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

Percentage Accurate: 99.7% → 99.7%

Time: 11.4s

Alternatives: 15

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

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 99.8%

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

   4. pow1/2 99.8%

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

3. Applied egg-rr 99.8%

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

   1. unpow1/2 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 94.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+32} \lor \neg \left(y \leq 2.5 \cdot 10^{+72}\right):\\ \;\;\;\;1 + \frac{y}{-\sqrt{x \cdot 9}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.5e+32) (not (<= y 2.5e+72)))
   (+ 1.0 (/ y (- (sqrt (* x 9.0)))))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.5e+32) || !(y <= 2.5e+72)) {
		tmp = 1.0 + (y / -sqrt((x * 9.0)));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.5d+32)) .or. (.not. (y <= 2.5d+72))) then
        tmp = 1.0d0 + (y / -sqrt((x * 9.0d0)))
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.5e+32) || !(y <= 2.5e+72)) {
		tmp = 1.0 + (y / -Math.sqrt((x * 9.0)));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.5e+32) or not (y <= 2.5e+72):
		tmp = 1.0 + (y / -math.sqrt((x * 9.0)))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.5e+32) || !(y <= 2.5e+72))
		tmp = Float64(1.0 + Float64(y / Float64(-sqrt(Float64(x * 9.0)))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.5e+32) || ~((y <= 2.5e+72)))
		tmp = 1.0 + (y / -sqrt((x * 9.0)));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.5e+32], N[Not[LessEqual[y, 2.5e+72]], $MachinePrecision]], N[(1.0 + N[(y / (-N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.5e32 or 2.49999999999999996e72 < y

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. +-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/ 99.5%

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

      9. fma-neg 99.5%

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

      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 93.2%

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

      1. *-commutative 93.2%

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

      2. associate-*l* 93.2%

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

      3. *-commutative 93.2%

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

   6. Simplified 93.2%

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

      1. expm1-log1p-u 49.1%

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

      2. expm1-udef 49.1%

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

      3. *-commutative 49.1%

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

      4. sqrt-div 49.1%

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

      5. metadata-eval 49.1%

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

      6. un-div-inv 49.1%

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

      7. *-commutative 49.1%

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

   8. Applied egg-rr 49.1%

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

      1. expm1-def 49.1%

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

      2. expm1-log1p 93.2%

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

      3. associate-/l* 93.2%

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

   10. Simplified 93.2%

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

       1. div-inv 93.4%

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

       2. metadata-eval 93.4%

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

       3. metadata-eval 93.4%

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

       4. metadata-eval 93.4%

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

       5. distribute-rgt-neg-in 93.4%

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

       6. sqrt-prod 93.4%

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

       7. add-sqr-sqrt 93.1%

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

       8. distribute-rgt-neg-in 93.1%

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

       9. pow1/2 93.1%

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

       10. sqrt-pow1 93.2%

           \[\leadsto 1 + \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{\left(\frac{0.5}{2}\right)}} \cdot \left(-\sqrt{\sqrt{x \cdot 9}}\right)} \]

       11. metadata-eval 93.2%

           \[\leadsto 1 + \frac{y}{{\left(x \cdot 9\right)}^{\color{blue}{0.25}} \cdot \left(-\sqrt{\sqrt{x \cdot 9}}\right)} \]

       12. pow1/2 93.2%

           \[\leadsto 1 + \frac{y}{{\left(x \cdot 9\right)}^{0.25} \cdot \left(-\sqrt{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}}\right)} \]

       13. sqrt-pow1 93.2%

           \[\leadsto 1 + \frac{y}{{\left(x \cdot 9\right)}^{0.25} \cdot \left(-\color{blue}{{\left(x \cdot 9\right)}^{\left(\frac{0.5}{2}\right)}}\right)} \]

       14. metadata-eval 93.2%

           \[\leadsto 1 + \frac{y}{{\left(x \cdot 9\right)}^{0.25} \cdot \left(-{\left(x \cdot 9\right)}^{\color{blue}{0.25}}\right)} \]

   12. Applied egg-rr 93.2%

       \[\leadsto 1 + \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.25} \cdot \left(-{\left(x \cdot 9\right)}^{0.25}\right)}} \]
   13. Step-by-step derivation

       1. distribute-rgt-neg-out 93.2%

          \[\leadsto 1 + \frac{y}{\color{blue}{-{\left(x \cdot 9\right)}^{0.25} \cdot {\left(x \cdot 9\right)}^{0.25}}} \]

       2. pow-sqr 93.4%

          \[\leadsto 1 + \frac{y}{-\color{blue}{{\left(x \cdot 9\right)}^{\left(2 \cdot 0.25\right)}}} \]

       3. metadata-eval 93.4%

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

       4. unpow1/2 93.4%

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

   14. Simplified 93.4%

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

   if -1.5e32 < y < 2.49999999999999996e72

   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* 99.6%

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

      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 97.4%

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

      1. cancel-sign-sub-inv 97.4%

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

      2. metadata-eval 97.4%

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

      3. associate-*r/ 97.4%

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

      4. metadata-eval 97.4%

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

      5. +-commutative 97.4%

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

   6. Simplified 97.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 42.1%

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

      3. frac-times 42.1%

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

      4. metadata-eval 42.1%

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

      5. metadata-eval 42.1%

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

      6. frac-times 42.1%

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

      7. sqrt-unprod 42.1%

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

      8. add-sqr-sqrt 42.1%

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

      9. clear-num 42.1%

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

      10. div-inv 42.1%

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

      11. metadata-eval 42.1%

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

      12. inv-pow 42.1%

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

   8. Applied egg-rr 42.1%

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

      1. unpow-prod-down 42.1%

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

      2. metadata-eval 42.1%

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

      3. metadata-eval 42.1%

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

      4. sqrt-pow1 42.1%

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

      5. metadata-eval 42.1%

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

      6. sqrt-prod 42.1%

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

      7. *-commutative 42.1%

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

      8. metadata-eval 42.1%

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

      9. metadata-eval 42.1%

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

      10. pow-prod-up 42.1%

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

      11. inv-pow 42.1%

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

      12. inv-pow 42.1%

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

      13. swap-sqr 42.1%

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

      14. div-inv 42.1%

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

      15. div-inv 42.1%

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

      16. sqrt-unprod 0.0%

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

      17. add-sqr-sqrt 97.4%

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

      18. metadata-eval 97.4%

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

      19. distribute-neg-frac 97.4%

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

      20. clear-num 97.5%

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

      21. distribute-neg-frac 97.5%

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

      22. metadata-eval 97.5%

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

      23. div-inv 97.6%

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

      24. metadata-eval 97.6%

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

   10. Applied egg-rr 97.6%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+32} \lor \neg \left(y \leq 2.5 \cdot 10^{+72}\right):\\ \;\;\;\;1 + \frac{y}{-\sqrt{x \cdot 9}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]

Alternative 3: 92.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+93} \lor \neg \left(y \leq 1.55 \cdot 10^{+74}\right):\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5.5e+93) (not (<= y 1.55e+74)))
   (* (sqrt (/ 1.0 x)) (* y -0.3333333333333333))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5.5e+93) || !(y <= 1.55e+74)) {
		tmp = sqrt((1.0 / x)) * (y * -0.3333333333333333);
	} 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 <= (-5.5d+93)) .or. (.not. (y <= 1.55d+74))) then
        tmp = sqrt((1.0d0 / x)) * (y * (-0.3333333333333333d0))
    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 <= -5.5e+93) || !(y <= 1.55e+74)) {
		tmp = Math.sqrt((1.0 / x)) * (y * -0.3333333333333333);
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -5.5e+93) or not (y <= 1.55e+74):
		tmp = math.sqrt((1.0 / x)) * (y * -0.3333333333333333)
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5.5e+93) || !(y <= 1.55e+74))
		tmp = Float64(sqrt(Float64(1.0 / x)) * Float64(y * -0.3333333333333333));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -5.5e+93) || ~((y <= 1.55e+74)))
		tmp = sqrt((1.0 / x)) * (y * -0.3333333333333333);
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5.5e+93], N[Not[LessEqual[y, 1.55e+74]], $MachinePrecision]], N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.5000000000000003e93 or 1.55000000000000011e74 < y

   1. Initial program 99.7%

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

      1. *-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 93.6%

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

      1. *-commutative 93.6%

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

      2. associate-*l* 93.5%

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

   8. Simplified 93.5%

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

   if -5.5000000000000003e93 < y < 1.55000000000000011e74

   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* 99.6%

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

      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 93.4%

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

      1. cancel-sign-sub-inv 93.4%

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

      2. metadata-eval 93.4%

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

      3. associate-*r/ 93.4%

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

      4. metadata-eval 93.4%

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

      5. +-commutative 93.4%

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

   6. Simplified 93.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 40.7%

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

      3. frac-times 40.7%

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

      4. metadata-eval 40.7%

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

      5. metadata-eval 40.7%

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

      6. frac-times 40.7%

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

      7. sqrt-unprod 40.8%

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

      8. add-sqr-sqrt 40.8%

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

      9. clear-num 40.8%

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

      10. div-inv 40.8%

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

      11. metadata-eval 40.8%

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

      12. inv-pow 40.8%

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

   8. Applied egg-rr 40.8%

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

      1. unpow-prod-down 40.8%

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

      2. metadata-eval 40.8%

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

      3. metadata-eval 40.8%

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

      4. sqrt-pow1 40.7%

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

      5. metadata-eval 40.7%

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

      6. sqrt-prod 40.7%

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

      7. *-commutative 40.7%

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

      8. metadata-eval 40.7%

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

      9. metadata-eval 40.7%

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

      10. pow-prod-up 40.7%

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

      11. inv-pow 40.7%

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

      12. inv-pow 40.7%

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

      13. swap-sqr 40.7%

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

      14. div-inv 40.7%

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

      15. div-inv 40.7%

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

      16. sqrt-unprod 0.0%

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

      17. add-sqr-sqrt 93.4%

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

      18. metadata-eval 93.4%

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

      19. distribute-neg-frac 93.4%

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

      20. clear-num 93.5%

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

      21. distribute-neg-frac 93.5%

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

      22. metadata-eval 93.5%

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

      23. div-inv 93.6%

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

      24. metadata-eval 93.6%

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

   10. Applied egg-rr 93.6%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+93} \lor \neg \left(y \leq 1.55 \cdot 10^{+74}\right):\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]

Alternative 4: 92.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+93} \lor \neg \left(y \leq 3.9 \cdot 10^{+72}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.2e+93) (not (<= y 3.9e+72)))
   (* -0.3333333333333333 (* y (sqrt (/ 1.0 x))))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.2e+93) || !(y <= 3.9e+72)) {
		tmp = -0.3333333333333333 * (y * sqrt((1.0 / x)));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-4.2d+93)) .or. (.not. (y <= 3.9d+72))) then
        tmp = (-0.3333333333333333d0) * (y * sqrt((1.0d0 / x)))
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -4.2e+93) || !(y <= 3.9e+72)) {
		tmp = -0.3333333333333333 * (y * Math.sqrt((1.0 / x)));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.2e+93) or not (y <= 3.9e+72):
		tmp = -0.3333333333333333 * (y * math.sqrt((1.0 / x)))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.2e+93) || !(y <= 3.9e+72))
		tmp = Float64(-0.3333333333333333 * Float64(y * sqrt(Float64(1.0 / x))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4.2e+93) || ~((y <= 3.9e+72)))
		tmp = -0.3333333333333333 * (y * sqrt((1.0 / x)));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.2e+93], N[Not[LessEqual[y, 3.9e+72]], $MachinePrecision]], N[(-0.3333333333333333 * N[(y * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.1999999999999996e93 or 3.89999999999999992e72 < y

   1. Initial program 99.7%

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

      1. *-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 93.6%

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

      1. *-commutative 93.6%

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

   8. Simplified 93.6%

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

   if -4.1999999999999996e93 < y < 3.89999999999999992e72

   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* 99.6%

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

      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 93.4%

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

      1. cancel-sign-sub-inv 93.4%

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

      2. metadata-eval 93.4%

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

      3. associate-*r/ 93.4%

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

      4. metadata-eval 93.4%

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

      5. +-commutative 93.4%

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

   6. Simplified 93.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 40.7%

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

      3. frac-times 40.7%

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

      4. metadata-eval 40.7%

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

      5. metadata-eval 40.7%

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

      6. frac-times 40.7%

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

      7. sqrt-unprod 40.8%

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

      8. add-sqr-sqrt 40.8%

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

      9. clear-num 40.8%

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

      10. div-inv 40.8%

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

      11. metadata-eval 40.8%

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

      12. inv-pow 40.8%

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

   8. Applied egg-rr 40.8%

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

      1. unpow-prod-down 40.8%

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

      2. metadata-eval 40.8%

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

      3. metadata-eval 40.8%

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

      4. sqrt-pow1 40.7%

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

      5. metadata-eval 40.7%

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

      6. sqrt-prod 40.7%

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

      7. *-commutative 40.7%

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

      8. metadata-eval 40.7%

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

      9. metadata-eval 40.7%

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

      10. pow-prod-up 40.7%

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

      11. inv-pow 40.7%

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

      12. inv-pow 40.7%

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

      13. swap-sqr 40.7%

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

      14. div-inv 40.7%

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

      15. div-inv 40.7%

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

      16. sqrt-unprod 0.0%

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

      17. add-sqr-sqrt 93.4%

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

      18. metadata-eval 93.4%

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

      19. distribute-neg-frac 93.4%

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

      20. clear-num 93.5%

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

      21. distribute-neg-frac 93.5%

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

      22. metadata-eval 93.5%

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

      23. div-inv 93.6%

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

      24. metadata-eval 93.6%

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

   10. Applied egg-rr 93.6%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+93} \lor \neg \left(y \leq 3.9 \cdot 10^{+72}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]

Alternative 5: 94.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+32} \lor \neg \left(y \leq 6 \cdot 10^{+72}\right):\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.2e+32) (not (<= y 6e+72)))
   (+ 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 <= -2.2e+32) || !(y <= 6e+72)) {
		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 <= (-2.2d+32)) .or. (.not. (y <= 6d+72))) 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 <= -2.2e+32) || !(y <= 6e+72)) {
		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 <= -2.2e+32) or not (y <= 6e+72):
		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 <= -2.2e+32) || !(y <= 6e+72))
		tmp = Float64(1.0 + Float64(y * Float64(-0.3333333333333333 / sqrt(x))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.2e+32) || ~((y <= 6e+72)))
		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, -2.2e+32], N[Not[LessEqual[y, 6e+72]], $MachinePrecision]], N[(1.0 + N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.20000000000000001e32 or 6.00000000000000006e72 < y

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. +-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/ 99.5%

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

      9. fma-neg 99.5%

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

      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 93.2%

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

      1. *-commutative 93.2%

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

      2. associate-*l* 93.2%

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

      3. *-commutative 93.2%

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

   6. Simplified 93.2%

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

      1. expm1-log1p-u 49.1%

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

      2. expm1-udef 49.1%

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

      3. *-commutative 49.1%

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

      4. sqrt-div 49.1%

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

      5. metadata-eval 49.1%

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

      6. un-div-inv 49.1%

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

      7. *-commutative 49.1%

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

   8. Applied egg-rr 49.1%

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

      1. expm1-def 49.1%

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

      2. expm1-log1p 93.2%

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

      3. *-commutative 93.2%

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

      4. associate-*l/ 93.2%

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

      5. metadata-eval 93.2%

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

      6. distribute-neg-frac 93.2%

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

      7. distribute-lft-neg-in 93.2%

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

      8. *-commutative 93.2%

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

      9. distribute-rgt-neg-in 93.2%

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

      10. distribute-neg-frac 93.2%

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

      11. metadata-eval 93.2%

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

   10. Simplified 93.2%

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

   if -2.20000000000000001e32 < y < 6.00000000000000006e72

   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* 99.6%

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

      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 97.4%

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

      1. cancel-sign-sub-inv 97.4%

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

      2. metadata-eval 97.4%

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

      3. associate-*r/ 97.4%

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

      4. metadata-eval 97.4%

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

      5. +-commutative 97.4%

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

   6. Simplified 97.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 42.1%

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

      3. frac-times 42.1%

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

      4. metadata-eval 42.1%

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

      5. metadata-eval 42.1%

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

      6. frac-times 42.1%

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

      7. sqrt-unprod 42.1%

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

      8. add-sqr-sqrt 42.1%

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

      9. clear-num 42.1%

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

      10. div-inv 42.1%

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

      11. metadata-eval 42.1%

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

      12. inv-pow 42.1%

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

   8. Applied egg-rr 42.1%

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

      1. unpow-prod-down 42.1%

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

      2. metadata-eval 42.1%

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

      3. metadata-eval 42.1%

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

      4. sqrt-pow1 42.1%

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

      5. metadata-eval 42.1%

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

      6. sqrt-prod 42.1%

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

      7. *-commutative 42.1%

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

      8. metadata-eval 42.1%

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

      9. metadata-eval 42.1%

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

      10. pow-prod-up 42.1%

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

      11. inv-pow 42.1%

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

      12. inv-pow 42.1%

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

      13. swap-sqr 42.1%

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

      14. div-inv 42.1%

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

      15. div-inv 42.1%

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

      16. sqrt-unprod 0.0%

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

      17. add-sqr-sqrt 97.4%

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

      18. metadata-eval 97.4%

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

      19. distribute-neg-frac 97.4%

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

      20. clear-num 97.5%

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

      21. distribute-neg-frac 97.5%

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

      22. metadata-eval 97.5%

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

      23. div-inv 97.6%

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

      24. metadata-eval 97.6%

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

   10. Applied egg-rr 97.6%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+32} \lor \neg \left(y \leq 6 \cdot 10^{+72}\right):\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]

Alternative 6: 94.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+33} \lor \neg \left(y \leq 5 \cdot 10^{+72}\right):\\ \;\;\;\;1 + \frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.5e+33) (not (<= y 5e+72)))
   (+ 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 <= -2.5e+33) || !(y <= 5e+72)) {
		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 <= (-2.5d+33)) .or. (.not. (y <= 5d+72))) 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 <= -2.5e+33) || !(y <= 5e+72)) {
		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 <= -2.5e+33) or not (y <= 5e+72):
		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 <= -2.5e+33) || !(y <= 5e+72))
		tmp = Float64(1.0 + Float64(y / Float64(sqrt(x) * -3.0)));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.5e+33) || ~((y <= 5e+72)))
		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, -2.5e+33], N[Not[LessEqual[y, 5e+72]], $MachinePrecision]], N[(1.0 + N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.49999999999999986e33 or 4.99999999999999992e72 < y

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. +-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/ 99.5%

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

      9. fma-neg 99.5%

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

      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 93.2%

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

      1. *-commutative 93.2%

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

      2. associate-*l* 93.2%

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

      3. *-commutative 93.2%

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

   6. Simplified 93.2%

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

      1. expm1-log1p-u 49.1%

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

      2. expm1-udef 49.1%

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

      3. *-commutative 49.1%

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

      4. sqrt-div 49.1%

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

      5. metadata-eval 49.1%

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

      6. un-div-inv 49.1%

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

      7. *-commutative 49.1%

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

   8. Applied egg-rr 49.1%

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

      1. expm1-def 49.1%

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

      2. expm1-log1p 93.2%

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

      3. associate-/l* 93.2%

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

   10. Simplified 93.2%

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

       1. div-inv 93.4%

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

       2. metadata-eval 93.4%

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

       3. metadata-eval 93.4%

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

       4. metadata-eval 93.4%

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

       5. distribute-rgt-neg-in 93.4%

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

       6. sqrt-prod 93.4%

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

       7. neg-sub0 93.4%

          \[\leadsto 1 + \frac{y}{\color{blue}{0 - \sqrt{x \cdot 9}}} \]

       8. sqrt-prod 93.4%

          \[\leadsto 1 + \frac{y}{0 - \color{blue}{\sqrt{x} \cdot \sqrt{9}}} \]

       9. metadata-eval 93.4%

          \[\leadsto 1 + \frac{y}{0 - \sqrt{x} \cdot \color{blue}{3}} \]

   12. Applied egg-rr 93.4%

       \[\leadsto 1 + \frac{y}{\color{blue}{0 - \sqrt{x} \cdot 3}} \]
   13. Step-by-step derivation

       1. neg-sub0 93.4%

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

       2. distribute-rgt-neg-in 93.4%

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

       3. metadata-eval 93.4%

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

   14. Simplified 93.4%

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

   if -2.49999999999999986e33 < y < 4.99999999999999992e72

   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* 99.6%

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

      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 97.4%

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

      1. cancel-sign-sub-inv 97.4%

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

      2. metadata-eval 97.4%

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

      3. associate-*r/ 97.4%

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

      4. metadata-eval 97.4%

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

      5. +-commutative 97.4%

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

   6. Simplified 97.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 42.1%

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

      3. frac-times 42.1%

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

      4. metadata-eval 42.1%

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

      5. metadata-eval 42.1%

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

      6. frac-times 42.1%

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

      7. sqrt-unprod 42.1%

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

      8. add-sqr-sqrt 42.1%

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

      9. clear-num 42.1%

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

      10. div-inv 42.1%

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

      11. metadata-eval 42.1%

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

      12. inv-pow 42.1%

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

   8. Applied egg-rr 42.1%

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

      1. unpow-prod-down 42.1%

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

      2. metadata-eval 42.1%

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

      3. metadata-eval 42.1%

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

      4. sqrt-pow1 42.1%

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

      5. metadata-eval 42.1%

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

      6. sqrt-prod 42.1%

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

      7. *-commutative 42.1%

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

      8. metadata-eval 42.1%

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

      9. metadata-eval 42.1%

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

      10. pow-prod-up 42.1%

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

      11. inv-pow 42.1%

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

      12. inv-pow 42.1%

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

      13. swap-sqr 42.1%

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

      14. div-inv 42.1%

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

      15. div-inv 42.1%

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

      16. sqrt-unprod 0.0%

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

      17. add-sqr-sqrt 97.4%

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

      18. metadata-eval 97.4%

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

      19. distribute-neg-frac 97.4%

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

      20. clear-num 97.5%

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

      21. distribute-neg-frac 97.5%

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

      22. metadata-eval 97.5%

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

      23. div-inv 97.6%

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

      24. metadata-eval 97.6%

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

   10. Applied egg-rr 97.6%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+33} \lor \neg \left(y \leq 5 \cdot 10^{+72}\right):\\ \;\;\;\;1 + \frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]

Alternative 7: 94.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+28}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+72}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8e+28)
   (+ 1.0 (/ -0.3333333333333333 (/ (sqrt x) y)))
   (if (<= y 2.4e+72)
     (+ 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 <= -8e+28) {
		tmp = 1.0 + (-0.3333333333333333 / (sqrt(x) / y));
	} else if (y <= 2.4e+72) {
		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 <= (-8d+28)) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) / (sqrt(x) / y))
    else if (y <= 2.4d+72) 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 <= -8e+28) {
		tmp = 1.0 + (-0.3333333333333333 / (Math.sqrt(x) / y));
	} else if (y <= 2.4e+72) {
		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 <= -8e+28:
		tmp = 1.0 + (-0.3333333333333333 / (math.sqrt(x) / y))
	elif y <= 2.4e+72:
		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 <= -8e+28)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 / Float64(sqrt(x) / y)));
	elseif (y <= 2.4e+72)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(1.0 + Float64(y * Float64(-0.3333333333333333 / sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8e+28)
		tmp = 1.0 + (-0.3333333333333333 / (sqrt(x) / y));
	elseif (y <= 2.4e+72)
		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, -8e+28], N[(1.0 + N[(-0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.4e+72], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.99999999999999967e28

   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/ 99.5%

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

      9. fma-neg 99.5%

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

      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 90.8%

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

      1. *-commutative 90.8%

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

      2. associate-*l* 90.8%

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

      3. *-commutative 90.8%

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

   6. Simplified 90.8%

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

      1. expm1-log1p-u 85.7%

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

      2. expm1-udef 85.7%

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

      3. *-commutative 85.7%

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

      4. sqrt-div 85.7%

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

      5. metadata-eval 85.7%

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

      6. un-div-inv 85.7%

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

      7. *-commutative 85.7%

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

   8. Applied egg-rr 85.7%

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

      1. expm1-def 85.7%

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

      2. expm1-log1p 90.9%

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

      3. associate-/l* 90.9%

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

   10. Simplified 90.9%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 6.8%

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

       3. frac-times 6.8%

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

       4. add-sqr-sqrt 6.8%

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

       5. metadata-eval 6.8%

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

       6. div-inv 6.8%

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

       7. metadata-eval 6.8%

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

       8. expm1-log1p-u 5.6%

          \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y}{\sqrt{x \cdot 9}}\right)\right)} \]

       9. expm1-udef 5.6%

          \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{y}{\sqrt{x \cdot 9}}\right)} - 1\right)} \]

   12. Applied egg-rr 85.7%

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

       1. expm1-def 85.7%

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

       2. expm1-log1p 90.9%

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

       3. *-commutative 90.9%

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

       4. associate-*l/ 90.9%

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

       5. associate-/l* 90.9%

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

   14. Simplified 90.9%

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

   if -7.99999999999999967e28 < y < 2.4000000000000001e72

   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* 99.6%

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

      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 97.4%

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

      1. cancel-sign-sub-inv 97.4%

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

      2. metadata-eval 97.4%

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

      3. associate-*r/ 97.4%

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

      4. metadata-eval 97.4%

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

      5. +-commutative 97.4%

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

   6. Simplified 97.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 42.1%

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

      3. frac-times 42.1%

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

      4. metadata-eval 42.1%

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

      5. metadata-eval 42.1%

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

      6. frac-times 42.1%

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

      7. sqrt-unprod 42.1%

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

      8. add-sqr-sqrt 42.1%

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

      9. clear-num 42.1%

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

      10. div-inv 42.1%

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

      11. metadata-eval 42.1%

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

      12. inv-pow 42.1%

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

   8. Applied egg-rr 42.1%

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

      1. unpow-prod-down 42.1%

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

      2. metadata-eval 42.1%

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

      3. metadata-eval 42.1%

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

      4. sqrt-pow1 42.1%

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

      5. metadata-eval 42.1%

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

      6. sqrt-prod 42.1%

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

      7. *-commutative 42.1%

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

      8. metadata-eval 42.1%

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

      9. metadata-eval 42.1%

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

      10. pow-prod-up 42.1%

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

      11. inv-pow 42.1%

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

      12. inv-pow 42.1%

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

      13. swap-sqr 42.1%

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

      14. div-inv 42.1%

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

      15. div-inv 42.1%

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

      16. sqrt-unprod 0.0%

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

      17. add-sqr-sqrt 97.4%

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

      18. metadata-eval 97.4%

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

      19. distribute-neg-frac 97.4%

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

      20. clear-num 97.5%

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

      21. distribute-neg-frac 97.5%

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

      22. metadata-eval 97.5%

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

      23. div-inv 97.6%

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

      24. metadata-eval 97.6%

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

   10. Applied egg-rr 97.6%

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

   if 2.4000000000000001e72 < 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/ 99.5%

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

      9. fma-neg 99.5%

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

      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 96.4%

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

      1. *-commutative 96.4%

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

      2. associate-*l* 96.3%

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

      3. *-commutative 96.3%

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

   6. Simplified 96.3%

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

      1. expm1-log1p-u 1.8%

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

      2. expm1-udef 1.8%

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

      3. *-commutative 1.8%

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

      4. sqrt-div 1.8%

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

      5. metadata-eval 1.8%

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

      6. un-div-inv 1.8%

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

      7. *-commutative 1.8%

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

   8. Applied egg-rr 1.8%

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

      1. expm1-def 1.8%

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

      2. expm1-log1p 96.1%

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

      3. *-commutative 96.1%

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

      4. associate-*l/ 96.3%

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

      5. metadata-eval 96.3%

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

      6. distribute-neg-frac 96.3%

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

      7. distribute-lft-neg-in 96.3%

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

      8. *-commutative 96.3%

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

      9. distribute-rgt-neg-in 96.3%

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

      10. distribute-neg-frac 96.3%

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

      11. metadata-eval 96.3%

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

   10. Simplified 96.3%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+28}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+72}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \]

Alternative 8: 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 9: 99.6% 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 99.8%

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

   4. pow1/2 99.8%

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

3. Applied egg-rr 99.8%

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

   1. unpow1/2 99.8%

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

5. Simplified 99.8%

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

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 10: 67.3% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \frac{-0.1111111111111111}{x}\\ \mathbf{if}\;y \leq -4.8 \cdot 10^{+128}:\\ \;\;\;\;\frac{-1 + \frac{\frac{0.1111111111111111}{x} \cdot -0.1111111111111111}{x}}{t_0}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+116}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{-0.012345679012345678}{x \cdot \left(-x\right)}}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ -1.0 (/ -0.1111111111111111 x))))
   (if (<= y -4.8e+128)
     (/ (+ -1.0 (/ (* (/ 0.1111111111111111 x) -0.1111111111111111) x)) t_0)
     (if (<= y 2e+116)
       (+ 1.0 (/ -1.0 (* x 9.0)))
       (/ (+ -1.0 (/ -0.012345679012345678 (* x (- x)))) t_0)))))

C

double code(double x, double y) {
	double t_0 = -1.0 + (-0.1111111111111111 / x);
	double tmp;
	if (y <= -4.8e+128) {
		tmp = (-1.0 + (((0.1111111111111111 / x) * -0.1111111111111111) / x)) / t_0;
	} else if (y <= 2e+116) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (-1.0 + (-0.012345679012345678 / (x * -x))) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-1.0d0) + ((-0.1111111111111111d0) / x)
    if (y <= (-4.8d+128)) then
        tmp = ((-1.0d0) + (((0.1111111111111111d0 / x) * (-0.1111111111111111d0)) / x)) / t_0
    else if (y <= 2d+116) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = ((-1.0d0) + ((-0.012345679012345678d0) / (x * -x))) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = -1.0 + (-0.1111111111111111 / x);
	double tmp;
	if (y <= -4.8e+128) {
		tmp = (-1.0 + (((0.1111111111111111 / x) * -0.1111111111111111) / x)) / t_0;
	} else if (y <= 2e+116) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (-1.0 + (-0.012345679012345678 / (x * -x))) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -1.0 + (-0.1111111111111111 / x)
	tmp = 0
	if y <= -4.8e+128:
		tmp = (-1.0 + (((0.1111111111111111 / x) * -0.1111111111111111) / x)) / t_0
	elif y <= 2e+116:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = (-1.0 + (-0.012345679012345678 / (x * -x))) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(-1.0 + Float64(-0.1111111111111111 / x))
	tmp = 0.0
	if (y <= -4.8e+128)
		tmp = Float64(Float64(-1.0 + Float64(Float64(Float64(0.1111111111111111 / x) * -0.1111111111111111) / x)) / t_0);
	elseif (y <= 2e+116)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(Float64(-1.0 + Float64(-0.012345679012345678 / Float64(x * Float64(-x)))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = -1.0 + (-0.1111111111111111 / x);
	tmp = 0.0;
	if (y <= -4.8e+128)
		tmp = (-1.0 + (((0.1111111111111111 / x) * -0.1111111111111111) / x)) / t_0;
	elseif (y <= 2e+116)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = (-1.0 + (-0.012345679012345678 / (x * -x))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.8e+128], N[(N[(-1.0 + N[(N[(N[(0.1111111111111111 / x), $MachinePrecision] * -0.1111111111111111), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 2e+116], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 + N[(-0.012345679012345678 / N[(x * (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.8000000000000004e128

   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/ 99.6%

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

      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 2.5%

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

      1. cancel-sign-sub-inv 2.5%

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

      2. metadata-eval 2.5%

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

      3. associate-*r/ 2.5%

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

      4. metadata-eval 2.5%

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

      5. +-commutative 2.5%

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

   6. Simplified 2.5%

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

      1. flip-+ 2.5%

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

      2. metadata-eval 2.5%

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

      3. sub-neg 2.5%

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

      4. div-inv 2.5%

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

      5. div-inv 2.5%

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

      6. swap-sqr 2.5%

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

      7. metadata-eval 2.5%

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

      8. inv-pow 2.5%

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

      9. inv-pow 2.5%

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

      10. pow-prod-up 2.5%

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

      11. metadata-eval 2.5%

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

      12. metadata-eval 2.5%

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

      13. sub-neg 2.5%

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

      14. metadata-eval 2.5%

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

   8. Applied egg-rr 2.5%

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

      1. metadata-eval 2.5%

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

      2. metadata-eval 2.5%

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

      3. pow-prod-up 2.5%

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

      4. inv-pow 2.5%

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

      5. inv-pow 2.5%

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

      6. swap-sqr 2.5%

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

      7. div-inv 2.5%

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

      8. div-inv 2.5%

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

      9. associate-*r/ 2.5%

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

   10. Applied egg-rr 33.2%

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

   if -4.8000000000000004e128 < y < 2.00000000000000003e116

   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/ 99.7%

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

      9. fma-neg 99.7%

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

      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* 99.6%

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

      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 83.9%

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

      1. cancel-sign-sub-inv 83.9%

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

      2. metadata-eval 83.9%

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

      3. associate-*r/ 83.9%

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

      4. metadata-eval 83.9%

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

      5. +-commutative 83.9%

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

   6. Simplified 83.9%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 36.6%

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

      3. frac-times 36.6%

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

      4. metadata-eval 36.6%

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

      5. metadata-eval 36.6%

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

      6. frac-times 36.6%

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

      7. sqrt-unprod 36.7%

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

      8. add-sqr-sqrt 36.7%

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

      9. clear-num 36.7%

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

      10. div-inv 36.7%

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

      11. metadata-eval 36.7%

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

      12. inv-pow 36.7%

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

   8. Applied egg-rr 36.7%

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

      1. unpow-prod-down 36.7%

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

      2. metadata-eval 36.7%

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

      3. metadata-eval 36.7%

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

      4. sqrt-pow1 36.6%

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

      5. metadata-eval 36.6%

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

      6. sqrt-prod 36.6%

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

      7. *-commutative 36.6%

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

      8. metadata-eval 36.6%

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

      9. metadata-eval 36.6%

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

      10. pow-prod-up 36.6%

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

      11. inv-pow 36.6%

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

      12. inv-pow 36.6%

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

      13. swap-sqr 36.6%

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

      14. div-inv 36.6%

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

      15. div-inv 36.6%

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

      16. sqrt-unprod 0.0%

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

      17. add-sqr-sqrt 83.9%

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

      18. metadata-eval 83.9%

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

      19. distribute-neg-frac 83.9%

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

      20. clear-num 83.9%

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

      21. distribute-neg-frac 83.9%

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

      22. metadata-eval 83.9%

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

      23. div-inv 84.0%

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

      24. metadata-eval 84.0%

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

   10. Applied egg-rr 84.0%

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

   if 2.00000000000000003e116 < 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/ 99.5%

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

      9. fma-neg 99.5%

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

      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 3.1%

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

      1. cancel-sign-sub-inv 3.1%

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

      2. metadata-eval 3.1%

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

      3. associate-*r/ 3.1%

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

      4. metadata-eval 3.1%

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

      5. +-commutative 3.1%

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

   6. Simplified 3.1%

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

      1. flip-+ 13.3%

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

      2. metadata-eval 13.3%

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

      3. sub-neg 13.3%

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

      4. div-inv 13.3%

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

      5. div-inv 13.3%

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

      6. swap-sqr 13.3%

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

      7. metadata-eval 13.3%

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

      8. inv-pow 13.3%

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

      9. inv-pow 13.3%

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

      10. pow-prod-up 13.3%

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

      11. metadata-eval 13.3%

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

      12. metadata-eval 13.3%

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

      13. sub-neg 13.3%

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

      14. metadata-eval 13.3%

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

   8. Applied egg-rr 13.3%

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

      1. metadata-eval 13.3%

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

      2. metadata-eval 13.3%

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

      3. pow-prod-up 13.3%

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

      4. inv-pow 13.3%

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

      5. inv-pow 13.3%

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

      6. swap-sqr 13.3%

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

      7. div-inv 13.3%

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

      8. div-inv 13.3%

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

      9. frac-2neg 13.3%

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

      10. metadata-eval 13.3%

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

      11. frac-times 13.3%

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

      12. metadata-eval 13.3%

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

   10. Applied egg-rr 13.3%

       \[\leadsto \frac{\color{blue}{\frac{-0.012345679012345678}{\left(-x\right) \cdot x}} + -1}{\frac{-0.1111111111111111}{x} + -1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+128}:\\ \;\;\;\;\frac{-1 + \frac{\frac{0.1111111111111111}{x} \cdot -0.1111111111111111}{x}}{-1 + \frac{-0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+116}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{-0.012345679012345678}{x \cdot \left(-x\right)}}{-1 + \frac{-0.1111111111111111}{x}}\\ \end{array} \]

Alternative 11: 64.8% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+116}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{-0.012345679012345678}{x \cdot \left(-x\right)}}{-1 + \frac{-0.1111111111111111}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2e+116)
   (+ 1.0 (/ -1.0 (* x 9.0)))
   (/
    (+ -1.0 (/ -0.012345679012345678 (* x (- x))))
    (+ -1.0 (/ -0.1111111111111111 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2e+116) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (-1.0 + (-0.012345679012345678 / (x * -x))) / (-1.0 + (-0.1111111111111111 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2d+116) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = ((-1.0d0) + ((-0.012345679012345678d0) / (x * -x))) / ((-1.0d0) + ((-0.1111111111111111d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2e+116) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (-1.0 + (-0.012345679012345678 / (x * -x))) / (-1.0 + (-0.1111111111111111 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2e+116:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = (-1.0 + (-0.012345679012345678 / (x * -x))) / (-1.0 + (-0.1111111111111111 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2e+116)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(Float64(-1.0 + Float64(-0.012345679012345678 / Float64(x * Float64(-x)))) / Float64(-1.0 + Float64(-0.1111111111111111 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2e+116)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = (-1.0 + (-0.012345679012345678 / (x * -x))) / (-1.0 + (-0.1111111111111111 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2e+116], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 + N[(-0.012345679012345678 / N[(x * (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.00000000000000003e116

   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/ 99.7%

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

      9. fma-neg 99.7%

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

      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* 99.6%

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

      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 66.8%

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

      1. cancel-sign-sub-inv 66.8%

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

      2. metadata-eval 66.8%

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

      3. associate-*r/ 66.8%

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

      4. metadata-eval 66.8%

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

      5. +-commutative 66.8%

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

   6. Simplified 66.8%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 35.9%

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

      3. frac-times 35.9%

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

      4. metadata-eval 35.9%

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

      5. metadata-eval 35.9%

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

      6. frac-times 35.9%

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

      7. sqrt-unprod 30.1%

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

      8. add-sqr-sqrt 30.1%

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

      9. clear-num 30.1%

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

      10. div-inv 30.1%

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

      11. metadata-eval 30.1%

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

      12. inv-pow 30.1%

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

   8. Applied egg-rr 30.1%

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

      1. unpow-prod-down 30.1%

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

      2. metadata-eval 30.1%

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

      3. metadata-eval 30.1%

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

      4. sqrt-pow1 35.9%

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

      5. metadata-eval 35.9%

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

      6. sqrt-prod 35.9%

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

      7. *-commutative 35.9%

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

      8. metadata-eval 35.9%

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

      9. metadata-eval 35.9%

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

      10. pow-prod-up 35.9%

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

      11. inv-pow 35.9%

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

      12. inv-pow 35.9%

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

      13. swap-sqr 35.9%

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

      14. div-inv 35.9%

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

      15. div-inv 35.9%

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

      16. sqrt-unprod 0.0%

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

      17. add-sqr-sqrt 66.8%

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

      18. metadata-eval 66.8%

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

      19. distribute-neg-frac 66.8%

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

      20. clear-num 66.8%

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

      21. distribute-neg-frac 66.8%

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

      22. metadata-eval 66.8%

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

      23. div-inv 66.9%

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

      24. metadata-eval 66.9%

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

   10. Applied egg-rr 66.9%

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

   if 2.00000000000000003e116 < 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/ 99.5%

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

      9. fma-neg 99.5%

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

      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 3.1%

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

      1. cancel-sign-sub-inv 3.1%

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

      2. metadata-eval 3.1%

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

      3. associate-*r/ 3.1%

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

      4. metadata-eval 3.1%

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

      5. +-commutative 3.1%

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

   6. Simplified 3.1%

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

      1. flip-+ 13.3%

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

      2. metadata-eval 13.3%

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

      3. sub-neg 13.3%

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

      4. div-inv 13.3%

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

      5. div-inv 13.3%

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

      6. swap-sqr 13.3%

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

      7. metadata-eval 13.3%

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

      8. inv-pow 13.3%

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

      9. inv-pow 13.3%

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

      10. pow-prod-up 13.3%

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

      11. metadata-eval 13.3%

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

      12. metadata-eval 13.3%

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

      13. sub-neg 13.3%

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

      14. metadata-eval 13.3%

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

   8. Applied egg-rr 13.3%

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

      1. metadata-eval 13.3%

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

      2. metadata-eval 13.3%

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

      3. pow-prod-up 13.3%

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

      4. inv-pow 13.3%

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

      5. inv-pow 13.3%

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

      6. swap-sqr 13.3%

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

      7. div-inv 13.3%

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

      8. div-inv 13.3%

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

      9. frac-2neg 13.3%

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

      10. metadata-eval 13.3%

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

      11. frac-times 13.3%

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

      12. metadata-eval 13.3%

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

   10. Applied egg-rr 13.3%

       \[\leadsto \frac{\color{blue}{\frac{-0.012345679012345678}{\left(-x\right) \cdot x}} + -1}{\frac{-0.1111111111111111}{x} + -1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+116}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{-0.012345679012345678}{x \cdot \left(-x\right)}}{-1 + \frac{-0.1111111111111111}{x}}\\ \end{array} \]

Alternative 12: 62.8% 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/ 99.7%

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

   9. fma-neg 99.7%

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

   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* 99.6%

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

   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 55.4%

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

   1. cancel-sign-sub-inv 55.4%

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

   2. metadata-eval 55.4%

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

   3. associate-*r/ 55.4%

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

   4. metadata-eval 55.4%

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

   5. +-commutative 55.4%

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

6. Simplified 55.4%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 29.6%

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

   3. frac-times 29.6%

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

   4. metadata-eval 29.6%

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

   5. metadata-eval 29.6%

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

   6. frac-times 29.6%

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

   7. sqrt-unprod 24.9%

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

   8. add-sqr-sqrt 24.9%

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

   9. clear-num 24.9%

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

   10. div-inv 24.9%

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

   11. metadata-eval 24.9%

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

   12. inv-pow 24.9%

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

8. Applied egg-rr 24.9%

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

   1. unpow-prod-down 24.9%

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

   2. metadata-eval 24.9%

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

   3. metadata-eval 24.9%

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

   4. sqrt-pow1 29.6%

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

   5. metadata-eval 29.6%

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

   6. sqrt-prod 29.6%

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

   7. *-commutative 29.6%

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

   8. metadata-eval 29.6%

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

   9. metadata-eval 29.6%

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

   10. pow-prod-up 29.6%

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

   11. inv-pow 29.6%

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

   12. inv-pow 29.6%

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

   13. swap-sqr 29.6%

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

   14. div-inv 29.6%

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

   15. div-inv 29.6%

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

   16. sqrt-unprod 0.0%

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

   17. add-sqr-sqrt 55.4%

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

   18. metadata-eval 55.4%

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

   19. distribute-neg-frac 55.4%

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

   20. clear-num 55.4%

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

   21. distribute-neg-frac 55.4%

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

   22. metadata-eval 55.4%

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

   23. div-inv 55.5%

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

   24. metadata-eval 55.5%

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

10. Applied egg-rr 55.5%

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

11. Final simplification 55.5%

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

Alternative 13: 61.6% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.195:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.195) (/ -0.1111111111111111 x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.195) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.195d0) then
        tmp = (-0.1111111111111111d0) / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.195) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.195:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.195)
		tmp = Float64(-0.1111111111111111 / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.195)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.195], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.19500000000000001

   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 58.2%

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

   if 0.19500000000000001 < 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/ 99.7%

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

      9. fma-neg 99.7%

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

      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 50.8%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.195:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14: 62.8% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{-0.1111111111111111}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return 1.0 + (-0.1111111111111111 / x);
}

Python

def code(x, y):
	return 1.0 + (-0.1111111111111111 / x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
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/ 99.7%

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

   9. fma-neg 99.7%

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

   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* 99.6%

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

   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 55.4%

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

   1. cancel-sign-sub-inv 55.4%

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

   2. metadata-eval 55.4%

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

   3. associate-*r/ 55.4%

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

   4. metadata-eval 55.4%

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

   5. +-commutative 55.4%

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

6. Simplified 55.4%

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

7. Final simplification 55.4%

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

Alternative 15: 31.0% accurate, 113.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. 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/ 99.7%

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

   9. fma-neg 99.7%

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

   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* 99.6%

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

   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 24.7%

   \[\leadsto \color{blue}{1} \]

5. Final simplification 24.7%

   \[\leadsto 1 \]

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 2023310 
(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)))))