111111 \]

      4. pow-prod-up 97.3%

         \[\leadsto 1 - \color{blue}{\left({x}^{-0.5} \cdot {x}^{-0.5}\right)} \cdot 0.1111111111111111 \]

      5. metadata-eval 97.3%

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

      6. swap-sqr 97.2%

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

      7. metadata-eval 97.2%

         \[\leadsto 1 - \left({x}^{-0.5} \cdot \color{blue}{\frac{1}{-3}}\right) \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right) \]

      8. div-inv 97.3%

         \[\leadsto 1 - \color{blue}{\frac{{x}^{-0.5}}{-3}} \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right) \]

      9. metadata-eval 97.3%

         \[\leadsto 1 - \frac{{x}^{-0.5}}{-3} \cdot \left({x}^{-0.5} \cdot \color{blue}{\frac{1}{-3}}\right) \]

      10. div-inv 97.2%

          \[\leadsto 1 - \frac{{x}^{-0.5}}{-3} \cdot \color{blue}{\frac{{x}^{-0.5}}{-3}} \]

      11. frac-times 97.2%

          \[\leadsto 1 - \color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5}}{-3 \cdot -3}} \]

      12. pow-prod-up 97.5%

          \[\leadsto 1 - \frac{\color{blue}{{x}^{\left(-0.5 + -0.5\right)}}}{-3 \cdot -3} \]

      13. metadata-eval 97.5%

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

      14. inv-pow 97.5%

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

      15. metadata-eval 97.5%

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

   6. Applied egg-rr 97.5%

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

   if 2.8500000000000001e63 < y

   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. +-commutative 99.4%

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

      3. associate--r+ 99.4%

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

      4. sub-neg 99.4%

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

      5. distribute-frac-neg 99.4%

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

      6. associate-+r- 99.4%

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

      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. associate-*l/ 99.3%

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

      9. fma-neg 99.3%

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

      10. associate-/r* 99.5%

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

      11. metadata-eval 99.5%

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

      12. distribute-neg-frac 99.5%

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

      13. metadata-eval 99.5%

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

      14. *-commutative 99.5%

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

      15. associate-/r* 99.5%

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

      16. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 93.7%

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

      1. *-commutative 93.7%

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

      2. associate-*l* 93.8%

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

      3. *-commutative 93.8%

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

   6. Simplified 93.8%

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

      1. *-commutative 93.8%

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

      2. sqrt-div 93.9%

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

      3. metadata-eval 93.9%

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

      4. un-div-inv 93.9%

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

      5. *-commutative 93.9%

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

   8. Applied egg-rr 93.9%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+55}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 2.85 \cdot 10^{+63}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \]

Alternative 9: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-/r* 99.7%

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

   5. metadata-eval 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 10: 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 99.7%

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

   1. *-commutative 99.7%

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

   2. metadata-eval 99.7%

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

   3. sqrt-prod 99.7%

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

   4. pow1/2 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

3. Applied egg-rr 99.7%

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

   1. unpow1/2 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in x around 0 99.7%

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

7. Final simplification 99.7%

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

Alternative 11: 92.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+55} \lor \neg \left(y \leq 6.2 \cdot 10^{+83}\right):\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -8.6e+55) (not (<= y 6.2e+83)))
   (/ -0.3333333333333333 (/ (sqrt x) y))
   (+ 1.0 (/ (/ -1.0 x) 9.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -8.6e+55) || !(y <= 6.2e+83)) {
		tmp = -0.3333333333333333 / (sqrt(x) / y);
	} else {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-8.6d+55)) .or. (.not. (y <= 6.2d+83))) then
        tmp = (-0.3333333333333333d0) / (sqrt(x) / y)
    else
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -8.6e+55) || !(y <= 6.2e+83)) {
		tmp = -0.3333333333333333 / (Math.sqrt(x) / y);
	} else {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -8.6e+55) or not (y <= 6.2e+83):
		tmp = -0.3333333333333333 / (math.sqrt(x) / y)
	else:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -8.6e+55) || !(y <= 6.2e+83))
		tmp = Float64(-0.3333333333333333 / Float64(sqrt(x) / y));
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -8.6e+55) || ~((y <= 6.2e+83)))
		tmp = -0.3333333333333333 / (sqrt(x) / y);
	else
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -8.6e+55], N[Not[LessEqual[y, 6.2e+83]], $MachinePrecision]], N[(-0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.5999999999999998e55 or 6.19999999999999984e83 < 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. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

   3. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around inf 88.1%

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

      1. associate-*r* 88.1%

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

      2. *-commutative 88.1%

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

      3. *-commutative 88.1%

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

   8. Simplified 88.1%

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

      1. associate-*r* 88.1%

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

      2. sqrt-div 88.1%

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

      3. metadata-eval 88.1%

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

      4. div-inv 88.2%

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

      5. associate-*l/ 88.2%

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

      6. *-commutative 88.2%

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

      7. associate-/l* 88.2%

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

   10. Applied egg-rr 88.2%

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

   if -8.5999999999999998e55 < y < 6.19999999999999984e83

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--r+ 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. associate-*l/ 99.8%

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

      9. fma-neg 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-/r* 99.8%

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

      16. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 97.3%

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

      1. *-commutative 97.3%

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

      2. inv-pow 97.3%

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

      3. metadata-eval 97.3%

         \[\leadsto 1 - {x}^{\color{blue}{\left(-0.5 + -0.5\right)}} \cdot 0.1111111111111111 \]

      4. pow-prod-up 97.2%

         \[\leadsto 1 - \color{blue}{\left({x}^{-0.5} \cdot {x}^{-0.5}\right)} \cdot 0.1111111111111111 \]

      5. metadata-eval 97.2%

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

      6. swap-sqr 97.1%

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

      7. metadata-eval 97.1%

         \[\leadsto 1 - \left({x}^{-0.5} \cdot \color{blue}{\frac{1}{-3}}\right) \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right) \]

      8. div-inv 97.3%

         \[\leadsto 1 - \color{blue}{\frac{{x}^{-0.5}}{-3}} \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right) \]

      9. metadata-eval 97.3%

         \[\leadsto 1 - \frac{{x}^{-0.5}}{-3} \cdot \left({x}^{-0.5} \cdot \color{blue}{\frac{1}{-3}}\right) \]

      10. div-inv 97.1%

          \[\leadsto 1 - \frac{{x}^{-0.5}}{-3} \cdot \color{blue}{\frac{{x}^{-0.5}}{-3}} \]

      11. frac-times 97.2%

          \[\leadsto 1 - \color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5}}{-3 \cdot -3}} \]

      12. pow-prod-up 97.4%

          \[\leadsto 1 - \frac{\color{blue}{{x}^{\left(-0.5 + -0.5\right)}}}{-3 \cdot -3} \]

      13. metadata-eval 97.4%

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

      14. inv-pow 97.4%

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

      15. metadata-eval 97.4%

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

   6. Applied egg-rr 97.4%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+55} \lor \neg \left(y \leq 6.2 \cdot 10^{+83}\right):\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \end{array} \]

Alternative 12: 92.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+55}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+83}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.6e+55)
   (/ y (* (sqrt x) -3.0))
   (if (<= y 4.6e+83)
     (+ 1.0 (/ (/ -1.0 x) 9.0))
     (/ -0.3333333333333333 (/ (sqrt x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.6e+55) {
		tmp = y / (sqrt(x) * -3.0);
	} else if (y <= 4.6e+83) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = -0.3333333333333333 / (sqrt(x) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-8.6d+55)) then
        tmp = y / (sqrt(x) * (-3.0d0))
    else if (y <= 4.6d+83) then
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    else
        tmp = (-0.3333333333333333d0) / (sqrt(x) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8.6e+55) {
		tmp = y / (Math.sqrt(x) * -3.0);
	} else if (y <= 4.6e+83) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = -0.3333333333333333 / (Math.sqrt(x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.6e+55:
		tmp = y / (math.sqrt(x) * -3.0)
	elif y <= 4.6e+83:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	else:
		tmp = -0.3333333333333333 / (math.sqrt(x) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.6e+55)
		tmp = Float64(y / Float64(sqrt(x) * -3.0));
	elseif (y <= 4.6e+83)
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	else
		tmp = Float64(-0.3333333333333333 / Float64(sqrt(x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8.6e+55)
		tmp = y / (sqrt(x) * -3.0);
	elseif (y <= 4.6e+83)
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	else
		tmp = -0.3333333333333333 / (sqrt(x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.6e+55], N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.6e+83], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.5999999999999998e55

   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. *-commutative 99.7%

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

      2. metadata-eval 99.7%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

   3. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around inf 87.4%

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

      1. associate-*r* 87.4%

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

      2. *-commutative 87.4%

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

      3. *-commutative 87.4%

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

   8. Simplified 87.4%

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

      1. associate-*r* 87.4%

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

      2. sqrt-div 87.5%

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

      3. metadata-eval 87.5%

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

      4. div-inv 87.4%

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

      5. associate-/r/ 87.4%

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

      6. div-inv 87.8%

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

      7. metadata-eval 87.8%

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

   10. Applied egg-rr 87.8%

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

   if -8.5999999999999998e55 < y < 4.5999999999999999e83

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--r+ 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. associate-*l/ 99.8%

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

      9. fma-neg 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-/r* 99.8%

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

      16. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 97.3%

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

      1. *-commutative 97.3%

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

      2. inv-pow 97.3%

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

      3. metadata-eval 97.3%

         \[\leadsto 1 - {x}^{\color{blue}{\left(-0.5 + -0.5\right)}} \cdot 0.1111111111111111 \]

      4. pow-prod-up 97.2%

         \[\leadsto 1 - \color{blue}{\left({x}^{-0.5} \cdot {x}^{-0.5}\right)} \cdot 0.1111111111111111 \]

      5. metadata-eval 97.2%

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

      6. swap-sqr 97.1%

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

      7. metadata-eval 97.1%

         \[\leadsto 1 - \left({x}^{-0.5} \cdot \color{blue}{\frac{1}{-3}}\right) \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right) \]

      8. div-inv 97.3%

         \[\leadsto 1 - \color{blue}{\frac{{x}^{-0.5}}{-3}} \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right) \]

      9. metadata-eval 97.3%

         \[\leadsto 1 - \frac{{x}^{-0.5}}{-3} \cdot \left({x}^{-0.5} \cdot \color{blue}{\frac{1}{-3}}\right) \]

      10. div-inv 97.1%

          \[\leadsto 1 - \frac{{x}^{-0.5}}{-3} \cdot \color{blue}{\frac{{x}^{-0.5}}{-3}} \]

      11. frac-times 97.2%

          \[\leadsto 1 - \color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5}}{-3 \cdot -3}} \]

      12. pow-prod-up 97.4%

          \[\leadsto 1 - \frac{\color{blue}{{x}^{\left(-0.5 + -0.5\right)}}}{-3 \cdot -3} \]

      13. metadata-eval 97.4%

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

      14. inv-pow 97.4%

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

      15. metadata-eval 97.4%

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

   6. Applied egg-rr 97.4%

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

   if 4.5999999999999999e83 < y

   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. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

   3. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around inf 88.6%

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

      1. associate-*r* 88.7%

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

      2. *-commutative 88.7%

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

      3. *-commutative 88.7%

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

   8. Simplified 88.7%

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

      1. associate-*r* 88.6%

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

      2. sqrt-div 88.7%

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

      3. metadata-eval 88.7%

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

      4. div-inv 88.8%

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

      5. associate-*l/ 88.8%

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

      6. *-commutative 88.8%

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

      7. associate-/l* 88.8%

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

   10. Applied egg-rr 88.8%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+55}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+83}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \end{array} \]

Alternative 13: 92.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+55}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+83}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.6e+55)
   (/ y (* (sqrt x) -3.0))
   (if (<= y 4.8e+83)
     (+ 1.0 (/ (/ -1.0 x) 9.0))
     (/ (* y -0.3333333333333333) (sqrt x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.6e+55) {
		tmp = y / (sqrt(x) * -3.0);
	} else if (y <= 4.8e+83) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = (y * -0.3333333333333333) / sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-8.6d+55)) then
        tmp = y / (sqrt(x) * (-3.0d0))
    else if (y <= 4.8d+83) then
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    else
        tmp = (y * (-0.3333333333333333d0)) / sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8.6e+55) {
		tmp = y / (Math.sqrt(x) * -3.0);
	} else if (y <= 4.8e+83) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = (y * -0.3333333333333333) / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.6e+55:
		tmp = y / (math.sqrt(x) * -3.0)
	elif y <= 4.8e+83:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	else:
		tmp = (y * -0.3333333333333333) / math.sqrt(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.6e+55)
		tmp = Float64(y / Float64(sqrt(x) * -3.0));
	elseif (y <= 4.8e+83)
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	else
		tmp = Float64(Float64(y * -0.3333333333333333) / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8.6e+55)
		tmp = y / (sqrt(x) * -3.0);
	elseif (y <= 4.8e+83)
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	else
		tmp = (y * -0.3333333333333333) / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.6e+55], N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.8e+83], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * -0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.5999999999999998e55

   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. *-commutative 99.7%

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

      2. metadata-eval 99.7%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

   3. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around inf 87.4%

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

      1. associate-*r* 87.4%

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

      2. *-commutative 87.4%

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

      3. *-commutative 87.4%

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

   8. Simplified 87.4%

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

      1. associate-*r* 87.4%

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

      2. sqrt-div 87.5%

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

      3. metadata-eval 87.5%

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

      4. div-inv 87.4%

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

      5. associate-/r/ 87.4%

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

      6. div-inv 87.8%

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

      7. metadata-eval 87.8%

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

   10. Applied egg-rr 87.8%

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

   if -8.5999999999999998e55 < y < 4.79999999999999982e83

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--r+ 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. associate-*l/ 99.8%

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

      9. fma-neg 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-/r* 99.8%

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

      16. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 97.3%

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

      1. *-commutative 97.3%

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

      2. inv-pow 97.3%

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

      3. metadata-eval 97.3%

         \[\leadsto 1 - {x}^{\color{blue}{\left(-0.5 + -0.5\right)}} \cdot 0.1111111111111111 \]

      4. pow-prod-up 97.2%

         \[\leadsto 1 - \color{blue}{\left({x}^{-0.5} \cdot {x}^{-0.5}\right)} \cdot 0.1111111111111111 \]

      5. metadata-eval 97.2%

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

      6. swap-sqr 97.1%

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

      7. metadata-eval 97.1%

         \[\leadsto 1 - \left({x}^{-0.5} \cdot \color{blue}{\frac{1}{-3}}\right) \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right) \]

      8. div-inv 97.3%

         \[\leadsto 1 - \color{blue}{\frac{{x}^{-0.5}}{-3}} \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right) \]

      9. metadata-eval 97.3%

         \[\leadsto 1 - \frac{{x}^{-0.5}}{-3} \cdot \left({x}^{-0.5} \cdot \color{blue}{\frac{1}{-3}}\right) \]

      10. div-inv 97.1%

          \[\leadsto 1 - \frac{{x}^{-0.5}}{-3} \cdot \color{blue}{\frac{{x}^{-0.5}}{-3}} \]

      11. frac-times 97.2%

          \[\leadsto 1 - \color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5}}{-3 \cdot -3}} \]

      12. pow-prod-up 97.4%

          \[\leadsto 1 - \frac{\color{blue}{{x}^{\left(-0.5 + -0.5\right)}}}{-3 \cdot -3} \]

      13. metadata-eval 97.4%

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

      14. inv-pow 97.4%

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

      15. metadata-eval 97.4%

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

   6. Applied egg-rr 97.4%

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

   if 4.79999999999999982e83 < y

   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. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

   3. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around inf 88.6%

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

      1. associate-*r* 88.7%

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

      2. *-commutative 88.7%

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

      3. *-commutative 88.7%

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

   8. Simplified 88.7%

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

      1. associate-*r* 88.6%

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

      2. sqrt-div 88.7%

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

      3. metadata-eval 88.7%

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

      4. div-inv 88.8%

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

      5. associate-*l/ 88.8%

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

   10. Applied egg-rr 88.8%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+55}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+83}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \]

Alternative 14: 92.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+55}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+83}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{-3}}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.5e+55)
   (/ y (* (sqrt x) -3.0))
   (if (<= y 1.1e+83) (+ 1.0 (/ (/ -1.0 x) 9.0)) (/ (/ y -3.0) (sqrt x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.5e+55) {
		tmp = y / (sqrt(x) * -3.0);
	} else if (y <= 1.1e+83) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = (y / -3.0) / sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-8.5d+55)) then
        tmp = y / (sqrt(x) * (-3.0d0))
    else if (y <= 1.1d+83) then
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    else
        tmp = (y / (-3.0d0)) / sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8.5e+55) {
		tmp = y / (Math.sqrt(x) * -3.0);
	} else if (y <= 1.1e+83) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = (y / -3.0) / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.5e+55:
		tmp = y / (math.sqrt(x) * -3.0)
	elif y <= 1.1e+83:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	else:
		tmp = (y / -3.0) / math.sqrt(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.5e+55)
		tmp = Float64(y / Float64(sqrt(x) * -3.0));
	elseif (y <= 1.1e+83)
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	else
		tmp = Float64(Float64(y / -3.0) / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8.5e+55)
		tmp = y / (sqrt(x) * -3.0);
	elseif (y <= 1.1e+83)
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	else
		tmp = (y / -3.0) / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.5e+55], N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.1e+83], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision], N[(N[(y / -3.0), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.50000000000000002e55

   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. *-commutative 99.7%

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

      2. metadata-eval 99.7%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

   3. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around inf 87.4%

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

      1. associate-*r* 87.4%

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

      2. *-commutative 87.4%

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

      3. *-commutative 87.4%

         \[\leadsto y \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot -0.3333333333333333\right)} \]

   8. Simplified 87.4%

      \[\leadsto \color{blue}{y \cdot \left(\sqrt{\frac{1}{x}} \cdot -0.3333333333333333\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 87.4%

         \[\leadsto \color{blue}{\left(y \cdot \sqrt{\frac{1}{x}}\right) \cdot -0.3333333333333333} \]

      2. sqrt-div 87.5%

         \[\leadsto \left(y \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}\right) \cdot -0.3333333333333333 \]

      3. metadata-eval 87.5%

         \[\leadsto \left(y \cdot \frac{\color{blue}{1}}{\sqrt{x}}\right) \cdot -0.3333333333333333 \]

      4. div-inv 87.4%

         \[\leadsto \color{blue}{\frac{y}{\sqrt{x}}} \cdot -0.3333333333333333 \]

      5. associate-/r/ 87.4%

         \[\leadsto \color{blue}{\frac{y}{\frac{\sqrt{x}}{-0.3333333333333333}}} \]

      6. div-inv 87.8%

         \[\leadsto \frac{y}{\color{blue}{\sqrt{x} \cdot \frac{1}{-0.3333333333333333}}} \]

      7. metadata-eval 87.8%

         \[\leadsto \frac{y}{\sqrt{x} \cdot \color{blue}{-3}} \]

   10. Applied egg-rr 87.8%

       \[\leadsto \color{blue}{\frac{y}{\sqrt{x} \cdot -3}} \]

   if -8.50000000000000002e55 < y < 1.09999999999999999e83

   1. Initial program 99.8%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. associate--l- 99.8%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. +-commutative 99.8%

         \[\leadsto 1 - \color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)} \]

      3. associate--r+ 99.8%

         \[\leadsto \color{blue}{\left(1 - \frac{y}{3 \cdot \sqrt{x}}\right) - \frac{1}{x \cdot 9}} \]

      4. sub-neg 99.8%

         \[\leadsto \color{blue}{\left(1 + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)\right)} - \frac{1}{x \cdot 9} \]

      5. distribute-frac-neg 99.8%

         \[\leadsto \left(1 + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}}\right) - \frac{1}{x \cdot 9} \]

      6. associate-+r- 99.8%

         \[\leadsto \color{blue}{1 + \left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

      7. neg-mul-1 99.8%

         \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

      8. associate-*l/ 99.8%

         \[\leadsto 1 + \left(\color{blue}{\frac{-1}{3 \cdot \sqrt{x}} \cdot y} - \frac{1}{x \cdot 9}\right) \]

      9. fma-neg 99.8%

         \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot \sqrt{x}}, y, -\frac{1}{x \cdot 9}\right)} \]

      10. associate-/r* 99.8%

          \[\leadsto 1 + \mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, y, -\frac{1}{x \cdot 9}\right) \]

      11. metadata-eval 99.8%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, y, -\frac{1}{x \cdot 9}\right) \]

      12. distribute-neg-frac 99.8%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \color{blue}{\frac{-1}{x \cdot 9}}\right) \]

      13. metadata-eval 99.8%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{\color{blue}{-1}}{x \cdot 9}\right) \]

      14. *-commutative 99.8%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{-1}{\color{blue}{9 \cdot x}}\right) \]

      15. associate-/r* 99.8%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \color{blue}{\frac{\frac{-1}{9}}{x}}\right) \]

      16. metadata-eval 99.8%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{-0.1111111111111111}{x}\right)} \]

   4. Taylor expanded in y around 0 97.3%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   5. Step-by-step derivation

      1. *-commutative 97.3%

         \[\leadsto 1 - \color{blue}{\frac{1}{x} \cdot 0.1111111111111111} \]

      2. inv-pow 97.3%

         \[\leadsto 1 - \color{blue}{{x}^{-1}} \cdot 0.1111111111111111 \]

      3. metadata-eval 97.3%

         \[\leadsto 1 - {x}^{\color{blue}{\left(-0.5 + -0.5\right)}} \cdot 0.1111111111111111 \]

      4. pow-prod-up 97.2%

         \[\leadsto 1 - \color{blue}{\left({x}^{-0.5} \cdot {x}^{-0.5}\right)} \cdot 0.1111111111111111 \]

      5. metadata-eval 97.2%

         \[\leadsto 1 - \left({x}^{-0.5} \cdot {x}^{-0.5}\right) \cdot \color{blue}{\left(-0.3333333333333333 \cdot -0.3333333333333333\right)} \]

      6. swap-sqr 97.1%

         \[\leadsto 1 - \color{blue}{\left({x}^{-0.5} \cdot -0.3333333333333333\right) \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right)} \]

      7. metadata-eval 97.1%

         \[\leadsto 1 - \left({x}^{-0.5} \cdot \color{blue}{\frac{1}{-3}}\right) \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right) \]

      8. div-inv 97.3%

         \[\leadsto 1 - \color{blue}{\frac{{x}^{-0.5}}{-3}} \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right) \]

      9. metadata-eval 97.3%

         \[\leadsto 1 - \frac{{x}^{-0.5}}{-3} \cdot \left({x}^{-0.5} \cdot \color{blue}{\frac{1}{-3}}\right) \]

      10. div-inv 97.1%

          \[\leadsto 1 - \frac{{x}^{-0.5}}{-3} \cdot \color{blue}{\frac{{x}^{-0.5}}{-3}} \]

      11. frac-times 97.2%

          \[\leadsto 1 - \color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5}}{-3 \cdot -3}} \]

      12. pow-prod-up 97.4%

          \[\leadsto 1 - \frac{\color{blue}{{x}^{\left(-0.5 + -0.5\right)}}}{-3 \cdot -3} \]

      13. metadata-eval 97.4%

          \[\leadsto 1 - \frac{{x}^{\color{blue}{-1}}}{-3 \cdot -3} \]

      14. inv-pow 97.4%

          \[\leadsto 1 - \frac{\color{blue}{\frac{1}{x}}}{-3 \cdot -3} \]

      15. metadata-eval 97.4%

          \[\leadsto 1 - \frac{\frac{1}{x}}{\color{blue}{9}} \]

   6. Applied egg-rr 97.4%

      \[\leadsto 1 - \color{blue}{\frac{\frac{1}{x}}{9}} \]

   if 1.09999999999999999e83 < y

   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. *-commutative 99.4%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\sqrt{x} \cdot 3}} \]

      2. metadata-eval 99.4%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\sqrt{x} \cdot \color{blue}{\sqrt{9}}} \]

      3. sqrt-prod 99.7%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\sqrt{x \cdot 9}}} \]

      4. pow1/2 99.7%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

   3. Applied egg-rr 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]
   4. Step-by-step derivation

      1. unpow1/2 99.7%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\sqrt{x \cdot 9}}} \]

   5. Simplified 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\sqrt{x \cdot 9}}} \]

   6. Taylor expanded in y around inf 88.6%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 88.7%

         \[\leadsto \color{blue}{\left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right) \cdot y} \]

      2. *-commutative 88.7%

         \[\leadsto \color{blue}{y \cdot \left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)} \]

      3. *-commutative 88.7%

         \[\leadsto y \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot -0.3333333333333333\right)} \]

   8. Simplified 88.7%

      \[\leadsto \color{blue}{y \cdot \left(\sqrt{\frac{1}{x}} \cdot -0.3333333333333333\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 88.6%

         \[\leadsto \color{blue}{\left(y \cdot \sqrt{\frac{1}{x}}\right) \cdot -0.3333333333333333} \]

      2. sqrt-div 88.7%

         \[\leadsto \left(y \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}\right) \cdot -0.3333333333333333 \]

      3. metadata-eval 88.7%

         \[\leadsto \left(y \cdot \frac{\color{blue}{1}}{\sqrt{x}}\right) \cdot -0.3333333333333333 \]

      4. div-inv 88.8%

         \[\leadsto \color{blue}{\frac{y}{\sqrt{x}}} \cdot -0.3333333333333333 \]

      5. metadata-eval 88.8%

         \[\leadsto \frac{y}{\sqrt{x}} \cdot \color{blue}{\frac{1}{-3}} \]

      6. div-inv 88.9%

         \[\leadsto \color{blue}{\frac{\frac{y}{\sqrt{x}}}{-3}} \]

      7. associate-/l/ 88.8%

         \[\leadsto \color{blue}{\frac{y}{-3 \cdot \sqrt{x}}} \]

      8. associate-/r* 88.9%

         \[\leadsto \color{blue}{\frac{\frac{y}{-3}}{\sqrt{x}}} \]

   10. Applied egg-rr 88.9%

       \[\leadsto \color{blue}{\frac{\frac{y}{-3}}{\sqrt{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+55}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+83}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{-3}}{\sqrt{x}}\\ \end{array} \]

Alternative 15: 62.5% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{-1}{x \cdot 9} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (/ -1.0 (* x 9.0))))

C

double code(double x, double y) {
	return 1.0 + (-1.0 / (x * 9.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
end function

Java

public static double code(double x, double y) {
	return 1.0 + (-1.0 / (x * 9.0));
}

Python

def code(x, y):
	return 1.0 + (-1.0 / (x * 9.0))

Julia

function code(x, y)
	return Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + (-1.0 / (x * 9.0));
end

Wolfram

code[x_, y_] := N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. associate--l- 99.7%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

   2. +-commutative 99.7%

      \[\leadsto 1 - \color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)} \]

   3. associate--r+ 99.7%

      \[\leadsto \color{blue}{\left(1 - \frac{y}{3 \cdot \sqrt{x}}\right) - \frac{1}{x \cdot 9}} \]

   4. sub-neg 99.7%

      \[\leadsto \color{blue}{\left(1 + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)\right)} - \frac{1}{x \cdot 9} \]

   5. distribute-frac-neg 99.7%

      \[\leadsto \left(1 + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}}\right) - \frac{1}{x \cdot 9} \]

   6. associate-+r- 99.7%

      \[\leadsto \color{blue}{1 + \left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

   7. neg-mul-1 99.7%

      \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

   8. associate-*l/ 99.6%

      \[\leadsto 1 + \left(\color{blue}{\frac{-1}{3 \cdot \sqrt{x}} \cdot y} - \frac{1}{x \cdot 9}\right) \]

   9. fma-neg 99.6%

      \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot \sqrt{x}}, y, -\frac{1}{x \cdot 9}\right)} \]

   10. associate-/r* 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, y, -\frac{1}{x \cdot 9}\right) \]

   11. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, y, -\frac{1}{x \cdot 9}\right) \]

   12. distribute-neg-frac 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \color{blue}{\frac{-1}{x \cdot 9}}\right) \]

   13. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{\color{blue}{-1}}{x \cdot 9}\right) \]

   14. *-commutative 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{-1}{\color{blue}{9 \cdot x}}\right) \]

   15. associate-/r* 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \color{blue}{\frac{\frac{-1}{9}}{x}}\right) \]

   16. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{-0.1111111111111111}{x}\right)} \]

4. Taylor expanded in y around 0 61.6%

   \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
5. Step-by-step derivation

   1. div-inv 61.7%

      \[\leadsto 1 - \color{blue}{\frac{0.1111111111111111}{x}} \]

   2. clear-num 61.7%

      \[\leadsto 1 - \color{blue}{\frac{1}{\frac{x}{0.1111111111111111}}} \]

   3. div-inv 61.7%

      \[\leadsto 1 - \frac{1}{\color{blue}{x \cdot \frac{1}{0.1111111111111111}}} \]

   4. metadata-eval 61.7%

      \[\leadsto 1 - \frac{1}{x \cdot \color{blue}{9}} \]

6. Applied egg-rr 61.7%

   \[\leadsto 1 - \color{blue}{\frac{1}{x \cdot 9}} \]

7. Final simplification 61.7%

   \[\leadsto 1 + \frac{-1}{x \cdot 9} \]

Alternative 16: 62.5% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{\frac{-1}{x}}{9} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (/ (/ -1.0 x) 9.0)))

C

double code(double x, double y) {
	return 1.0 + ((-1.0 / x) / 9.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
end function

Java

public static double code(double x, double y) {
	return 1.0 + ((-1.0 / x) / 9.0);
}

Python

def code(x, y):
	return 1.0 + ((-1.0 / x) / 9.0)

Julia

function code(x, y)
	return Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + ((-1.0 / x) / 9.0);
end

Wolfram

code[x_, y_] := N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. associate--l- 99.7%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

   2. +-commutative 99.7%

      \[\leadsto 1 - \color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)} \]

   3. associate--r+ 99.7%

      \[\leadsto \color{blue}{\left(1 - \frac{y}{3 \cdot \sqrt{x}}\right) - \frac{1}{x \cdot 9}} \]

   4. sub-neg 99.7%

      \[\leadsto \color{blue}{\left(1 + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)\right)} - \frac{1}{x \cdot 9} \]

   5. distribute-frac-neg 99.7%

      \[\leadsto \left(1 + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}}\right) - \frac{1}{x \cdot 9} \]

   6. associate-+r- 99.7%

      \[\leadsto \color{blue}{1 + \left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

   7. neg-mul-1 99.7%

      \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

   8. associate-*l/ 99.6%

      \[\leadsto 1 + \left(\color{blue}{\frac{-1}{3 \cdot \sqrt{x}} \cdot y} - \frac{1}{x \cdot 9}\right) \]

   9. fma-neg 99.6%

      \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot \sqrt{x}}, y, -\frac{1}{x \cdot 9}\right)} \]

   10. associate-/r* 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, y, -\frac{1}{x \cdot 9}\right) \]

   11. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, y, -\frac{1}{x \cdot 9}\right) \]

   12. distribute-neg-frac 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \color{blue}{\frac{-1}{x \cdot 9}}\right) \]

   13. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{\color{blue}{-1}}{x \cdot 9}\right) \]

   14. *-commutative 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{-1}{\color{blue}{9 \cdot x}}\right) \]

   15. associate-/r* 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \color{blue}{\frac{\frac{-1}{9}}{x}}\right) \]

   16. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{-0.1111111111111111}{x}\right)} \]

4. Taylor expanded in y around 0 61.6%

   \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
5. Step-by-step derivation

   1. *-commutative 61.6%

      \[\leadsto 1 - \color{blue}{\frac{1}{x} \cdot 0.1111111111111111} \]

   2. inv-pow 61.6%

      \[\leadsto 1 - \color{blue}{{x}^{-1}} \cdot 0.1111111111111111 \]

   3. metadata-eval 61.6%

      \[\leadsto 1 - {x}^{\color{blue}{\left(-0.5 + -0.5\right)}} \cdot 0.1111111111111111 \]

   4. pow-prod-up 61.6%

      \[\leadsto 1 - \color{blue}{\left({x}^{-0.5} \cdot {x}^{-0.5}\right)} \cdot 0.1111111111111111 \]

   5. metadata-eval 61.6%

      \[\leadsto 1 - \left({x}^{-0.5} \cdot {x}^{-0.5}\right) \cdot \color{blue}{\left(-0.3333333333333333 \cdot -0.3333333333333333\right)} \]

   6. swap-sqr 61.5%

      \[\leadsto 1 - \color{blue}{\left({x}^{-0.5} \cdot -0.3333333333333333\right) \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right)} \]

   7. metadata-eval 61.5%

      \[\leadsto 1 - \left({x}^{-0.5} \cdot \color{blue}{\frac{1}{-3}}\right) \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right) \]

   8. div-inv 61.6%

      \[\leadsto 1 - \color{blue}{\frac{{x}^{-0.5}}{-3}} \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right) \]

   9. metadata-eval 61.6%

      \[\leadsto 1 - \frac{{x}^{-0.5}}{-3} \cdot \left({x}^{-0.5} \cdot \color{blue}{\frac{1}{-3}}\right) \]

   10. div-inv 61.5%

       \[\leadsto 1 - \frac{{x}^{-0.5}}{-3} \cdot \color{blue}{\frac{{x}^{-0.5}}{-3}} \]

   11. frac-times 61.5%

       \[\leadsto 1 - \color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5}}{-3 \cdot -3}} \]

   12. pow-prod-up 61.7%

       \[\leadsto 1 - \frac{\color{blue}{{x}^{\left(-0.5 + -0.5\right)}}}{-3 \cdot -3} \]

   13. metadata-eval 61.7%

       \[\leadsto 1 - \frac{{x}^{\color{blue}{-1}}}{-3 \cdot -3} \]

   14. inv-pow 61.7%

       \[\leadsto 1 - \frac{\color{blue}{\frac{1}{x}}}{-3 \cdot -3} \]

   15. metadata-eval 61.7%

       \[\leadsto 1 - \frac{\frac{1}{x}}{\color{blue}{9}} \]

6. Applied egg-rr 61.7%

   \[\leadsto 1 - \color{blue}{\frac{\frac{1}{x}}{9}} \]

7. Final simplification 61.7%

   \[\leadsto 1 + \frac{\frac{-1}{x}}{9} \]

Alternative 17: 61.3% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 190000:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 190000.0) (/ -0.1111111111111111 x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 190000.0) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 190000.0d0) then
        tmp = (-0.1111111111111111d0) / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 190000.0) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 190000.0:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 190000.0)
		tmp = Float64(-0.1111111111111111 / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 190000.0)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 190000.0], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.9e5

   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. *-commutative 99.6%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\sqrt{x} \cdot 3}} \]

      2. metadata-eval 99.6%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\sqrt{x} \cdot \color{blue}{\sqrt{9}}} \]

      3. sqrt-prod 99.6%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\sqrt{x \cdot 9}}} \]

      4. pow1/2 99.6%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

   3. Applied egg-rr 99.6%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]
   4. Step-by-step derivation

      1. unpow1/2 99.6%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\sqrt{x \cdot 9}}} \]

   5. Simplified 99.6%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\sqrt{x \cdot 9}}} \]

   6. Taylor expanded in x around 0 58.2%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x}} \]

   if 1.9e5 < x

   1. Initial program 99.8%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. associate--l- 99.8%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. +-commutative 99.8%

         \[\leadsto 1 - \color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)} \]

      3. associate--r+ 99.8%

         \[\leadsto \color{blue}{\left(1 - \frac{y}{3 \cdot \sqrt{x}}\right) - \frac{1}{x \cdot 9}} \]

      4. sub-neg 99.8%

         \[\leadsto \color{blue}{\left(1 + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)\right)} - \frac{1}{x \cdot 9} \]

      5. distribute-frac-neg 99.8%

         \[\leadsto \left(1 + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}}\right) - \frac{1}{x \cdot 9} \]

      6. associate-+r- 99.8%

         \[\leadsto \color{blue}{1 + \left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

      7. neg-mul-1 99.8%

         \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

      8. associate-*l/ 99.8%

         \[\leadsto 1 + \left(\color{blue}{\frac{-1}{3 \cdot \sqrt{x}} \cdot y} - \frac{1}{x \cdot 9}\right) \]

      9. fma-neg 99.8%

         \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot \sqrt{x}}, y, -\frac{1}{x \cdot 9}\right)} \]

      10. associate-/r* 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, y, -\frac{1}{x \cdot 9}\right) \]

      11. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, y, -\frac{1}{x \cdot 9}\right) \]

      12. distribute-neg-frac 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \color{blue}{\frac{-1}{x \cdot 9}}\right) \]

      13. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{\color{blue}{-1}}{x \cdot 9}\right) \]

      14. *-commutative 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{-1}{\color{blue}{9 \cdot x}}\right) \]

      15. associate-/r* 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \color{blue}{\frac{\frac{-1}{9}}{x}}\right) \]

      16. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{-0.1111111111111111}{x}\right)} \]

   4. Taylor expanded in x around inf 64.1%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 190000:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 18: 62.5% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{-0.1111111111111111}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (/ -0.1111111111111111 x)))

C

double code(double x, double y) {
	return 1.0 + (-0.1111111111111111 / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + ((-0.1111111111111111d0) / x)
end function

Java

public static double code(double x, double y) {
	return 1.0 + (-0.1111111111111111 / x);
}

Python

def code(x, y):
	return 1.0 + (-0.1111111111111111 / x)

Julia

function code(x, y)
	return Float64(1.0 + Float64(-0.1111111111111111 / x))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + (-0.1111111111111111 / x);
end

Wolfram

code[x_, y_] := N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. associate--l- 99.7%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

   2. +-commutative 99.7%

      \[\leadsto 1 - \color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)} \]

   3. associate--r+ 99.7%

      \[\leadsto \color{blue}{\left(1 - \frac{y}{3 \cdot \sqrt{x}}\right) - \frac{1}{x \cdot 9}} \]

   4. sub-neg 99.7%

      \[\leadsto \color{blue}{\left(1 + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)\right)} - \frac{1}{x \cdot 9} \]

   5. distribute-frac-neg 99.7%

      \[\leadsto \left(1 + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}}\right) - \frac{1}{x \cdot 9} \]

   6. associate-+r- 99.7%

      \[\leadsto \color{blue}{1 + \left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

   7. neg-mul-1 99.7%

      \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

   8. associate-*l/ 99.6%

      \[\leadsto 1 + \left(\color{blue}{\frac{-1}{3 \cdot \sqrt{x}} \cdot y} - \frac{1}{x \cdot 9}\right) \]

   9. fma-neg 99.6%

      \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot \sqrt{x}}, y, -\frac{1}{x \cdot 9}\right)} \]

   10. associate-/r* 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, y, -\frac{1}{x \cdot 9}\right) \]

   11. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, y, -\frac{1}{x \cdot 9}\right) \]

   12. distribute-neg-frac 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \color{blue}{\frac{-1}{x \cdot 9}}\right) \]

   13. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{\color{blue}{-1}}{x \cdot 9}\right) \]

   14. *-commutative 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{-1}{\color{blue}{9 \cdot x}}\right) \]

   15. associate-/r* 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \color{blue}{\frac{\frac{-1}{9}}{x}}\right) \]

   16. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{-0.1111111111111111}{x}\right)} \]

4. Taylor expanded in y around 0 61.6%

   \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
5. Step-by-step derivation

   1. cancel-sign-sub-inv 61.6%

      \[\leadsto \color{blue}{1 + \left(-0.1111111111111111\right) \cdot \frac{1}{x}} \]

   2. metadata-eval 61.6%

      \[\leadsto 1 + \color{blue}{-0.1111111111111111} \cdot \frac{1}{x} \]

   3. associate-*r/ 61.7%

      \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111 \cdot 1}{x}} \]

   4. metadata-eval 61.7%

      \[\leadsto 1 + \frac{\color{blue}{-0.1111111111111111}}{x} \]

   5. +-commutative 61.7%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]

6. Simplified 61.7%

   \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]

7. Final simplification 61.7%

   \[\leadsto 1 + \frac{-0.1111111111111111}{x} \]

Alternative 19: 31.6% accurate, 113.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. associate--l- 99.7%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

   2. +-commutative 99.7%

      \[\leadsto 1 - \color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)} \]

   3. associate--r+ 99.7%

      \[\leadsto \color{blue}{\left(1 - \frac{y}{3 \cdot \sqrt{x}}\right) - \frac{1}{x \cdot 9}} \]

   4. sub-neg 99.7%

      \[\leadsto \color{blue}{\left(1 + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)\right)} - \frac{1}{x \cdot 9} \]

   5. distribute-frac-neg 99.7%

      \[\leadsto \left(1 + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}}\right) - \frac{1}{x \cdot 9} \]

   6. associate-+r- 99.7%

      \[\leadsto \color{blue}{1 + \left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

   7. neg-mul-1 99.7%

      \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

   8. associate-*l/ 99.6%

      \[\leadsto 1 + \left(\color{blue}{\frac{-1}{3 \cdot \sqrt{x}} \cdot y} - \frac{1}{x \cdot 9}\right) \]

   9. fma-neg 99.6%

      \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot \sqrt{x}}, y, -\frac{1}{x \cdot 9}\right)} \]

   10. associate-/r* 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, y, -\frac{1}{x \cdot 9}\right) \]

   11. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, y, -\frac{1}{x \cdot 9}\right) \]

   12. distribute-neg-frac 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \color{blue}{\frac{-1}{x \cdot 9}}\right) \]

   13. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{\color{blue}{-1}}{x \cdot 9}\right) \]

   14. *-commutative 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{-1}{\color{blue}{9 \cdot x}}\right) \]

   15. associate-/r* 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \color{blue}{\frac{\frac{-1}{9}}{x}}\right) \]

   16. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{-0.1111111111111111}{x}\right)} \]

4. Taylor expanded in x around inf 33.7%

   \[\leadsto \color{blue}{1} \]

5. Final simplification 33.7%

   \[\leadsto 1 \]

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 2023336 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (- (- 1.0 (/ (/ 1.0 x) 9.0)) (/ y (* 3.0 (sqrt x))))

  (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))